The Electron

Fundamental Theories of Physics

An International Book Series on The Fundamental Theories of Physics:

Their Clarification, Development and Application

Editor: ALWYN VANDER MERWE

University of Denver, U.S.A.

Editorial Advisory Board:

ASIM BARUT, University of Colorado. U.S.A.
HERMANN BONDI, University of Cambridge. UK.
BRIAN D. JOSEPHSON, University of Cambridge. U.K.
CLIVE KILMISTER, University of London. UK.
GONTER LUDWIG, Philipps-Universitiit. Marburg, F.R.G.
NATHAN ROSEN,lsraellnstitute of Technology, Israel
MENDEL SACHS, State University of New York at Buffalo. U.S.A.
ABDUS SALAM, International Centre for Theoretical Physics. Trieste, Italy
HANS-JURGEN TREDER, Zentralinstitut fur Astrophysik der Akademie der
Wissenschaften, GD.R.

Volume 45
The Electron
New Theory and Experiment

edited by

David Hestenes
Physics Department.
Arizona State University.
Tempe. Arizona. U.S.A.

and

Antonio Weingartshofer
Laser-Electron Interactions Laboratory.
Department 01 Physics.
St. Francis Xavier University.
Antigonish. Nova Scotia. Canada

..
SPRINGER SCIENCE+BUSINESS MEDIA, RV_
Library of Congress Cataloging-in-Publication Data
The Electron new theory and experi~ent I edited by David Hestenes
and Antonio ~elngartshofer.
p. cm. -- (Fundamental theories of phySiCS ; v. 45)
Proceedings of the 1990 Electron Workshop held at St. Francls
Xavier University, Antigonish, Nova Scotia, Aug. 3-8.
Inc 1udes 1ndex.
ISBN 978-94-010-5582-6 ISBN 978-94-011-3570-2 (eBook)
DOI 10.1007/978-94-011-3570-2
1. Electrons--Congresses. 2. Quantum electrodynamics--Congresses-
3. Dirac equation--Congresses. 1. Hestenes, David, 1933-
II. Weingartshofer, Antonio, 1925- III. Electron Workshop (1990
St. Francis Xavier Unlversityl IV. Serles.
QC793.5.E62E42 1991
539.7'2112--dc20 91-22205
CIP
ISBN 978-94-010-5582-6

Printed an acid-free paper

Ali Rights Reserved

© 1991 Springer Science+Business Media Dordrecht
Original1y published by Kluwer Academic Publishers in 1991
Softcover reprint ofthe hardcover Ist edition 1991
No part of the material protected by this copyright notice may be reproduced or
utiIized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
 This book is dedicated to the memory of
Reverend Dr. E. M. Clarke
In him there was a comfortable and congenial convergence
between Dr. Clarke the physicist and Father Clarke the priest
 CONTENTS

Preface . ix

PART I: Theory

Scattering of Light by Free Electrons as a Test of

Quantum Theory . . . . . . . . . . 1
E. T. Jaynes

Zitterbewegung in Radiative Processes . . . . . . . . . . . . . . . 21

D. Hestenes

Charged Particles at Potential Steps 37

S. F. Gull

New Solutions of the Dirac Equation for Central Fields 49

H. Kruger

The Role of the Duality Rotation in the Dirac Theory . . . . . 83

R. Boudet

Brief History and Recent Developments in Electron Theory

and Quantumelectrodynamics . . . . . . . . . . . . . . . . 105
A. O. Barut

The Explicit Nonlinearity of Quantum Electrodynamics . . . . . . . 149

W. T. Grandy, Jr.

On the Mathematical Procedures of Selffield

Quantumelectrodynamics . . . . . . . . . . . . 165
A. O. Barut

Non-Linear Gauge Invariant Field Theories of the Electron

and Other Elementary Particles . . . . . . . . . . 171
F. I. Cooperstock

How to Identify an Electron in an External Field . . . . . . . . . 183

A. Z. Capri

The Electron and the Dressed Molecule . . . . . . . . . . . . . . . 191

A. D. Bandrauk

Scattering Chaos in the Harmonically Driven Morse System . . . . . 219

C. Jung
viii CONTENTS

PART II: Experiment

Experiments with Single Electrons . . . . . . . . . . . . . . . . . 239

R. S. Van Dyck, Jr., P. B. Schwinberg and H. G. Dehmelt

Experiments on the Interaction of Intense Fields with Electrons . . 295

A. Weingartshofer

Ionization in Linearly and Circularly Polarized Microwave

Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
T. F. Gallagher

MUltiple Harmonic Generation in Rare Gases at High

Laser Intensity . . . . . .. . . . . . . . . 321
A. L'Huillier, L.-A. Lompre and G. Hainfray

Absorption and Emission of Radiation During Electron Excitation

of Atoms . . . . . . . . . . . . . . . . . . . . . . 333
B. Wallbank

Penning Ionization in Intense Laser Fields . . . . . . . . . . . . 341

H. Horgner

Microwave Ionization of H Atoms: Experiments in Classical and

Quantal Dynamics . . 353
L. Hoorman

The Electron 1990 Workshop

List of Participants 391

Index . 393
 PRE F ACE

The importance of conducting experiments on this particle

resides in our intrinsic interest in understanding the
relatively simple system we call "the electron".

Robert S. Van Dyck, Jr.

Paul B. Schwinberg
Hans G. Dehmelt

This opening sentence to a comprehensive review article by

scientists from the University of Washington is also a fitting opening to
these Proceedings of the 1990 ELECTRON WORKSHOP for three reasons: Their
history-making experiments occupy a central place in this book. (2) Their
words express the spirit that brought together scientists from two
continents to discuss and debate current conceptions of the electron.
(3) Hopefully, this will remind the theorists that experiment has the
last word.

The first word has been allotted to the eminent theorist Edwin T.
Jaynes, who observes about the Electron Workshop: "It seems strange that
this is the first such meeting, since for a century electrons have been
the most discussed things in physics". This decade marks the centennial
for both the discovery of the electron by J. J. Thomson and the invention
of classical electron theory by H. A. Lorentz, but there are better
reasons for the Workshop than a centennial celebration. First, new
theoretical insights into the structure and self -interaction of the
electron hold promise for resolving the nagging difficulties of quantum
electrodynamics and achieving a deeper theory of electrons. Second, new
experimental techniques provide powerful new probes for electron
properties. The Electron Workshop was organized to achieve a confluence
of these theoretical and experimental streams, bringing together, for
mutual intellectual enrichment, physicists animated by a common will to
know the electron. The success of this endeavor can be seen in the
contributions to these Proceedings, which have been organized into
theoretical and experimental parts.

The contributions by the theorists differ widely in approach but can

be grouped into five categories. (1) Jaynes discusses the possibility of
experimental tests for an old idea of Schrodinger's that the wave
function really describes an extension of the electron in space. (2) A
cluster of papers by Hestenes, Gull, KrUger and Boudet deals with a
radical new formulation of the Dirac theory which reveals hidden
geometric structure in the theory, provides powerful new computational
ix
x PREFACE

techniques, and raises new issues of physical interpretation as well as

possibilities for deepening the theory. (3) Barut contributes a
comprehensive review of his own ambitious program in electron theory and
quantum electrodynamics. Barut's work is rich with ingenious ideas, and
the interest it provokes among other theorists can be seen in the
cri tique by Grandy. Cooperstock takes a much different approach to
nonlinear field-electron coupling which leads him to conclusions about
the size of the electron. (4) Capri and Bandrauk work within the
standard framework of quantum electrodynamics. Bandrauk presents a
valuable review of his theoretical approach to the striking new
photoelectric phenomena in high intensity laser experiments. (5) Jung
proposes a theory to merge the ideas of free-free transitions and of
scattering chaos, which is becoming increasingly important in the
theoretical analysis of nonlinear optical phenomena.

For the last half century the properties of electrons have been
probed primarily by scattering experiments at ever higher energies.
Recently, however, two powerful new experimental techniques have emerged
capable of giving alternative experimental views of the electron. We
refer to (1) the confinement of single electrons for long term study, and
(2) the interaction of electrons with high intensity laser fields.
Articles by outstanding practitioners of both techniques are included in
Part II of these Proceedings.

The precision experiments on trapped electrons by the Washington

group quoted above have already led to a Nobel prize for the most
accurate measurements of the electron magnetic moment. Sadly, there is
a dearth of good theoretical ideas to be tested by this sensitive
technique. Theorists, attention!

The interaction of intense laser fields with electrons and atoms is

currently the subject of vast research activity. Experiment has far
outstripped theory in this domain, so much of the data can be given only
a qualitative explanation. This circumstance has unfortunate reper-
cussions. Without sharply drawn theoretical issues, much experimental
effort is wasted. Thus, there is a plethora of experiments on
mul tiphoton ionization while more subtle, possibly more informative,
phenomena are overlooked. In particular, the possibility that these
phenomena might tell us something new about the electron is seldom
considered. That is a point we especially wish to emphasize in these
Proceedings.

In Part II, Weingartshofer provides an overview of experimental

problems with intense field-electron interactions. Gallager discusses
experimental results of ionization with circularly polarized microwave
fields and compares them with observations in the optical regime. The
outstanding research group at Saclay, represented at the Workshop by Anne
L'Huillier, reviews the very informative quantitative results of their
experiments on harmonics of the driving laser frequency in the radiation
that accompanies ionization. The complications of multiphoton ionization
are totally avoided in the next two papers discussing electron scattering
PREFACE xi

techniques in the presence of intense laser fields. Wallbank describes

simultaneous electron-multiphoton excitation of atoms and Morgner makes
some predictions about the behavior of Penning electrons in intense laser
fields. In the final paper Moorman deals with microwave ionization of
hydrogen atoms as the prototype system for studying quantum chaos.

The 1990 ELECTRON WORKSHOP was held at St. Francis Xavier University
in Antigonish. Nova Scotia (August 3 to 8). supported by funds from the
Natural Sciences and Engineering Research Council of Canada. The
Workshop organizer. Antonio Weingartshofer, wishes to thank Heinz Kruger
for initiating contact with other theoreticians friendly to the electron
and his colleague Barry Wallbank for invaluable help and encouragement.
The enthusiastic support of University officials greatly enhanced the
atmosphere of the Workshop. The Academic Vice-President. Dr. John J.
MacDonald. reached deep into the empty coffers of the University to help
out with expenses. and the President. Dr. David Lawless. opened the
scientific sessions with a warm welcome to the University.

One of Canada's most distinguished scientists and public figures.

Dr. Larkin Kerwin, returned to his Alma Mater to present the Keynote
Address for this international meeting. Some thirty-six years before.
his inspiring research at Laval University attracted the Reverend
Dr. E. M. Clarke. and together they developed the prototype of the first
practical 127 electrostatic electron selector. an instrument that soon
0

found its way into many laboratories throughout the world as a vital
component of the modern electron spectrometer. The history of the Laser-
Electron Interaction Laboratory at St. Francis Xavier University began
with that event while Dr. Clarke was Chairman of the Physics Department.
In his address entitled "AND SO AD INFINITUM. the Continuing Evolution of
the Electron", Dr. Kerwin spoke about cultural and philosophical
implications as well as the scientific implications of our evolving
concepts of the ultimate constituents of matter. We hope as well that
this book will serve as a road sign to the physics of the 21st century.

David Hestenes
Antonio Weingartshofer
SCATTERING OF LIGHT BY FREE ELECTRONS
AS A TEST OF QUANTUM THEORY

E. T. JAYNES
Arthur Holly Compton Laboratory of Physics
Washington University
St. Louis MO 63130, U.S.A.

ABSTRACT: Schrodinger and Heisenberg gave two very different views about the physical
meaning of an electron wave function. We argue that SchrOdinger's view may have been
dismissed prematurely through failure to appreciate the stabilizing effects of forces due to
Zitterbewegung, and suggest experiments now feasible which might decide the issue.

1. Introduction.
We are gathered here to discuss the present fundamental knowledge about electrons and
how we might improve it. On the one hand it seems strange that this is the first such
meeting, since for a Century electrons have been the most discussed things in physics.
And for all this time a growing mass of technology has been based on them, which today
dominates every home and office. But on the other hand, this very fact makes it seem
strange that a meeting like this could be needed. How could all this marvelously successful
technology exist unless we already knew all about electrons?
The answer is that technology runs far ahead of real understanding. For Centuries
practical men grew better varieties of grapes and bred faster horses without any conception
of chromosomes and DNA. The most easily perceived facts give sufficient knowledge to
start a technology, and trial-and-error experimentation takes over from there. Because of
this, the practical men who give us our technology sometimes see no need for fundamental
knowledge, and even deprecate it.
This happens even within a supposedly scientific field. The mathematics of epicycles was
a successful 'technology' found by trial-and-error for describing and predicting the motion
of planets, and because of this success the idea that all astronomical phenomena must be
described in terms of epicycles captured men's minds for over 1000 years. The efforts of
Copernicus, Kepler, and Galileo to find the 'chromosomes and DNA' underlying epicycles
were not only deprecated, but violently opposed by the practical men who, being concerned
only with phenomenology, found in epicycles all they needed.
We know today that the mathematical scheme of epicycles was flexible enough (a po-
tentially unlimited number of epicycles available, whose size and period could be chosen
at will) so that however the planets moved, it could always have been 'accounted for' by
invoking enough epicycles. But a mathematical system that is flexible enough to represent
any phenomenology, is empty of physical content. Indeed, the real content of any physi-
cal theory lies precisely in the constraints that it imposes on phenomena; the stronger the
constraints, the more cogent and useful the theory.
1
D. Hestenes and A. Weingartshofer (eds.J, The Electron, 1-20.
© 1991 Kluwer Academic Publishers.
2 E. T.IAYNES

In the next two Sections, we summarize the historical background of the puzzled thinking
that motivates our present efforts. The reader who wants to get on with the job currently
before us may turn at once to Section 4 below.

2. Is Quantum Theory a System of Epicycles?

Today, Quantum Mechanics (QM) and Quantum Electrodynamics (QED) have great prag-
matic success - small wonder, since they were created, like epicycles, by empirical trial-and-
error guided by just that requirement. For example, when we advanced from the hydrogen
atom to the helium atom, no theoretical principle told us whether we should represent the
two electrons by two wave functions in ordinary 3-d space, or one wave function in a 6-d
configuration space; only trial-and-error showed which choice leads to the right answers.
Then to account for the effects now called 'electron spin', no theoretical principle told
Goudsmit and Uhlenbeck how this should be incorporated into the mathematics. The
expedient that finally gave the right answers depended on Pauli's knowing about the two-
valued representations of the rotation group, discovered by Cartan in 1913.
In ad vancing to QED, no theoretical principle told Dirac that electromagnetic field modes
should be quantized like material harmonic oscillators; and for reasons to be explained here
by Asim Bamt, we think it still an open question whether the right choice was made.
It leads to many right answers but also to some horrendously wrong ones that theorists
simply ignore; but it is now known that virtually all the right answers could have been
found without, while some of the wrong ones were caused by, field quantization.
Because of their empirical origins, QM and QED are not physical theories at all. In
contrast, Newtonian celestial mechanics, Relativity, and Mendelian genetics are physical
theories, because their mathematics was developed by reasoning out the consequences of
clearly stated physical principles which constrained the possibilities. To this day we have
no constraining principle from which one can deduce the mathematics of QM and QED;
in every new situation we must appeal once again to empirical evidence to tell us how we
must choose our mathematics in order to get the right answers.
In other words, the mathematical system of present quantum theory is, like that of
epicycles, unconstrained by any physical principles. Those who have not perceived this have
pointed to its empirical success to justify a claim that all phenomena must be described in
terms of Hilbert spaces, energy levels, etc. This claim (and the gratuitous addition that it
must be interpreted physically in a particular manner) have captured the minds of physicists
for over sixty years. And for those same sixty years, all efforts to get at the nonlinear
'chromosomes and DNA' underlying that linear mathematics have been deprecated and
opposed by those practical men who, being concerned only with phenomenology, find in
the present formalism all they need.
But is not this system of mathematics also flexible enough to accommodate any phe-
nomenology, whatever it might be? Others have raised this question seriously in connection
with the BCS theory of superconductivity. We have all been taught that it is a marvelous
success of quantum theory, accounting for persistent currents, Meissner effect, isotope ef-
fect, Josephson effect, etc. Yet on examination one realizes that the model Hamiltonian
is phenomenological, chosen not from first principles but by trial-and-error so as to agree
with just those experiments.
Then in what sense can one claim that the BCS theory gives a physical explanation of
superconductivity? Surely, if the Meissner effect did not exist, a different phenomenological
SCATIERING OF LIGHT BY FREE ELECTRONS 3

model would have been invented, that does not predict it; one could have claimed just as
great a success for quantum theory whatever the phenomenology to be explained.
This situation is not limited to superconductivity; in magnetic resonance, whatever
the observed spectrum, one has always been able to invent a phenomenological spin-
Hamiltonian that "accounts" for it. In high-energy physics one observes a few facts and
considers it a big advance - and great new triumph for quantum theory - when it is al-
ways found possible to invent a model conforming to QM, that "accounts" for them. The
'technology' of QM, like that of epicycles, has run far ahead of real understanding.
This is the grounds for our suggestion (Jaynes, 1989) that present QM is only an empty
mathematical shell in which a future physical theory may, perhaps, be built. But however
that may be, the point we want to stress is that the success - however great - of an
empirically developed set of rules gives us no reason to believe in any particular physical
interpretation of them. No physical principles went into them.
Contrast this with the logical status of a real physical theory; the success of Newtonian
celestial mechanics does give us a valid reason for believing in the restricting inverse-square
law, from which it was deduced; the success of relativity theory gives us an excellent reason
for believing in the principle of relativity, from which it was deduced. Theories need not
refer specifically to physics: the success of economic predictions made from the restricting
law of supply and demand gives us a valid reason for believing in that law.

3. But What is Wrong With It?

Of course, finding a successful empirical equation can be an important beginning of real
understanding; perhaps even the necessary first step. In this sense, the mathematics of
QM does contain some very important and fundamental truth; but the process by which it
was found reveals nothing about its meaning, and it remains not only logically undefined,
but pragmatically incomplete. It can, for example, predict the relative time of decay of
two Co60 nuclei only with a probable error of about five years; but the experimentalist can
measure this interval to a fraction of a microsecond.
Contemplating this, we understand why Bohr once remarked that the 'deep truths'
are ones for which the opposite is also true; repeatedly, the attempt to present a unified
front on questions of interpretation forces QM into schizoid positions. In spite of the fact
that the experimenter can measure details about individual decays that the theory cannot
predict, those who speculate about the deeper immediate cause of each individual decay
are considered incompetent, and the currently taught physical interpretation claims that
QM is already a complete description of reality.
Then it contradicts itself by its inability to describe reality at all. For example, we are
not allowed to ask: "What is really happening when an atom emits light?" We may ask
only: "What is the probability that, if we make a measurement, we shall find that a photon
has been emitted?" As Bohr emphasized repeatedly, the Copenhagen interpretation of
quantum theory cannot, as a matter of principle, answer any question of the form: "What
is really happening when - - -?" Yet we submit that such questions are exactly the ones
that a physicist ought to be trying to answer; for the purpose of science is to understand
the real world. If there were no such thing as a reality that exists independently of human
knowledge, then there could be no point to physics or any other science.
It is not only in radioactivity that QM is pragmatically incomplete; for example, the
data record from a Stern-Gerlach experiment can tell not only the number of particles in
4 E. T.JAYNES

each beam, but the time order in which 'spin up' and 'spin down' occurred. Indeed, as we
noted long ago (Jaynes, 1957), in every real experiment the experimenter can observe things
that the theory cannot predict. Always, official quantum theory takes a schizoid position,
admitting that the theory is observationally incomplete; yet persisting in the claim that it
is logically complete.
Even the EPR paradox failed to force retraction of this claim, and so currently taught
quantum theory still contains the schizoid elements of local acausality on the one hand -
and instantaneous action at a distance on the other! We find it astonishing that anyone
could seriously advocate such a theory.
In short, the currently taught physical interpretation has elements of nonsense and mys-
ticism which have troubled thoughtful physicists, starting with Einstein and Schrodinger,
for over sixty years. The more deeply one thinks about these things, the more troubled he
becomes and the more convinced that the present interpretive scheme of quantum theory
is long overdue for a drastic modification. We want to do everything we can to help find it.
Not surprisingly, there has been no really significant advance in basic understanding since
the 1927 Solvay Congress, in which this schizoid mentality became solidified into physics.
Theoretical physics can hardly hope to make any further progress in such understanding un-
til we learn how to separate the permanent mathematical truths from the physical nonsense
that now obscures them.
To be fair, we should add that some of these contradictions disappear when we note
that "currently taught" quantum theory is quite different from the "Copenhagen theory",
defined as the teachings of Niels Bohr. The latter is much more defensible than the former if
we recognize that Bohr's intention was never to describe reality at all; only our information
about reality. This is a legitimate goal in its own right, and it has a useful - indeed,
necessary - role to play in physics as discussed further in Jaynes (1986, 1990).
The trouble is that this is far from the only legitimate goal of physics; yet for 60 years
Bohr's teachings have been perverted into attempts to deprecate and discourage any fur-
ther thinking aimed at finding the causes underlying microphenomena. Such thinking is
termed 'obsolete mechanistic materialism', but those who hurl such epithets then reveal
their schizoid mentality when they ascribe unquestioning ontological reality - independent
of human information - to things such as 'quantum jumps' and 'vacuum fluctuations'. This
does violence to Bohr's teachings; yet those who commit it claim to be disciples of Bohr.
For a time we were optimistic because it appeared that the new thinking of John Bell
(1987) might show us the way. It was refreshing to see from his words that he was not
brainwashed by the conventional muddled thinking and teaching, but was able to discern
the real difficulty. But his recent work (Bell, 1990) shows him apparently at the end of his
rope, reduced to destructive criticism of the ideas of everybody else but offering nothing to
replace them. Therefore it is up to us to find the new constructive ideas that theoretical
physics needs.

4. Our Job for Today

Theoretical work of the kind presented at this meeting is sometimes held to be "out of
the mainstream" of current thinking; but that is quite mistaken. There is no mainstream
today; it has long since dried up and our vessel is grounded. We are trying rather to start
a new stream able to carry science a little further. Indeed, our efforts are much closer
to the traditional mainstream of science than much of what is done in theoretical physics
seATIERING OF LIGHT BY FREE ELECI'RONS 5

today. Talk of tachyons, superstrings, worm holes, the wave function of the universe, the
first 10- 40 second after the big bang, etc., is speculation vastly more wild and far-fetched
than anything we are doing.
In the present discussion we want to look at the problems of QM from a very elementary,
lowbrow physical viewpoint in the hope of seeing things that the highbrow mathematical
viewpoint does not see. I want to suggest, in agreement with David Hestenes, that Zit-
terbewegung (ZBW) is a real phenomenon with real physical consequences that underlie
all of quantum theory; indeed, such important consequences that without ZBW the world
would be very different and we would not be here. But my ZBW differs from his in some
basic qualitative respects, and so our first order of business is to describe this difference and
see whether it could be tested experimentally. Then we can proceed to some speculations
about the role of ZBW in the world.

5. The Puzzle of Space-Time Algebra

It is now about 25 years since I started trying to read David Hestenes' work on space-time
algebra. All this time, I have been convinced that there is something true, fundamental,
and extremely important for physics in it. But I am still bewildered as to what it is, because
he writes in a language that I find indecipherable; his message just does not come through
to me. Let me explain my difficulty, not just to display my own ignorance, but to warn
those who work on space-time algebra: nearly all physicists have the same hangup, and
you are never going to get an appreciative hearing from physicists until you learn how to
explain what you are doing, in plain language that makes physical sense to us.
Physicists go into a state of mental shock when we see a single equation which purports
to represent the sum of a scalar and a vector. All of our training, from childhood on, has
ground into us that one must never even dream of doing such an absurd thing; the sin is
even worse than committing a dimensional inhomogeneity. How can David get away with
this when the rest of us cannot?
If u and v are vectors, then in Hestenes' equation (uv = U· v + u 1\ v) the symbol
'+' must have a different meaning than it does in conventional mathematics. But then the
symbol '=' must also have some different meaning, and he does not choose to enlighten us,
so the above equation remains incomprehensible to me. The closest I can come to making
sense out of it is to note that we cannot speak of a sum of apples and oranges; yet we may
place an apple and an orange side by side and contemplate them together. Perhaps this is
something like the intended meaning.
There is another possibility. Perhaps '+' and '=' have their conventional meanings after
all, but u and 11 do not. In my view it simply does not make sense to speak of the sum of
a scalar and a vector, any more than of the proverbial square circle; but it makes perfectly
good sense to speak, as Cartan does, of the sum of two matrices, interpreted as abstract
mathematical representations of a scalar and a vector. If this is what David really means, he
could have prevented decades of confusion by a slight change in verbiage. Physicists are very
touchy about the distinction between an physical or geometrical object and a mathematical
representation of that object, because the mathematical representation usually holds only
in some restricted domain that does not apply to the object itself.
But I suspect that what he "really means" is something more abstract than either of
these two suggestions, and neither his writings nor his talks provide enough clues for me to
decide what it is. I have never been able to get past that equation, with any comprehension
of what is being said.
6 E. T.JAYNES

Passing on to the next part of Hestenes' work, we encounter a physical difficulty. In

discussing the Dirac equation we read that some symbol "stands for a rotation". Now it
is evident that from a mathematical standpoint there is an abstract correspondence with
rotations (composition law of the rotation group). But from a physical standpoint, rotation
of what? When I look at the Dirac equation, I see just what SchrOdinger did: a four-
component wave function {1jJ,,( x, t)} representing a continuous distribution of something
(perhaps probability or charge); but that something itself has no further internal directional
structure that could 'rotate'.
But just at this meeting, I finally picked up the first clue as to what David is talking
about here. It seems that when he looks at the Dirac equation, he sees not a continuous
distribution of anything, but a tangle of all the different possible trajectories of a point
particle! Presumably, these are the things that are rotating. This must be the most
egregious example of a hidden-variable theory ever dreamt of, and nothing of his that I
have ever read prepared me for this revelation.
I would never, in 1000 years, have thought of looking at the Dirac equation in that way.
For, in any theory where the underlying reality is conceived to be a single particle hiding
somewhere in the wave packet, the behavior of the packet is determined, not by what that
particle actually does, but by the range of possible things that it might have done, but
did not. Thus the wave packet cannot itself describe any reality; it represents only a state
of knowledge about reality. Indeed, this is the view that Heisenberg (1958) stated very
explici tly.
You can, of course, account for many facts by such a picture - namely, those so unsharp
in time and space that what the experimentalist observes can be regarded as some kind of
average over an ensemble of many different trajectories. But when the experimental result
ought to depend on one such motion, I think that the point particle trajectory picture will
surely fail, because there is no such thing in Nature as a point particle.
Put more constructively, a point particle theory can be confirmed only by an experi-
ment which actually sees that particle, removed from its ensemble or wave packet, doing
something as a particle, all by itself. In an exactly similar way, Louis Pasteur's microbe
theory of diseases could be confirmed only by developing the instruments by which a single
microbe could be seen and its behavior observed.
Of course, we must be as demanding as Pasteur about the resolving power of our instru-
ments. A continuous structure that is small compared to our resolution distance will appear
to us as a point particle. That a microbe is not a point particle, but a continuous structure
with definite shape and internal moving parts, can be learned only with a microscope that
has a resolution distance small compared to the dimensions of the microbe.
We must accomplish something like this, in order to check the reality of those tangled
trajectories which were supposed to be the 'chromosomes and DNA' hiding in the Dirac
wave function. But is there any such experiment where our observation is so sharp in both
time and space that it depends on a single trajectory? It seems to me that there are simple
(in concept) experiments now on the borderline of feasibility, which are capable of testing
this rather fundamental issue. In fact, they do not differ in principle - or even in the
relevant dimensions and resolving power - from Pasteur's microscopes.

6. What is a Free Electron?

We have long been intrigued with the fact that in applications of quantum theory, in our
equations we write only wave functions 1jJ, either explicitly or implicitly (as matrix elements
seATIERING OF LIGHT BY FREE ELECTRONS 7

between such wave functions). But in the interpretive words between those equations we
use only the language of point particles. Even the Feynman diagrams are a part of that
inter-equation language, depicting particles rather than waves.
Thus the wave-particle duality is partly an artifact of our own making, signifying only
our own inability to decide what we are talking about. But the predictions of observable
facts come entirely from wave functions 1/;(x, t) = r(x, t) exp[i4>(x, t)]; and not merely the
magnitudes 11/;1 2 = r2, but even more from the phases 4>(x, t). Then if anything in the
mathematics of QM could be held to represent some kind of reality, it is surely the complex
wave function itself, not that point particle imagined to be hiding somewhere in it, but
which plays no part in our calculations. This is just Schrodinger's original viewpoint.
The idea that a free electron is something more like an amoeba than a point particle
was suggested by David Bohm (1951); but there he was only trying to help us visualize the
mathematics of wave packets. Here we want to endow that suggestion with physical reality
and suppose, with Schrodinger, that a wave packet 1/;(x, t) is not merely a representation
of a state of information about an electron; but a physically real thing in its own right,
with a shape and internal moving parts that are capable of being changed by external
interactions and observed by us. The arguments that were raised against this picture long
ago (spreading of the wave packet, etc.) are easily answered today, as we shall see. The
spreading wave packet solution does not, in our theory, describe the physical free electron;
among all solutions of the Dirac equation it represents a set of measure zero.
This thinking -long on the back burner, so to speak - was moved to the front burner by
an incident that occurred at the 1977 meeting on free electron lasers at Telluride, Colorado.
I gave a talk (Jaynes, 1978) on the general principles for generating light from electrons,
which contained some speculation about possible explanations of the then much discussed
'blue electron' effect reported by Schwartz & Hora (1969). Here 50 kev free electrons are
irradiated by blue light from an Argon laser, then drift 20 cm and were reported to emit
the same color blue light on striking an alumina screen which is normally not luminescent.
It requires very little thought to see that such an effect cannot be accounted for by mutual
coherence properties of different electrons, which would require impossible collimation of the
electron beam; somehow, each individual electron must be made to carry the information
about the light wavelength, for a million light cycles after irradiation.
My calculation showed that interaction with laser light can perturb the wave packet of
a free electron, 1/;0 ( x, t) -+ 1/;0 + 1/;1, so that it is partially separated into a linear array of
smaller lumps, making the transverse density profile

look somewhat like a comb, with teeth [those internal moving parts] separated by the light
wavelength .Ai here A is proportional to the light amplitude and the cosine of a polarization
angle. Then when all these lumps, moving in the z-direction, strike a screen simultaneously,
there is in effect a pulse of simultaneous current elements with that separation, which can
act like an end-fire array and radiate light in the x-direction with the original wavelength
,\ (but, of course, with a much broader spectrum, whose width indicates the lateral size of
the electron wave packet 11/;01 2 in the x-direction).
To my astonishment, Willis Lamb objected strongly to this, saying "You don't under-
stand quantum theory. The electron is not broken up into many little electrons; the lumps
g E. T.JAYNES

are only lumps of probability for one electron. There is no interference effect in the radia-
tion emitted when the electron is suddenly decelerated, because the electron is in reality in
only one of those lumps."
This is a beautiful example of that schizoid attitude toward reality that believers in QM
are obliged to develop; for he believed at the same time that if an electron wave packet is
broken up into separated lumps by passing through the standard two slit apparatus, there
is interference between those lumps, making the standard electron diffraction pattern (and
showing, at least to me, that the electron was in both lumps simultaneously).
I was so taken aback by Willis's objection that I then did a conventional QED calculation,
which showed to my satisfaction that standard QED does predict the interference effect that
I had obtained so much more easily from a semiclassical picture. In fact, my calculation
predicted just the dependence on polarization and drift distance that Schwartz and Rora
reported seeing.
Other experimentalists have insisted that the effect does not exist (or at least could
not have been observed because of impossible requirements for collimation of the electron
beam) and it is not considered respectable to mention it today. But according to our wave
packet theory, the collimation should not matter; perhaps their failure to confirm the effect
was due just to their concentration on the irrelevancy of collimation, while the essential
thing was that Schwartz's electron gun, with its small holes, produced electrons in wave
packets of the right size.
If the effect does not exist, this would seem to be a major embarrassment for quantum
theory; surely, predicting so easily an effect that does not exist is just as bad as failing to
predict an effect that does exist [as someone put it at the time, if the effect is confirmed
it will become known as der Schwartz-Hora Effekt; if not, it will be die schwarze Aura].
In any event, if the effect was not reproducible at the time - for whatever reason - the
technology available today might overcome the old difficulties.
What has remained from this incident is the picture of the wave packet of a free electron
as something with a physical meaning, its size and shape in principle measurable. For
example, an electron in a wave packet ten microns long is physically different from one
in a wave packet two microns long; a spherical wave packet electron is physically different
from one a cigar-shaped one, and those differences should be observable in experiments
now becoming feasible.
This issue now suggests our proposed experiment. Eventually we shall return to Zitter-
bewegung, but for the time being the only issue before us is: Does interference exist between
light waves scattered from different parts of a free electron wave packet? If it does not, then
something like the Lamb-Hestenes picture must be correct; if it does, then a whole new
world of observable physical phenomena is opened up. As the old 1977 calculation showed
in one case, quantum theory predicts a mass of very detailed, observable, new phenomena.
They would make a free electron, when prepared in various sizes and shapes by previous
irradiation, even more versatile in behavior than an amoeba.
In fact, that technology noted in the Introduction has already indicated something of
the possibilities here. The electron diffraction microscope is able to reveal to us an amazing
variety of fine detail. All this information is contained somehow in the electron wave
functions; but it is surely not in coherence properties of different electrons, which could
never be collimated well enough for that. Each individual electron must be carrying, in
its phases (i.e. the distortion of its wave fronts) a vast amount of information about what
SCAITERING OF LIGHT BY FREE ELECTRONS 9

it has passed through. It seems to us that if electrons lacked this plastic, amoeba-like
character, electron microscopes would not work.
Consider, then, the exact analog of Pasteur's observation of a microbe: scattering of
light of wavelength A by a free electron which we represent by a wave packet of dimensions
perhaps 2A - lOA. If we are to see any interference between different parts of the wave
packet, then that hypothetical point particle must be in two different places at the same
time (or at least, at two places with spacelike separation). But it is easy for a continuous
wave structure to give such interferences.

7. Relativistic Basis of the Nonrelativistic (NR) Schrodinger Equation.

To define the proposed experiment it will be sufficient to use the non-relativistic spinless
Schrodinger equation, if we note first how and under what circumstances it can arise from
the relativistic Klein-Gordon equation satisfied by each Dirac wave component:

V24> _ ~ 8 24> = (~)2 4> (1)

c2 84>2 n
This has plane wave solutions of the form

4>(x, t) = exp(ik. x - wt) (2)

with
(3)

and thus Iwl ~ mc 2 In. The possible frequencies of propagating waves lie into two ranges sep-
arated by 2mc 2 In, which is just the ZBW frequency. Frequencies Iwl < mc 2 In correspond
to waves evanescent in space. Now given any solution of (1) which has only frequencies in
one of those propagating ranges, say exp( -iwt) with mc2 In ::; w, we can view the solution
as a rapid oscillation at frequency mc 2 In, modulated by an envelope function which may
be slowly varying. To separate them, make the substitution

2
4>(x, t) = 1jJ(x, t) exp ( - imc
-n-t ) (4)

whereupon 1jJ(x, t) is found to satisfy the equation

.. n2 2 n2 ••
tn1jJ = - 2m V 1jJ + 2mc21jJ (5)

which is exact. Now if 1jJ(x,t) contains frequencies up to order II, the two time derivative
terms are in approximately the ratio

(6)

and if this is small compared to one, we have the NR Schrodinger equation

(7)
10 E. T.JAYNES

but we now understand when it applies and what it means. It describes the slow 'sec-
ular' variations in the envelope when the relativistic wave function has only components
exp( -iwt) with no admixture of terms exp( +iwt).
The point we stress is that solving the NR Schrodinger equation does not give us an ap-
proximation to an arbitrary relativistic solution; but only to those particular solutions which
contain no ZBW effects. But the solutions we have discarded in making this approximation
comprise in a sense the 'vast majority' of all possible relativistic solutions!
With this perhaps 'new' understanding, we may reexamine the conventional solutions of
(7) for our problem. It has the general initial-value solution

1/Jo(X,t) = J G(x - X') 1/Jo(x', 0) d3x' (8)

with the Green's function

G( x - X'; t)
m
= ( 27riht
)3/2 exp {imr2}
2ht (9)

corresponding to diffusion with an imaginary diffusion coefficient. In particular, with an

initial packet of RMS radius a,

1 )3/4 ( 2)
1/Jo(X,O) = ( -27ra
-2 exp-~
4a
(10)

we find the canonical spreading wave packet solution of our textbooks:

(11)

for which the probability density or normalized charge density is given by

1
I1/Jo (x , t)12 = ( 27r0'2
)3/2 exp {- ;0'22} (12)

with
(13)

so the packet grows with an ultimate spreading velocity which 'remembers' its initial size
compared to the Compton wavelength:

(~) final = (h~;c) (14)

Doubtless all of us were, as students, assigned the homework problem of calculating from
(13) how long it would require for some object like a marble to double its size (which now
seems to me a totally wrong conception of what quantum theory is and says).
SCATIERING OF LIGHT BY FREE ELECTRONS 11

8. Non-Aristotelian Scattering of Light by Free Electrons

We want to look at the wave packets of free electrons in the same way that Pasteur looked at
microbes. But conventional quantum mechanical scattering theory does not contemplate
this, being based on an older idea, namely Aristotle's theory of Dramatic Unity. This
prescribes that "The action must be complete, having
(1) a beginning which implies no necessary antecedent, but is itself a natural
antecedent of something to come,
(2) a middle, which requires other matters to precede and follow, and
(3) an end, which naturally follows upon something else, but implies nothing
following it." (Taylor, 1913)
Conventional scattering theory follows this plan perfectly, presupposing an initial state in
which the particles involved are propagating as free particles but we ask not whence they
come, an intermediate state in which the action takes place, and a final state in which they
again propagate as free particles and we ask not where they go.
Note that the conventional textbook arguments (the Heisenberg ,-ray microscope, etc.)
warning us that there is an uncertainty principle making it impossible to do so many
things, always presuppose Aristotelian scattering and draw those conclusions from overall
momentum conservation, coupled with a naIve 'buckshot' picture of a photon and ascribing
a separate ontological reality to the scattering of each individual photon. But the actual
mathematical formalism of QED, as developed afterward, does not have anything in it
corresponding to the concept of 'a given photon'.
It seems to us that these things should be pointed out in elementary QM courses; the
Heisenberg conclusion, far from being a firmly established foundation principle of physics,
is not an experimental fact at all, only an unverified conjecture which presupposes just the
things that we want to test here. It is high time that we found out whether it is true, and
our technology is just now coming to the point where the experiments are possible.
But the scattering theory that we and Pasteur need is non-Aristotelian, in that the
action does not have a beginning or an end at any times that are relevant to what we
observe; rather we have something akin to a slowly changing nearly steady state, the incident
light constantly bathing the electron or microbe and the scattered radiation constantly
proceeding from it.
This makes an important technical difference, in that principles like overall momentum
conservation, although we do not deny them, do not have the same application that they
have in conventional scattering theory. This is not a handicap; on the contrary, it means
that we, like electron microscopists, shall be able to see details that Aristotelian scattering
theory does not describe. Requiring that the action be complete before we observe anything
greatly restricts what one could see.
In the above we have used the notation '1f;o to denote solutions of the free-particle equation
(7). In a transverse EM field, the minimal coupling ansatz makes the replacement p ->
p - (e/e)A, giving in first order
. p2 e
ili'1f; = 2m - me (A· p)'1f; (15)
To see what this interaction term means physically, consider a classical point electron in
the same EM field: e .
mv = eE = --A e
(16)
12 E. T.JAYNES

or, in a gauge where A = 0 when v = 0,

e
v(t)=--A(t) (17)
me
as in the London theory of superconductivity (where this equation accounts for a great deal
of that phenomenology noted above). Therefore, we have

~ A· p'I/J = -v· p'I/J = in (v. 'i1) 'I/J (18)

me
and two terms of (15) combine thus

in'I/J. + -e A· p'I/J
me
(a)
= in -at + v· 'i1 'I/J (19)

which we recognize as the 'convective derivative' of hydrodynamics, so we have simply

(20)

which is exactly equivalent to (15). The change in 'I/J as seen by a local observer moving at the
classical electron velocity is just the free particle spreading! The external perturbation, in
first order, merely translates the wave function locally by the classical motion. Recognizing
this, we can write the solution and the current response immediately without any need to
work out perturbation solutions of (15). A fairly good solution of the initial value problem
for (15) is simply
'I/J(x, t) = 'l/Jo(x - J vdt, t) (21)

which we may visualize as in (Fig. 1).

C=___ A __ ~

Fig. 1. (A) A cigar-shaped wave packet 5 microns long, unperturbed. (B) The
same packet with light of wavelength), = 0.5 micron incident from the left, polar-
ized vertically. The undulations are moving to the right at the velocity of light.
SCATIERING OF LIGHT BY FREE ELECTRONS 13

To fix orders of magnitude, think of a wave packet a few microns in size, optical wave-
lengths of perhaps a half micron. Whenever the optical wavelength is small compared to
the size of the wave packet, then the perturbation converts an initially smooth wave packet
into something more like a caterpillar than an amoeba, with undulations moving forward
at the speed of light (of course, the size of the undulations is greatly exaggerated in Fig. 1;
they are the same size as the displacement (xo = eA/mcw) of a classical point electron,
and amount only to perhaps 10- 9 cm in rather intense laser light at optical frequencies).
Then the incident wave Ainc(X, t) induces a local current response

e e 2
J(x, t) = -Il/I(x,
c
tW v(t) = - -2 Il/II2 Ainc(X, t)
mc
emu (22)

and we note that the original Klein-Nishina derivation of the Compton cross-section, and
the Dyson (1951) calculation of vacuum polarization started from this same local current
response. To calculate the field scattered by it they used standard classical EM theory (thus
providing two early examples of the fact that QED does not actually use field quantization);
and we shall do the same.
Resolving the current and field into time fourier components:

J(x, t) = Jdw J(x,w) exp( -iwt)

211"
(23)

etc., the scattered field at position x' is

Asc(X',W) = J d3xJ(x,w) (
eiwr/C)
-r- (24)

where r == Ix - xii. Using (23) and making the far-field approximation this becomes

(25)

where R is the distance from the center of gravity Xo of the wave packet to the field point
x', and k is a vector of magnitude w/c pointing from Xo to x'. Finally, noting that the
coefficient is the classical electron radius TO = e2 / mc2 and taking the incident field as a
plane wave with propagation vector ko:

(26)

we have the scattered wave in the direction of k:

(27)

where
(28)
14 E. T.JAYNES

is the space fourier transform of the wave packet density. Note that p(O) = 1 is the
statement that the wave function is normalized. The differential scattering cross-section
into the element of solid angle df! is then

(29)

where (J = (Ao, k) is a colatitude polarization angle. As a check, if the wave packet size
a« A, then p(ko - k) -> 1, and we have for the total scattering cross-section

(Y = f 2· 2
df! ro sm (J
81[" 2
= ""3 ro , (30)

the usual classical Thompson cross-section. Put most succinctly, the scattering experiment
measures

[ Cross section of wave packet ] = I (k -kW

(31)
Cross section of classical electron p 0

so if the experiment is feasible and the wave function is something physically real, one
should get information about the size and shape of the wave packet from the directional
properties of the scattered light. If the true object is only a point particle and the wave
function represents only a state of knowledge about its possible positions, then at optical
frequencies one should see only the classical cross-section at all scattering angles, whatever
the size of the wave packet.
Scattering oflaser light by free electrons has been observed experimentally - as long ago as
1963 experimenters were reporting it - but to the best of our knowledge they have not looked
for this effect. The first order of business is simply to verify whether these interference effects
are or are not observable; but we have not yet specified what size wave packets are to be
expected. One experimental clue is provided by electron interference downstream from a
fine Wollaston wire; how thick can the wire be while we still see interference? The answer
appears to be about 4 microns; thus we would conclude that the wave packet must be about
this size.
The result is interpreted differently by those experimentalists who talk in terms of col-
limation of electron beams instead of wave packets; they represent an electron by a plane
wave and view the 4 microns as the coherence range of plane waves with slightly different
directions. Of course, a wave packet 1j;(x) can be fourier analyzed and the components 'lI(k)
would have that meaning; but the spread of k-values in a wave packet has nothing to do
with the spread of velocities of the different electrons in the beam; it appears to us that
they fail to see this distinction. In our view, the 4 microns must be seen as a property of a
single electron, not an indication of lack of parallelism of trajectories of different electrons.

9. Back to Zitterbewegung
We propose that, while ZBW is such a high-frequency effect that it does not play any great
role in the scattering of optical frequency light, nevertheless ZBW is the origin of forces
that modify the electron wave packet, so that the 'spreading wave packet' solution (11)
does not describe the real free electron. Indeed, we have seen from the above derivation
that the conventional spreading wave packet solution describes only the NR approxima.tion
SCAITERING OF LIGHT BY FREE ELECTRONS 15

to a very special relativistic motion in which ZBW effects are absent. But the description
of a real free electron must use the relativistic Dirac equation, and include the effects of
the interaction of the electron with its own electromagnetic field.
Mathematically, in early QED this interaction diverged, and in no theory can it be
considered a negligibly small perturbation. Conceptually, as Einstein warned us long ago
(Jaynes, 1989), neither the electron nor the field can exist without the other and their
interaction is never turned off, so it is not possible to describe either correctly if we ignore
the other. But whenever ZBW oscillations are present this represents a current oscillating
at frequency w '" 2mc2 In, whose electromagnetic field reacts back on the electron and
modifies its behavior.
A current J(x',t') generates a field A(x,t) = J D(x - x',t- t')J(x',t')d3 x'dt'. This in
turn exerts a local force density on the current J(x, t) given by F = J(x, t) x [V'x A(x, t)]. If
the currents at both positions have the same ZBW frequency, then there is a time-average
secular force that depends on their relative phases and can be either attractive or repulsive.
We emphasize again that these speculations are quite modest and respectable compared to
those utterly wild ones noted above, which dominate present theoretical physics. The effect
we are proposing is not strange or new; it is predicted by standard relativistic Quantum
Theory (which does not forbid the use of wave packets in our calculations); only it was not
heretofore overtly mentioned. We are only trying to anticipate, by physical reasoning, what
observable effects this secular force might have.
One reason why the effect was not seen clearly before is that the current-current interac-
J
tion J J(x)D(x - y)J(y) was perceived as only an energy term. That it also represents a
force could, of course, have been found by carrying out a variation {ja of the size of the wave
packet, but here the physical outlook of the calculators prevented them from doing this.
One did not think of a free electron wave packet as a physically real thing that might be
distorted by local forces; scattering theories calculated only matrix elements between those
Aristotelian plane-wave states. Thus the actual size and shape of a wave packet never got
into the calculations.
Why do we include only transverse fields here? We think the answer is that, because of
fine features of the Dirac equation not presently in view, these are the only fields actually
generated by the Dirac current. In any event, to include a longitudinal interaction in the
present calculations would introduce a Coulomb repulsion between different parts of the
wave packet, which would probably be much stronger than the transverse forces studied
here. That would cause the wave packet to explode in a time far shorter than the Gaussian
spreading time of (13).
But we knew from the start that we must never include a coulomb interaction of an
electron with itself. For example, if in the hydrogen atom we interpret p(x, t) = eJ1jJ(x, t)J2,
we must still use the Hamiltonian H = p2/2m + V( x) including only the coulomb field
Vprot = -e 2 /r of the proton. We must not include a term Vint = e J(p(x')/r)d 3 x', or our
hydrogen atom would be completely disrupted into something qualitatively different from
the atom that we know experimentally. In quantum theory, longitudinal and transverse
fields have quite different properties (and, perhaps, different physical origins). This issue
requires more study, both in our theory and in conventional QED (where we have also
managed only to get around it in a pragmatic sense, not actually resolve it theoretically).
16 E.T.JAYNES

10. Forces due to Zitterbewegung

Our calculation is not different in principle from those that Asim Barut and Tom Grandy
do, trying to find solutions of the coupled Maxwell and Dirac equations. However, we are
looking at a different phenomenon for which specific solutions may be much harder to find,
so in the present work we concentrate on getting a clear picture of the physical mechanisms
at work; only after this is well understood would we be ready to tackle the explicit solutions
that we shaIl demand eventually.
Our current comes from the Dirac equation in the standard way: jf-L = e {yyf-L'Ij;. But if the
wave function is an admixture of what are usuaIly called positive and negative energy solu-
tions [which we think should be called only positive and negative frequency solutions], this
has high frequency oscillations at the ZBW frequency w = 2mc 2 /n. For a monochromatic
component of the current, the time component of jI' is determined by charge conservation
from the ordinary vector components J(x, t) = {j1, i, P}, which therefore determine the
entire radiated field. In the present case, we consider the oscillating current solenoidal:
'V . J = 0, so only transverse fields are generated.
Consider now two smaIl regions of space d3xI, d3x2 both inside the wave packet but
separated by a distance r == IX2 -xII large compared to the ZBW wavelength n/2mc ~ 10- 10
cm. Denote the current elements in these by

J(x1,t)d 3x1 = It cos(wt+ 4>1)

(32)
J(X2' t) d3X2 = lz cos(wt + 4>2)
The current element II generates at the position X2 an EM field

(33)
where the 'propagator' is

A.) = cos(wt - kr + 4>1)

I( r,t,'!-'1 (34)
- .
r

and, as usual, k == w/c. This exerts on d3x2 an element of force

But
'VI =aI 'Vr = k sin(wt - kr + 4>t} n
ar r (36)
where n == 'Vr is a unit vector pointing from Xl -+ X2, and we used the aforementioned long
distance condition kr > > 1 to discard a near field term that is appreciable only at points
within a ZBW wavelength of the current.
Now the force density at X2 takes the form

k sin(wt - kr + 4>d cos(wt + 4>2) I ( I)

F ( X2, t ) = 2 X nx 1 (37)
r
This has terms oscillating at frequency 2w and constant ones: to get the time average over
a ZBW cycle, note that

sin(wt - kr + 4>1) cos(wt + tP2) = (1/2) sin(tPl - tP2 - kr) (38)

SCATTERING OF LIGHT BY FREE ELECTRONS 17

so the time average force density seen at X2 due to the current element at Xl is

(39)

Its component along n depends only on the product of transverse components of the cur-
rents:
n . F = k sine <PI - <P2 - kr) [( h . 12) - (n . It) (n . h)1 ( 40)
2r
which can be either positive (repulsive) or negative (attractive) depending on the relative
phases. Thus ZBW currents generate secular force terms which must modify the wave
packet of a free electron, and give it a tendency to expand or contract, in addition to the
conventional spreading in (11).
There is also a secular term from the ZBW electric field; but this is purely transverse
to n and does not contribute to attraction or repulsion. Indeed, since it is unnatural to
suppose that two isolated current elements can produce a net torque on themselves, the
time average of the electric field forces must just cancel the transverse component of (39):
n X Ftotal = 0, and (40) gives the entire force. Note that the phases <P can vary with position
along r in such a way that there would be virtually no external radiation in the n direction,
while the total phase term (<PI - <P2 - kr) remains mostly at values which give attractive
internal forces; this hints at a more general stability property.
But this is of no interest unless the ZBW forces are large enough to compete with the
spreading tendency exhibited in (11). Now it is evident already that these forces are orders
of magnitude stronger than the corresponding ones due to the same currents at optical
frequencies, because from (36) the magnetic field in the radiation zone generated by a given
current is proportional to the frequency. Let us estimate their general magnitude, very
crudely.
Supposing a wave packet with dimensions a, a normalized wave function will have a
magnitude indicated by 1'1/'112 a 3 ~ 1. Therefore the current densities J = e ifi"fiJ'I/'I might
conceivably be as large as about e/a 3 ; let us suppose they are a tenth of that, about
e/lOa 3 • Then consider the wave packet broken up into two volume elements d3 x ~ a 3 /2,
separated by an average distance of about a/2. The current coefficients II, 12 above are of
the order of (e/10a 3 )(a3 /2) = e/20. If phases are optimal, the attractiveforce (40) between
them could be as large as F ~ kP /a = ke 2 /400a. The binding energy of these parts to
each other would then be of the order of Fa/2 = (1/800) mce 2 /n = mc 2 /(800 . 137), or
about 4.5 ev. The parts would also have some internal binding energy of their own.
We see that the ZBW forces are easily strong enough to do the job we require of them;
one can depart considerably from the 'optimum' conditions and still have 0.5 ev of binding
energy, enough to stabilize the packet. We might put it thus: in a kind of bootstrap
operation, the wave packet digs its own potential well and is confined in it (unable to
spread), while remaining free to move about and carry the well along with it, like a turtle
trapped in its own shell.
There is a close analogy - perhaps more than an analogy - to some well known things
in solid-state theory. As in Fig 2, the propagating frequency ranges are the 'conduction
bands' of space, while the 'forbidden band' in which solutions oscillatory in time are spa-
tially evanescent, lies between them. But in the solid-state case, any additional potential
which perturbs the periodic lattice potential, can make additional solutions possible in the
18 E. T.JAYNES

forbidden band; the localized bound states lying just outside the conduction band, caused
by donor or acceptor impurity atoms. This is depicted io Fig 2; and having seen it, we find
it hard to avoid supposing that free positrons must be represented by short lines just above
the lower conduction band. Some physical arguments in support of this view of things are
given below.

Figure 2. The solid-state analogy. The 'conduction' or propagating regions

have frequencies w > 11, w < - 11, wlIile the 'forbidden' region lies between
them. Then the physical free electron is thought of as corresponding to local-
ized but mobile states (A, B, C), with wave function oscillating at just below
the propagating frequency 11 == mc 2 /h.

Given this picture, let us close by indulging in some perhaps wild and free speculation.
Evidently, there is a bewildering variety of possible solutions here; presumably, many dif-
ferent initial conditions can be realized, for which the subsequent motions do almost every
conceivable thing. But attractive forces always win over repulsive ones because they lower
the energy, and then there is no way to escape from them; the energy needed to expand the
packet against attractive forces has been radiated away. So after some time the solutions
must, inexorably, settle down to some final stable nonradiating limit cycle.
This, we suggest, means that the original arguments against Schri:idinger's interpre-
tation of the wave function no longer hold; the real free electron wave packet does not
spread indefinitely, but settles down into some steady state of definite size (perhaps about
4 microns) in which the attractive ZBW forces just balance the spreading tendency in (ll).
This would be the experimental 'aged' free electron, loosely analogous to the 'dressed'
particle of current theory, but quite different in essential properties.
The final stable solution will be easier to find than any particular transient one
because we have some guiding principles for this. Firstly, the balancing of attractive and
spreading tendencies calls for an analysis rather like Einstein's original argument relating
mobility and diffusion constant. Secondly, mechanical stability against arbitrary small
deformations gives us a variational principle. Thirdly, the condition of no net external
radiation imposes conditions on the possible stable current distributions. Finally, all this
must satisfy the Dirac equation, in such a way as to just maintain this steady current
oscillation.
In fact, a very general electro--mechanical theorem tells us that the second and
third principles are closely related, and gives us the additional information that the actual
SCATIERING OF LIGHT BY FREE ELECTRONS 19

oscillation frequencies ±n in the final steady packet will be slightly less than mc 2 In, the
difference indicating the 'binding energy' of the wave packet relative to a plane wave as
depicted in Fig. 2.
The condition for mechanical stability is the same as the condition for no net external
radiation. To state this, let a current distribution j(x, t) be confined to a finite spatial
volume, and take the fourier transform

J(k,w}=. J d3 xdtj(x,t)exp[(i(wt-k.x)]

Then from standard EM theory, the necessary and sufficient condition for no external
radiation is k X J(k,w) = 0 whenever clkl = w. When this condition is not met, the
radiation can exert relatively strong forces on external objects, inducing changes in their
states; thus stability is a joint property of a system and its surroundings.
Why do we not see fragments of electrons (leptoquarks?) flying about, produced by
collisions, slits, etc? Perhaps we do, but are not mentally prepared to recognize them. But
in those experiments there are always very many electrons involved, and in those fragments
the nonradiation condition is far from satisfied; they interact resonantly with neighboring
fragments to bring them back together into the stable 'aged' electrons before they arrive at
our detectors (and indeed, very few electron detectors actually measure the charge of the
things they are detecting). Again, even though the initial phases are random, attractive
forces always win out because their effects reinforce, drawing the pieces together and thus
increasing the attractive forces. Note also that the ZBW forces are long range compared to
Coulomb forces, falling off only as lIT; thus fairly distant fragments can interact strongly.
On any slow perturbation of an oscillating system, there is a principle of stationary
action, which makes changes of energy proportional to changes of frequency. This was
known to Lord Rayleigh in acoustical systems, and was used by Wien in the theory of
black-body radiation, and by Einstein and Born as the 'adiabatic principle' in the early
days of quantum theory.
In the writer's Neoclassical theory of electrodynamics (Jaynes, 1973) we showed a
classical Hamiltonian for which this result is not merely an approximation for slow perturba-
tions, but is an exact conservation law; and our present equations of motion can be derived
from a Hamiltonian of the same form. In effect, it is a 'new' constant of the motion never
contemplated in classical statistical mechanics because it is not a conservation of energy
or momentum. It is a law of conservation of action E I wand much of the phenomenology
of quantum theory (which Bohr saw as revealing 'the inadequacy of classical concepts') is
explained by it, very easily.

11. Conclusions
The full story that we have started here is much too long to tell in a single article, but
let us take a glimpse at what lies ahead, if these speculations prove to have some truth in
them. Of course, it is too much to expect that every detail of our present thinking will be
correct; but that is unnecessary. Indeed, even if all our speculations prove to be wrong,
these ideas may stimulate new constructive thought in the right direction, which would not
have happened without them.
We have developed a physical picture of ZBW as performing two essential functions
in the world. Its transient solutions provide the "dither" that initiates changes of state;
20 E. T.JAYNES

while its limit cycles provide the stabilizing forces that hold particles together. As early as
1904 Poincare perceived the need for these forces, but to the best of our knowledge every
previous attempt to find them has sought them in static rather than oscillatory models.
The great 'advantage' of high frequency oscillations is that a given current generates much
stronger EM fields, so that quite large energies and forces result.
Once one has seen this much, a great variety of new effects can be seen, which
begin to suggest simple causal explanations for many of those mysterious "quantum effects"
previously thought to defy explanation in classical terms. But it is evident that a very large
amount of hard work remains to be done before much of this picture can be realized.

12. References
Bell, J. (1987), Speakable and Unspeakable in Quantum Mechanics, Cambridge University
Press, U.K.
Bell, J. (1990), "Against 'Measurement"', Physics World, Aug. 1990, pp. 33-40.
Bjorken, J. D. & Drell, S. D. (1964), Relativistic Quantum Mechanics, Vol. 1, McGraw-Hill
Book Co., N. Y.
Bohm, D. (1951), Quantum Theory, Prentice-Hall, Inc., Englewood Cliffs, N. J.
Cartan, E. (1966), Theory of Spinors, Hermann, Paris.
Dyson, F. J. (1951), Notes on QED, Cornell University
Heisenberg, W. (1958), Daedalus, 87, 100.
Hestenes, D. (1966), Space-Time Algebm, Gordon & Breach, N. Y.
Jaynes, E. T. (1957), "Information Theory and Statistical Mechanics II", Phys. Rev. 118,
pp. 171-190.
Jaynes, E. T. (1973) "Survey of the Present Status of Neoclassical Radiation Theory", in
Coherence and Quantum Optics, L. Mandel and E. Wolf, Editors, Plenum Publishing
Company, N. Y., pp. 35-81.
Jaynes, E. T. (1978) "Ancient History of Free-Electron Devices", in Novel Sources of Co-
herent Radiation, S. F. Jacobs, Murray Sargent II, & M. O. Scully, Editors, Addison-
Wesley Publishing Company, Reading MA; pp. 1-39.
Jaynes, E. T. (1986) "Predictive Statistical Mechanics", in Frontiers of Nonequilibrium
Statistical Physics, G. Moore & M. Scully, Editors, Plenum Press, N. Y., pp. 33-55.
Jaynes, E. T. (1989) "Clearing up Mysteries: the Original Goal", in Maximum Entropy
and Bayesian Methods, J. Skilling, Editor, Kluwer Academic Publishers, Dordrecht-
Holland, pp. 1-27.
Jaynes, E. T. (1990) "Probability in Quantum Theory", in Complexity, Entropy, and the
Physics of Information, W. H. Zurek, Ed., Addison-Wesley, Redwood City CA,
pp.38-403.
Schwartz, H. & Hora, H. (1969); App. Phys. Lett. 15, 349. For a summary and extensive
bibliography oflater developments, see H. Hora, II Nuovo Cimento 26, 295 (1975).
Taylor, H. O. (1913), Ancient Ideals, MacMillan, London; p. 288.
ZITTERBEWEGUNG IN RADIATIVE PROCESSES

David Hestenes
Physics Department
Arizona State University
Tempe, Arizona 85287
USA

ABSTRACT. The zitterbewegung is a local circulatory motion of the electron presumed

to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac
theory shows that this interpretation can be sustained rigorously, with the complex phase
factor in the wave function describing the local frequency and phase of the circulatory
motion directly. This reveals the zitterbewegung as a mechanism for storing energy in a
single electron, with many implications for radiative processes.

1. INTRODUCTION

Schrodinger was never satisfied with quantum mechanics. He was especially

disturbed by the absence of a clear physical mechanism for radiative processes. This issue
is of renewed interest today, for powerful new experimental techniques make it possible to
investigate radiative processes with unprecedented resolution and precision. Indeed, the
discovery of multiphoton ionization and related phenomena has already upset the
conventional wisdom about the photoelectric effect [1]. It seems though, that the initial
confusion has been cleared up to the satisfaction of most theorists in the field, and
extensive theoretical work has produced explanations for most of the new phenomena
brought to light in high intensity laser-atom interactions. All this has been accomplished
without any fundamental changes in theory, so some regard it as another triumph of
quantum electrodynamics. However there are good reasons to be doubtful.

Most explanations in quantum optics are phenomenological in the sense that each is
based on some ad hoc hamiltonian tailored to the problem at hand. A truly fundamental
explanation, of course, must be derived from the Dirac equation. To be sure, that is not
always possible or practical. But it is essential if anything truly new about radiating
electrons is to be learned. Phenomenological models for laser-electron interactions are
incapable of distinguishing collective activity from the radiative behavior of single
electrons. Consequently, I believe, opportunities for discovering fundamentally new
knowledge about radiative processes have been missed.

In this note I will argue that a generally overlooked feature of the Dirac theory, the
zitterbewegung (ZBW), is the key to understanding radiative processes, and genuinely new
physics is to be expected from studying its implications theoretically and experimentally.
The argument has three main steps. The first step is a purely mathematical reformulation of
21
D. Hestenes and A. Weingartshofer (eds.), The Electron, 21-36.
© 1991 Kluwer Academic Publishers.
22 D.HESTENES

the Dirac theory, so it should be uncontroversial. Nevertheless, the results are so surprising
and unfamiliar that most physicists are taken aback. Briefly, the reformulation eliminates
superfluous degrees of freedom and reveals a hidden geometric structure in the Dirac
theory. The imaginary factor in in Dirac's equation automatically becomes identified with
the electron spin, and the electron wave function has a geometrical interpretation wherein
the spin is the angular momentum of a local circulatory motion, that is, the ZBW. This
raises serious questions about the interpretation of quantum mechanics which are discussed
in the second step of the argument. The fmal step is concerned with a qualitative discussion
of the ZBW in radiative processes and the prospects for new physics.

The key to recognizing the geometric structure of the Dirac theory is reformulating it
in terms of spacetime algebra, a Clifford algebra providing the optimal encoding of
spacetime geometry in algebraic form. Only a "bare bones" account of the spacetime
algebra and the ZBW structure of the Dirac theory can be given here. Much more is
provided in [2] and the many references therein. Hopefully, the present account can serve
as an intelligible introduction to the other articles in these proceedings which employ the
spacetime algebra. However, there is no getting around the fact that genuine insight into
any mathematical system requires a good deal of time and effort.

2. SPACETIME ALGEBRA.
The spacetime algebra (STA) is generated from spacetime vectors by introducing a
suitable rule for mUltiplying vectors. We begin with the usual Minkowski model of
spacetime as a 4-dimensional vector space 9r{4. In mathematical parlance, STA is the real
Clifford algebra of the Minkowski metric on 9r{4. More specifically, STA is a real
associative algebra generated from 9r{4 by defining an associative product on 9r{4 with the
special property that the square of every vector is scalar-valued. I call this product the
geometric product to emphasize the fact that it has a definite geometric interpretation which
fully characterizes the geometrical properties of spacetime. The geometric product uv of
vectors u and v can be interpreted by decomposing it into symmetric and skew symmetric
parts; thus,
UV=U'V+UI\V, (1)
where two new products have been introduced and defined by

U'V = t(uv + vu) = V· U, (2)

U 1\ V = t(uv - vu) = -v 1\ U. (3)
It follows from the definition of the geometric product that u·v is scalar-valued; indeed, it is
the usual inner product defined on Minkowski space. The quantity U 1\ v is called a
bivector, and it represents a directed plane segment in the same way that a vector represents
a directed line segment.
In these proceedings, Ed Jaynes [3] confesses to a long-standing "hang-up" over
Eq. (1) which has prevented him from getting into STA. As I have the greatest respect for
Ed's intellect and other physicists may suffer the same hang-up, I shall attempt a cure
forthwith. But first, some further discussion will be helpful in preparation.
ZIITERBEWEGUNG IN RADIATIVE PROCESSES 23

Note that for orthogonal vectors (as defined by u·v = 0), Eq. (1) gives
uv = u" v = -vu. Thus, the geometric relation of orthogonality is expressed algebraically
by an anticommutative geometric product. Similarly, collinearity is expressed by a
commutative geometric product. For in that case u" v = 0, Eqs. (1) and (2) gives
uv = U· v =vu. In general, (1) shows that the geometric product represents the relative
direction of any two vectors by a combination of commutative and anticommutative parts.
To facilitate comparison with the Dirac matrix algebra, it is convenient to

characterize the structure of STA in terms of a basis. Let {YJl; J1. = 0,1,2,3} be a
righthanded orthonormal basis for 9I{4 with timelike vector Yo in the forward (future)
lightcone. In terms of this basis the spacetime metric is expressed by the equations

Y02 = 1 = -Y/ for k = 1,2,3, (4)

and

YJl . Yv =0 for J1. "* v. (5)

Other basis elements of STA, each with a definite geometric interpretation, can be generated
from the YJl by multiplication. For example, Y2 Y1 = Y2 " Y1 is a bivector of unit magnitude,
as expressed by

(6)
Returning now to Ed's hang-up, he believes that the validity of Eq. (1) requires
some new concept of addition. On the contrary, the concept of addition in Eq. (1) is
identical to the one physicists are familiar with in complex numbers. Indeed, Eq. (1) can be
read as a separation of a complex number z = uv into real and imaginary parts. To make
that obvious, suppose that u and v are spacelike unit vectors subtending an angle e,
so that
U· v = -cose, where the minus sign is due to the negative signature. The area of the
parallelogram determined by u and v is given by sin e,so we can write u" v = i sin e,
where i is the unit bivector for the spacelike plane containing u and v.If Yl and Y2 compose
an orthonormal basis for that plane, then i = Y2 Yl and i 2 =-1. Thus, Eq. (1) assumes the
familiar form
e e
z = uv = - cos + i sin = -e-;8. (7)
Of course, this gives a much richer concept of complex numbers than the ordinary one. The
i has a twofold geometrical meaning: It is the generator of rotations in the plane, as can be
seen by solving (7) for v; thus,

(8)
It is also the unit directed area element for the plane. Of great importance to us later on will
be the fact that all Lorentz rotations are generated by the bivectors of STA. Although STA
enriches the concept of complex number, it employs the same old concept of addition.
24 D.HESTENES

Fonnally, addition is defined by the associative and commutative rules. When

adding complex numbers, these rules enable us to collect and concatenate real and
imaginary parts separately. The same is true when adding combinations of scalars and
vectors in STA. I suspect that the underlying cause of Ed's hang-up is the worry that
scalars and vectors will get inextricably mixed-up under addition. But addition doesn't mix-
up real and imaginary parts of complex numbers. Why? Because they are linearly
independent! That, I believe, is the concept that Ed overlooked in this context. Scalars and
vectors don't get mixed-up under addition because they are linearly independent. Indeed,
the addition of scalars to vectors is equivalent to augmenting the vectors with an additional
component. But if scalars and vectors cannot be concatenated, why add them at all? The
answer is the same as for complex numbers: Because multiplication intennixes them,
creating valuable new entities such as the spin representations of the rotation group.
There is a certain historical irony that a discussion like this should be necessary in
this day and age. The matter was already cleaned up more than a century ago. More than
150 years ago, William Rowan Hamilton worried that a complex number written as the
sum of a real and imaginary parts can have no meaning, because unlike things cannot be
added. The concept of linear independence had not been invented yet, but it was implicit in
his resolution of the problem: He showed that complex algebra is equivalent to a system of
operations relating pairs of real numbers. That insight helped gain general acceptance for
complex numbers. A decade later, when Hamilton invented the quaternions, the adding of
unlike things didn't bother him any more. Our terms scalar and vector were coined by
Hamilton to denote the two unlike parts of a quatemion (though Hamilton's vectors actually
correspond to bivectors in STA). Thus, as originally conceived, scalars and vectors were
added. Most physicists became familiar with quaternions from Maxwell's great Treatise on
Electricity and Magnetism (1873). This included J. Williard Gibbs, who developed the
standard vector calculus of today primarily by dismantling quaternions into separate scalar
and vector parts. A few generations later the physics community had forgotten about
quatemions, and young physicists like Ed were inculcated with the dogmatic proscription
against adding scalars and vectors. Trained incapacity! Fortunately, the true relation of
vector algebra to quaternions (which Gibbs and everyone else at the time had failed to see)
is perfectly clear within the broader perspective of STA. Indeed, as demonstrated in detail
elsewhere, both these algebraic systems are fully encompassed and integrated by STA.
Now let us return to discussing the structure of STA. The unit pseudoscalar for
spacetime is so important that the special symbol i will be reserved to represent it. Its
generation by the vector basis is expressed by

(9)
Geometrically, it represents the unit oriented 4-volume element for spacetime. Its algebraic
properties

? =-1, (10)

(ll)
ZITfERBEWEGUNG IN RADIATIVE PROCESSES 25

make it easy to manipulate. Multiplication of (9) by J() yields the pseudovector

(12)

Geometrically, this is the (directed) unit 3-volume element for a hyperplane with normal
Yo.
By forming all distinct products of the y~ we obtain a complete basis for STA
consisting of the 24 = 16 linearly independent elements

(13)

It follows that a generic element M in STA, called a multivector, can be written in the
expandedform

M = a + a + B + bi + /3i, (14)
where a and /3 are scalars, a and b are vectors and (with summation over repeated indices)

B=.lBJlVy
2 Jl I\y v (15)

is a bivector with scalar components BJlv. Ed should note that (13) implies that the STA is a
16-dimensionallinear space, so (14) is equivalent, with respect to addition, to a vector with
16 components. But multiplication is a different story.
The multivector Min (14) can be decomposed into an even part M+ and an odd part
M., as expressed by
M =M+ +M., (16a)

M+=a+B+/3 (16b)
M. = a + bi. (16c)
A multivector is said to be even (odd) if its odd (even) part vanishes.
For M in the expanded form (14), the operation of reversion in STA is defined by

M= a + a - B - bi + /3i. (17)

It follows that for any multivectors M and N,

(MN)" =NM. (18)

Essentially, reversion amounts to reversing the order of geometric products.

The relation of STA to the Dirac algebra is now easy to state. The Dirac matrices,
commonly denoted by the symbols ~, can be put into one-to-one correspondence with the
basis vectors denoted by the same symbols above. Then the algebra generated by the Dirac
matrices over the reals is isomorphic to STA. It follows that the geometric meaning
attributed to the vectors ~ and their products above is inherent in the Dirac algebra, though
it is scarcely recognized in the literature. This isomorphism completely defines the
geometric content of the Dirac algebra with respect to spacetime. It suggests also that the
26 D. HESTENES

representation of the ~ by matrices is irrelevant to their function in physical theory. This

suggestion is confinned in the next Section by casting the Dirac theory in terms of STA
with no reference at all to matrices.
The full Dirac algebra is generated by the Yfl. over a complex instead of a real
number field. The fact that the real field suffices to express the full geometric content of the
algebra suggests that the 16 additional degrees of freedom introduced by employing a
complex field instead are physically irrelevant. This suggestion is also confirmed in the
next Section by fonnulating the Dirac theory without them. Elimination of the irrelevant
H in the complex number field opens up the possibility of discovering a geometric
meaning for the H which occurs so prominently in the equations of quantum mechanics.
Indeed, equations (4), (6) and (8) show that STA contains many different roots of minus
one, including three geometrically different types. Each type plays a different role in the
Dirac theory.

3. GEOMETRY OF THE DIRAC THEORY.

In the language of STA, the Dirac equation can be written in the form

(19)

where

(20)

A = A,uJiL is the usual electromagnetic vector potential, and i is the unit bivector

(21)

The Dirac wave function ljI = 1jI(x) at each spacetime point x = x,uYfl. is an even multivector
with the invariant canonical form

lfI = (peifJ)t R, (22)

where i is the unit pseudoscalar, p and f3 are scalars and R satisfies

(23)
A brief proof that the above STA representation of the Dirac equation and wave function is
mathematically equivalent to the conventional matrix representation is given in the appendix
to Gull's article [4].
Equation (19) is Lorentz invariant, despite the explicit appearance of the constants
Yo and i= Y2Yl in it. These constants are arbitrarily specified by writing (19). They need
not be identified with the vectors of a particular coordinate system, though it is often
convenient to do so. The only requirement is that Yo be a fixed timelike unit vector, while i
is a spacelike unit bivector which commutes with Yo. Of course, the yo and i = Y2Yl in
ZITIERBEWEGUNG IN RADIATIVE PROCESSES 27

(19) are the same constants that appear in the expressions (25) and (27) below for the Dirac
current and the spin.
The most striking thing about (19) is that the role of the unit imaginary in the matrix
version of the Dirac equation has been taken over by the unit bivector i, and this reveals
that it has a geometric meaning. Indeed, equations (27) and (28) below show that in is to
be identified with the spin.
Equation (19) may look more complicated than the conventional matrix form of the
Dirac equation, but it actually simplifies and enriches the analysis of solutions by making
their geometric structure manifest, as is shown in the detailed calculations of KrUger [5].
The key result of the STA formulation is the invariant decomposition (22) of the Dirac
wave function. Its geometrical and physical significance is determined by its relation to
observables of the Dirac theory, which we specify next.
At each point x, the function R = R(x) in (22) determines a Lorentz rotation (i.e. a
proper, orthochronous Lorentz transformation) of a given fixed frame of vectors (Y,u} into
a frame (e,u = e,u(x)} given by

ell = RyJ?. (24)

In other words, R determines a unique frame field on spacetime. This frame field has a
physical interpretation.
First, the vector field

(25)
is the Dirac current, which according to the Born interpretation, is to be interpreted as a
probability current. Accordingly, at each point x, the timelike vector v = vex) = eo(x) is
interpreted as the probable (proper) velocity of the electron, and p =p(x) is the relative
probability (i.e. proper probability density) that the electron actually is at x.
Second, the vector field
n _ n
-IfIY31f1 = p-e3=ps (26)
2 2
is the spin (or polarization) vector density. The spin angular momentum S = Sex) is actually
a bivector quantity, related to the spin vector s by

n. n
·
S =!sv=2!e3eO =2ezel =2n RY Y R-
2 I (27)
Multiplying this on the right by (22) and using (23), one easily obtains

SIfI =! IfIY2 Y/l, (28)

which relates the spin S to the bivector Y2 Yin.

28 D. HESTENES

In general, six parameters are needed to specify an arbitrary Lorentz rotation. Five
of the parameters in the Lorentz rotation (24) are needed to specify the direction of the
electron velocity v and spin s. This also determines the plane containing ej and ez' as
shown in (27). The remaining parameter lP determines the orientation of e j and e2 in the
e2e j, plane. This can be expressed by factoring R into the form

(29)

where Ro is determined by the first 5 parameters just mentioned. The parameter lP is the
phase of the wave function, and here we have a geometrical interpretation of the phase. The
vectors ej, and e2 are not given a physical interpretation in the conventional formulation of
the Dirac theory, because the matrix formalism suppresses them completely. But they will
be given a kinematical interpretation when the ZBW interpretation is introduced below.

The factorization (22) of the wave function ljf can now be seen as a decomposition
into a 6-parameter kinematical factor R and a 2-parameter statistical factor (pe ifJ )1. The
parameter p is clearly a probability density. The physical interpretation of f3raises problems
which are yet to be fully resolved. Important insights into this issue are supplied by other
articles in these Proceedings. Boudet [6] describes formal properties of f3 in the geometry
of the Dirac theory. Kriiger [5] finds new solutions for the hydrogen atom with f3 = 0, in
sharp contrast to the strange properties of f3 in the Darwin solution. Gull [4] discusses the
essential role of f3 and its relation to spin in matching boundary conditions at a potential
step, including the Klein paradox. My own expectation is that a full understanding of f3 will
come only from elaborating the statistical interpretation of the the Dirac theory discussed
below. That is why I have relegated f3 to the statistical factor in the wave function.

The physics (in contrast to the statistics) in the wave function appears to be in the
kinematical factor R. Support for this assertion comes from examining the "free particle"
solutions of the Dirac equation. There are two distinct types of plane wave solutions with
momentump = mev, an electron solution and a positron solution. The electron solution has
the form

(30)

where p and Ro are constant, but the phase lP has the spacetime dependence

hlP = p' x =mcv· x =mc 2 r. (31)

Here r is the proper time along "streamlines" of the Dirac current, which are straight lines
with tangent v orthogonal to the I-parameter family of hyperplanes with constant phase
constituting a moving plane wave. According to (25) and (26) the electron velocity v and
spin s are constant everywhere. But along a streamline lP increases uniformly, so the phase
factor in (30) rotates e j and e2 in the plane of the spin S with the circular zitterbewegung
frequency

(32)
ZITIERBEWEGUNG IN RADIATIVE PROCESSES 29

A similar rotation takes place along the streamlines of every solution of the Dirac equation,
though, in general, with a variable frequency. Indeed, the decomposition (29) tells us that
generally the phase t/J = t/J(x) at each spacetime point x determines a well-defined rotation,
not just in some abstract complex plane, but in a definite physical plane, the plane of the
spin S atx.

Jaynes [3] tells us that when he looks at the standard Dirac wave function he
doesn't see anything that could rotate. This is a striking illustration of how crucially the
interpretation of a theory depends on the form of its mathematical representation. The STA
formulation makes the rotation inherent in the wave function absolutely explicit. But, as
physicists, we are not satisfied with a "mere" mathematical rotation. With Jaynes, we
demand to know "What physically is rotating?" Here again the plane wave solution helps
gain a vital insight.
The kinematical factor in (29) can be written in the form

(33)
where n is the constant bivector

(34)

with e1e2 =RY1 y)? =-RoiRo' Accordingly, n is the angular velocity of the frame {eJl =
eJl (x( r»} as it moves along a streamline. Both eo = v and e3 =S are constants of the
motion, but
e 1 (r) = en-re1 (0) = e 1 (O)coswo 't' + e 2 (0)sin wo't',

(35)

where Wo =Inl is the ZBW frequency. These equations describe the rotation of the frame
explicitly.
Now, it has often been suggested on heuristic grounds the electron spin and
magnetic moment may be generated by some kind of local circular motion of the electron.
This idea cannot be maintained if the electron velocity is identified with the streamline
velocity v of the Dirac current, because v is orthogonal to the spin. If the idea is physically
correct, the true electron velocity must have a component in the spin plane. Our geometrical
representation of the electron plane wave presents us with an obvious choice.We suppose
that the velocity of the electron can be identified with the null vector

(36)

Of course, this means that the electron moves with the speed of light, as in Schrooinger's
original ZBW model. This hypothesis defines what I call the zitterbewegung interpretation
of the Dirac theory [2]. It is more general than SchrMinger's idea of the ZBW, for (36) is
30 D. HESTENES

obviously applicable to any solution of the Dirac equation. In the plane wave case, however
it is easy to integrate.

With the time dependence of e2 given by (35) and u = c-1i, Eq. (36) is easily
integrated to get the history z = z('r) of the electron; thus,

z( r) = vcr + (e nT -l)ro + ZOo (37)

This is a parametric equation for a lightlike helix z(r) = x(r) + r(r) centered on the
streamline x( r) =vcr + Zo - ro with radius vector

(38)

The radius of the helix is half the electron Compton wavelength

Ao =c/ OJo = fi/2mc =1. 9 X 10- 13 m.. (39)

The Dirac current describes the mean velocity over a zbw period:

u=eo =v, (40)

so the Compton wavelength is the diameter of ZBW fluctuations about this mean.
From (34) and (38) , which imply r = Or, we find
(41)

Thus, the spin angular momentum can be regarded as the angular momentum of ZBW
fluctuations.

With rexpressed as a function of spacetime position by (31), Eq. (37) describes a

spacetime-filling congruence of lightlike helixes centered on Dirac streamlines, with exactly
one helix through each spacetime point. In accord with the statistical interpretation of the
Dirac wave function, each helix is a possible worldline for the electron, and the modulus of
the wave function determines the probability that the electron traverses any particular helix.
All these conclusions about the geometry of the plane wave solutions apply generally to
every solution of the Dirac equation, though, of course, in the presence of external fields
the helixes are bent and distorted. Jaynes [3] has described this view of Dirac solutions
aptly as a "tangle of all the different possible trajectories of a point particle," and he
exclaimed "I would never in 1,000 years have thought of looking at the Dirac equation in
that way!" Never without the STA formulation! Mark again how crucial the mathematical
representation is to physical interpretation! The tangled geometry of helixes has been
inherent in the Dirac theory all the time; only a suitable representation and definition was
necessary to reveal it. Mark that the ZBW interpretation attributes a purely kinematical
meaning to the phase factor, so the entire factor R in (29) and (22) has a purely kinematical
interpretation. This gives the interpretation of the Dirac wave function a maximum degree
of coherence.
ZI1TERBEWEGUNG IN RADIATIVE PROCESSES 31

Berry [7] has given the quantum phase factor a general geometrical interpretation.
According to the ZBW interpretation, the phase factor is more literally geometrical than
anyone had imagined.

IV. WHAT IS AN ELECTRON, REALLY?

Is the electron a particle always, sometimes, or never? Theorists have come down
on every side of this question. A definitive answer is essential to any sort of objectivity
attributed to quantum mechanics. I am pleased that Ed Jaynes [3] has come down on the
side opposite mine, for the comparison of contrasting interpretations helps highlight the
critical issues. I am equally pleased that he has placed Willis Lamb on my side.

The contrasting interpretations that Ed and I defend should not be regarded as

dogmatic stances, nor should it overshadow the great extent to which we agree. The
participants at this conference know that Ed is well established as one of the world's
leading practitioners of quantum mechanics, especially in the domain of quantum optics.
All the while, though, he has been one of the most astute critics of quantum mechanics. His
criticism has always been based, not on sterile philosophical speculation or mathematical
formalism, but on cogent physical reasoning born of his intimate knowledge of both
classical and quantum electrodynamics and how they relate to real experimental data. The
criticism he presents in these proceedings is only part of the extensive critical evaluation he
has presented on other occasions. I find myself in whole-hearted agreement with the entire
body of his criticism, and I commend it to any serious student of the foundations of
quantum electrodynamics. Ed and I agree that there is great truth in standard quantum
mechanics, but the problem is to separate the truth from the fiction. We also agree that the
interpretation of the electron wave function is a critical issue. We part company on what to
do about it. Though I am sure that Ed agrees that the Dirac theory is somehow more
fundamental, like most other theorists in quantum optics, he is content to base his analysis
on the Klein-Gordon and Schrodinger approximations to it. I regard that as a grave
mistake, for the ZBW structure of the Dirac theory could never be discovered in these
approximate theories, even though it is inherent in the phase factor of the wave function,
and they thereby inherit a ZBW interpretation from the Dirac theory. With this understood,
let us return to the particle issue.

Ed Jaynes, like Asim Barnt [8], wants to interpret the electron wave function as
describing a real physical entity, rather than just a state of knowledge about the electron as I
wist. While I believe that that viewpoint faces insuperable difficulties, I applaud Ed's
objective to bring the matter to decisive experimental test, and I agree that this is feasible. I
note that the main reason for Ed's stance is that he believes, along with most other
physicists, that electron diffraction can only be explained as due to " interference of the
electron with itself," so with admirable consistency, Ed maintains that the electron must be
extended in space like the wave function. This issue of how to explain diffraction is one of
the great bugaboos of quantum mechanics, so I will address it from the particle perspective
below.

Ed seems to have a hang-up about point particles as well as STA. Let me attempt
some therapy. The question "Is the electron a particle?" can and must be addressed at
different levels, where different physical issues are at stake. Let me call the first level the
interpretation level. Here the question is "Does the Dirac theory admit to a coherent particle
interpretation which is superior to alternative interpretations?" My answer to this question
is, of course, yes! Indeed, I maintain that the ZBW interpretation is the only one which
comes close to giving a coherent account of all details of the Dirac theory. It is not
32 D. HESTENES

maintained at this level that the electron really is a point particle, but only that the Dirac
theory says it is, in the sense that it ascribes to the electron no internal structure and no
finite dimensions. The electron spin and magnetic moment are features of electron
kinematics rather than internal structure.
In the spirit of Jaynes, it might be suggested that the electron is an extended body
and the helixes are world lines of its component parts. This suggestion faces difficulties
which seem to rule it out. First, there is an absence of evidence for any interaction among
the parts which would be needed to make the body cohere. Second, the dimensions of the
body would have to be on the order of a Compton wavelength (-lO- 13m). But this is much
too big! Scattering experiments limit the size of the electron (Le. the size of the domain in
which momentum transfer takes place) to less than 1O- 18 m [9]. Only the particle
interpretation appears to be consistent with this experimental evidence. Additional evidence
for the particle interpretation ([2], [10]) is less direct. For example, the explanation for Van
der Waals forces requires that atoms are fluctuating dipoles, which they certainly are if
electrons are particles orbiting the nucleus rather than Jaynesian amoebas enveloping the
nucleus in static charge clouds. Moreover, the time dilatation in the decay of I.r particles
captured in atomic s-states indicates that they really are moving with the Bohr velocity in
those states [11]. So must electrons move also.
It seems to me that the Born statistical interpretation is essential for understanding
scattering data, and this demands the particle interpretation. How then do we explain the
structure in a diffraction pattern? "Interference" is the standard answer! But there is a lesser
known alternative which has been propounded vigorously by David Bohm [12] and others
for years. This puts Bohm firmly on my side, though Jaynes cites him as a precursor to his
amoebic viewpoint. Bohm maintains that the electron is a particle with a definite trajectory
and that the wave function determines a family of possible trajectories, just as I do in my
ZBW interpretation. On this point, we differ only in details of how the trajectories are
determined by the wave function. Bohm uses SchrOdinger theory rather than Dirac theory.
The trajectories have actually been calculated from the SchrOdinger equation for the double
slit experiment [13], and the Dirac equation would surely yield essentially the same result.
The trajectories flow uniformly through both slits, but thereafter they spread out, bunching
up at diffraction maxima and thinning out at minima. When a single electron has been
detected on the "diffraction screen," one can (in principle to any desired precision)
determine which of the trajectories it actually followed and trace the trajectory backwards to
determine where the electron passed through one of the slits. In this sense, quantum
mechanics allows us to measure definite electron trajectories.

This description of electron diffraction is a self-consistent interpretation of the

equations of quantum mechanics. It has the great advantage of preserving a consistent
particle interpretation, allowing us to maintain that every electron has a continuous (albeit
indirectly observable) trajectory. But physicists want more. They want an explanation of
diffraction, not just a description. They want to identify a causal mechanism underlying
diffraction. I don't believe that standard quantum mechanics has achieved that, but I
suggest below where the missing mechanism might be found. On the contrary, standard
quantum mechanics purports to explain diffraction as a consequence of interference. The
possibility of such an explanation is a mathematical consequence of the fact that the QM
wave equation is linear, so it can be argued in the double slit experiment that the diffraction
pattern is caused by interference in the superposition of particular solutions with each slit as
source. Accepting this mathematical possibility as physical reality has the strange
consequence that the electron must somehow pass through both slits in order to interfere
with itself. I maintain that this interpretation buys nothing but trouble, since it is obviously
inconsistent with the factually grounded particle interpretation, but it has no greater
ZITIERBEWEGUNG IN RADIATIVE PROCESSES 33

predictive power. It is as awkward as it is unnecessary. There is actually only one valid

solution of the wave equation which matches the boundary conditions in a diffraction
experiment, and only that solution is used in the above particle interpretation of the
experiment. The subdivision of that solution into interfering particular solutions which
separately do not satisfy the boundary conditions can therefore be safely dismissed as a
mere mathematical artifice. Accordingly, the interference explanation of particle diffraction
can be dismissed as an artifice introduced in an attempt to manufacture an explanation out
of a description.
Now let us address the particle question at a second, more fundamental level. At
this level, I agree with Einstein, Rosen and Cooperstock [14] that the electron, as a particle,
must not be treated independently of the electromagnetic field but as part of it. The electron
in the Dirac theory is an emasculated charged particle, stripped of its own electromagnetic
field, like a classical test charge. The central problem of quantum electrodynamics, as
recognized by Barut [8] and many others, is to restore the electron's field and deduce the
consequences. This is the self-interaction problem. Whether, in the ultimate solution to this
problem the electron will emerge as a true singularity in the field or some kind of soliton
[14] is anybody's guess. One thing is certain, though, the problem is nonlinear. And if
quantization is a consequence of this nonlinearity, as I have suggested elsewhere [10], then
the self-interaction problem can never be solved with standard quantum mechanics; a more
fundamental starting point must be found.

Though the Dirac theory omits the electron's field, it appears to contain vestiges of
self-interaction which are valuable clues to a deeper theory. It is widely believed that the
electron mass and spin are consequences of self-interaction. But these are properties of the
ZBW, so the ZBW itself must derive from self-interaction. Already this suggests [2] that
the electron self-field is of magnetic type to produce the spin, and the electron mass comes
from a kind of self-inductance of the circular motion.

Interpreted literally, the ZBW motion should be reflected in the electron's

electromagnetic field. Specifically, the electron should be the seat of a nonradiating field
that oscillates with the ZBW frequency. Call it the ZBW field. The usual Coulomb and
magnetic dipole fields of the electron are then averages of the ZBW field over a ZBW
period. The ZBW frequency is much too high to detect experimentally. However, it has
been suggested [10] that many familiar quantum phenomena might be explained as
consequences of ZBW resonances. Here are three examples:

(1) Electron Diffraction. The ZBW field broadcasts the electron's deBroglie
frequency and wavelength to the environment. I submit that in diffraction it is the ZBW
field, rather than the electron itself, that feels out the topology of the target and by feedback
produces a shift in the phase of the ZBW motion which alters the electron's trajectory. In
crystal diffraction, the Bragg angles must then be conditions for resonance between the
broadcasted ZBW wave and the feedback wave scattered off the crystal. They are thus
conditions for resonant momentum transfer between the electron and the crystal. An
attractive feature of this explanation is that it includes a mechanism for momentum transfer
which is missing from conventional explanations of diffraction.

(2) Atomic States. An electron bound in an atom is in a ZBW resonant state,

wherein the frequency of the orbital motion is a harmonic of the ZBW frequency. The
principal quantum number indexes the harmonics. Now, if the above explanation of
diffraction is correct, the electron must be broadcasting a ZBW wave which is scattered
resonantly off the nucleus and back to the electron. An atomic state is thus a state of
34 D.HESTENES

resonant momentum exchange between the electron and itself. This is to say that an electron
accelerated by the field of the nucleus is always radiating continuously, but it is also
continuously absorbing its own radiation. In the ground state, all the radiated energy must
be absorbed, since the state is stable. However, in an excited state the radiation rate must
exceed the absorption rate so the state decays. It should therefore be possible to calculate
the lifetime of the state from the mismatch between these two rates, thus to explain
spontaneous emission. All this is just another example of diffraction, with atomic states
corresponding to diffraction peaks and "quantum conditions" corresponding to the Bragg
law. The main difference between transient diffraction by a crystal and continuous
diffraction within an atom is that the momentum transfer is between two different objects in
the first case but between an object and itself in the second.

(3) Pauli Principle. Two electrons in the same atomic state will certainly have
resonant ZBW frequencies, so momentum exchange via their ZBW fields is to be expected.
Thus we have here a natural mechanism for explaining the Pauli principle, Evidently, then,
if the electrons have antiparallel spins the ZBW interaction produces a stable two electron
state, while if the spins are parallel the state is unstable and so never seen.

Though suggested by the Dirac theory, all this goes well beyond it. It is conceivable
that, besides spontaneous emission, the ZBW is responsible for other phenomena, such as
the Lamb shift and the anomalous magnetic moment, which are attributed to quantization of
the electromagnetic field. Clearly the ZBW idea is pregnant with possibilities for new
physics.

V. RADIATIVE PROCESSES.
Ed Jaynes is quite right to assert that if the electron really is a particle but quantum
mechanics describes only the behavior of an ensemble, then it must be possible to extract
the particle from the ensemble and study it all by itself. The problem with such an
extraction, of course, is to ascertain suitable equations of motion for the particle alone,
because they might differ significantly from equations for the particle behavior within the
ensemble. Nevertheless, as a first approach to the problem, I propose to extract a single
ZBW worldline from the Dirac theory and interpret it literally as the worldline of an
individual electron. Even without equations of motion, we can reason qualitatively about
the electron's behavior from what we know about solutions of the Dirac equation. Such
reasoning can be quite provocative. Even if it cannot be refined by calculations from exact
equations of motion, it may prove useful in guiding the solving and interpreting of the
Dirac equation.

When an electron is placed in an external field, energy can be absorbed by the ZBW
field, producing an increase in ZBW frequency and hence a decrease in the ZBW radius.
We know that from solutions of the Dirac equation where binding energies appear in the
complex phase factor. Indeed, the Minimal Coupling Ansatz can be interpreted as
specifying that external fields produce shifts in the ZBW frequency. As Steve Gull puts it,
the electron is a parametric oscillator with frequency modulated by external fields. A shift in
the ZBW frequency OJ, is also a shift in electron mass m, because nOJ = mc 2 holds
generally. The so-called electron rest mass is therefore only a lower bound to the electron
mass. The electron mass is actually variable and changing all the time in interactions.
However, if no external field is present to induce radiation, it may be that the electron can
retain a mass greater than its empirical rest mass. In other words, it may be that energy can
be stored in the ZBW of a single free electron. This possibility can surely be put to
ZITTERBEWEGUNG IN RADIATIVE PROCESSES 35

experimental test. Indeed, the basic mechanism may have been probed already by recent
experiments in quantum optics.

For example, the ZBW mechanism can be deployed to explain multiphoton

ionization [1]. When a bound atomic electron is irradiated by an intense laser field, the
ZBW may absorb a harmonic of the laser frequency, with an attendant increase of electron
mass and shrinking of its atomic orbit. Evidently this excited ZBW state is metastable and
may persist for some time after the laser field is off. Then the stored energy is liberated
either by reradiation or ionization. The phenomenon of above threshold ionization [1]
shows that the electron (ZBW) may absorb much more than the minimum necessary for
ionization. If, indeed, the ZBW is the mechanism for multiphoton and above-threshold
ionization, then it must be possible to demonstrate these phenomena in experiments with
single atoms. According to the standard explanations, such experiments should not work.
It may be added that final state interactions in ionization should be significantly affected by
ZBW mass shifts.

This ZBW explanation for the new photoelectric phenomena may appear to be
incompatible with conventional explanations. An excellent and accessible explanation
grounded in standard quantum electrodynamics is given by Andre' Bandrauk [15]. The
idea is that embedding a molecule in a laser alters the effective electronic potential to create a
new set of bound states which can be observed with electron probes. This is not
necessarily inconsistent with the ZBW explanation, but the putative physical mechanism is
quite different. Other experiments will probably be necessary to distinguish between the
two possibilities.

Evidence that irradiated single free electrons can absorb harmonics of the laser
frequency exists already in the pioneering "stimulated bremsstrahlung" experiments of
Tony Weingartshofer [16]. These experiments have been regarded as anomalous in the
high intensity laser field, because they cannot be explained by standard arguments.
However, I submit that they are just further examples of the ZBW mechanism at work.

To establish unequivocally that energy can be stored in the ZBW of a single free
electron, we need cleaner experiments on single electrons. The prediction is that an
electron can absorb an n-th order harmonic to put it in a metastable state with mass m given
by mc 2 = I1loC 2 + nnwt , where I1lo is the rest mass and wt is the laser frequency. Then,
under suitable conditions, the electron can be released in this excited state to transport the
additional energy until the electron is induced to release it by a collision or some other
means. This phenomenon may actually have been observed already in the infamous
Schwartz-Hora effect described briefly by Jaynes [3]. I hold with Jaynes that this effect is
probably real and the possibility deserves to be investigated thoroughly. Our explanations
for the effect may appear to be quite different, but remember, I attribute the standard QM
phase factor to the ZBW, and the phase factor plays the key role in Jaynes' argument. The
main difference is that Jaynes sees the effect as due to coherent action of parts of the
electron spread out over a wave packet. The issues are clear. The truth will be found out.

Acknowledgement. The idea that energy can be stored in the ZBW of a single free
electron was developed jointly with Heinz Kriiger in several conversations.
36 D.HESTENES

REFERENCES

[1] P. Agostini and G. Petite (1988), Photoelectric effect under strong irradiation,
CONTEMP. PHYS.l, 57-77.

[2] D. Hestenes (1990), The Zitterbewegung Interpretation of Quantum Mechanics, Found.

Phys.20, 1213-1232.

[3] E. T. Jaynes (1991), Scattering of Light by Free Electrons as a Test of Quantum

Theory, (these Proceedings).

[4] S. F. Gull (1991), Charged Particles at Potential Steps, (these Proceedings).

[5] H. KrUger, New Solutions of the Dirac Equation for Central Fields, (these
Proceedings) .

[6] R. Boudet (1991), The Role of Duality Rotation in the Dirac Theory, (these
Proceedings).

[7] A. Shapere & F. Wilczek (1989), GEOMETRIC PHASES IN PHYSICS, World

Scientific.
[8] A. O. Bamt (1991), Brief History and Recent Developments in Electron Theory and
Quantum Electrodynamics, (these Proceedings).

[9] D. Bender et. ai.(1984), Tests of QED at 29 GeV center-of-mass energy, Phys. Rev.
D30, 515 .

[10] D. Hestenes (1985), Quantum Mechanics from Self-Interaction, Found. Phys.15, 63-
87.

[11] M. Silverman (1982), Relativistic time dilatation of bound muons and Lorentz
invariance of charge, Am. 1. Phys. 50, 251-254.

[12] D. Bohm & B. Hiley (1985), Unbroken Quantum Realism, from Microscopic to
Macroscopic Levels, Phys. Rev. Letters 55,2511.

[13] I.-P. Vigier, C. Dewdney, P.R. Holland & A. Kypriandis (1987), Causal particle
trajectories and the interpretation of quantum mechanics. In Quantum Implications,
B.J.Hiley & F.D. Peat (eds.), Routledge and Kegan Paul, London.

[14] F. I. Cooperstock (1991), Non-linear Gauge Invariant Field Theories of the Electron
and other Elementary Particles, (these Proceedings).

[15] A. D. Bandrauk (1991), The Electron and the Dressed Molecule, (these Proceedings).

[16] A. Weingartshofer, I. K. Holmes, G. Caudle, E. M. Clarke & H. KrUger (1977),

Direct Observation of Multiphoton Processes in Laser-Induced Free-Free Transition,
Phys. Rev. Let., 39, 269-270. A. Weingartshofer, I. K. Holmes, J. Sabbagh & S. L.
Chin (1983), Electron scattering in intense laser fields, I. Phys B 16, 1805-1817.
CHARGED PARTICLES AT POTENTIAL STEPS

S. F. Gull
Mullard Radio Astronomy Observatory
Cavendish Laboratory, Madingley Road
Cambridge CB3 OHE, U.K.

Abstract. The behaviour of charged particles at electromagnetic steps is analysed using the powerful
mathematical tools provided by the SpaceTime Algebra. Currents predicted in the evanescent region
of a Dirac wavefunction strongly suggest that the electron "zitterbewegung" (ZBW) represents a
real circulation. At higher potentials, the Klein paradox reveals a crucial difficulty of interpretation
of "positronic" wavefunctions that must be overcome before Hestenes' ZBW model can be taken
seriously. The problem of radiation reaction is still not solved. Solutions of the Lorentz-Dirac
equation for a potential step show crazy teleological features: certain input velocities have no possible
future output states. The prospects for realistic electron models are briefly discussed.

1. Introduction
I can safely say to this audience that, as a radio astronomer, I have observed more
electrons than anyone else present. A medium-sized radio galaxy displays radio synchrotron
emission from about 1063 electrons as jets of relativistic electrons and positrons shoot out
into space from a black hole deep inside a galactic nucleus. Our interest in these, the most
violent objects in the Universe, requires a corresponding understanding of the humble lepton
and its radiation mechanisms, particularly at the highest energies.
Like some of the other participants here, I have for a long time been deeply unhappy
about the accepted theories of this little particle and I have to admit that I have been
unable to heed Feynman's (1967) excellent advice not to ask oneself

" 'But how can it be like that?' because you will get 'down the drain',
into a blind alley from which no one has yet escaped.".

Two years ago, however, an unexpected event brightened up my view of this particular drain,
when I became aware of the work by David Hestenes on Geometric Algebra. He claimed
(Hestenes 1986) that our mathematical language is seriously incomplete, because physicists
do not know how to multiply vectors together, and that they are thereby missing the
geometrical content of the equations of physics, particularly the Dirac equation. I rapidly
became convinced that Geometric Algebra, which includes the algebra of our spacetime,
the SpaceTime Algebra (STA), as a special case, is an essential ingedient in correcting our
misconceptions about the nature of quantum mechanics.
As a beginner in a strange new field, I have modest aims in this paper and focus attention
on the behaviour of a charged particle encountering an electromagnetic potential step, all
37
D. Hestenes and A. Weingartshofer (eds.), The Electron, 37-48.
© 1991 Kluwer Academic Publishers.
38 S. F. GULL

the time using the powerful mathematical tools provided by the STA. This situation forms
a convenient "theoretical laboratory" for the electron, allowing us to examine critically the
predictions of presently-available models. After a brief review of the STA and the Dirac
equation, I consider the riddle of the charge current in the Dirac theory and the extent to
which the behaviour of an electron wave at an electromagnetic potential step throws light
upon the new "zitterbewegung" (ZBW) interpretation (Hestenes 1990). Klein's paradox is
briefly mentioned, but not resolved. I believe that the difficulties of the Dirac current are
clearly exposed by this paradox and that our present ways of overcoming it (essentially due
to Feynman) are unsatisfactory. "The cure is worse than the disease", as Ed Jaynes is fond
of saying.
In the second part of this paper we return to the potential step to examine the mystery of
the radiation reaction. This is still an important problem, because it is extremely desirable
to have a self-consistent equation of motion for an accelerated charge that takes into account
its own radiation. The case of a finite-si~ed charge distribution is tractable if sufficient care
is used in any approximations, but the difficulties for a point charge seem insuperable.
The most famous attempt, the Lorentz-Dirac equation (Dirac 1938), fails miserably when
confronted with the problem of the potential step, showing unacceptable overall scattering
states that prohibit certain ranges of input condition. The STA suggests an alternative
equation.
Inspired by the ZBW interpretation of quantum mechanics, we have a brief look at
realistic electron models (Barut & Zhanghi 1984). Translated into the STA, these models
begin to look very interesting, but certain defects become obvious.

2. The SpaceTime Algebra and the Dirac equation

This paper embraces the ideas and notations developed by Hestenes over the last 30
years (Hestenes 1966, 1986). The SpaceTime Algebra (STA) is a real, Geometric (Clifford)
algebra developed on a 4-dimensional flat spacetime with a standard Minkowski metric.
The basic ingredients of this algebra are an orthonormal frame of vectors {JO'll'12'13},
where 16 = -I~ = 1. The time-like vector 10 defines a Lorentz frame, that we can think
of as representing the laboratory frame. The {II,} satisfy the same algebraic relations as
the Dirac I-matrices, but it must be stressed that they here represent 4 unit vectors in
spacetime and not the 4 components of a single vector. We shall have no use for a matrix
representation of the {J/l} here, but a translation table is given in the Appendix.
From this basic set of vectors we build up the 16 (= 24) geometric elements of the STA:

{J/l} {ak,iakl {it/l} i

1 scalar 4 vectors 6 bivectors 4 pseudovectors 1 pseudoscalar .
The time-like bivectors ak == IklO obey the same algebraic relations as the Pauli spin-
matrices, but in the STA they represent an orthonormal frame of vectors in space relative
to the laboratory time vector 10. The unit pseudoscalar of spacetime is defined as

i == 1011/213 = al a2 a 3·
The STA is the 16-dimensional real linear space formed from these geometric objects. We
will not need to consider the coefficients to be complex scalars, because the STA already con-
tains 10 geometrically distinct square roots of -1, which is plenty enough for our purposes.
CHARGED PARTICLES AT POTENTIAL STEPS 39

Indeed, the STA representation of the complex imaginary is different from one application
to another, so that the use of complex numbers as scalars may be unnecessary in physics.
A particularly important subalgebra of STA is the set of spinors, which, for spacetime,
is simply the even subalgebra, comprising the scalar, bivectors and pseudoscalar. We can
write a general spinor in the form

'Ij;=ao+a+i(bo+b),

where a == ak(Jk and b == bk(Jk are relative vectors. A very important point of interpretation
is the fact that spinors of the STA are geometrical objects in their own right, rather than
(as in matrix versions) residing in a complex "spin-space". The notion of "spin-space" is
not needed here.
The spinors are important because they define the transformation properties of frames
in spacetime. A spinor 'Ij; defines a Lorentz rotation through the transformation

where the {ell} are a new frame of orthogonal vectors and ~ is the reverse of 'Ij;, formed by
reversing the order of all geometric products. To see this explictly we write a Dirac spinor
'Ij; in canonical form:

'Ij; (pei(J) t L(u) R(O,</>,X)·

spinor complex Lorentz Rotation by
amplitude transformation Euler angles
to velocity v = eU')'o (O,</>,X)
Explicit forms for L( u) and R( 0, r/J, X) are

Writing the Dirac equation in the STA (see Appendix) we have

This admits plane-wave solutions for 4-momentum p (with p2 = m 2),

'Ij; L(p)R(O,r/J,x)e- iu 3P"X

or L(p)R(O,r/J,X)i e+iU3P-X.

The Dirac current 'Ij;')'o~ == pv and the spin current !'Ij;')'3~ == ps can then be interpreted
using the STA. The Dirac wavefunction represents a change of frame, providing a Lorentz
boost of /0 to the comoving velocity v and a rotation of ')'3 to align it with the spin axis
of the electron. The phase of the wavefunction gives the rotation of the ')'2')'1 plane about
the spin axis. Of the two other factors contained in 'Ij;~ == pei(J, p can be taken to represent
the probability of finding the electron at position x, but the role of fJ is uncertain. We can
see that it distinguishes between positive (fJ = 0) and negative (fJ = 1r) frequency (energy)
states.
40 S.F. GULL

In the ZBW interpretation of quantum mechanics (Hestenes 1990), the Dirac current is
-0,
redefined as pv == 1/;,_ where ,_ == ,0 -,2,
This would make the worldline of a particle
into a light-like helix, making manifest a transverse ZBW as the source of the electron's spin
and magnetic moment. The positive frequency solutions above are interpreted as electrons,
and the negative frequencies as positrons, so that (3 measures the extent to which we have
a pure particle/ antiparticle state. It is safe to say, however, that no entirely satisfactory
interpretation of (3 is yet available.

3. The Dirac electron at a potential step

An elementary application of the Dirac equation is to consider an electron wave incident
normally upon a simple electromagnetic potential step of magnitude <jJ, so that (A = 0, Z <
0) and (A = <jJ,o,z > 0). In the STA we write the incident, reflected and transmitted waves
as follows:
Incident Reflected Transmitted
1/;1 = eU(J3/2if!e- i(J3pr' x 1/;T =r e- u(J3/ 2 i!Je- i(J3Pr' X 1/;t = t e'/(J3/2i!Je- i(J3P"X
PI = E,o + P/3 PT = E,o - P,3 Pt = E,o + P',3,
where i!J is a Pauli spinor describing the spin state, if! = corresponding to longitudinal
spin "up" and i!J = -iCT2 to spin "down". Matching the spinors at Z = 0 we find

cosh(u/2)(1 + r) = cosh(u'/2) t,
sinh( u/2)(1 - r) = sinh( u' /2) t,
sinh( u - u')/2
r=
sinh(u + u')/2'
This is plausible, the reflection coefficient increasing to unity as the step height approaches
the classical reflection point at e<jJ = E - m.
For steps higher than this, the reflection coefficient is unity and the wavefunction inside
the step is evanescent. We have to take some care when the solutions are written in terms
of the STA, because we cannot just let P' become imaginary. The evanescent wave is

where 1/Jo is a constant Dirac spinor. We can find a matching condition if " . P' = 0 and
(p')2 _ ,,2 = m 2. The solution displays very interesting general features that may give us a
clue about the nature of the ZBW. For the case of longitudinal spin we find

where E - e<jJ = m cos (3 and K = ±m sin (3 for the two polarisation states, spin "up" taking
the positive sign. The evanescent wave has a non-zero (3, which flips to ±1!" as e<jJ approaches
E+m.
If the spin of the incident wave is tranverse (polarised in the x - y plane), then there
is a non-zero Dirac current in the evanescent region, the Dirac "velocity" having the value
K/m. The direction of this velocity is perpendicular both to the incident spatial momentum
CHARGED PARTICLES AT POTENTIAL STEPS 41

and to the spin. Similar effects can be seen in solutions of the Dirac or the non-relativistic
Pauli equations whenever the amplitude of 'Ij; varies with position. Other examples are
the hydrogen wavefunction or a Gaussian wave-packet (Hestenes 1979). It is tempting the
interpret this phenomenon in terms of the non-cancelling of "Amperian" currents due to
the circulation of the ZBW. It seems that the evanescent wave is giving us a "tomographic"
view of the ZBW, implying again that the spin of the electron represents a real circulation.
There is a need, however, for a sensible interpretation of the longitudinal behaviour, which
shows a non-zero value of (3.
The Klein paradox and Feynman's resolution
When the size of the potential step exceeds E + m the solution becomes propagating.
Defining E' = e¢ - m, (p')2 = (E'? - m 2 and tanh u' = p' / E', we find

cosh( u - u')/2
r = h(
cos U + u' 2
)/ ; r < 1: (3 = 7L

This case needs some care in order to match on to a solution with a positive group velocity
inside the step. Many standard books give a solution with a negative group velocity, a
notable example being Bjorken & Drell (1964).
This result for the reflection coefficient is actually very odd indeed, and I suspect that
the reason some books give the wrong solution (with r > 1) is simply wishful thinking. We
could perhaps understand r > 1 as representing the creation of electron/positron pairs at
the step, with the positrons continuing into the step. In fact we seem to have an entirely
different situation, and the transmitted wave looks more like a "positronic hole", having
negative mass and negative charge.
The central problem is, as everybody knows, that the Dirac current e'lj;,o-¢ is positive
definite, and thus cannot represent a positronic current without some modification. The
problem does not arise for the (non-STA) Klein-Gordon equation

(n2 + m2) 'Ij; = 0, where D == '\7+ ieA

(the i is now just the uninterpreted A of common usage). The Klein-Gordon reflection
coefficients are ,
r=P-P <1 (e¢<E-m),
p+P'
p+ p'
r=-->l (e¢>E+m).
P-P'
The Klein-Gordon charge current behaves sensibly, showing a strong resonance at p = p',
which one could interpret as stimulated emission of pairs.
The reason for the striking difference between these solutions is easy to find, because
the Dirac equation does not lead to the minimally-coupled Klein-Gordon equation:
42 S.F.GULL

where F = \7 A = E + iB (\7 . A = 0). There is an extra spin term eF'lj;iu3, which couples
into the very strong E field in the step, effectively changing the matching conditions. We
see that the spin is crucial to this longitudinal ZBW phenomenon, which again cries out for
a better physical interpretation.
Why should we bother with the Klein paradox of the Dirac equation when we are going
to use field theory? The very simple case of an electromagnetic step has revealed a stark
contradiction about the nature of the Dirac current. In almost any other branch of science
we would conclude at this point that we have already done something wrong, and that we
should go back and correct it. It seems, however, that in quantum theory we conclude that
we have to do something else as well. This is more than a little odd and, in any case, it is
unreasonable to suppose that such a contradiction will go away just by making the theory
more complicated.
I am not able to understand or willing to repeat the arguments that lead to the standard
resolution of the Klein paradox; a representative reference is Nikishov (1970). It appears
that somewhere in the quantum fog the ideas of "exclusion principle" and "no stimulated
emission offermions" are invoked. The conclusion, on the other hand, is crystal-clear: r = 1
and there is no longer any paradox.

I know an old lady who swallowed a fly,

I don't know why she swallowed a fly, perhaps she'll die
(Traditional nursery rhyme).

There seems to be a fundamental truth in the Dirac equation; we can interpret it ge-
ometrically. I believe that the same is probably true of the Weinberg/Salam electroweak
model (Abers & Lee 1973, Hestenes 1982). But the Dirac theory doesn't quite work even
when, with apologies to David Hestenes, it is translated into the STA. How should we patch
it up? QED? Families of leptons? Quarks? Higgs particles? I suspect that we are in the
same unfortunate position as the the old lady in the nursery rhyme who swallowed a fly and,
finding it not to her liking, followed it with a series of increasingly indigestible remedies.
My diagnosis is that we physicists have now reached the stage where we are attempting to
swallow a "goat"and should beware the "horse" that must be waiting for us soon. I honestly
believe that most "unifications" (except Weinberg/Salam) are headed in the wrong direc-
tion and that we (or at least some of us) should try to turn the clock back and start again
using the proper mathematical tools. The SpaceTime Algebra is an essential ingredient in
this programme.
Pushing the analogy a little further, the place to start is, I suppose, at the point we
swallowed the fly. My guess is that we went wrong when the first infinity appeared in
physical theory: the self-energy of a charge. To this end I am sympathetic to the view
expressed by Jaynes (1990) about the reality or otherwise of the electromagnetic field,
following the lead given by Wheeler & Feynman (1949). In this view the electromagnetic
field does not actually exist in spacetime, but represents instead an information storage
device. The electromagnetic field at any point might be a summary of what we need to
know about distant charges in order to predict the behaviour of a charge if there should
be one at that point. But I have already been sufficiently radical in this paper and I shall
defer any further heresy. There is, however, a closely related problem concerning radiation
CHARGED PARTICLES AT POTENTIAL STEPS 43

reaction. Is it possible to have a self-consistent equation of motion for a particle that takes
account of its electromagnetic radiation?
4. Radiation reaction at a step
The radiation reaction on an accelerated charge is a century-old problem that is still not
resolved, despite the valiant efforts of generations of theoretical physicists. The Lorentz-
Dirac equation (Dirac 1938) is "self-consistent" in the sense that it correctly accounts for
the energy budget of a radiating particle, but it suffers from strange pre-acceleration effects
and admits runaway solutions. Dirac derived the equation by expanding the field near a
point charge as a power-series in retarded time and collecting the parts that did not diverge
at the origin. The standard form of this equation (in SI units) is

mv - ~
67rfoc
(v + v2 v) = eF . v.
The derivation of this equation is badly flawed: the power-series in retarded time has only a
finite radius of convergence. Burke (1970) shows that, if the same mathematical techniques
are applied to the case of spherical oscillations of a rubber ball in air, pre-acceleration and
runaway solutions again appear. Careful analysis of finite-sized charged distributions do
not show such peculiar effects, and do not yield the Lorentz-Dirac equation (Jaynes 1980
(unpublished notes), Grandy & Aghazadeh 1983). Although these analyses are fine, I do
not believe that the electron has a finite size (though see Jaynes' paper in this volume for
a different point of view), and it is still very attractive to have an equation of motion that
self-consistently accounts for the radiation of an accelerated point charge. Consequently,
the Lorentz-Dirac equation is still used (see, for example, Barut 1988, Barut 1990), usually
with the added boundary condition (due to Dirac) that the velocity remains finite as t --T 00.
Unfortunately, even with Dirac's condition, the solutions of this equation are particularly
strange and physically unacceptable for the apparently innocuous electromagnetic potential
step, which we now study.
The usual form of the Lorentz- Dirac equation does not fully reveal its geometrical mean-
ing. Expressing the equation in the STA, we note that v 2 = 1, v . v = 0 implies

v.. + v.2 v = v.. - (

V· v.. ) v = 2"1 (..v - ..)
vvv ..
== V-L,

which is the component of v projected perpendicular to v. Because the acceleration v

and the Lorentz force eF . v are both perpendicular to v, the reactive term must also be
perpendicular. Multiplying by v we find

-1( vv
. .- .vv
. ) = -1d(
- .
vv - .
vv ) = -d(.)
vv ,
2 2dT dT
where we have again used v . v = O. Defining e 2 j(67rfoC3) == TO, we can then rewrite the
equation in terms of the rest-frame acceleration bivector l1v == vv = v 1\ v:
dl1v l1v e eEv
- - - = - - - ( F - vFv) = - -
dT TO 2mTo mTo'

where Ev == t(F - vFv) is the electric field in the rest frame.

44 S. F.GULL

+ +

tl) If)

a ./ 0
+ +
"- /
cU
>- C)
"- '50
a . I
>0
"-
tl)
If!
'\
0 0
I I

(0) (b)

I I
-1 -0.5 0.0 +0.5 +1 -1 -0.5 0.0 +0.5 +1
Vau\ Vin
Figure l. Input and output velocities for the Lorentz-Dirac equation at a potential step of
e¢ = O.lm. (a) Vin is a single-valued function of Vaut. (b) Vaut is a multi-valued function of Vin. The
dashed curves show the behaviour of a non-radiating particle.
The field-free solutions can be expressed using the STA as !1 v = Bexp(r/ro) for any
simple time-like bivector B, so that vCr) = exp[Bexp(r/ro)Jv(O). The solutions clearly
have very undesirable properties as r -> 00 unless we employ Dirac's boundary condition.
To apply the Lorentz-Dirac equation to the potential step problem we search all possible
finite output velocities Vaut and integrate the equation backwards in time to see what the
appropriate input velocities Vin must have been.
For a particle incident on a step having higher potential for x > 0 there are three distinct
cases to consider.
1. Transmission left to right (Vin and Vout both positive). Note that, if Vaut is large,
then the particle, which has pre-accelerated up to Vaut before it reached the step, goes
through the step quickly, thereby receiving a small influence from it, so that Vin must
have been large. However, if Vaut is small the particle spends a longer time in the step
and the change in velocity due to the step (not at it, though) is also large, so that Vin
is agai n large.

2. Transmission right to left (Vin and Vout both negative). The particle, which has Vout <
Vin, has pre-accelerated by an amount that increases as Vaut is reduced. When Vaut
is sufficiently small, the long-term value of Vin is positive, so that the step is re-
encountered.

3. Anomalous reflection (Vin > 0; Vout < 0). The particle approaches the step from
the left, pre-decelerates towards the step and crosses it. On the other side the pre-
deceleration of its next step-crossing changes the sign of its velocity and drags it back
into the step, from which it emerges, having been reflected.

In Figure 1 the values of Vin and Vaut are plotted for the particular case e<jJ = 0.1 m.
The dashed curve shows, for comparison, the behaviour of a non-radiating particle. Even if
CHARGED PARTICLES AT POTENTIAL STEPS 45

the pre-acceleration properties of the equation are forgiven, the solutions are totally crazy.
Figure 1(a) plots Vin as a singled-valued function of Vout, but a time-dweller's view of Figure
1(b) shows that for Vin > 0 there are ranges of Vin with either 2 or 4 values of Vout. In case
that is not bad enough, for low values of Vin (corresponding in the non-radiating case to
simple reflection) there are NO allowed values of Vout. We have, apparently, lost control
of the input velocity! That is the sort of thing which is almost bound to happen when
complicated equations are integrated backwards in time.
The firm conclusion of this little investigation is that Dirac's suggested boundary con-
dition is not correct; merely making v finite at T - t 00 is not sufficient to rescue the
Lorentz-Dirac equation. It should also be noted that there are no mathematical difficulties
associated with the idealised case of a sharp step, which can be approached as a well-
behaved limit of a finite-width step of any shape. An extra condition on the Lorentz-Dirac
equation (Sawada, Kawabata & Uchiyama 1983), which apparently prohibits such steps,
cannot remove the difficulty, therefore.
A recent suggestion by Barut (1990) of "renormalising" the Lorentz-Dirac equation,
taking solutions without pre-acceleration and removing the runaway part by decree is, I
believe, even worse, because it does not conserve energy. Applying it to the finite-width
step problem, and letting the step width go to zero whilst keeping the height constant, we
find Vout - t Vin. That is extremely unfortunate if the step is a decelerating one as we have
supposed, because the particle is now in a region of higher potential, which we could use to
accelerate the particle in another, wider, downward step. This leads to an interesting new
design of linear accelerator!
Modified Lorentz-Dirac equations
We can throw some light on the problems of the Lorentz-Dirac equation by writing the
equation of motion as m v = e(F + Fs) . v, where Fs is a "self-field" representing the inner
workings of the particle. It would be natural to suppose that these internal mechanisms
would introduce a time lag, so that the effective force might be due to some retarded average
of the applied field, for example Fret == *" f~oo e(T'-T)!TO F( T')dT' ~ F( T - TO). A simple
example shows why this suggestion is unsatisfactory.
Imagine a classical electron orbiting a nucleus. The instantaneous force is always to-
wards the nucleus, so that the particle is kept in a circular orbit. The retarded force
is directed ahead of the nucleus so that it would tend to increase the radius of the or-
bit, and hence the energy. Dirac solved this problem by using the advanced force Fadv ==
*" fTOO e(T-T')!TO F( T')dT', so that the response to any impulse precedes its cause. This can
be easily seen by rewriting the Lorentz-Dirac equation in terms of Fs:
P_"!"F __ dEv
s TO s - dT·
In this form the equation looks extremely dangerous, as we have already shown it to be in
practice. An obvious causal modification is
. 1 dEv
Fs+-Fs = -d '
TO T
which uses (2F - Fred instead of Fadv. It responds to an impulse like a mass of m/2 and then
remembers the rest of its mass over a time ~ TO. Applied to the step problem, the modified
46 S. F. GULL

equation has properties very similar to the sensible branches of the Lorentz-Dirac solutions
in Figure l(b). The disallowed values of Vin now have Vout = 0, and we take the upper branch
for large Vin. However, although the modification produces results indistinguishable to the
accuracy of plotting from those of Figure l(b), they are not identical, and the suggestion is
ad hoc. To make further progress we must have a more detailed model of electron behaviour,
but I conclude from this study that we probably have to revise our notions about the rate
of radiation from accelerated charges.

5. Realistic classical models

Making classical models of particles with spin is an interesting game, all the more so
when the STA is available. An example of such a model has been given by Barut &
Zhanghi (1984); in addition to its position x, the particle has an internal mechanism or
"clock" determined by a Dirac spinor 1/J. Using the STA, and returning to natural units
(li = c = 1), we translate their Lagrangian as
1 . - - -
I: = '2 (1/J ia31/J) + (p(i: -1/J101/J)) + e (A(x)1/J,01/
J) ,

where ( ) means "scalar part of". The equations of motion are

ir = eF· v,
where 7r == P - eA. The Hamiltonian is 7r • V = m and the angular momentum bivector is
~1/Jia30 + x /\ p.
The solution of these equations for zero field can be written in terms of

The model can display longitudinal znw through interference of positive and negative
frequencies but, unlike solutions of the Dirac equation, the negative frequency solution has
the same charge/mass ratio as the positive energy one. To reverse the charge/mass ratio,
it is necessary for the momentum 7r to point into the backward light-cone, thereby making
the Hamiltonian negative. When the particle encounters a potential step there is a sudden
change in its momentum 7r, but the spinor1/J is continuous. Energy is conserved; the time
component of 7r changing by e¢. The subsequent motion inside the step shows longitudinal
ZBW and analysis of the mean velocity shows that the appa.rent rest mass has decreased.
The behaviour of the model in a magnetic field F = Bia3 shows that it has no magnetic
moment (g = 0), because a stationary solution (v = 10, 7r = mlo, 1/J = e- i0'3 mT ) can
always be found. Following the suggestion by Hestenes (1990) for the Dirac current, we
can make the ZBW of this model manifest by redefining the velocity v = 1/J1- 0, where
1_ == 10 - 12 again. Examination of the stationary solutions shows that the negative
frequencies have been eliminated and the positive frequencies moved to 2m. There is now a
magnetic moment; computer studies show that 9 = 1 for this modification, with the ZBW
orbit plane precessing at one half the gyro frequency eB /m.
CHARGED PARTICLES AT POTENTIAL STEPS 47

I conclude that, whilst Barut's little model is not yet satisfactory to describe the electron,
there is still great promise for realistic models, particularly when the STA is employed.

6. Conclusions
We now have a natural language for spacetime physics that simplifies manipulations
and gives equations which are fully Lorentz invariant and coordinate-free. The SpaceTime
Algebra is able to provide important insights into the geometrical content of Dirac theory.
Hestenes' ZBW interpretation of quantum mechanics seems promising, and there are im-
portant clues suggesting that the electron has a helical motion associated with its spin, but
the longitudinal ZBW remains a mystery.
The problem of radiation reaction will not go away, but we can at least hope that the
Lorentz-Dirac equation will finally be laid to rest. The crazy behaviour of this equation is
well illustrated by its disastrous failure for the potential step problem.

Acknowledgements
I have greatly benefited from discussions with Anthony Lasenby and David Hestenes.

References

Abers, E. & Lee, B. (1973). Gauge Theories. Phys. Reports, 9, 1-141.

Barut, A. O. (1988). Lorentz-Dirac Equation and Energy Conservation for Radiating Elec-
trons. Phys. Lett., 131, 11-12.
Barut, A. O. (1990). Renormalisation and Elimination of Preacceleration and Runaway
Solutions of the Lorentz-Dirac Equation. Phys. Lett., A131, 11-12.
Barut, A. O. & Zhanghi, N. (1984). Classical Model of the Electron. Phys. Rev. Lett., 52,
2009-2012.
Bjorken, J. D. & DreH, S. D. (1964). Relativistic Quantum Mechanics, Vol. 1. McGraw-Hill,
New York.
Burke, W. L. (1970). Runaway Solutions: Remarks on the Asymptotic Theory of Radiation
Damping. Phys. Rev., A2, 1501-1505.
Dirac, P. A. M. (1938). Classical Theory of Radiating Electrons. Proc. Roy. Soc. Lond.,
A47,148-169.
Feynman, R. P. (1967). The Character of Physical Law. MIT Press, Cambridge MA.
Grandy, W. T., Jr. & Aghazadeh, A. (1983). Radiative Corrections for Extended Charged
Particles in Classical Electrodynamics. Ann. Phys., 142, 284-298.
Hestenes, D. (1966). Space-Time Algebra. Gordon & Breach, New York.
Hestenes, D. (1979). Spin and Uncertainty in the Interpretation of Quantum Mechanics.
Am. J. Phys., 47, 399-415.
Hestenes, D. (1982). Space-Time Structure of Weak and Electromagnetic Interactions.
Found. Phys., 12, 153-168.
Hestenes, D. (1986). A Unified Language for Mathematics and Physics. In Clifford Algebras
and Their Application in Physics, (ed. J. S. R. Chisholm & A. K. Common), pp. 1-23.
D. Reidel, Dordrecht.
Hestenes, D. (1990). The Zitterbewegung Interpretation of Quantum Mechanics. Found.
Phys., 20, 1213-1232.
48 S.F.GULL

Jaynes, E. T. (1990). Probability in Quantum Theory. In Complexity, Entropy, and the

Physics of Information, (ed. W. H. Zurek), pp. 33-55. Addison-Wesley, Redwood City
CA.
Nikishov, A.1. (1970). Barrier Scattering in Field Theory and Removal of the Klein Paradox.
Nuclear Phys., B21, 346-358.
Sawada, T., Kawabata, T. & Uchiyama, F. (1983). Derivation of an extra condition on the
Lorentz-Dirac equation from a conservation law. Phys. Rev., D27, 454-455.
Wheeler, J. A. & Feynman, R. P. (1949). Classical Electrodynamics is Terms of Direct
Interparticle Action. Rev. Mod. Phys., 21, 425-433.

Appendix. A translation table for Dirac spinors

In this Appendix the conventional and STA versions of the Dirac spinor are written
down explicitly, so that one can translate freely between them. For the standard column
spinor we employ the form of the Dirac matrices given by Bjorken & Drell (1964), the
components of which are themselves 2-component Pauli spin matrices:

io = (~ ~l); ik = (-~k ~k).

We identify a 4-component complex column vector iii in the standard treatment with an even
multi vector '¢ of the STA. To make it quite clear that iii is a column vector in "spin-space",
we will use Dirac notation. Thus
1iI=1,¢) f--T '¢.
conventional STA
The STA form of the Dirac spinor can be written in terms of its components:
'¢ = ao + akO"k + boi + bkiO"k·
This translates into a matrix as above, and can be converted into the conventional spinor
by taking its first column:

I'¢) = r -~~: ~::o 1·

a3 +lb
at + ia2
We translate the operators of matrix theory as follows:
il,l'¢) i--+ "'(111/1"'(0,

iiV') ..... ,¢i0"3'

We are now in a position to translate the Dirac equation from the conventional form
ii I1 811 1'¢) - ei l1 AI1I,¢) = ml'¢) ~ ",(1181',¢,oi0"3 - ql1 AI'I/J,o = m'¢.
Multiplying by ,0
and re-assembling the vectors A = ,11 AI1 and V = "'(11811 we obtain the
STA version of the Dirac equation
V'¢i0"3 - eA'¢ = m,¢,o.
NEW SOLUTIONS OF THE DIRAC EQUATION
FOR CENTRAL FIELDS

HEINZ KRUGER
Fachbereich Physik der Universitat
Postfach 9049
6750 Kaiserslautern
West Germany

ABSTRACT. A modified form of Hestenes's space-time version of Dirac's equation

is separated in spherical polar coordinates. The angular part of the solution spinor
is derived in its most general form in terms of Gegenbauer polynomials and exponen-
tials. One- and two-valued (multi-valued) spinors are obtained as a generalization
of Weyl's spherical harmonics with spin. The radial spinor equation defines a redu-
ced real space-time algebra with one time and two space dimensions and displays
symmetries which are hidden in the conventional matrix form. The superiority of
this real valued Clifford algebraic formulation of the radial problem is demonstrated
for the bound states of hydrogen-like atoms. The single-valued solutions turn out
to be the ones of Darwin in a different representation.
One- and two-valued solutions together imply the unexpected discovery that
the hydrogen atom in its states of lowest energy may realize the following values of
HI
its magnetic moment: J.L = J.LD and J.L = J.LD /11", where J.LD = J.LDarwin = + 21)J.LB ,
1 = V1- 0: = ~2, J.LB = J.LEohr = ~. Presumably it is an effect of the
0: 2 ,
selfinteraction which s~ems to prefer the singl~-valued Darwin solution of the Dirac
equation in the normal state of the hydrogen atom.

1. Introduction

°
When I presented my new f3 = solutions for the first time at the Antigonish
workshop 1990, I was asked the following question:
Can these f3 = °
solutions be generalized to also include magnetic quantum
numbers different from zero?
This article gives an answer in terms of a new and complete separation of the
real Dirac equation with respect to both angles of spherical polar coordinates.
49
D. Hestenes and A. Weingartshofer (eds.), The Electron, 49-8\.
© 1991 Kluwer Academic Publishers.
50 H.KRUGER

The separation is carried through in section 2. In section 3 the general solution

of the angle equations is derived, leading to multi-valued, square-integrable spherical
harmonics with spin. Section 4 is reserved for the study of algebraic properties of the
radial spinor equation and their symmetries. The special (J = 0 solutions mentioned
above are identified within the general context in section 5. The investigation of
the transformation properties of the general central field spinor with respect to the
space inversion - or parity operation is the content of section 6. It is shown that
the repeated application of the parity operation P provides a simple coordinate-
free criterion for the multi-valuedness of a spinor on the 3-dimensional euclidean
subspace IE3 in the Minkowski space 1M(1,3): A spinor is single valued iff (if and
only if) the twofold application of P is the identity, i.e. p 2 = 1. So, a spinor is
two-valued on IE 3 , iff p 2 = -1, four-valued, iff p 4 = -1, etc ..
Hydrogen-like bound states and their normalization are derived in section 7 for
nuclear charge numbers Z :c:; 137. The intent of this section only is to demonstrate
in detail the practical feasibility of this representation-free Clifford algebraic spinor
calculus. It is evident, that for instance the case Z > 137 (Klein paradox) and
Coulomb scattering can be treated with the same efficiency. These investigations
as well as their applications to radiative transitions between bound- and scattering
states (bremsstrahlung and pair creation) are subject to forthcoming publications.
In section 8, a brief formulation of the anomalous Zeeman effect by means
of space-time algebra is presented for the first time. The states of lowest energy
are discussed in detail and the magnetic moments announced in the abstract are
calculated.

2. Separation of Variables

The real Clifford algebraic formulation of Dirac's equation developed by David

Hestenes [1]

(2.1)

explicitely depends on the basis vector 10, 16 = 1 defining the time axis and on
the spacelike bivector i3 = 1211 = iu3 , which at the first view may seem to prefer
the 3-axis in IE3. That this preference in fact is as spurious as the dependence of
a matrix formulation of Dirac's theory on the choice of a special representation has
been discussed in [1]. We are free to rotate the axis u3 with the help of a unitary
spinor U to an arbitrary axis ii, ii 2 = 1 by
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 51

t/;H = t/;U , t/; = t/;H U t

(2.2)

The sign if means the reverse of a multivector M and is obtained by reversing the
order of all vector factors in all the products of which M is composed. From (2.2)
and (2.1) we find the Dirac equation

tt8t/;iii - t{ At/; = mct/;'Yo

c
it/; = t/;i . (2.3)

Just as the 3-axis in Hestenes's equation (2.1), the direction ii may be interpreted
as the unit normal on the spin plane in which the" Zitterbewegung" of the lepton
takes place. For the standard chart of the spherical polar coordinates discussed
below there is a special choice of ii, which greatly simplifies the task of separating
the variables, namely

(2.4)

k = 1,2,3 (2.5)

The standard chart of spherical polar coordiates

(2.6)
O<r<oo o < tJ < 11" , 0 < 'P < 211" ,

can be factorized in spinor form according to

r = r (sin tJ U1 ei • 'P + Us cos tJ) = re-" 'P /2 [ii1 sin tJ + Us cos tJJ eis 'P /2
= re- is 'P/ 2 e- i .{J/2 Us ei .{J/2 eis 'P/2 (2.7)
=rSusS=r,
S-l = S = st , S· = S. (2.8)

This factorization of r induces a spinor factorization of the invariant derivative

(gradient operator) both on lEs
52 H.KRUGER

Orv'sin,'} = S (2.9)

8
8~
r
== -8cp , (2.10)

and on the Minkowski space 11\1(1,3)

ro = ct . (2.11)

For a central field

qA=1'oV(r), r=lrl , (2.12)

equation (2.3) with the choice (2.4) and (2.11) becomes

(2.13)

which permits a separation of the ro- and cp-dependence with the two separation
constants E and .x

.x, E E 1R , (2.14)

(2.15)

Separation of rand,'} however is less trivial because of the noncommutativity of the

Clifford product and the post multiplication factors i2 and 1'0 in (2.15). For solving
this problem I put

?jJdr,,'}) = l1(r)g(,'}) +i3 11(r)f(,'}) , tl1=l1t • (2.16)

where f and g have to fulfill the conditions

NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 53

1=/* = "Ioho g = g* (2.17)

Equation (2.15) then leads to

0= - ""I3'1'2g
f< I • n"lt
- - ' 1 (, .
g '2 >'1 ) + "10 -1 (E -
+ ---;----:Q V )'1g - mC'1g"lo
r Slllu C

- In3'S
f< • . I
'1, '2 - ' 1 (I" '2
+ -nis"It - >.g ) + "10 -1 (E - V)''3'1 I -
---;----:Q
. I "10 = 0
mC'3'1 ,
r Slllu C

where the primes stand for the derivatives with respect to the corresponding va-
riables r and t'J. Introducing the separation constant It E IR according to

I ' '.2 - >.g = It I

---;----:Q , g
' '2. + ---;----:Q
>'1 = - Itg ,
Slllu Slllu (2.18)
1= 1(t'J) , g = g(t'J) , It E IR ,

I achieve a complete separation of the angle spinors 1(t'J) and g(t'J) from the radial
spinor '1(r), which has to solve the radial equation

f<
-/~"I3 '1
I •
'2 + (E -V nit) '1 = mc'1"1o
"10 - c - + "It --;: '1 = '1(r) , i'1 = '1i . (2.19)

3. Spherical Harmonics with Spin

Before proceeding to derive the general solution of the angle equations (2.18),
it is worth to study the special case>. = 0 first. For>. = 0 the spinors I and g are
decoupled and one finds two types of linear independent solutions

(3.1)

(3.2)

The general solution then is a linear combination of (3.1) and (3.2). In perhaps
more familiar physical terms, (3.1) and (3.2) comprise a complete set of two normal-
vibration solutions into which any>. = 0 solution of (2.18) may be decomposed. In
54 H.KROGER

section 5 it is shown that each of (3.1) and (3.2) separately, when inserted into
(2.16), lead to a solution of the Dirac equation with a vanishing Yvon-Takabayasi
parameter (3 = O. Just these solutions I have presented at the Antigonish workshop
1990.
A first step towards the general solution of (2.18) for A i= 0 is to decouple I
and g. For this sake let me introduce

(3.3)

instead of I, which leads to

I 1, + 12/t
. I
1
Ag A(g)-
= ~ = ---=----=0 , (3.4)
sIn v sIn v

and after a reversion of the equation (2.18) for g, one notes the suggestive form

(3.5)

Thus 9 is up to a constant spinor factor equal to 11, viz.,

c>O , 6EIR. (3.6)

Inserting (3.6) into (3.5) and (3.4), one notes that c = 1 and that

(3.7)

has to fulfill

,. AW
W +t 2 I<:w= ~. (3.8)
smv

Choosing the phase 6 = 0, I obtain

(3.9)

and from (2.16)

(3.10)
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 55

There is a close connection between the spinor equation (3.8) and the spherical har-
monics with spin P(D) and Q(D), P, Q E JR, defined by Hermann Weyl eventually
before 1929 [21. In fact,

(3.11)

and the replacement of It by k and .\ by -m - t lead to equations (7.13) in [21 for

the variable z = cos D.
Since I am interested in the general solution of (3.8), I make no use of Weyl's
single-valued special solutions. As the discussion of the case It = 0 already shows,
there are square-integrable but multi-valued solutions of (3.8) for 1.\1 <i
sinD o"w =.\w = (Z2 -1)ozw , z = cost? , (3.12)

given by

W = Cl (-
1-Z)~/2 .
- + '2C2
(1+Z)~/2
--
1
1.\1 < 2 . (3.13)
1+z 1-z

i
Inspection of (2.14) however shows that the factor i'~'P for 1>.1 < gives rise to a
solution (2.14) whose valuedness is greater than 2! Therefore, the finally adopted
restriction [9} to one- and two-valued spinors (2.14) leads to the conclusion, that
It = 0 is no admissible eigenvalue of (9.8).
In order to derive the general solution of (3.8) in the case .\ =I- 0, I put

(3.14)
C E JR , a = ao + i2 a2

where the constant spinors a and b may be adjusted such that

(3.15)

whence the real function C(t?) has to solve the differential equation of Gegenbauer
functions

(3.16)
56 H.KROGER

With the scalar integration constants Cl, C2 the general solution of (3.16) is [41

C(.1) = clCllAI 1.1 (cos .1)

"+. -IAI
F
+C221 (11t+~I+IAI
2 ' 11t+~I+IAI+1
2 I 11 2.1)
jlt+2"+ljcos, (3.17)

AiO , AEIR ,

from which a square-integrable spinor w results if C2 = 0 and only if

11t+ ~I-IAI = p = 0,1,2, ... E INo• (3.18)

Under these conditions (3.17) degenerates to a Gegenbauer polynomial C~AI(cos.1)

of degree pin z = cos .1, and from (3.15), (3.14) one infers for A> 0

while for A < 0

(3.20)

Before proceeding to calculate the normalization constant Ibl in (3.19), it is

necessary to study the case p = 0 in (3.18) first. For It + ~ = 1>.1 one notes the
result w = w+ = 0 and for It + i = -IAI, w+ = -21Allbl sin lAI .1 ei .(J/2. Imposing the
normalization condition
00

411" / dt1 ww = 1, (3.21)

o

one finds

(3.22)
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 57

For p = 1,2, ... E IN, the relation

(3.23)

allows to write (3.19) in the form

w+ = Ibl sin l>·l1J e,·6/2 [ (It + ~ -IAI) C~>'I (z) - 2i21AI~ C~~~l (z)]
z=cos1J ,
(3.24)

such that the necessary integrals in (3.21) are reduced to normalization integrals of
Gegenbauer polynomials. By use of r(x + 1) = xr(x) and comparison with (3.22),
one obtains the normalization conditions

It
Equation (3.8) shows that walso is a solution if is replaced by -It, and (3.25)
shows a shift p --+ P + 1 with respect to (3.26). So, w> obtained from (3.26), (3.24)
and (3.20) after p --+ p+1 has to be proportional to w<, as derived from (3.26), (3.24)
and (3.20), and, the factor of proportionality must be scalar. Thus for p --+ p + 1 in
w> , the relation must hold

c =c . (3.27)

In fact, making use of

and
2IAlzC~>"+ 1 (z) - (p + l)C!~ 1 (z) = 2IAIC~~~ 1 (z) ,
58 H.KROGER

one notes the result c = -1, provided the Gegenbauer polynomials C; (z) for a =f:. 0
are defined according to

C~ 1 (z) = 0 , C; (z) = 1 ,
(p+l)C:+ 1 (z)=2(p+a)zC;(z)-(p+2a-l)C:_ 1 (z) , (3.28)
p = 0,1, ... E /No •

Without loss of generality, the factor c = -1 can be absorbed into a convenient

choice of phase, such that, for A =f:. 0 the normalized solutions w of (3.8) can be
written in the following final form:
i, r(111-1)
W = w< = Wp). (11) e 'I"~
1 (3.29)
It = -k , k = p+ IAI +"2 ' p = 0,1,2, ... E /No ,

(3.30)

w (11) =21'·1 r(IAi) ( p! )1/2 sinl).1 11 e'.{}/2

p), 4'11" r(p + 21AI + 1)
[(p + 2IAI)C~),1 (cos 11) + 2i21AI sin 11 C!>:.!: 1 (cos 11)]

C~l(cos11) = 0 , C;(cos11) =1 , A =f:. 0 , (3.31)

(3.32)

4'11" I
fr

d11 [wp.\ (11)w q .\ (17)]0 = 6pq = { I for p = q

0 for p =f:. q (3.33)
o

Those readers who are interested in a connection between w(d) and spherical mono-
genics [5] will find a brief discussion of this obvious relation at the end of section 6.
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 59

4. Radial Algebra

The radial spinor equation (2.19)

- 1'1/"3 a f]i2 + ( '10 E ~ V + '11 hrlt) f] = mCf]'1o

r

(4.1)
a
if] = f]i , f] = f](r) ar = ar '

defines a radial algebra 9(1,2) as a subalgebra of the full space-time algebra 9(1,3).
Its generic vectors of grade 1 are the timelike vektor '10 and the two spacelike vectors
'13 and '11 which span a 3-dimensional Minkowski space. For all grades of 9 (1,2)
the generic elements are displayed in the following table:

TABLE 1. Generic elements of 9(1,2) C 9(1,3)

Name grade elements

Scalar o 1
Vector 1 '10, '13, '11
Bivector 2
Pseudoscalar
3
Trivector in 9(1,3)

According to if] = f]i the radial spinor f] is composed of all even elements of 9(1,3)
and, at the very least, has to be a linear combination of all the underlined even
elements of 9(1,2) which span 9+ (1,2) ~ 9(2), i.e.,

It should be emphasized however that the scalar coefficients may be generalized

to include pseudoscalar parts of 9(1,3) as well. In that case the multivector (4.2)
would comprise all even products of 9(1, 3) and hence represent a general spinor of
8 real components.
In this article I shall employ scalar coefficients f],. only, such that f] E 9(2).
The restriction then induces the following form of the space inversion or parity
operation on 9(2)

(4.3)
60 H.KRUGER

Equation (4.1) is forminvariant under a reflection at the 8 2 8 3 - plane with the

normal 8 1

(4.4)

Therefore the general solution 17 E 9(2) can always be decomposed into an odd part

170
1(17 - 17 R) = 2
=2 1(17 + 0"1170"1
~~) ~
= 170 + 0"1171 (4.5)

and an even part

17.
1(17 + 17 R) = 2
=2 1(17 - ~~)
0"1170"1
~).'2
= (172 + 0"1173 . (4.6)

Since the even spinor 17., up to the phase factor i 2 , is of the same form as the
odd spinor 170' a further restriction from 17 E 9(2) to 170 = 170 + 8 1I'Ji is no loss of
generality. With 17 = 170 = 170 + 8 1171 the radial equation (4.1) yields the following
set of coupled differential equations for the scalar functions 17o(r) and 17dr),

nCK
lic8r 171 = --1'J1 + (E - mc2 ) 170
- V
r
(4.7)
2)
lic8 r 170 = (V - E - mc 1'J1
liCK
+ -170 •
r

After a replacement of m by -m, this system is identical with the conventional radial
equations [6]. In this sense, the new radial spinor equation (4.1) may be understood
as a natural hypercomplex generalization of the radial equations (4.7) with a well
defined geometrical structure. Both the structural geometric insight and the gain of
practicability by avoiding matrices make (4.1) superior to (4.7).

5. Selection of f3 = O(mod 1l") Solutions

The general form of a spherically separated central field solution of the Dirac
equation (2.3) follows from (2.4), (2.8), (2.9) and (2.16),

r,
1/J(ro, d, IP) = n(17g + i317/) exp {i2 (AlP _ ~:o )} (5.1)

The spinor (5.1) may be brought into the canonical polar form
NEW SOLUTIONS OF THE DIRAC EQUAnON FOR CENTRAL FIELDS 61

t/J = l/2 eLf- R , iR = Ri , RR = 1 , e> 0 , f3 E 1R, (5.2)

in order to allow the determination of the parameter f3 of Yvon [7] and Takabayasi
[8]. From equation (5.2) one finds

t/J~ = e ei{J = ecosf3 + ie sin f3 (5.3)

One notes in particular that f3 = O(mod 11") iff t/J~ = [t/J~]o = scalar. From (5.1) and
the fact that because of (2.17) 1j = 1/12 and gg = Igl2 are scalars, one obtains
t/J~ = n(1Jg + i31JI)(gii - jiii3)O
(5.4)
= n(gg1Jii -: Iji 31Jii i 3 + i31J19ii -1Jgjiji3)O

and equation (4.2)

'1 = a + iA b , a = 1Jo + i21J2 , b = 1J1 + i 2 1J3 ,

(5.5)
'1,. E 1R , I-' = 0,1,2,3 ,

leads to

(5.6)

Therefore, equation (5.4) can be rearranged according to

The decomposition 19 = [19]0 - i 2 [idg]0 = (gj)~ implies

iJ31JHii - '1gjiiiJ3 = -(U31Ji2ii + 1J i 2iiiJ3)[idg]0

(5.8)
= 2(ab + ab)[i2Ig]0

by making use of (5.5). So, together with (2.8) and (2.9) one finds the result
62 H.KROGER

where

2a = Tf + Tf* (5.10)

There are only two possibilities for the pseudoscalar in (5.8) to vanish:
either ab = -ab = -(abt, or, [i2f9JO = O. The first condition leads to ab = i 2 clal 2 ,
c E JR, i.e., b = -i2ca, which implies that the radial spinor Tf must be of the
very particular form Tf = (1 + Cif3 )a. With the exception of e.g. Ii: = ±1, It = 0
and m = 0 (charged neutrino with no centrifugal repulsion) the particular spinor
Tf = (1 + cif3 )(Tfo + i2Tf2) can not be a solution of the radial equation (4.1).
The only possibility to find {3 = O(mod 11') solutions for a massive lepton the-
refore is to fulfill the condition [i2 f910 = 0, or, jg = j g. One notes that equation
(3.1) or equation (3.2) is sufficient to satisfy this restriction. That either equation
(3.1) or equation (3.2) are also necessary, is seen by inspection of (3.9), which then
leads to the constraint [w 2 Jo = O.
My final conclusion concerning the existence of {3 = O{mod 11') solutions of the
central field Dirac equation therefore is, that the only ones are those defined by
equation (3.1), or ,equation (3.2). Hence, we fall back upon those, I have presented
at the Antigonish workshop 1990.

6. Parity and Multi-valuedness

The parity operation

(6.1)

leaves equation (2.3) with potential (2.12) invariant. So, if t/J is a solution of (2.3),
then, t/JP, the higher iterates (t/JP)P , etc., and their linear combinations are also
solutions, provided the potential A is even, i.e., A = A P •
In order to investigate the parity of the various solutions t/J obtained via
(5.1), (3.1), (3.2), (3.9), (3.10) and (3.29) to (3.32), let me start with the case
{3 = O(mod 11'), or, equivalently, >. = O. Equations (5.1) and (3.1) then yield

(6.2)

and equation (3.2) with a convenient choice of phase leads to

NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 63

(6.3)

From equations (2.8) and (2.9) one finds

(6.4)

and with (4.3)

(6.5)

such that finally

(6.6)

In the case of the hydrogen atom (see section 7), the constant It is confined to integer
values with the exception of the value zero. Wether this also holds for the valence
electron of alkaliatoms remains to be seen! At any rate, for It E Z\ {O}, tP+ and tP-
have opposite parities and for It E 1R one obtains

(6.7)

Turning now to the>. 1= 0 solutions (5.1), (3.9) and (3.10), two cases must be
distinguished according to (3.29) and (3.30), namely,

tP. = O(fJ.w - i3fJ.illw) exp {ill (>'IP _ ~o)} , (6.8)

1
It = -k , k = p + 1>'1 + 2 '

and

1
, It = k = p + 1>'1 + 2' (6.9)

where
64 H.KROGER

(6.10)

and "I., "Ig are solutions of (2.19) and (4.2) for the respective values of It = -k or
It = k, k = p + 1>'1 + t,

(6.11)

One might be tempted to generate the solutions YJg from YJ. by applying the linear
mapping

(6.12)

since this obviously transforms (up to phase factors) YJ. into YJg and vice versa. It
should be emphasized however, that for a spinor YJ satisfying the restriction (4.2),
the image I(YJ) no longer lies in the domain (4.2). Consequently the map (6.12) is
not disposable to generate such a simple equivalence between "I. and YJg. This aspect
is quite important when degeneracies between (6.8) and (6.9) are to be classified.
Again, with the help of (6.4), (6.5) and (3.22), the parities of tP. and tPg are
easily found, viz.,

P E!No (6.13)

As seen from (2.8) and (2.9), the valuedness of 0 for the cycle cp -4 cp + 2". is
O(cp + 2".) = -O(cp) . The valuedness of (6.8) and (6.9) in addition is determined
by the phase factor ei • ~ 'P. So, one finds

(6.14)

which proves the conjecture stated in the introduction.

At the end of section 3, I promised to briefly establish the relation between
spherical harmonics with spin and spherical monogenics. Here it is! Going back
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 65

from (2.14) to (2.3) and putting m = 0, q = 0, or, V = 0, and E = 0, the spinors

(6.8) and (6.9)

(6.15)

(6.16)

are seen to be solutions of

at/J • = 0 (6.17)
g

The radial equations (6.11)

(6.18)

according to (4.5), have the solutions

(6.19)

and

(6.20)

which generate multi-valued spherical monogenics with the aid of (6.16) and (6.15).
In the same way one may construct f3 = O{mod '11") monogenics by making use of
(5.1), (3.1), or, (3.2).

7. Hydrogen-like Bound States

In this section, square integrable solutions of type (4.5) are derived for the
radial equation

-1i ,3 o. '7'2. + (E -V iiI\.) '7 =

10 - c - + 11 -;- mc'7/o
(7.1)
'7 = '70 + 01 '71 , '70,1 E IR ,

with the attractive Coulomb potential

66 H.KRUGER

1 he
rV = -heZa , - = - = 137,036
a q2
, (7.2)

under the conditions, that the charge number Z of the nucleus satisfies

0< Za < k , 1~ k = IKI , (7.3)

and the energy E is in the range

0< E < mc 2 (7.4)

Introducing a reduced energy g, an effective fine-structure constant a

a=Za (7.5)

and instead of r = if' 1 the variable

me r;---;;
Y = 2r-v1- g2 O<g<1 , (7.6)
h '

the radial equation (7.1) becomes

(7.7)

Provided condition (7.3) is fulfilled, the vector a10 + K11 is spacelike and can be
made proportional to 11 with the help of Lorentz transformations. The spinors for
all these transforms can be traced back to the following definitions

1~ k = IKI , a < 1 , (7.8)

(7.9)

(7.10)

The spinor L has the properties

NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 67

L- 1 = L = L' (7.11)

and

(7.12)

So, for II: > 0 one finds the Lorentz transformation

(7.13)

and for II: <0

Making use of (7.14) the radial equation (6.11) in the form (7.7) for the II: < 0
solution (6.8) (with the suffix s = smaller) may be written

(7.15)

and the corresponding one for II: > 0 (suffix g = greater)

(7.16)

As seen from (7.8) - (7.10), both equations depend on 111:1 and not on the sign of 11:.
Therefore the value

1
111:1 = k = p + 1>'1 + 2" ' p = 0,1,2, ... , E /No (7.17)

may be taken in both (7.15) and (7.16).

For solving (7.15) and (7.16), I follow the standard method, see e.g. section 28
in [6], and split from t7 its behaviour at y = 0 and y = 00. For y ~ 0+ (7.15) and
(7.16) behave like (6.18) with II: replaced by /. So, the regular solutions are
68 H.KRUOER

(7.18)

It should be noted that, as distinguished from the incorrect nonrelativistic limit

(Schrodinger), the Coulomb potential (7.2) in Dirac's theory acts as a singular po-
tential, which modifies the free-particle behaviour to a potential-dependent different
form.
At y - t 00 the spinor L cancels out, since its importance is confined to the
origin, and equations (7.15) and (7.16) behave asymptotically like

(7.19)

Looking for square integrable solutions, I write

(7.20)

Where H is a constant spinor of the form (4.5), i.e., with HO,l E JR,

Equation (7.19) then imposes on the unimodular spinor K the condition

(7.22)

which because of K2{K2t = 1 can only hold if 4/1,2 = 1, or, for J1. = ~ in (7.20)
and (7.22). From 1 + K4 = 2K2 Ie one infers (up to a sign)

Equations (7.18) and (7.20) suggest to put

(7.24)

(7.25)
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 69

where a and b are constant spinors and .c, M are scalar functions of the variable
y. Inserting (7.24) and (7.25) in (7.15) and (7.16) one finds after some straightfor-
ward algebra that .c and M have to solve the following confluent hypergeometric
differential equation

[ya~ + (2, + 1 - y)a y + V(e)].c(y) = 0 , M(y) = .c(y), (7.26)

where

lie
V(e) = ~-' , (7.27)
1- e
2

provided the spinors a and b are determined by

(7.28)

With integration constants C1, cz, the general solution of (7.26) may be written, see
e.g. [4], pp. 252-253,

Square integrable solutions exist if C2 = 0 and only if the Sommerfeld condition

lie
V(e) = ~-' = 0,1,2, ... ,E INo (7.30)
2 1- e

is fulfilled, which leads to his famous fine-structure formula

- 1/2

e=~=
mc
1+(~)2
£1+,
2
[ ]

(7.31)
v,p E INo

For C2 = 0 and £lEINo, .c(y) according to (7.29) is proportional to the Laguerre

polynomial L~2.t) (y), [4], p.268, and the unnormalized spinors (7.24), (7.25) become
70 H.KRUGER

which up to a scalar factor may as well be written in the form

11. =y" e-II/2kh(1-K2L2)+yaIlJL~2")(y) =11t ,

(7.32)
11g =y" e-II/2kh(1+K2L2)+yaIlJL~2")(y) =11t

The derivative of the Laguerre polynomial can be eliminated with the help of

which yields

A further simplification is achieved by following Bethe and Salpeter [l1J and elimi-
nating c in favour of the apparent principal quantum-number

N = vv(v + 21) + k2 = Vn2 + 2vb - k) ,

1 (7.36)
n=v+k, k=p+I..\I+'2 '

whence

cN=V+1, NVl-c 2 =a=Zcx. , v(v + 21) = N2 - k2 , (7.37)

and
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 11

(/I + "Y)K2 = N + ifl ii. ,

/I + "y - "y~ i} = (N - k)K2 , (7.38)
/I + "y + "y~ L2 = (N + k)K2

So, (7.34) and (7.35) can be written in the simpler form

'1. = '1! = N.y7 e- II / 2 [(N - k)KL~27) (y) - (/I + 2"Y)kL~2':!<y)], (7.39)

'1g = '1! = Nfly7 e- II / 2 [(N + k)KL~27) (y) - (/I + 2"Y)kL~2':! (y)] (7.40)

The normalization constants N. and Nfl are now determined such that the spinors
(6.8) and (6.9) are normalized according to

(7.41)

which restricts the total probability in the 1E3 to the value 1, or, the total charge
to the value q. Equation (3.33) then implies for the spinor '1 the normalization
condition
co co

/ dr ['1'1 flo = 1 = ~: / dy ['1 2]0 , (7.42)

o o

where ao is the Bohr radius. The integrals in (7.42) are easily done by making use
of

co d 27 _II L(2 7 ) ( )L(2 7 ) ( ) = o""r(/I + 2"Y + 1) (7.43)

/ YY e " y" y r(/I + 1)
o

Before displaying the final results in Table 2, it should be noted that '1. = 0 for
/I = O. Hence, square-integrable nontrivial solutions '1. only exist for /I ~ 1, /I E IN.
Together with (7.40) they form a twofold degenerated set of solutions of (7.1) for the
energy eigenvalues (7.31). Only for /I = 0, equation (7.40) defines nondegenerate
solutions, which include the state of lowest energy and Darwin's [10] IS.l ground
state solution. The degeneracy for /I ~ 1 is dynamical, because it originates • in
the shape of the potential. A variation of the Coulomb potential, e.g., by the
selfinteraction removes this degeneracy!
72 H.KROGER

TABLE 2. Normalized one- and two-valued bound state spinors

for hydrogen-like atoms

;"2
Yflo = 2Zr , flo =-2 .
mq

IV=1,2,3, ... ElN I "(=v'k2 _Z2a.2 , N = v'v(v + 2,,() + k2 ,

EN = mc (v 2 + "() , k = 1,2,3, ... E lN

'1. = ~ [ZV!(V+"()(N-k)]1/2 y1 e-u/2

[KL~21)(y) - ~(N +k)L~2_11(y)]
N flo Nr(v + 2"( + 1)
= ~ [ZV!(V+"()(N+k)]1/2 y1 e- II / 2
[KL~21)(y) - ~(N - k)L~2_11(Y)]
'1g N aoNr(v+2,,(+1)
2Zr;,,2 ~
y= -N' flo = - 2 ' Ky2(v+"() = v'N+V+"(+U1v'N-v-"(,
flo mq
(v + 1)L~2:Uy) = (2v + 2"( + 1- y)L~2~) (y) - (v + 2"()L~2_~1(y) ,
v~O , L~2;)(y)=0 , L~2~)(y)=1 .

lone-valued spinors = Darwin[9] I 1

k = p+ 1>'1 + 2' p,m = 0,1,2, ... E lNo,

1>'1 = m + ~, tP. = n('1.w - i3'1.i2W) exp {i2 (>.cp _ ~o ) } ,

tPg = n('1gW- i 3'1 gi 2W) exp {i2 (>.cP _ ~:o )}

21>'lr(l>.1) ( p! )1/2 sinl>'1 iJ e,,(J/2
411" r(p + 21>'1 + 1)
[(p + 21>.I)C~>'1 (cos iJ) + 2i 2 1>'1 sin iJ C~~~ 1 (cos iJ)]
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 73

Comparing the one-valued solutions with those of Bethe and Salpeter [11], one
finds after deriving their matrix form according to equations (2.1)-(2.4) and [lJ,
that they become identical with Darwin's solutions [9] in a nonstandard matrix
representation.

8. Anomalous Zeeman Effect, Magnetic Moments

Richard Gurtler [12], to my knowledge, has given the first variational formula-
tion of the real space-time version of Dirac's theory. His action functional, a mapping
from a sufficiently well behaved set of real spinor fields t/J into the scalars I E IR,

!
ct.

I(t/J) = dro (8.1)

corresponding to equation (2.3) then is defined in terms of the Lagrange density

(8.2)

If for time-independent potentials A = A(r) this set of spinor fields is restricted to

square-integrable stationary ones satisfying

(8.3)

equation (8.1) gives rise to the hamiltonian action

A= ! tfr [(Et/J - )I (t/J))t/JtL

t
t/J =
-
10 t/J,O =
-.
t/J (8.4)
lE 3

with the linear and spinor-valued hamiltonian operator

(8.5)

which includes non-commutative pre-and postmultiplication by multivectors. It

should be noted, that the scalar product of the two spinors t/J and cp, defined by
[cpt/J t]o = [t/Jcp t]o, is just the natural one in IR8 to which the space of Dirac-Hestenes
74 H.KROGER

spinors 9(3) s:: 9+ (1,3) is isomorphic. The euclidean magnitude ItPl = V[tPtP tl o of
a spinor tP fulfills all laws of a Minkowski-Banach metric. So, on the linear space of
(Lebesgue) square-integrable spinor fields (on 1E3 ), the bilinear functional (into IR
and not into ([;' as in conventional quantum mechanics)

(8.6)

leads to the norm J< tP,tP > (and vice versa) and endows this linear space with
the structure of a complete normed space, i.e., of a Banach space. By application
of Stokes's theorem, one notes that (8.5) is symmetric, viz.

<)I(tP),tP>=<tP,)I(tP» , (8.7)

whence equation (8.4) alternatively may be written in the form

.If = < EtP-)l(tP),tP > = < tP,EtP-}l(tP) > = E < tP,tP > - < }I(tP),tP > . (8.8)

In the case of the Zeeman effect the potential A is composed of a central field
and the vector potential i of a constant magnetic field B = Ii x i = BiJ3 ,

(8.9)

(8.10)

The hamiltonian (8.5) may be decomposed according to (8.9)

(8.11)

into the unperturbed part

(8.12)

and the perturbation due to the magnetic field

NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 75

)(1 (1/1) = -qA1/I , (8.13)

which entails a corresponding splitting of the action (8.8)

A = Ao + Al ,
Ao = E < 1/1,1/1 > - < )(0(1/1),1/1 >
J J
(8.14)
=E d3 r [1/I1/I t ]0 - d3 r [)(0(1/I)1/It]0

(8.15)

In order to establish a simple relation between Al and the magnetic dipole moment
of Dirac's current j = 1/I,0~' one may rearrange the term [A1/I1/I t ]0 for A = x r, iE
aE = 0 as follows:
[A1/I1/I t ]0 = boA1/I,o~]0 = -[A,01/l,0~]0 = -(A,o) . j ,

j = 1/1,0 ~ = (J~ +; ho ,
Now because of (A,o) ·,0 = 0 one finds
-t -I -I 1 __ __ -f-l

-(A,o) . j = -(A,o) . (ho) = 2(Aj + jA ) == A· j

(8.16)
1 ....
= -(B
-11-1 -
x r).j = -B· (,xi) = [A1/I1/I ]0
- t
2 2

So, defining the magnetic dipole moment j1 of the Dirac charge current cq; according
to

j1=~Jd3rrx; , ;=(1/I,0~)1\,0=[1/I1/Ith, (8.17)

suffix 1 in [1/11/1 th means: grade 1 part in g(3) = g+ (1, 3) ,

equation (8.16) implies that (8.15) may be written in the wellknown simple form

(8.18)

It is obvious, that if the special magnetic field E is replaced by a time dependent

general electromagnetic field F = E + iE, the interaction term in (8.2) may be
76 H.KRUGER

brought into a form, which generalizes (8.18) in terms of the full electromagnetic
polarization bivector of the Dirac current. Applications of this modified action [13]
in quantum electrodynamics will be published elsewhere.
The application of the principle of least action on the Zeeman effect now is
conventional and straightforward. Assume that a finite number of unperturbed
solutions 1/;, is known,

1 = 1, ... ,lmax (8.19)

and extremize (minimize) the action (8.14) for the trial spinor

(8.20)

with respect to the scalar, linear variation parameters C,. (A generalization to

nonlinear spinor-valued variations is evident.) In this way, one finds

A == A(c) = L [(E - E" )~"" + jj. iii, I, ] C" C" ~0 , (8.21)

1,1,

where the overlap matrix ~"" is

(8.22)

and the ii"" are the magnetic dipole moments of the current matrix

-
1-'1,1, = 2'qjd 3
r r- X JI,I,
'"'!
= 1-'1,1,
- (8.23)
suffix 1 means: grade 1 part in 9(3)

The characteristic system of Euler-Lagrange equations then is

1 = 1,2, ... ,lma .. (8.24)

and the energy eigenvalues E are obtained by solving the characteristic equation
NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 77

det [( E - E,) .6./11 + jj . il/ll] = ° (8.25)

Equations (8.24) and (8.25) are now applied to a few states in TABLE 2. As
representative two- and one-valued spinors I select the following solutions for the
lowest energy Eo = mc2 " Z = 1, ,= VI - a 2 with v = 0, k = 1, A = 0,
or, p = 0, IAI = h N = 1:

two-valued solutions

tPa 7rV2 = 0" exp {i2 (tJ _ E~;o ) } ,

(8.26)
tPb 7rV2 = Oi3" exp { -i2 (tJ + E~;o ) }
OrVsintJ = S = e- ia 'P/ 2 e- i ,{!/2 (8.27)

one-valued solutions (Darwin)

Apart from fooo dr r[i71,,2]0 < 00, and the normalization condition

(8.29)

the spinor " may be left arbitrary in principle. In the case of the hydrogen atom"
is given by " = "9' i.e.,

;,,2 (8.30)
aoy = 2r , ao = - -
mq2

As a consequence of the normalization (8.29) the overlap matrix for the set
and the set tPD +, tPD _ is the unitmatrix respectively
tPa, tPb
78 H.KRUGER

~aa= 1 = ~bb , ~ab = 0 = ~ba

(8.31)
~D++ = 1 = ~D-- ~D+- = 0 = ~D-+

Evaluating the current matrix one finds that the currents 1aa are poloidal
bb

2~,-2 sint11aa = ±[Ol1l2 ]OSU1 S , (8.32)

bb

and the transition current Lb = J"a is equatorial

(8.33)

which implies that

..
JL =- qfd rrx1 S " '1
= 0" , (8.34)
:b 2 ::

and

f.lSa-r r.. X f dr r [.. 1l2]

00

.. .. q
JLab = JLba = 2
'1
1ab
qus
= s;- Ul 0
- -JLUs
= .. (8.35)
o

The result (8.34) may be interpreted that a hydrogen atom in one 0/ the two-valued
states (8.26) has a vanishing magnetic dipole moment as long as there is no external
magnetic field (or an internal selffield) which magnetizes it. If the presence of the
external Zeeman field jj = Bus, B ~ 0, is taken into account by the trial spinor
tP = catPa + CbtPb, equation (8.24) leads to the characteristic system

From the characteristic equation (8.25) one obtains for an electron (JL > 0)

E = E± = Eo ± IJLBI = Eo ± JLB , -q = Iql , JL >0 , B >0 , (8.37)

NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 79

and the corresponding normalized solutions of (8.36), (8.26) are

(8.38)

which yield the following magnetic dipole moments and energies

i1±± = ±i1ab = =fp,if3

(8.39)
E± = Eo ± p,B = Eo - jj . i1± ±

In the same way one finds for the one-valued Darwin solutions (8.28) the results

(8,40)
o

Therefore in the case of the Zeeman field jj = BU3' B ~ 0, the characteristic

matrix is diagonal and hence one obtains the following energies and corresponding
normalized spinors

(8,41)

For a hydrogen electron (-q = Iql) the quantity p, in (8.40) is easily calculated with
the help of (8.30). One derives the result

p, = ~~ j dr r[U17]2]0 = I~~~ j dy y[U17]2]0

o 0 (8,42)

where

Jl-B
Iqlh
=-- (8,43)
2mc

is the magneton of Bohr. Equations (8.39)-(8,43) confirm the announcement made

in the abstract Jl-7r = Jl-Darwin, which means that the magnetic dipole moment for
the one-valued Darwin solutions (8.28) is a factor of 7r times larger than for the
80 H.KROGER

two-valued realizations (8.38) of this state of lowest energy. Also, it should be em-
phasized that although tPa and tPb according to (8.26) and (8.27) are (J = O(mod 71")
solutions, the two-valued spinors (8.38) no longer are of this type. This fact indi-
cates a sensitive dependence of the Yvon-Takabayasi parameter {J on the magnetic
properties of a lepton.

9. Summary

By means of a few selected examples it has been shown that Hestenes's space-
time formulation of Dirac's theory provides an efficient geometric alternative to the
traditional relativistic quantum theory.
 NEW SOLUTIONS OF THE DIRAC EQUATION FOR CENTRAL FIELDS 81

REFERENCES

1. Hestenes, D. (1975) 'Observables, Operators, and Complex Numbers in the

Dirac Theory', J. Math. Phys. 16, 556-572.
2. Weyl, H. (1930) The Theory of Groups and Quantum Mechanics, Dover, U.S.A ..
3. Bopp, F. and Haag, R. (1950) 'fIber die Moglichkeit von Spinmodellen', Z.
Naturforschg. 5a,644-653.
4. Erdelyi, A. (1953) Bateman Manuscript Project, Higher Transcendental Func-
tions, Volume I, pp. 178-179, McGraw-Hill, New York.
5. Delanghe, R. and Sommen, F. (1986) 'Spingroups and spherical monogenics' ,
in J.S.R. Chisholm and A.K.Commen (eds.), Clifford Algebras and Their Ap-
plications in Mathematical Physics, D. Reidel Publishing Company, Dordrecht,
pp. 115-132.
6. Rose, M.E. (1961) Relativistic Electron Theory, John Wilney, New York, p.
159, equation 5.5 ..
7. Yvon, J. (1940) 'Equations de Dirac-Madelung',J. Phys. et Ie Radium 1, 18-24.
8. Takabayasi, T. (1957) 'Relativistic Hydrodynamics of the Dirac Matter', Prog.
Theor. Phys. Suppl. 1, 1-80.
9. Hestenes, D. (1967) 'Real Spinor Fields', J. Math. Phys. ~, 798-808, p. 805,
equations (5.7)-(5.11).
10. Darwin, C. G. (1928) 'The Wave Equations of the Electron', Proc. Roy. Soc.
Lond., A118, 654-680.
11. Bethe, H. A. and Salpeter, E. E. (1957) Quantum Mechanics of One- and Two-
Electron Atoms, Springer, Berlin, pp. 68-70.
12. Gurtler, R.W. (1972) 'Local observables in the Pauli and Dirac formulations of
quantum theory', Dissertation, Arizona State University, Appendix A2.
13. Babiker, M. and Loudon, R. (1983) 'Derivation of the Power-Zienau-Woolley
Hamiltonian in Quantum Electrodynamics by Gauge Transformation', Proc.
Roy. Soc. Lond. A385, 439-460.
THE ROLE OF THE DUALITY ROTATION IN THE DIRAC THEORY. COM-
PARISON BETWEEN THE DARWIN AND THE KRUGER SOLUTIONS FOR
THE CENTRAL POTENTIAL PROBLEM.

Roger BOUDET
Universite de Provence
PI. Hugo
13331 Marseille
France

ABSTRACT. The Dirac particle is mostly to be represented by a spacelike plane

P(x), the "spin plane" considered at each point x of the Minkowski spacetime M.
The infinitesimal motion of this plane expresses, after multiplication by the con-
stant fIc/2, the local energy of the particle.
The spin plane is subjected to a transformation of an euclidean nature, but
particular to the geometry of M and situated outside our usual understanding of the
geometry of the euclidean spaces, the duality rotation by an "angle" f3. Such a
transformation can act on planes but leaves invariant the straight lines. It plays a
fundamental role in the passage from the equation of the particle to the one of the
antiparticle: this passage is achieved by the change f3 -I f3+'If which reverses the
orientation of planes without changing the one of straight lines. It brings arguments
against the physical interest of the PT transform.
The solutions of the Dirac equation for a central potential problem are
studied from the point of view of the behaviour of the energy-momentum tensor
and of the angle f3. All would be clear if one would have, everywhere, f3 = 0 for the
electron and f:J = 'If for the positron! But it is not the case in the Darwin solutions.
A comparison between these solutions and the ones recently established by H.
Kruger, for which f3 = 0 or 'If everywhere, is carried out.

1. Introduction

This paper is mostly concerned with the role played in the Dirac theory by the
"angle" f3. The "mysterious" angle of Yvon-Takabayasi (the qualification "myste-
rious" is due to L. de Broglie), recognized in the Dirac theory in 1940 (or even
before) has found in 1967 its geometrical meaning with the use of the multivector
Clifford algebra C(Jl), or Space Time Algebra (STA), associated with the Minkowski
spacetime .II. = !Ri'3.
In this algebra, introduced for the first time in Quantum Mechanics, in a
convenient way by D. Hestenes [2], f3 is an "angle" of a "rotation", the duality
rotation, which acts on the antisymmetric tensors of rank two (bivectors) of .11., but
leaves the vectors of .II. invariant. Its meaning is beyond the field of properties which
can be reached by means of the orthogonal group O(Jl) of .11..
83
D. Hestenes and A. Weingartshofer (eds.), The Electron, 83-104.
© 1991 Kluwer Academic Publishers.
84 R.BOUDET

The role of f3 in Quantum Mechanics is both mathematically clear and

physically obscure.
It is clear that the change f3 -+ f3+'If plays an essential role in the passage from
the equation of a particle to the one of its associated antiparticle. But the angle f3
has never been measured in an experiment.
However, one might deduce, as an indirect result, from the Darwin solutions
of the Dirac equation for the central potential problem (see the article of P. Quili-
chini or the Gurtler thesis [3]) that this angle could have a value distinct from zero
for an electron, or 'If for a positron, and depending on the point x of ){ where the
wave function 1/J is considered.
But the solutions, recently established by H. Kruger (IiNew solutions of the
Dirac Equation ll , in the present IIProceedings"), in which f3 = 0 or f3 = 'If everywhere
give to the question a new aspect. Should not the values f3 = 0 or f3 = 'If be consi-
dered as eigenstates of the 1/J function, which assign to a Dirac particle the quality of
particle or antiparticle?
This is a crucial question with implications for the theory of all elementary
particles.
In part I, we recall some results previously established about the geometrical
interpretations of the local energy of a Dirac particle, and the role played by the
duality rotation, both in this interpretation and in the passage from the electron
equation to the positron one ([4] - [9]). Furthermore, using a pure geometrical
construction, the Dirac equation is presented here as a simple generalization of the
equality E = ± mc 2, relative to the particle or antiparticle at rest, which takes into
account its spin.
The aim of this part is, in particular, to avoid a widespread misunderstan-
ding: the use of ST A is more than a change of formalism (with respect to the usual
matrices and operators representation of Quantum Mechanics); it also brings an
important clarification and Simplification in Quantum Theory.
ST A is to be considered as the fundamental structure of the geometry of
spacetime, and the euclidean geometric properties that it allows one to describe
cannot be obtained by the usual methods of the matric representations on complex
spaces.
Hence, to the extent that it could not be possible that there exists disagree-
ment between the laws of physics and the geometry of spacetime, STA is to be
considered as a fundamental structure for Quantum Mechanics.
In part II, using the form given by P. Quilichini [3] and myself [6] to the
Darwin solutions, we achieve a comparison between these solutions and Kruger I s.

The reading of the first part needs, for its complete comprehension, some knowled~e
about STA (see Ref. [2] or the article of D. Hestenes in the present IIProceedings").
Readers non acquainted with STA may read the second part, only by using
the usual matricial formalism. The sympols 'YJI. and Gk='Yk'Yo are to be thought as
representing vectors of the euclidean spaces ){ = !R I '3 and E3 = !R 3'O (the latter being
to be considered as a space of bivectors inside the former). They are employed in
part II only with the help of operating rules which are the same as those used for the
Dirac and Pauli matrices.
ROLE OF DUALITY ROTATlON IN DIRAC THEORY 85

2. The Dirac Particle and the Duality Rotation.

2.1. THE MOMENTUM-ENERGY TENSOR OF A DIRAC PARTICLE AND
THE DUALITY ROTATION. A GEOMETRICAL INTERPRETATION OF THE
QUANTAL ENERGY.
2.1.1. Kinematical dkfinition. If I had to propose a definition of the energy associa-
ted with a Dirac particle, I would say first that such a definition implies a) the
mathematical knowledge of the Minkowski spacetime J( = !RI,a, b) the physical
notions (which are not pecular to Quantum Mechanics) of the local energy and
momentum-energy of a system.
Then I would continue to suggest: "We will define a Dirac particle as being
an oriented plane P(x) of J(, called the "spin plane", passing through a fOint x of J(.
P(x) is orthogonal, at each point x E J(, to an unit time-like vector v(x of J(, called
the "velocity unit vector" of the particle.
Furthermore, we will suppose that the plane P(x) satisfies the following
properties that we will call "Some of the Geometrical Principles of Quantum
Mechanics. "

First Principle (Principle of Inertia).

"If the energy associated with the particle would be null, v would be a con-
stant vector, and furthermore the plane Plx) would keep the same direction when x
varies, without having any movement of rotation on itself."

Second Principle.
"The kinetic part (Le. the part which does not imply a potential energy) of
the momentum-energy tensor T associated with the particle is equal to the product
T = kL of a scalar constant k by a linear mapping
L: n E J(.- L(n) E J(
which expresses the infinitesimal motion in J( of the plane P(x) when the point x
varies.
The constant k is equal to fl,c(.2, where h = 211-tL is the so--ealled Planck
constant, and c the velocity of the light I.

How is the tensor L to be defined? We have both to express the proper motion of
P(x) on itself, and the motion in J( of the direction of the plane P(x).
Suppose that P(x) is defined by a couple (nl'n2) of two orthonormal (space-
like) vectors, each one orthogonal to v.
Let us complete the set S(x) = {v,nt,n2} by a fourth unit (space-like) vector
s in such a way that the moving frame 1(x) = {v,nt,n2,s}, or "proper frame of the
particle" is orthonormal. We suppose furthermore that 1(x) has the same orien-
tation as a fixed frame of J(, B = h'O,It,12,,3} on which are defined the components
xl-' of x:

x = XP.,p. = xp.1' (,p.'1' E J(, 1p.·1' = g~)

The infinitesimal motion of P(x) may be described by the four following
mappings (App. I):
86 R.BOUDET

n E Ji-l Lin) = OJ.£0(i(n AS)) E IR (1)

in which
- the symbol XO Y means the inner (or "contracted") product, associated with the
euclidean structure of Ji, of two tensors of Ji;
- 0 J.£ is a bivector of Ji which defines the infinitesimal rotation of the frame 1(x)
when x moves in the direction of the vector "Y (App. I,ll);
J.£
- the symbol i(aAb) represents the bivector which is the "dual" of the bivector aAb
(a,b E Ji).
For example one has

(2)

We will define

L(n) = LJ.£(n) ~ E Ji (3)

in such a way that the spacetime vector L(n) is invariant in all change offrame B.
The spacetime vector w:

w= L(v) (4)

is such that the scalar Now = NJ.£wJ.£ expresses the infinitesimal proper rotation of
the plane P(x) on itself when x moves in the direction of a vector N.
If one uses the formalism of STA (see Ref [2]) i(nAs) may be interpreted as
the Clifford product of the unit pseudoscalar of Ji

(5)

by the bivector nAs.

However, we do emphasize that the use of STA is not a necessity for the
construction of the tensor L, and that the traditional tensor algebra is sufficient.

Note. Relation with the Zitterbewegung.

What we have called "the infinitesimal proper rotation of the plane P(x)",
represented by the spacetime vector w, corresponds to what is called "Zitterbe-
wegung" in the articles of D. Hestenes and H. Kruger, of the present Proceedings.

The duality rotation. The definition of the tensor L is to be completed by taking into
account a transformation, the duality rotation, which concerns the planes of Ji, and
consequently may affect the plane P(x) (App. 11).
We have said that the duality rotation feaves all the vectors of Ji invariant
and acts on the bivectors. In particular, by using such a mapping, it is possible to
change the orientation of planes without changing the one of straight lines.
In this way, such a transformation is beyond the traditional understanding of
the geometry of the euclidean spaces because:
ROLE OF DUALITY ROTATION IN DIRAC THEORY 87

a) one uses to consider mostly the transformations which acts on vectors,

b) or the ones acting on multi vectors which can be only deduced from the ortho-
gonal group.
Nevertheless, the duality rotation is a pure euclidean transformation, and
such a mapping is a trivial fact for mathematicians who are acquainted with the
modern approach of the study of the euclidean spaces [10].
For the purpose of taking into account the geometry of planes in spacetime,
we consider that the bivector u = nlAn2, associated with the direction of P(x), may
be submitted to a duality rotation by an angle ,8(x):

u = n 1An 2 --! u' = ei,8u = cos ,8 n1An2 + sin ,8 sAv (6)

Then Lin) becomes (App. II)

Lp,(n) = (lp,.(i(nAs)) + (n.s)8;. (7)

2.1.2. Introduction of the Planck constant. One can notice that, up to now, we have
said nothing about physics. We have used only the euclidean properties of space-
time.
Physics is introduced now by means of the relation

T(n) = ¥L(n) (8)

as defining the momentum-energy tensor of the particle. One deduces from T the
momentum-energy spacetime vector p (reduced to its kinetic part),

(9)

We are far away from the obscure "Rule of Correspondence" pp, -I - iftap,!
We use the expression "obscure'" because in such a relation, the links which
could exist between the symbol "i" and the constant ft on one side, and the
euclidean structure of spacetime on the other, are in a total obscurity.
Certainly, such an obscurity has been the source of some aberrations one can
find [11] in the Theory of Quantum Fields.

If one takes into account the effect of an electromagnetic potential A E ){ on

the particle; furthermore if one mUltiplies T by the density p(x) associated, at each
-point x, with the particle, one obtains the tensor of density of momentum energy
lTetrode tensor), in the form we have established in [6]:

where q is the charge of the particle.

We recall that, as a consequence of the Dirac equation, the divergence of the
Tetrode tensor is equal to the density of the Lorentz force.
The momentum-energy vector p becomes
88 R.BOUDET

b.c
p = T(v) = 2""" w-qA. (11)

The electromagnetic gauge invariance of p is obtained by replacing the

couple (nhn2) by an other couple ~I' ,n2 / ), defining a same direction and orien-
tation for P I (x) as for P(x), and Ap. y

A' = A + ftc 0 X (12)

p. P. '2q P.
where X(x) is the angle of ni' and nl'

Note. Topological implicatioru due to the geometrical nature 0/ the

momentum-energy vector.
In a survey devoted to monopoles [12], F. Gliozzi has introduced the infini-
tesimal proper rotation of a space-like plane 1r(x) for expressing the electromagnetic
gauge potential. Some of his remarks, concerning the topological defects of such a
rotation, might be applied to the momentum-energy p of the Dirac particle.
The question of the physical incidence of such defects is all the more impor-
tant as what we call a photon is nothing else but a discontinuity of the vector b.cw/2
associated with the proper rotations of the spin plane in the Dirac theory.

2.1.3. Introduction o/the mass o/the particle The definition of the energy of a Dirac
particle proposed above is to be related to the definition E = mc 2 of the energy of a
material particle whose mass is m. So we will add to the two first principles, the
following one.

" Third principle"

"We will say that the Dirac particle is at rest if v, the direction of the plane
P(x), and the proper rotation on itself of P(x) are constant.
For a Dirac particle at rest one has T(v)·v = mc2; for a Dirac antiparticle at
rest one has T(v)·v = - mc 2."
In this particular case, one has w = wov wo = (dnddr)·n2 where ric is the
Tt
proper time of the particle and one can write v) = p = (t.c./2)wov. Furthermore if
one links the quality of particle or antiparticle to the duality rotation, the two
above relations may be unified in a single one:

(13)

in which /30 = 0 for a particle and /3 = 1r for an anti-particle.

But the duality rotation has been related to a transformation of the spin
plane.
Multiplying eq. (13) from the right by the bivector u which represents the
direction of P(x), it is convenient to write eq. (13) in the form

(14)
ROLE OF DUALITY ROTATION IN DIRAC THEORY 89

Eq. (14) is exactly the Dirac equation (see eq. (17) and the relation (0), App. II) in
the particular case where the potential is null; v, (1, the density p, are constant, and
/3 = 0 or /3 = 11" everywhere.
So the Dirac equation appears as an immediate generalization of the relations
p = (hc(2)WQV, P = ± mc 2v, which defines the momentum-energy of a material
particle anti-particle) at rest.

Note. TM Louis de Broglie cOfPUScule.

We recall the L. de Broglie has used a comparison with a clock to describe
the stationary state associated, as the foundation of Quantum Mechanics, with his
corpuscule.
Here, one can imagine that the dial-plate of de Broglie's clock is the spin
plane P(x), and the vector nt is the hour-hand of the clock.
The duality rotation by an angle 11" may be seen as a returning of the gradua-
tions of the dial-plate of the clock, in the sense inverse of the one which corresponds
to the increasing proper time of the corpuscule.
Thus, the notions of spin and anti-particle, the germ of Dirac equation were
already included in the de Broglie clock-<:orpuscule!
2.2. THE PASSAGE FROM THE EQUATION OF THE ELECTRON TO THE
ONE OF THE POSITRON, AND THE DUALITY ROTATION. A GEOMETRI-
CAL INTERPRETATION OF THE DIRAC EQUATION.

2.2.1. The invariant form of the Dirac equation. In Ref. [2], D. Hestenes has written
the Dirac equation for an electron in the form

(15)

in which e is the charge of the electron and

(16)
R corresponds to a Lorentz rotation such that

In Ref. [5], we have put this equation in a form independent of the frame B:
Multiplying eq. (15) from the right by

1/J-1 = .J. e-i/3/2 R-1

.fP
one finds immediately the intrinsic equation

(17)
90 R.BOUDET

Here

is the invariant infinitesimal operator associated with the mapping X ~ 'l/JXr:w (App.
II).
All the terms of eq. (17) have an intrinsic geometrical meaning. The presence
of the angle /3 in the mass term, and of its gradient in the infinitesimal operator of
the left hand side of the equation, is to be related to what we said about the con-
struction of the momentum--energy tensor, and the generalization of the equality
p = ±mc 2v verified by an (anti-)particle at rest.
The scalar part of the equation which may be deduced from eq. (17) after
multiplication from the right by (TV gives the relation Tr(T) = mc 2cos/3 (see Ref.
[7]). This equality cancels the Lagrangian density which is usually associated with
the Dirac particle (see Ref. [1-d] , and eq. (8-2) in Kriiger's article of these "Pro-
ceedings").
2.2.2. The fundamental role of the duality rotation. The intrinsic form (17) of the
Dirac equation allows us to discuss the passage from the electron equation to the
positron one.
There are in actual two possible orientations of the spin bivector (1, the "spin
up" and the "spin down" ones. The equation corresponding to one of these
orientations is transformed into the other by changing (1 into -(1 in eq. (17) (or 'Yt'Y2
into 'Y2'Y! in eq. (15)).
Suppose we want to pass from the equation of an electron to the
corresponding one of a positron. In any way, the sign of the charge e is to be
changed.
It is convenient, for the conservation of the spin in the creation or an
nihilation of a pair, to associate a "spin down" positron with a "spin up" electron,
and vice-versa. Thus, the sign in front of (1 is to be changed.
As a result, we note that the sign of the charge term of the right hand side of
eq. (17) (or eq. (15)) is unchanged. Thus the sign of the mass term is to be
unchanged.
Because of the change of sign of (1, we have to change the sign of one of the
other factors of the mass term.
Because mc 2 is positive, we have the choice between the changes v ~ -v or /3
~/3+1I"([1-d], [8]).
Let us notice that the change v ~ -v conforms to the PT transform. The P
and T transformations have separately no sense in the consideration of e1' (17)
because they need the choice of a particular frame of Jf and because eq. 17) is
invariant. But PT changes the sense of the vectors of Jf independently of all rames
and may be applied to the vectors associated with the particle in eq. (17).
This implies that one has -v for the positron, and its physical meaning is
that this particle runs from the future to the past (Stiickeberg, Feynmann)!
The other possible change /3 ~ /3+11" (Takabayasi) corresponds simply to a
duality rotation by an angle 11".
If we do not accept that the positrons which are observed come from the
future, we are obliged to consider the change of the orientation of the planes of J/,
due to a duality rotation by an angle 11", as the key of the explanation one can give
to the difference between the world of the particles and the one of the antiparticles.
But such a point of view is an infirmation of the physical interest of the PT
transform.
ROLE OF DUALITY ROTATION IN DIRAC THEORY 91

Historical note
All the results established in the article [21 of Hestenes and in my paper [5]
may be considered as already included in the works achieved by "Louis de Broglie's
School", concernin~ the "Relativistic Hxdrodynamics Theory of the Dirac Matter".
The angle {j was introduced in L1-&]. But furthermore, for example, a syste-
matic use of the multivectors of .Il appears in [I-b] j the form (16) given to the
Dirac wave function is included (in the matricial formalism) in [1-c] j the notion of
"proper frame" in [1-e] j the one of "spin plane" in [I-f]. The invariant equation
(17) is strictly equivalent to the set of invariant equations of [1--d]. And the point
of view of Hestenes by which only real quantities are to be considered is the basis of
this "hydrodynamics".

(Note. The identification of the P parameter of Hestenes with the angle of

Yvon-Takabayasi was made by G. Casanova [1-e]. The relation between the
equation (17) and the ones of Takabalasi was pointed out by O. Costa de
Beauregard. [private communication, 1968J).

But with respect to three points, the Hestenes approach has supplied an
indispustable progress:
- The use of STA. For example, without employing the Clifford product, it is
impossible to express the Dirac equation in an invariant form by a single equation
as eq. [17].
The elimination of a useless "spinor unity" of the traditional formalism
which is the source of a lof of complications and ambiguities in relativistic Quantum
Mechanics (see No. 3.1).
- But, beyond these conveniences, the fact that STA contains a specific property of
spacetime, the duality rotation, allows one to understand the geometrical links
between a particle and its antiparticle.

3. Comparison Between the Darwin and the KrUger Solutions of the Coulomb
Problem.

3.1. QUATERNIONIC SOLUTIONS OF THE DIRAC EQUATION

The readers non acquainted with the formalism of STA have simply to consider the
symbols 1p.' Ok, i, as representing the Dirac, Pauli and 15 matrices, that they are
used to employ, instead of vectors, time-like bivectors and pseudoscalar, respec-
tively, of.ll = 1Rl>3.

We recall the relations one can associate with a particular galilean frame B of
.Il,

and, as consequences of (19),

92 R. BOUDET

~ O'{k + O'kO'j) = t5jk , O'j = 1k10' (j,k = 1,2,3), (20)

. .2
1 = 10111213 = 0'10'20'3' 1 = - 1. (21)

The set

(22)

(which depends on the choice of 10) is isomorphic to the field of the Hamiltonian
quaternions.
The ring 1 + il of the biquaternions is isomorphic (independenly of 10) to
the direct sum of the vectorial spaces of the scalars, bivectors and pseudoscalars of
Jl, endowed with the Clifford product.

We recall the Hestenes presentation of the Dirac equation (see Ref. [2]).

The Dirac spinor wave function IJI is put in the form \II = 'I/Ju, in which u is a
"spinor unity" and 1/J E 1 + il. (As a consequence 1/J may be interpreted as in eq.
(16)).
The elimination of u leads to eq. (15). If the potential A is such that

Kc A = V(rho' V(r) E IR (23)

(the source of the potential is then at rest in the frame 8) the Dirac- Hestenes
equation may be written

(24)

in which iU3 = 1211, £0 = mc 2j(li.c).

· x = x 0 10+r-+ , r = -+r10
Denotmg = rn, n 2 = 1, we look for solutions in the
form

1/J = ~(r)ei0'3£xo, £ E IR, ~(r) E 1 + il, (25)

in which £ = Ej(li.c), E being a constant energy.

Because 1oi0'3 = iO'a1o one can write

ROLE OF DUALITY ROTATION IN DIRAC THEORY 93

Multiplying from the left by 'Yo and because 'Yo-YO = 1, 'Yo'Yk = - 'Yk'Yo = 'Yk'Yo = Ok,
one obtains
(26)

3.2. THE DARWIN SOLUTIONS

One writes

~1' ~2 E 1. (27)

Because iV = Vi, i 'Yo = - 'Yoi, 'Ya = 1, one obtains the equation

(28)

from which one deduces the system

(29)

Multiplying from the left (27)1 by n, one may solve this system by writing
~1 = g(r)S(n), ~2 = f(r)nS(n)0"3' n = r/r, (30)

in which g(r), f(r) E IR, and S(n) E 1 is such that (App. III)

i(rxV)S = (1+I»S, *,
I> E H. S E 1. (31)

Indeed: one has nV(gS) = ~ + gnVS, V(fnS) =~ + fV(nS), nVS = i(rxV)S/r,

Vn = 2/r, V(nS) = (Vn)S - nVS = (l-II;)S/r.
The functions g,f must satisfy the system

Qg + 1+1> g = (£ +£-V)f df + 1-11; f = (£ -£+ V)g. (32)

Or r 0 ' Or r 0
00
with the condition of normalization J (g2 +r)r 2dr = l.
o
The normalized solutions of eq. (29) are (App. III)

S = N0"3ei0"3mrp, N = L0"3 + Mu, (33)

with u = cos '1'0"1 + sin '1'0"2' n = cos e0"3 + sin eu,
94 R.BOUDET

m
and pm(x) = (_I)m [(i-m) ! ~2l+1)]t (l_x2)"2'"~(x2_1l1(l+m) ,
l (l +m ! 2 2 l!
l E IN, m E 71, -l ~ m ~ l, or pi(x) = 0 if Im I >l, and the alternative

[l-m
1'f K=l, L=- 2i(2l+1)]t Pm () ll+m+l]t m+l( cose,
l cose, M= 2'11'(2l+1) P l
)
(34)
. [l+m+l
IfK=-{l+1), L= 271'(21+1)]t Pl(cose),M=
m [2i(2l+1)
l - m ] t Pm+l( cose.
l
)
(35)
Thus, one obtains [6]
. (mrp+ n;c
10'3 E x0)
'IjJ = (gN0'3 + finN)e (36)

For example,

. E 0
1 1 10'3n;cx
Levels nS2' (l = 0, K = -1, m = 0): 'IjJ = - (g + fin0'3)e ,
.ffi
. E 0
1 1 10'3 n;c x
Levels nP2' (l = 1, /'i, = 1, m = 0): 'IjJ = - - (gn0'3 + file
.ffi
One passes from the IIspin Upll solutions to the IIspin down ll ones by changing i into
-i.

3.3. THE KRUGER SOLUTIONS

The Kruger solutions are based on the following transformation of the opera-
tor V:

.
-10: ~ -I. 0: e
~
-1 e 3 e 2 ~
V~ = OVO(O ~), 0 = t (37)
r(sin e)

1 1
Vo = 0'3 0r + i(O'l oe + sin e 0'2°crJ' (38)

One deduces from eq. (24)

(39)
ROLE OF DUALITY ROTATION IN DIRAC THEORY 95

Eq. (39) has exactly the same form as eq. (26) except that the operator Vo uses the
fixed frame (0"3,0"1,0"2). This allows one to write solutions in the form (see the article
of H. Kruger in the present "Proceedings" and App. IV)

. 1r

~ = n~O = - 1
mr(g(r)O"I + f(r))
.
eI 0"2/'iI!J e
-lO"I"4
, K,
*
E 71 • (40)
/l1r
in which the functions g,f satisfy the system (32).

3.4. INVARIANTS ASSOCIATED WITH THE WAVE FUNCTION.

We calculate pei/3 = 1/J~, the Dirac current j = 1/J'Yo~.

We recall (see Ref. [6]) that if one denotes

N2 = L 2 + M2, tg II =~, tg r = ¥ (41)

one obtains for the Darwin solutions

. . -+ . -+ du
In WhICh u = cos I{J'YI + SIn 1{J'Y2, V = ~.

After some calculations, one can obtain the spacetime vector (see also Ref.
[3-a]).

Ttc IB21-+ {B = - cos II sinr + i sinv sin(r-a)

p + eA = E'YO + r In e ZI v 2
s z = cos2v+i sin2v cos(2r-a)
(43)
whose time component is the energy E associated with the solution.
The calculation of the angle /3 gives, for example, for the fundamental level,

tg/3 = tga cose, a = e2/(Ttc)

The angle /3 is non null except for e = 1r/2.

One obtains for the Kruger solutions

/3 - 0 - gLf2 . _ (g2+f2~'YO + 2fg~ (44)

- , p - 21r2 SIlle' J - 21r SIne '
96 R.BOUDET

P + eA = E'YO + r1i.c f+f 2 ] -i

[ K + 2 g L£2) w, (45)

-i • -i-i
where n = cos a'Y3 + sm au, w = (Jij.
an
The transition Dirac current between two levels.
If 1/Jl' 1/J2 correspond to the two levels, one has

j = 1/J1'YO~2 + 1/J2'YO~1 l cos (w12xO) + jII sin(w12xO),

= w12 = Elfi~1 (46)
l = ~1'Y0~2 + ~2'Y0~1' P = ~lh3~2 -~2i'Y3~1
For the Darwins solutions one obtains (if m1 = m2)

l = 2 [(glg2+flf2)(LlL2+MlM2ho + (glf2+g2f1) (sina(LIL2-MIMl)

-cosa(M2Ll +L 2M1))v]

p = 2 [(glf2-g2fl)(LlL2+MlM2)ri + (g2fl+glf2)(MlL2-LlM2)w] (47)

For the Kruger solutions one has

j : 'lf2!ina [((g6g2+ f1 f2h~ + (glf2+g 2f1)w) cos e+ (g2fl-gl f2)ri sin ell
e- K12a - W12 X, K12 - Kl-~ .
(48)
3.5. COMPARISON BETWEEN THE DARWIN AND THE KRUGER
SOLUTIONS.

p, (J are not defined for a °

Contrary to Darwin's, the Kruger solutions 1/J and the associated invariants,
= and a = 'If. These singularities are to be added to the
ones which affect the functions g, f for r = 0, when I KI = l.
However the integration of p and j over any space volume is finite.
A remark is to be made about the transverse component (orthogonal to the

°
vector ri) of the Dirac transition current between two levels. In both Darwin and
Kriiger solutions, this component is not defined for a = or 'If. But these defects
disappear after integration over a space volume.

°
One can notice that, in both cases, the spacetime vector p+eA = 1i.cw/2 is
not defined on the poles axis (a = or 'If). To the extend that a) what we call the
energy of a real photon emitted in the passage from a level to another, is nothing
else but a discontinuity of the time component of the vector w, and b) no inte-
gration over a space volume is to be made for expressing this energy, this property
ROLE OF DUALITY ROTATION IN DIRAC THEORY 97

may confer to the poles axis a particular significance (perhaps related to the topo-
logical defects one can associate to the vector Wi see No. 2.1.2).
Note that in the Kriiger solutions, the vector w is endowed with an inter-
esting property which does not exist for the Darwin solution. The fundamental level
1St is the only level for which W is a gradient.

4. Conclusion

Certainly, from a mathematical point of view, the Kriiger solutions are as

convenient as the Darwin ones. But do they represent a physical reality?
The criterion cannot be the exactitude of the energy level values in the bare
problem, where only the exterior potential is considered since the values are the
same in both solutions. The verification must concern phenomena in which the
radiation of the electron is implied as the Lamb shift or the spontaneous emission.
In our opinion, it would be sufficient that the Kriiger solutions are in good agree-
ment with the experimental results of spontaneous emission, for having a strong
presumption that the agreement with the Lamb shift is good. (See in Ref. [14] the
Barut complex formula whose spontaneous emission is the imaginary part and the
Lamb shift the real part).
If it would appear that the Darwin and the Kriiger solutions equally account
for the phenomenas we are used to associate with an electron or a positron in a
central potential, we could abandon the former as non representative of a pure state
of particle or antiparticle, because of the non equality to zero or 11", everywhere, of (3.
The mystery of the angle (3 would be dissipated!
But if the Kruger solutions were not in accordance with these phenomenas,
we would be in the situation where one should have to consider exact solutions of
the Dirac equations, which are in actual fact much simpler than Darwin's, but
which would not have a physical meaning!
In any way these solutions impose a reconsideration of the traditional opi-
nions about the Dirac theory of the Coulomb problem.

Appendix I
Let us define, in agreement with the orthonormality relations

v2 =1, n~=-l, v'nk=O, n 1 ·n2 =O (k=1,2) (a)

the three scalars

They represent, at the point x, the infinitesimal proper rotation of the ortho-
normal set of vectors S( x) = {v,n t,n2} when the point x moves in the xJ1. direction.
This definition is exactly the same as the ones of the components of the
instantaneous rotation vector, associated with the movement of a three-dimensional
rigid body. But here, we are in a four dimensional space and it is convenient to
98 R.BOUDET

consider the set S(x) as a sub-set of the orthonormal moving frame l(x) =
{v,nt,n 2,s}.
Now, we define a linear mapping nEJi-l LJL(n)EIR, such that

A simple expression of this mapping may be given by the use of ST A.

Recalling that the associative Clifford product of vectors (at, ... ,ajl) - I al ... ap
is equal to their Grassmann product alA .. Aa p when these vectors are orthogonal, one
can write

(d)

because the frames B and l(x) are both orthonormal and have the same orientation.

The mapping X - I iX allows one to define the "dual" (in the tensor sense) of
a multivector X.
For example, one has

Because the Clifford square i2 ofthe pseudo-scalar i is equal to -1, one has

i(n 1An 2) = -vAs = sAv.

If 0. JL is the bivector which expresses the infinitesimal rotation of the moving frame

l(x) when x moves in the xJL direction (App. II):

(e)

one can write

(f)

Indeed, replacing n b~ v,nt,n2,s respectively and using the following relation

of the tensor algebra X·(aAb) = (X· a). b, (a,b E Jl), one can immediately verify (c).

Note that one can write

(g)
ROLE OF DUALITY ROTATION IN DIRAC THEORY 99

where [X] (0) means the scalar part of X.

Appendix II.

The duality rotation in .It belongs to the following type of mapping

X E C(E) -+ Y = ~X~ E C(E), ~ E C(E). (a)

C(E) is the Clifford algebra associated with some euclidean space E = IRq'n-q

and ~ -+ ~ means the operation of reversion, or principal anti automorphism in C(E).

One recalls that

The operation of reversion has a particular importance in C(E). For example

the mappings

XEE-+y=±UxtJ, U=u1... u p ' UkEE, lu~1 =1 (b)

generate the orthogonal group O(E).

If ~ = f(t), where t is a scalar parameter and f some differential function,

such that ~ is invertible for all t, one can associate with the mapping (a) the follo-
wing element of C(E) [8]

d<b ",-1 (c)

~=2at'l' .

We will call ~ the infinitesimal operator at t of the mapping (a). ~ allows one
to calculate dY /dt by means of the relation

(d)

We recall [2] that if E = .It, and if i means the unit pseudo-scalar of.lt (or
antisymmetric tensor of rank 4 associated with the orthonormal frame), we have

I.2 = -1, '"I = I,. aI. = -la,

. . '"
lal = a, . b'"I = -a,
la b Va, b E Jl1/ (e)
Thus, the mapping or IIduality rotation ll

(f)
is such that Y = X if X E .It, and Y = exp(i,B)X if X = aAb = (ab-ba)/2 is a bivec-
tor.
100 R.BOUDET

In this way, the duality rotation "rotates" the bivectors by an "angle" fl, but
leaves the vectors invariant.

Consider the mapping

if.l/2
N
X -+ Y = gXg, g = e f'. R (g)

in which R is the sum of a scalar, a bivector and a pseudo scalar, and such that :ti =
R-1. We recall that the mapping

(h)

transforms the orthonormal frame h,) into an orthonormal frame {e,) by a proper
Lorentz rotation (see Ref. [2]).

If g depends on x, the infinitesimal operator, at x, when x moves in the xp.

direction, associated with the mapping (h) is

(i)

Op. = 2(8p.R)R-l is the bivector (see Ref. [4]) which corresponds to the
infinitesimal rotation of the frame {e,,}. Indeed one has, using (d)

(j)

where the fundamental relations aX = a· X + aAX, Xa = X· a + XAa, aeJi have been

applied.
Now, if we denote eo=v, el=nh e2=n2, e3=S and if we replace in the right
hand side of eq. (g), App. I, the infinitesimal operator Op. by ep' as given in (i), one
obtains
(k)

The infinitesimal operator associated with the mapping

(1)
is

(m)
ROLE OF DUALITY ROTATION IN DIRAC THEORY 101

We recall the following relation (see Ref. [4], [5])

fOIL = - (fo/Lu + W + (;4) U (0)

in which w= ((O/Lv).s)f.
Appendix III

· w = 06'
Deno t mg du one can wn·t e
On v = ~

V = n 0r + '12" (W0a + sllla

v 0) I{Y'
Vn = !r (W2 + ~
sma v2) = ~r (a)

One has to solve the equation

rnVS = n(woa + s~na oq)S = )'S, ). E IR (b)

where S =(L(a)u3 + M(a)u) u3eiu3mcp.

Because i = U3UV = nwv, nWU3 = nw(v2)u3 = iVU3 = -u, nwu = ivu = ua,
nvu3iu3 = W, nvuiu3 = n, nv2 = n, one obtains

8L u + Qe
- ae dM m m+I (
u3 + sIne Lw + sIne Mn = ). LU3 + Mu
) (c)

from which one deduces the system

w + (1+m)cotg aM - mL = )'L, - ~+ mcotg aL + (1+m)M =).M (d)

whose solutions are, if ().-I). = l(l+1), l E IN,

m
m m+1 m
L = q p l(cOS e), M = Pi (cos e), p l(x) =
Clt 2)2 2 l (l)
[(x -1)] ,mEl. (e)
2 l!
The number q may be chosen in such a way that the compatibility of the system (d)
is achieved. Using the relation (see Gel'fand "Representations of the rotation and
Lorentz group" Pergamon Press, 1963) in which x = cos e:

one obtains
102 R.BOUDET

~+ (1+m)cotg a M = (l+m+1)(l-m)~ = (.Hm)L (g)

and thus q(.Hm) = (l+m+1)(l-m) which implies q = l-m if >. = l+l, q =

-{l+m+1) if>. = -l.
The case where q = 0 may be obtained by taking l = m = 0, >. = 1. But the
correspondent solution N = u/sin a is not acceptable because the relation (h) is then
impossible. So, if one denotes >. = 1 + It, one must have It E H*. Note that the case
>. = -l = 0 corresponds to the levels nSt.
The condition of normalization of the current

(h)

impose J~7r J~ N2 sina dcp da = 1. This is achieved by taking L, M as in (34) or

(35).
Note. For the general solution of eq. (b), see Ref. [13].
Note that the solution of the central potential problem described here (Ref. [3],
[6]) is very close to the one written by A. Sommerfeld [15].

Appendix IV

Multiplying from the right eq. (39) by /111" exp(-iu2K,e) exp(iul;f)' and
because

one obtains

i[u3((rg' + g)u1 + rf' +f) + K.{gu1 + f)u1iu2]

= ir[-EO(gu1 -f) + (£ - V)(gu1 + f)] iu2·

i.e. the system (32).

ROLE OF DUALITY ROTATION IN DIRAC THEORY 103

REFERENCES

[1] -a- Yvon, J. (1940) 'Equations de Dirac-Madelung', J. Phys. et Ie Radium'

VIII, 18.
a
-b- Costa de Beauregard, O. (1943) Contribution l'etude de la Theorie
de l'electron", Ed. Gauthiers-Vlliars, Paris.
--c- Jakobi, G. and Lochak, G. (1956) 'Introduction des parametres relati-
vistes de Cayley-Klein dans la representation hydrodynamique de
l'equation de Dirac', C.R. Acad. Sc. Paris 243, 234.
-d- Takabayasi, T. (1957~ 'Relativistic hydrodynamics of the Dirac
matter', Suppl. Prog. T eor. Phys. 1, l.
--e-- Halbwachs, F. (1960) Theorie relativiste de fluides a spin, Ed.
Gauthiers-Villars, Paris.
-f- Halbwachs, F. Souriau, J. M. and Vigier, J.R. (1961) 'Le groupe
d'invariance associe aux rotateurs relativistes et la theorie bilocale',
J. Phys. et Ie Radium, 22, 293.
--e-- Casanova, G. (1968) 'Sur l'angle de Takabayasi', C.R. Acad. Sc. Paris,
266 B, 155l.

m Hestenes, D. (1967) 'Real spinor fields', J. Math. Phys., .6., 798.

-a- Quilichini, P. (1971) 'Calcul de Pangle de Takabayasi dans Ie cas de
l'atome d'hydrogene', C.R. Acad. Sc. Paris 273 B, 829.
-b- Gurtler, R. (197~, Thesis, Arizona State University.
[4] Boudet, R. and uilichini, P. (1969) 'Sur les champs de multivecteurs
unitaires et les champs de rotations', C.R. Acad. Sc. Paris 268 A, 725.
[5] Boudet, R. (1971) 'Sur une forme intrinseque de l'equation de Dirac et
son interpretation geometrique', C.R. Acad. Sc. Paris 272 A, 767.
[6] Boudet,R. (1974) 'Sur Ie tenseur de Tetrode et Pangle de Takabayasi.
Cas du potential central', C.R. Acad. Sc. Paris 278 A, 1063.
[7] Boudet, R. (1985) 'Conservation laws in the Dirac theory', J. Math.
Phys., 26, 718.
[8] Boudet, R. (198~ 'La fieometrie des particules du groupe SU(2)',
Annales Fond. L. e Brog 'e (Paris), 13, 105.
[9] Boudet, R. 'The role of Planck's constant in Dirac and Maxwell theo-
ries' ("JounCes Relativistes" Tours 1989), Ann. de Physiques, Paris 14,
No.6 suppl. 1, 27.
[10] Micali, A. (1986) 'Groupes de Clifford et groupes des spineurs' in
Clifford Algebras and Their Applications in Mathematical Physics,
67-78, Ed. Chrisholm J. and Common, K., Reidel Publ. Co., Dord-
recht, Holland, The Netherland.
[11] Boudet, R. (1990) 'The role of Planck's constant in the Lamb shift
standard formulas' in Quantum Mechanics and Quantum Optics, Ed.
Barut, A.O. Plenum Press.
[12] Gliozzi, F. (1978) 'String-like topological excitations of the electromag-
netic field', Nucl. Phys., B 141, 379.
[13] Boudet, R. (1975) 'Sur les fonctions propres des operateurs differentiels
invariants des espaces euclidiens, et les fonctions speciales', C.R. Acad.
Sc. Paris, 280 A, 1365.
[14] -a- Barut, A.O. and Kraus, J. (1983), 'Nonperturbative Quantum Electro-
dynamics: The Lamb Shift', Found. of Phys., 13, 189.
104 R. BOUDET

-b-- Barut, A.O. and van Huele, J.F. (1985) 'Quantum electrodynamics
based on self-energy: Lamb shift and spontaneous emission without
field quantization', Phys. Rev. A, 323187.
[15] Sommerfeld, A. (1960), Atombau und Spektrallinien, Ed. Friedr.
Vieweg and Sohn, Braunschweig.
BRIEF HISTORY AND RECENT DEVELOPMENTS IN ELECTRON
THEORY AND QUANTUMELECTRODYNAMICS

A. O. Barut
Department of Physics
University of Colorado
Boulder, CO 80309, USA

ABSTRACT. Major steps in the hundred years history of the electron concerning its
selfenergy due to its own electromagnetic field are outlined and the present status and
revival of the selfenergy and radiative problems of the electron are discussed.

1. HISTORY OF THE ELECTRON

1.1. Concept and Discovery

The first and the foremost of all elementary particles, the electron, has not been
discovered suddenly, as the muon, for example. It was "in the air" for a long time.
1990 is a good date to remember the hundredth anniversary of its conception. The
evidence came from at least three widely different phenomena.
Its study in the cathode rays can be said to begin with the work of the mathematician
J .Plucker in 1858. Plucker is also a pioneer in differential geometry in introducing the
rays as coordinates rather than the points (Plucker coordinates) which anticipates
the space of all light rays or light cones. Wilhelm Hittorf (1869) talks about the
straightline trajectories of the negatively charged "glowrays". Other important names
in the early studies of cathode rays are W. Crookes (1879), H.Hertz (1881), P.Lenard,
J.Perrin, W.Wien and J.J.Thomson.
Secondly, from Faraday's equivalence law for electrolysis, Helmholtz in 1881 con-
cluded that each ion must carry a multiple of an elementary charge. Independently,
the irish physicist G.Johnstone Stoney in 1881 also talks of fundamental units of both
positive and negative electric charges. Already in 1874 Stoney had the idea of an
"atom" of electricity. And it was Stoney who introduced the name "electron" a bit
later in 1894.
The third phenomenon in which the electron appears independently was its identi-
fication in radioctivity by Jean Becquerel (1 March 1896).
The first precise determination of the ratio elm were made by Peter Zeeman (31
October 1896), E.Wiechert (7 Jan. 1897) and J.J.Thomson (30 April 1897). Zeemann
also gave an explanation of what is now called the Zeeman effect on the basis of
105
D. Hestenes and A. Weingartshofer (eds.!. The Electron. 105-148.
© 1991 Kluwer Academic Publishers.
106 A.O.BARUT

the electron-hypothesis. These early experimental developments we may call the first
period of electron's history.
The second phase, up to the discovery of the wave properties of the electron, is the
history of the electron as a relativistic particle, the electron according to Lorentz, and
the models by Abraham and Lorentz. The relativistic equations of motion derived
by Poincare and Einstein in 1904 and 1905 have been verified e.g the increase of the
effective mass with velocity.
The third period of electron's history opens with the unexpected wave and spin prop-
erties of the electron and leads to the picture according to Schrodinger and Dirac,the
electron described by a wave equation, and ends with the QED picture of the electron.
It is remarkable that such a seemingly simple object as an electron has produced so
many surprises. In view of the appearance of heavy leptons, like muon and tau which
are very much like the electron, I am tempted to conjecture a fourth period in which
we may be again surprised by a nonperturbative internal structure of the electron. The
perturbative treatment of electron interactions by Feynman graphs has somewhat di-
minished the preeminence of the electron; it is just one of the many "elementary"
particles. But these results have not solved the structure problem. In the last sen-
tence of his famous review article on Quantum Theory of Radiation l Fermi writes: "In
conclusion, we may therefore say that practically all the problems in radiation theory
which do not involve the structure of the electron have their satisfactory explanation;
while the problems connected with the internal properties of the electron are still very
far from their solution" .
Because the structure of the electron is the main topic of these lectures I would like
also to review briefly the history of the selfenergy of the electron.

1.2. History of Selfenergy

It is remarkable that the force law between two charged particles in motion including
the magnetic force was written down as early as 1875 by R. Clausius, and this in
"relativistic" form namely, what would correspond to an interaction Lagrangian of
the form
1
L = -mlv 2 1
+ -m2v2 2 el e2 (
1 - -1V I ' V2 ) 1
l - --
2 2 47rE"o c2 Irl - r21
This is not surprising because the laws of relativity were essentially deduced later from
electrodynamics. J.J.Thomson in 1881 confirmed this law offorce (up to a factor) from
Maxwell equations. In 1892 H.A.Lorentz began to combine the action of the particles
with that of the electromagnetic field culminating in his Theory of the Electrons as
described in his famous 1904 Encyclopedie article and in his book. This is essentially
the present "classical" electron theory, the first comprehensive selfconsistent treatment
of charged particles and their electromagnetic field in mutual interaction. vVe shall see
that the basic tenets of this theory remains unchanged today, only the way we describe
matter has undergone several changes over the years. Because the selfenergy of the
electron for a point particle is infinite at the position of the particles, Lorentz and
Abraham tried to model the electron as an extended charge distribution in order to
understand whether the mass of the electron is wholly of electromagnetic origin, that
is an electromagnetic origin of the concept of mass and an electromagnetic view of all
HISTORY OF THE ELECTRON 107

matter. The advent of the relativity principle, independently formulated by Poincare

and Einstein, put a temporary end to these endeavours by elevating the concept of the
rest mass to the level of an invariant, like the velocity of light. Also the covariant laws
of mechanics were promoted to the same footing as the Maxwell's equations; in fact
every classical theory could be relativized. Even if we accept an unexplained invariant
rest mass the problems of the infinite selfenergy and the structure of the electron
remained. Dirac in 1938 returned to the classical electron theory and extracted the
covariant form of the selfenergy contribution to the motion of the electron. This is the
radiation reaction force in addition to the external force and the resultant equation
is now known as the Lorentz-Dirac equation. But an infinite renormalization was
necessary by putting part of the sclfenergy into the rest mass. Dirac returned to
the classical electron theory in 1962 by modelling it as an extensible charged shell
held together by a surface tension. The parameter of the surface tension can be
eliminated in terms of the mass and charge so that this model has also only two
parameters, rest mass and charge, just like a point particle. Unfortunately,to my
knowledge, the relativistic motion of such a shell in space-time, its radiation reaction
and renormalization have not been studied at all.
The reason might be that meanwhile we have developed first a wave mechanics
for the electron, then a relativistic Dirac wave equation incorporating the new spin
property of the electron and finally a quantumclectrodynamics so that classical models
seemed to be obsolete. However, as exemplified by the above quote from Fermi the
selfenergy problem remained.At first it might be thought that the description of the
electron by an extended wave function instead of a point charge might alleviate the
selfenergy infinity. Schrodinger himself in 1926 tried to include additional radiation
reaction or selfenergy terms to his famous equation, but obtained, due to an incomplete
treatment, wrong results. In the meantime the statistical interpretation of the wave
function became the dominant paradigm; the self energy terms were dropped, an
independent quantized radiation field was introduced and the self energy infinities
were absorbed by renormalization into mass and charge. So the expected finite, closed
and selfconsistent electrodynamics did not materialize.

1.3. The Importance of the Structure and Selfenergy of the Electron

It is with this background that I wish to reexamine the problem of the selfenergy
of the electron. As Lorentz had already prophesied, "in speculating on the structure
of these minute particles we must not forget that there may be many possibilities not
dreamt of at present". It is not only a problem of having a mathematically sound
finite theory of the electron, but a question of curiosity which may have farreaching
consequences: what is really an electron? What is the structure which gives its spin?
Why must there be a positron? What is mass? Why and how does the electron
manifests wave properties? And what is the interaction between two electrons or
between an electron and a positron at short distances? The Dirac equation gives
a mathematical description of these questions, except the last one, but not a clear
intuitive picture.
These questions, in fact the history of the electron,show that although there has been
tremendous progress in details and applications of the electron theory, the progress in
the foundations of physics is very slow. \Ve are normally busy with our calculations
108 A.O.BARUT

and with building models. But at occasions like this, which I welcome, we may reflect
on these larger issues, fundamental ideas and unifying concepts. I think some of the
most fundamental and soluble problems of basic physics at the present time are: (1) a
clear understanding of the wave and particle properties of electron, photon and other
particles within simple logic, without paradoxes, as objective material properties; (2)
the completion of quantumelectrodynamics; a selfconsistent intutive rendering of the
interactions of matter and electromagnetic field without infinities; (3) different forms
of matter : how many really fundamental particles do we have, the electron-muon
puzzle, for example; and (4) how many distinct fundamental interactions do we have
to introduce, and how do we unify the four seemingly different forces ? Into a large
mathematical framework containing all these, or showing that they are just different
manifestations of a single interaction? For example, the chemical force and alpha
decay, although extremely week, have been shown to be different manifestations of
the electromagnetic force. It may seem incredulous, but I think we may have a much
better understanding of these four basic problems on the basis of electron's structure
and specially on the basis of the strong selfenergy effects between electron and positron
at short distances. This would be a fullfilment of Einstein's statement "You know,
it would be sufficient to really understand the electron". The framework for this
program,1 think,exists, but we must justify it with precise mathematical calculations.

2. SOME RECENT RESULTS FROM ELECTRON THEORY

2.1 Selfenergy of the Point Electron: The Lorentz-Dirac Equation

Because the basic postulate of both the classical electrodynamics and the selffield
quantumelectrodynamics , the action principle, is essentially the same, we begin with
the longstanding apparent problem of causality violation and runaway solutions of the
Lorentz-Dirac equation. The classical action for spinless charged point particles in
terms of invariant time parameters is given by

A= J dTppX p - J [eApjl' - ~FpvFpV] dx

J
where the current is
jp(x) = dT8(x - x(t)) X(T)
The action leads to Maxwell's equations and to the particle equations
F ,vI'll .
= -)1'
mxp = eFpvxll
Each particle sees the external field of all other particles plus its own selffield which is
a necessary consequence of the selfconsistent action principle. The selffield is obtained
from the Lienard- Wiechert potential

Ap(x) = e JdTXp(t)8 (x - x(t)) D (x - x(t))

HISTORY OF THE ELECTRON 109

A regularization or renormalization is necessary, because the selffield at the position

of the point particle is not defined, which can be done in different ways. The result is
the covariant equation

which contains all the radiative processes for point particles in a closed nonperturbative
manner which, I think, is one of the most important feature of this equation. However,
the complete consistency of this equation has been questioned. This comes about if
one tries to solve this equation in regions where the external field vanishes. The
nonlinear term by itself then leads to preacceleration and to runaway solutions, the
exponential increase of x with time. I have recently pointed out that renormalization
of the theory has two aspects: one is putting an infinite inertial term lim (l/u )x P , into
the mass term, the other to make sure that when the external field vanishes the charge
must move like a free particle with an experimental mass. In renormalizing a theory
we must know beforehand to what we are renormalizing. This means that in regions
where the external field is zero the correct solution must coincide with that for a free
particle. The socalled causality violation and runaway solutions all have been shown
in the examples where the external field is switched on and off at finite times. On the
basis of explicit solutions it has been shown that with the above physical requirement
no preacceleration or runaway solutions arise 2 . Furthermore, with the radiated power
and the finite change of mass correctly included the Lorentz Dirac equation conserves
energy3 removing some other doubts expressed in the literature. This does not mean
that we should be completely happy with the Lorentz Dirac equation. First of all
it does not contain spin which however will be added in the next Section. But the
infinity in the mass renormalization is still with us. But then the electron has wave
and spin properties which are not yet in the point particle model.

2.2. Classical Relativistic Spinning Electron

The spin properties of the electron are very well described by the quantum Dirac
equation. For many practical applications of the Dirac equation there is no need to
make a model of spin. But eventually it becomes important to understand the physical
mechanism underlying the spin or magnetic moment degree of freedom of the electron.
Dirac 4 has found (by chance as he says) his equation without quantizing of an
existing classical model. Ever since there was no lack of effort to find an intuitive
model of this remarkable relativistic spinning particle. A model of spin may help
to discuss posible excited states of the electron, the existence of antiparticles, heavy
leptons, and perhaps shed some light on Pauli exclusion principle, and short distance
extrapolation of electrodynamics.
If a spinning particle is not quite a point particle, nor a solid three dimensional
top, what can it be? What is the structure which can appear under probing with
electromagnetic fields as a point charge, yet as far as spin and wave properties are
concerned exhibits a size of the order of the Compton wave length? I want to describe
a model in which a point charge performs as its natural motion a helix which gives
an effective structure and size scale to the particle and accounts for the spin as the
110 A.O.BARUT

intrinsic angular momentum of the helix, and the frequency of the helical motion
determines an internal clock and attributes a mass to the particle. Furthermore the
sense of the orientation of the helix is related to the particle-antiparticle duality. And
all this already in a purely classical framework.

The Action

The phase space consists of the usual conjugate pair of variables (X Il( r), PIl( r)) plus
another conjugate pair of internal variables (z(r),z(r)). Here r is an invariant time
parameter, z and z are 4-component classical c-number spinors, thus in C 4 . \Ve could
have used the real and imaginary parts of z, but for a symplectic formulation the
spinor form is much more economical and elegant. The notation is such that

where nil is the normal to a space-like surface L' The action is the integral of Cartan's
symplectic I-form
w = pdq - 'Hdr

where'H is the "Hamiltonian" with respect to r or the mass operator. In our case we
have explicitly
w = pdx + i)..z,,( . nz - 'Hdr

with
'H = Z"(IlZ (Pll - eA Il )

so that the action including that of the coupled electromagnetic field A is 5

where we have introduced the kinetic momenta

Properties of the Particle and Solutions

1. There are only two fundamental constants in the theory: the coupling constant e,the
charge, and the constant).. of dimension of action (n) multiplying z for dimensional
reasons. The mass m will enter as the value of an integral of motion in the solutions.
2. The system is integrable. Two integrals of motion are

'H = 7r It Z"(lt Z

N = z"(' nz

\Ve can choose N = 1 (normalization), and 'H = m.

HISTORY OF THE ELECTRON 111

3. The equations of motion are

. z
z,· n = -Z7r
).
. z
,.nz=-~7rZ

*1'- = eFl'-vxv
xI'- = z'l'-z

4. The velocities xI'- are thus constrained by the internal variables (similar to the rolling
condition of a rigid body on a surface) by XI' = z'l'-z. In particular

. _ dx o
Xo = z,OZ = -
dT
which for n = (1000) is equal to N = 1. Hence T has the meaning of proper time
in the frame determined by n. It is also possible to rewrite the action so that r has
the meaning of the proper time of the center of mass.
5. Solutions for a free particle. We give only the solution for Xp.-6

. () . () .. ( ) sin 2pr sin pr v: 2

xI'- r = xI'- 0 + xI'- 0 ---+2--2- I'-
2p P

where

Thus • 2
Xo( r) = Xo(O) + .To(O) sm pr + 2 sin 2 pr
2p
• 2 ~

fer) = f(o) + ¥(O)sm pT + _P_ sin2 pT

2p Po + m
Compare this with the solution when the invariant parameter is taken to be the
proper time of the center of mass:

.
Xo - + (.xo(O) -
= Po PO) .To (0)
- cos2mr + - .
- sm2mr
m m 2m
:' p~ ( ;t P~) x
;teO) .
x = - + x(O) - - cos2mr + --sm2mr.
m m 2m

6. vVe see in either of the above forms the helical motion of the velocities, hence after
integration, of the coordinates x I'- • This is the natural motion of the particle and
the helix docs not radiate. The frequency of the helical; motion is 2m in the proper
time of the center of mass.
112 A.O.BARUT

7. Instead of z and z we can introduce the more physical velocity and spin variables.
Defining S"" = izh',.'Y"jz we have the dynamical system

XI' = v,,
vI' = 4SI'Il7r1l

7r1' = eFllllv ll
5,,,, = 7r Il U" - 7r v U Il

These equations are identical in form to the Heisenberg equations of the Dirac
equation.
S. The system is symplectic given by the Poisson brackets

{f g} = ( of og _ og Of) _ i (Of og _ og Of)

, ox a oPa ox a oPa OZ OZ oz oz
{z,iz}=l {xl',p,,}=t5~

Consequently the equations of motion have the Poisson bracket as well as the Hamil-
tonian forms
Xil = {x ll , H} = OH/Opil
ii = {iz, H} = -i oH/oz
similarly for all the other dynamical variables.
9. Quantization can be performed in three different forms.
(i) Canonical quantization: This is the replacement of Poisson brackets by the com-
mutators and of the dynamical variables by the Heisenberg operators. And as
we mentioned above the resultant equations of motion coincide with those of the
Dirac theory. Thus we have a correct classical model of the relativistic spinning
electron.
(ii) Path Integral quantization. There is a longstanding problem of how to obtain
discrete quantum spin values from continuous classical spin variables by path
integration. This problem has been solved with our classical spin variables z
and z. In fact it is possible to formulate precisely the whole of QED perturbation
theory directly from classical particle trajectories by path integration 7 • The Dirac
propagator, by the way, has the physical interpretation that its matrix element
corresponds to a path integral not only with the endpoints fixed, but also with
the initial and final spin components fixed at C\' and (3:

Ea{3 = ( . 1_
l' p
m) a{3
=
}
rb.~ V(x)V(p)V(z)V(z)e 1/h J drw
X.,a

(iii) Schrodinger quantization. Here we start from a function <p( z, X; 7) in the con-
figuration space and represent the conjugate momenta by first order differential
operators. The function <p satisfies

i :7<P(Z,X;7) = H<p(z, x; 7)
HISTORY OF THE ELECTRON 113

where 1-{ is now the differential operator

1-{ = z,,,~ (i~

az ax"
- eA )
"
We expand rP on both sides of this equation in powers of Zo as

Acting with the operator 1-{ and comparing the coefficients of Zo the first equation
is the Dirac equation
i :T '¢o = ," (ia" - eA,,) '¢o
10. Excited States of Zitterbewegung. The Schrodinger quantization gives in addition
to Dirac equation a set of higher spin equations 8 . The next one is

where the matrices (3 satisfy the Kemmer algebra

the irreducible parts of which are ,as is well known, the 5 and lO-dimensional
matrices representing spin 0 and spin 1 particles. The general (3-matrices are of the
form
(3I'Cr) = ," ® I ® 1. .. + I ® ," ® 1. .. + ... + I ® I ® ... ® ,"
and satisfy the commutation relations

where ",' are the elements of a Lie algebra.

There are also supersymmetric structure given by the maps, for example,

'¢o --t 1'oft

, " --t , " ® I + I ® ,"
(A --t A ® A)

11. Lorentz-Dirac equation with radiation reaction and spin. Now that we have included
the correct spin terms into our classical equation we can evaluate self-energies and
radiation reaction. The generalized Lorentz-Dirac equation so derived reads 9
114 A.O.BARUT

where

t
We have a new term coming from spin with coefficient- in addition to the standard
Larmore term. In the spinless limit v 2 --+ 1, v . V --+ 0, ir I' --+ mx 1" we obtain back
the Lorentz Dirac equation.
12. Generalization to curved spaces is obtained by adding to the momenta a spin con-
nection r lt ; 71"1' = PI' - eAJL +izT JLz. The action in terms of 71"1' is the same as before.
The equations of motion are (>. = 1) 10 .

h· n = i (zir + izr I'VIl)

'Y. nz = -i (71"Z + iv''rl,z)
XI' = z'YJL Z = VI'
.
71"1' -
r aJLII71"O/V II _ F " 1 R 0/(31'11 50/(3"
- e JL"v - 2 V

Papapetrou equations for spin follow.

13. Generalizations to spinning strings and membranes can be obtained by introducing
world sheet variables and functions XII (0"0/ , T)and z( 0"0/ , T).11
14. Internal Algebras SO(5) and SO(6). If one separates center of mass and relative
coordinates and momenta one can exhibit a remarkable algebraic structure of the
zitterbewegung which shows also the interesting geometry of the internal phase
space. Again this structure is the same for the classical and quantum case. The
brackets of the dynamical variables close to a SO(6) algebra. We only indicate here
how the brackets of the relative conjugate coordinates is related to spin 12 ;

15. Two and many-body equations with spin and radiation reaction. In analogy to
recently established covariant many-body equations in quantumelectrodynamics we
can also derive many -body equations for classical spinning particles. The method
consists of defining composite spinors from the tensor product of the spinors for
each particles, Z = Zl I8i Z2 rewrite the action in terms of these composite spinors
and derive their equations. For the two body problem one obtains in particular the
following covariant Hamiltonian 13 ;

where iJ = 1· 1 + 1 ·1 and 5's are the Dirac matrices.

HISTORY OF THE ELECTRON 115

2.3 Modelling Mass and Wave-Particle Duality

In the previous Section we have presented an intuitive model for spin, but the
mass entered the theory as a constant of integratioin for which we do not have a more
physical representation. Moreover after quantization we have the Heisenberg equations
of motion for the dynamical variables, but how do we picture the operator-valued
Heisenberg equations? We picture the classical equations of motion by trajectories, but
not so for the Heisenberg equations. In this Section we shall present a complementary
classical wave approach to the internal structure of the particle which gives us more
insight into the mass and wave properties of the quantum particle. At the end then we
can attempt a synthesis between the wave and particle approaches to pinpoint what
a "quantum" particle is.
The wave approach is based on the explicit construction of localized oscillating
non spreading wavelets which move like relativistic particles. This is best explained in
the simplest case of ordinary scalar linear wave equation

04> = 0

We look for a localized solution initially at a point Xo of the form

4>(x, t) = F(x - xo)e- lflt

The wave form oscillates with an internal frequency fl; it is not static and this is a
crucial point. The function F then satisfies the Helmholtz equation

t::J.F+(flle)2F=O

whose solutions in spherical coordinates are

F(x - xo) =L elm ~Jl+l/2 (~r) Pt(cos8)eim'f', r = v(x - x o )2

fm

As an example, the simplest spherically symmetric solution is

We can associate a size elfl to the large central region of the wavelet, although it has
a small but infinitely long oscillating tail (Fig.I). The moving solution is obtained by
a Lorentz transformation with parameters

and has the form 13

4>(x, t) = F (r ~(x - xo, )t) ei(k.x-wt)
116 A.O.BARUT

where

or

is the space-like distance on the surface 2:= perpendicular to n" .\Ve see now the de
Broglie phase with k = ~1i3 and w = ,S1 and the dispersion relation

w
The group velocity of the wave lump is v, whereas the phase velocity is u Ii
satisfying uv = e .
2
We have here the remarkable result that the dispersion relation for the localized
solution of the massless wave equation Op = 0 exactly coincides with that of the plane
wave solution of the massive Klein-Gordon equation (0 + "';2c 2) ¢ = O. Historically,
when wave properties of the electron were discovered one has added by hand a mass
term to the wave equation; it would have been equally possible to consider the localized
solutions of the massless equation of the above type. We thus obtain an identification
between the internal frequency S1 and the mass m:

me S1
=
h e
and it is here that, for dimensional reasons, the Planck's constant enters. Thus the
concept of mass is now related to the internal oscillations of a wave lump, the fre-
quency of an internal dock, S1 and without such internal oscillations we could neither
construct localized solutions, nor make a Lorentz transformation. We shall see again
this connection between mass and frequency when we calculate the total field energy
contained in the lump. It is given, in the rest frame of the lump, by

where N E is a normalization constant, for dimensional reasons since a free field has no
scale. In order to obtain a finite energy we take a superposition of different frequencies
S1 in the neighborhood of some basic frequency S1 with a distribution function 1(S1).
Using the orthogonality properties of the Bessel functions we then obtain a finite
energy which, normalizing the charge of the complex scalar field, becomes proportional
to the frequency S1 in the rest frame. In the moving frame we obtain

and the relativistic relation

HISTORY OF THE ELECTRON 117

In the nonrelativistic limit the wave equation goes over into the Schrodinger equation
and our solution into
.2k 2
01
of = F( X- - - - v-t)'
xo
'( __
e mv'x- -
2m
t e -,,,t
) 'n

One can verify directly that this localized solution satisfies the Schrodinger equation
with F given as before and with the dispersion relation

It is the Galilei-boosted rest frame solution '¢ = F( x - Xo )e- iflt The phase differs from
the usual Schrodinger phase by the constant rapid mass oscillations e- iflt which drops
out in the calculation of phase differences in interference experiments. The energy and
momentum calculated in the same manner as above are given by

E= nw
In the presence of a potential V (x) in the Schrodinger equation one can generalize the
above localized solutions into the form 14

'¢ = F (x - Xo - get)) w(x, t)eih(x,t)

where F again is the localization function and Wis the usual solution of the Schrodinger
equation in the given potential. In this form we obtain a new deterministic interpreta-
tion of a single quantum particle, as well as the statistical interpretation of in repeated
experiments as follow. In the limit n --+ 00 the localization function F approaches
o (x - Xo - g( t)) hence we get a classical trajectory of a point particle. On the other
hand if we average over the parameters of the solution, for example over Xo and Vo,
F approaches unity and we are left with W. Thus W represents the typical or aver-
aged behaviour of the particle in repeated experiments, and this is the standard Born
statistical interpretation of quantum theory applicable only to repeated experiments.
But now inbetween these two limits we get something new, namely a description of a
single particle which has both particle-like and wave-like behaviour. Further details
about these interpretation questions are given elsewhere 15 .
Similar localized solutions have been obtained for the electromagnetic field 16 , the
spinor Dirac field 17 and for the linearized gravity18. With these highly localized solu-
tions of wave equations we have modelled the mass in terms of the internal oscillations.
We also modelled the wave-particle duality because the wave lumps move like relativis-
tic particles and have the correct wave properties in phase and in dispersion relations.
The frequency of the internal oscillations is the same as that of the helical motion of
the spinning particle model of Section II. In fact if we imagine a point charge at a dis-
tance it/me from the center of our wavelet solution, it would perform a helical motion
during the time evolution of the solution. We may thus view the localized solution
as the "quantum" picture of the helical solution, or as the picture of the Heisenberg
equations of motion.
118 A. O.BARUT

References to Chapter 2

1. E.Fermi, Rev. Mod. Physics, 4, 87 (1932)

2. A.O.Barut, Physico Lett. A, 145,387 (1990)
3. A.O.Barut, Physics Lett. A, 131,11 (1988)
4. P.A.M.Dirac, Proc. Roy. Soc. (London), A117,610 and A1I8, 351 (21928)
5. A.O.Barut and N.Zanghi, Phys. Rev. Lett, 52,2009 (1984)
6. A.O.BaruL C.Onem and N.Unal, J. Phys. A, . 23,1113 (1990)
7. A.O.Barut and LH.Duru, Physics Reports, 172, 1 (1989)
8. A.O.Barut, Phys. LetterlJ B, 237,436 (1990)
9. A.O.Barut and N.Unal, Phys. Rev. A, 40, .j404 (1989)
10. A.O.Barut and M. Pavcic, Class. Quant. Gravity, 4, L41 and L131 (1987); 5, 707
(1988)
11. A.O.Barut and M.Pavcic, Lett. Math. Phys,. 16, 333 (1988)
12. A.O.Barut and W.D.Thacker, Phys. Rev., 31, 1386 (1985)
13. A.O.Barut, Phys. Lett. A, 143, 349 (1990)
14. A.O.Barut, Found. Phys., 20, 1233 (1990)
15. A.O.Barut, Quantum Theory of Single Events, in Symposium on the Foundations
of Modern PhysiclJ, 1990, 1,.Vorld Scientific
16. A.O.Barut and A.Grant, Found. Phys. Lett., 3, 303 (1990)
17. A.O.Barut, E.Okan and G.Akdeniz, to be published
18. A.O.Barut and Y. Sabouti, to be published
 HISTORY OF THE ELECTRON 119

3. SELF-FIELD QUANTUMELECTRODYNAMICS

3.1 Introduction

There are many approaches to radiative processes, or more generally, to electro-

magnetic interactions of charged particles. We should welcome this multitude because
different ways of looking at the same physical phenomena can only bring clarity and
hopefully enlightenment. I list those different formulations which are definite and
more or less complete:
(i) Second quantized quantum field theory, or the perturbative QEDl.
(ii) The S matrix theory of electromagnetic interactions, either from unitarity, analyt-
icity and successive pole approximation 2 , or from regularization of the product of
distributions 3 • Both of these lead to the renormalized perturbation theory with
particles on the mass shell.
(iii) Path integral method. Either path integrals of Maxwell-Dirac fields 4 , or path inte-
grals directly from the classical particle trajectories 5 .
(iv) Source theory 6.
(v) Selffield quantumelectrodynamics.
Of these only the selffield approach is in the long tradition of classical radiation
theory and classical electrodynamics and is the subject of these lectures.
It is often stated that a large number of radiative phenomena conclusively show that
the electromagnetic field, and further the electron's field, is quantized as a system of
infinitely many oscillators with their zero point energies. The radiative phenomena
are listed in Table 1. We shall show that all these processes can also be understood and
calculated in the selffield approach which does not quantize the fields. The quantum
properties of the electromagnetic field are reduced here to the quantum properties of
the source. One avoids thereby some of the difficulties of the quantized fields, such as
the infinite zero point energy and other infinities of the perturbative QED.
120 A.O.BARUT

TABLE I: RADIATIVE PROCESSES

Spontaneous emission
Lamb shift
Anomalous magnetic moment
Vacuum polarization
Casimir effect between parallel plates
Casimir Polder potentials
Planck-distribution law for blackbody radiation
Unruh effect
QED in cavities
e+ - e- system:
positronium spectrum
positronium annihilation
pair production and annihilation
e+ - e- scattering
Relativistic many body problem with retardation
Electron - photon system:
photoelectric effect
Compton effect
Bremsstrahlung

This lecture tells the story of the developments of selffield QED and it is good to
begin from the beginning, namely the classical electrodynamics.

3.2 Classical Electrodynamics

The selfconsistent treatment of coupled matter and electromagnetic field goes back
to H.A. Lorentz 7 . The electromagnetic field has as its source all the charged particles
which in turn move in this total electromagnetic field. We have thus the Maxwell's
equations coupled to the equations for matter:

1) F,,/ = -jl'
(1)
2) Equation of motion of matter in the field F }
HISTORY OF THE ELECTRON 121

These equations, both, can be derived from a single action principle. It has the general

J
from
TV = [Kinetic energy of matter - jJlAJl - 1/4 FJlvFJlV] (2)

The last term is the action density of the field and the middle term represents the
interaction of the matter current with the field.
We shall keep this general framework throughout also for quantum electrodynamics.
The only change will be in the specific form of the current or how we describe the
matter, the electron.
Classical electrodynamics per se is usually associated with the current of point
particles moving along wordlines. But we can have more general extended sources of
currents, as we shall see. For a number of point particles the current is given by

jJl(X) = Lei
,
J dSiXiJl(Si)§ (x - Xi(Si)) (3)

Hence the fundamental equations are

FJlV'V = -jJl = - L ei J
dSiXi(Si)§ (x - Xi(Si)) (4)
•
Here Si are invariant time-parameters on the worldlines of the particles, and dots
represent differentiation with respect to these times.
The equations of motions of the worldlines are

miXiJl = eiF,wxi , i = 1,2,3, ... (5)

It is essential for the selfconsistency of our system that the field F entering the last
equation is the field produced by all the particles including the particle i, namely the
selffield. Hence we divide F into two parts

(6)

The selffield can be obtained from the Lienard-Wiechert potential

AJl(x) = J dXJl(s)D (x - xes)) =e J dsxJl(s)D (x - xes)) (7)

but is formally infinite at the position of the particle. It must be treated properly,
for example, by analytic continuation onto the world lines. This leads to the final
Lorentz-Dirac equation for each particle (in natural units c = n = 1)

(8)
122 A.O.BARUT

This is the basic nonperturbative equation of classical electrodynamics. Here m is now

the renormalized mass. Furthermore we must find solutions of this equation which
have the property that whenever the external force is zero the electron moves like a free
particle,mx I' = 0, that is the second term must vanish together with the external field.
This is part of the renormalization program. The important feature of this equation
is that all radiative effects are now expressed in a closed, we repeat, nonperturbative
way. The price we pay for this is that the equation is not only nonlinear but also
contains the third derivatives. The selffield approach to quantumelectrodynamics has
the goal of finding the analogous nonlinear, nonperturbative equation in the case of
quantum currents. It is clear that radiative effects like the Lamb shift, anomalous
magnetic moment, spontaneous emission, etc. have their counterparts also in classical
electrodynamics.
As a second example of a classical current we consider the classical model of the
Dirac electron which describes a spinning and charged relativistic point particle. In
this model the worldline of the point particle is a helix, called zitterbewegung, and
the orbital angular momentum of the helix in the rest frame of the center of mass
accounts for the spin and the magnetic moment of the particle. The generalization of
the Lorentz-Dirac equation for this case has recently been given 9 :

. _ F ext
71"1' - e 1''' v
II
+ e2(91''' _VI'VII)[~ii"
V2 3 v2
_~(v.v)v"]
4 v4
(9)

where
71"1' = PI' - eAI" v = x and v2 =I 1 due to spin.
There are other classical models of the electron. A remarkable one is due to Lees 10
amd Dirac l l in which a charged shell is held stable with a surface tension. In the
equilibrium position the surface tension can be expressed in terms of the mass of the
electron so that this model has exactly again two parameters, mass and charge, like
the point worldline. The Lorentz-Dirac equation for this model to my knowledge has
not been worked out yet.

3.3 Schrodinger and Dirac Currents Quantumelectrodynamics

Quantumelectrodynamics has the same two basic equations (1). Only the form of
the current j is different. According to Schrodinger and Dirac the electron is described
not by a worldline but by a field1j;(x, t) and the basic coupled equations (1) become

and
(10)
for the relativistic Dirac case, and

i ~~ = (- 2~ [(fl - eA)2] + eAo) 1j; (11 )

HISTORY OF THE ELECTRON 123

for the nonrelativistic Schrodinger case. The currents for a number of electrons is

. '" - (i)
Jp,(X) = ~ eil/Ji(X)-y p,l/Ji(X) (12)

with a similar expression for the Schrodinger current.

Again the field Ap, is the sum of an external and a selffield parts:

(13)
With the choice of gauge A~~ = 0 the Maxwell equations become

(14)

so that the selffield can be expressed in terms of the current as

(15)

where D(x-y) is the appropriate Green's function corresponding to initial and bound-
ary conditions. Equation (15) is our generalized Lienard-Wiechert potential. Thus the
light emitted by a source depends on the nature and preparation of the current, and
also on the nature of the environment determining the Green's function. Furthermore
the whole light cone where l/J is different from zero contributes to the field at the field
point and not just a single intersection of the worldline with the light cone, as in the
case of a point particle.
Thus the selffield can be eliminated from the coupled Maxwell-Dirac equations.
Inserting Ap, into the equation of motion we obtain

{!'p, (i8p, - ekA~xt) - mk} l/Jk(X) = enP,l/Jk(X) J dyD(x - y) L e;1Pi(y}rp,l/Ji(Y)

,
(16)

Here Aext is a fixed external field whose sources are far away and not dynamically
relevant. In the next Section we shall treat two or many body systems in which
we shall eliminate completely all the fields in favor of the currents. Eq. (16) is a
nonlinear integral equation for l/J analogous to the nonlinear equation of the classical
electrodynamics. The corresponding equation for the Schrodinger case is (n = 1)

i 8l/J = [ __1_
at 2m
(p _eAext _ eAself) 2 + e (A ext + ASelf )]
0 0
l/J (17)

where the selfpotentials are

A~elf = JdyD(x - y) L ekl/J'k(y)l/Jk(Y),

k
Aself(x) = JdyD(x - y) L l/J'k(y) ~ l/Jk(y)
k
(18)
124 A.O. BARUT

In writing these equations we have assumed that the 1,&-current is an actual mate-
rial charge current, and not just a probability current. Thus we are inevitably led to
contemplate the interpretation and foundations of quantum theory. The foundations
of quantumelectrodynamics and that of quantum theory must be the same, for quan-
tum mechanics was invented to understand the interactions between light and matter.
Not surprisingly, it was Schrodinger who first formulated the selfconsistent coupled
Maxwell and matter field equations, i.e., the program of Lorentz, for the new wave
mechanics and insisted that for the selfconsistency of the theory the self field of the
electron must be included as a nonlinear term. Schrodinger however calculated only
the static part of the selfenergy and obtained unacceptable large selfenergies. Sub-
sequently quantum electrodynamics went into a different direction. The selffield was
dropped completely. Instead, one introduced a separate quantized radiation field with
its own new degrees of freedom and coupled this to the quantized matter field. In the
selffield approach the electromagnetic field has no separate degrees of freedom, they
are determined by the source's degrees of freedom, but then we must include the full
nonlinear selffield term. We shall come to this duality between the two approaches
and to the questions of interpretation of quantum theory after the developments of
the selffield QED.

3.4 Radiative Processes in an External (Coulomb) Field

The basis of selffield quantumelectrodynamics is conceptually very simple and is

completely expressed by the single equation (16). All QED processes in an external
field listed in Table I should be derived from this single equation. To perform actual
calculations it is much simpler and more direct to work with the action rather than
with the equations of motion. The action W can, up to an overall 5-function, be
related to the energies of the system for bound state problems, and to the scattering
amplitude for scattering problems.
The action for the system (10) is

Here we shall express AI'Cx) in terms of 1,& using (15). For bound state problems the
action of the electromagnetic field can be reexpressed by a partial integration, using
(10), as
-~ J dxFl'vFI'V = +~ J dxjl' (x)AI' (x) (20)

Putting all together we have the action underlying our nonlinear equation (16), namely

W = J dx [ibex) (,/l (WI' - eA~xt) - m) 1,&(x)

J
(21)
- e; dyib(xhl'(xN)(x)D(x - Y)ib(Yhl'1,&(y)]

We shall consider now the single electron problem in an external field.

HISTORY OF THE ELECTRON 125

We expand the classical field 1/J into a Fourier series

1/J(X) = t1/Jn(x)e- iEnt (22)

n

and shall try to determine the expansion coefficients 1/Jn(x) and the spectrum En-
discrete and continous. This expansion is quite different than the one used in standard
QED and quantumoptics, namely the Coulomb series expansion, for example, in the
Coulomb field,
1/J(X) = tcn(t)l/J~(x)
n

Here one derives equations for the time-dependent coefficients c n ( t). The idea behind is
that the system has definite levels and the perturbation will cause transitions between
these levels. In our formulation, due to selfenergy, there are no definite (discrete) levels
as exact eigenstates of the system to begin with, but the equations will determine the
spectrum. In fact it will turn out that only the ground state of the system will be
a stable eigenstate followed by a continuum with spectral concentrations around the
unperturbed spectrum.
If we insert the Fourier expansion into the action we obtain

w = 1:.1 dX{ ¢n(x)eiEnXO bl' (i0l' - eA~xt) - m]1/Jm(x)e-iEmXO_

- e; 1dY¢n(xhl'1/Jm(x)e i(En-E
(23)
m )x OD(x - Y)¢r(if)'YI'1/Js(if)e i (E r -E,)yO }

Time integrations can be performed using

1
D(x- Y )=-(27T)4
1 dk
e-ik(x-y)
k2 (24)

and we can write the interaction part of the action entirely in terms of the Fourier
components of the current

Wint = + "\f 8(En - Em + Er - E.) J¢n (xhl'1/Jm (x)

e; n1;::.s
(25)

For the exact solutions of our equations the action W will vanish identically. We
will now solve the system iteratively.
126 A.O.BARUT

To lowest order of iteration we take the field to be given by the solutions of the
external field problem without the selfenergy terms, and the energies to be shifted by
a small amount:
V'n(X) = 1/,~xt(x)
(26)
En = E~xt + b..En
The first term in (22) therefore gives simply, using the orthonormality of 1/;n's,

H'o = J d£!{!n Cl E~xt -;y. fi - m - eA:xt ) 1/;m b(En - Em)

:::} L b..Enb(En - Em)bnm
nm

In the second term we separate the terms according to En = Em, Er = E. and

according to En = E., Em = En the two ways of satisfying the overall o-function.
And since W = 0 to this order of iteration we can solve for b..En . The action and the
total energy of the system are related by a o-function. Cancelling this o-function and
also the sum over n to obtain the energy shift of a fixed level n, we obtain

This can be written in the form

2 ~ ·"(k)·mm( k) 2 ~ .
~E n =_~¥~Jnn J,. - -~1;,~.,. (k)"mn(_k) Z7r o(E -E -k)
2 (271-)3 k2 2 p
(27l" Jnm J,. 2 m n
<n

e
-2"1;:
2
'\f dk
-:
.,. (k) ·mn( k)
(27l")3 Jnm J,. - 2k
1[1 Em-En-k - Em-En+k
1]
(28)
Thus the energy shifts are entirely expressed in terms of the integrals over the Fourier
spectra of currents of all states. The first term corresponds to vacuum polarization,
the second to spontaneous emission, and the third term to the Lamb shift proper. In
HISTORY OF THE ELECTRON 127

arriving at these results we have used the causal Green's function and separated the
integrals into a principal and a imaginary part according to the formula

~ = P~ ±i7r8(x) (29)
x x

All the main QED effects are obtained here from a single expression. In fact one can
also read off the anomalous magnetic moment (g - 2) from this expression as we shall
show in Section VI.
The evaluation of these expressions is a rather laborious technical problem. We
have to use relativistic Coulomb wave functions for both the discrete and continous
spectrum and integrate the products of such functions and sum over the whole spec-
trum. We shall indicate some of these calculations and give results in Section VIII.
The most important feature of the present formulation is that there are no infrared
nor ultraviolet divergences.
The spontaneous emission term in Eq. (27) has been exactly evaluated 12 • We have
now complete relativistic spontaneous decay rates for all hydrogenic states l3 . Table
II shows some of these results.

TABLE II. Decay rates (8- 1) in hydrogen and muonium

Transition Hydrogen Muonium

251/ 2 --t 151/ 2 2.4964 X 10- 6 2.3997 X 10- 6
251/ 2 --t 1P1 / 2 5.194 X 10- 10 5.172 X 10- 10
2P1 / 2 --t 151/ 2 2.0883 X 10 8 2.0794 X 10 8
2P3 / 2 --t 151/ 2 4.1766 X 10 8 4.1587 X 10 8
2P --t 151 / 2 6.2649 X 10 8 6.2382 X 10 8

The vacuum polarization term has also been evaluated analytically14 to lowest order
term in 0:( Z 0:)4. This is the most divergent term in perturbative QED and vanishes
in the nonrelativistic limit.
The Lamb shift term which correctly reduces to the standard expressions in the
dipole approximation has also been shown to be finite and will be evaluated in closed
form l5 .
In all these calcuations, since we are using Coulomb wave fuctions instead of the
plane waves, the individual integrals are all finite. The summation over all the discrete
and continous levels are done by means of the relativistic Coulomb Green's functions.

3.5 Quantumeletrodynamics of the Relativistic Two-Body System

One of the most important and perhaps unexpected features of the selffield formu-
lation of quantumelectrodynamics turned out to be a nonperturbative treatment of
two and many body systems in closed from. It is well known that bound state prob-
lems cannot be treated in perturbative QED starting from first principles. Instead
128 A.O.BARUT

one begins from a Schrodinger or Dirac-like equation obtained from some approxima-
tion to the Bethe-Salpeter relations and then calculates the perturbation diagrams
to the bound state solutions of these equations. \Vhat one really needs is a genuine
two-body relativistic equation which includes all the radiative terms as well as all the
recoil corrections at once. \Ve shall now discuss the principles of this theory.
In nonrelativistic quantum theory the many body problem is formulated in config-
uration space by a wave equation with pair potentials Vij (x i - X j) of the form

p~ p~ 8~,
( --
2ml
+ --
2m2
+ ... V 12 + V13 + V23 + ...) I
1p(Xl,"" Xn; t) = zlio:;-
.

ut

This a priori not obvious. We may also think that each particle has its own fieldljJ( x)
and satisfy a wave equation with a potential coming from the charge distribution of the
other particles. For two particles, for example, we would have the coupled Hartre-type
equations

These two formulations are closely related but not identical. We shall see that they
correspond to two different types of variational principles and actually describe two
different types of physical situations. Quantum theory has a separate new postulate for
two or more particles, namely that the state space is the tensor product of one particle
state spaces. This leads immediately to the first formulation in configuration space.
Such combined systems are called in the axiomatic of quantum theory "nonseparated"
systems with all the nonlocal properties of quantum theory. But this postulate does
not apply universally. There are other systems, namely the "separated" systems, which
are described by the second type of equations. For example, for the system hydrogen
molecule the two protons are separated, whereas the two electrons are nonseparated.
The superposition principle holds for the nonseparated systems only. We shall now
see how all this comes about from two different basic variational princples in the
relativistic case (the nonrelativistic case is similar).
Consider a number of matter fields 1/>l (x), ljJ2 (x) ... The action of these fields inter-
acting via the electromagnetic field is

where the current jlt is the sum of Dirac currents for each field

P'(x) = L ek 1h(xhl'1/>l'(x) (31)

k
HISTORY OF THE ELECTRON 129

Again in the gauge AI' ,p. = 0 we obtain the equations for the electromagnetic field as

DAp. = jl' = Lj~k) (32)

k

with the solution

(33)

If we insert this into the action both in the j I' . AI' term as well as in the term
-(1j4)Fp.vFl'v, and using the identity (20), we obtain

w= J L 1fid
dx
k
/,1'iOl' - mk) '¢k - L
k,f
J
~ dxdyj~k)(x)D(x - Y)/(f)(Y) (34)

The interaction action is a sum of current-current interactions containing both the

mutual interaction terms, e.g.

and the self interaction terms like

If we vary this action with respect to each field '¢k separately we obtain coupled
nonlinear equations. For example for two particles

(35)

Next let us define a composite field II> by

(36)

This is a 16-component spinor field. We can rewrite our action (34) entirely in terms
of the composite field. This is straightforward in the mutual interaction terms. In the
130 A.O.BARUT

kinetic energy and selfinteraction terms we multiply suitable by normalization factors.

For example for the first kinetic energy term we get

where d0"2nll = dO"~' is a 3-dimensional volume element perpendicular to the normal

nit. Similarly for the other kinetic energy term. The selfenergy terms need two such
normalization factors. The resultant action in terms of the composite field is then

w = [J dX1d0"2~(X1X2) (-yll 11"1 II - m1) @ 'Y. n«P(X1X2)

+ JdX2dO"l ~(X2X1h· n @ (-y1l1l"211 - m2) «P(X2Xt) (37)

- ele2 JdX1dx2~(XIX2hll @'YIID(XI - X2)«P(XIX2)]

The generalized canonical momenta 11";11 are given further below. Here and through
the rest of the paper we shall write spin matrices in the form of tensor products @,
the first factor always referring to the spin space of particle 1, the second to particle
2. We shall give the selfenergy terms explicitely below.
Now our second variational principle is that the action be stationary not with re-
spect to the variations of the individual fields but with respect to the total composite
field only. This is a weaker condition than before and leads to an equation for «P
in configuration space. For bound state problems only the symmetric Green's func-
tion contributes and it contains a 8(x 2 )-function which we decompose relative to the
space-like surface with normal nil as follows

8(r2) = 8[(r· n) - r .L) ± h[(r· n) + r .L), r.L = [(r . n)2 _ r 2 F/ 2 (38)

2r .L
where r.L is a relativistic three dimensional distance which for n = (1000) reduces to
the ordinary distance r. All the integrals in the action (37) are 7-dimensional. For
covariance purposes it is necessary to have the vector nil. It tells us how to choose the
time axis. The vector n is also present, in principle, in the one-body Dirac equation
but we usally do not write it when discussing the solutions, but automatically choose
it to be n = (1000), i.e., the rest frame. The final form of our two-body equation is
then

where now the selfpotentials are inside the generalized momenta

II
11"; = PiII - ej
Aliself
i - ei
Allext
; (40)
HISTORY OF THE ELECTRON 13l

J
with
A~eg(x)= ed dzda u D(x-z)<I>(z,tt}r1'0,·n<1>(z,u)

A~~~(.r) = e; J
(41)
dazduD(x - u)<I>(z, u}r· n 0'1'<1>(z, u)

each particle; e2 l~," and tel le:

We note that the last term in (39) can also be put into the potential AI" one halffor
t l , respectively.
The self potentials are nonlinear integral expressions. The arguments of <1> consist
of seven variables because <1>( Xl, X2) is different from zero only if (Xl - X2) is lightlike;
only then there is a communication between the particles. This means that we have
one time-variable and three space variables for each particle. We see this more cleary
if we introduce center of mass and relative variables according to

IT = 7l"1 + 7l"2, 7l" = 7l"1 - 7l"2

(42)
X = Xl - X2, X = Xl + X2
Then equation (39), without the selffield terms for simplicity, becomes

(43)

where we have introduced

and
1
kI' = 0,· n -,. n 0,1')
-hI' (44)
2
We see now that k . n = 0, i.e., the component of kJL parallel to nl' vanishes which
means that the component of the relative momentum 7l" I' parallel to n I' drops out of
the equation automatically. For n = (1000), in particular, we have

(45)

Thus we have only one time variable conjugate to the center of mass energy ITo and
three degrees of freedom for the center of mass momentum IT and three degrees of
freedom for the relative momentum 77°; 7l"o does not enter, as it should be so on
physical grounds. In contrast the Bethe-Salpeter equation has two time coordinates.
Since ITo is the "Hamiltonian" of the system we obtain, by multiplying (45) by r;;-l
the Hamiltonian form of the two-body equation
132 A.O.BARUT

where we have defined

( 46)

Our two-body equation has the form of a generalized Dirac equation, now a 16-
component wave equation. In fact it reduces to the one-body Dirac equation in the
limit when one of the particles is heavy.
The above developments are completely relativistic and covariant. The physical
results are independent of the vector n although a vector n must appear for manifest
covariance. Thus recoil corrections are included to all orders. Further interesting
properties of the equation, beside being a one-time relativistic equation, are that
relative and center of mass terms in the Hamiltonian are additive, and radial and
angular parts of the relative equation are exactly separable. It has also a nonrelativistic
limit to the two-body Schrodinger equation. We shall discuss numerical results in
Section VIII.

3.6 The Interpretation of Negative Energy States

It is often stated that only in second quantized field theory can one have an adequate
description of antiparticles and negative energy solutions where one changes the roles of
the creation and annihilation operators for the negative energy solutions. We shall now
show that there is also a consistent way of dealing with the negative energy solutions
and antiparticles in the Dirac equation as a classical field theory and elaborate how
we obtain the annihilation potential in positronium, for example.
There are actually not one but two Diract equations

h· p - m)1/JJ = 0
(47)
h·p+m)1/JII=O

obtained from the factorization of the Klein-Gordon operator, for example. By con-
vention we just peak one and work with the complete set of solutions of this equation.
Now the negative energy solutions of 1/JJ coincide with the positive energy solutions of
1/JII. Furthermore in the presence of the electromagnetic field with minimal coupling
we have the two equations

h . (p - eA) - m) 1/JJ = 0
(48)
h . (p - eA) + m) 1/J II = 0

and we can easily prove that

1/JJ(-p,-e) = ~'II(p,e) (49)

that is the negative energy momentum solutions of·ljJ J coincide with the positive energy
solutions of ~)II of opposite charge. Therefore we should consider positive energy
solutions of both equations as physical particles. The total number of such physical
HISTORY OF THE ELECTRON 133

solutions is the same as the total number of both positive and negative energy solutions
of a single Dirac equation.
With this interpretation we obtain quite naturally the annihilation diagrams and
annihilation potentials between particles and antiparticles. Consider our interaction

J
action
dxdYlPl(xh ll 1/Jl(X)D(x - Y)lP2(yhll1/J2(Y)

Here the classical fields 1/Ji(X) contain all positive and negative energy solutions accord-
ing to our general expansion (22). Separating positive and negative energy solutions
as
~'n(x) = 1/JEN>O + 1/JEN<O == 1/J~ + 1/J-
and inserting it into the action we get 16 terms. In the limiting case of the lowest order

)x
scattering, where we replace the fields by plane wave solutions, we have essentially two
distinct types of vertices at each point x or Y, namely

lP+(xhll1/J+(x)
and (50)

V·p
x
In the second case we have used our interpretation of the negative energy solutions
as the antiparticles with reversed energy-momentum PI'" The complete interaction
action to this order consists of all combinations of these two vertices located at x and
Y for particles 1 and 2 multiplied with the Green's function D(x - y). Of these 16
terms some cannot be realized because of the overall energy-momentum conserving
6-function, 6(Pl + P2 - P3 - P4), and we are left with two disctinct types of terms

V XI

/\
(51)

plus the same terms with particles and antiparticles interchanged. This result agrees
with the standard QED. But we shall go a bit further and apply it to bound state
problems in Section VII after a discussion of the case of identical particles.
Identical Particles
For two identical particles we use the postulate of the first quantized quantum theory
that the field is symmetric or antisymmetric under the interchange of all dynamical
variables of identical particles. In our formulation we go back to the original action
principle and assume that the current ill is anti symmetric in the two fields

(52)
134 A.O.BARUT

This implies in the interaction action

~Vint = ~e2 [J dxdy1/;1(xhll¢2(X)D(x - y)1/;1(yh ll ¢2(Y)

-J
(53)
dxdy1/;nll¢2 D(X - y)1/;n ll ¢l + (1 +-t 2)]

and again when the fields are expanded we see that identical particles with exactly
the same wave functions i.e., the same quantum numbers or the same state, will not
interact and that in the lowest approximation we will get besides the direct interaction
also an exchange term as shown in the following diagrams
1=2 2=1

1= x
Finally we combine the two effects, identical particles and particle-antiparticle prop-
erties, to discuss systems like electron-positron complex and positronium. According
to our discussion this system is just a part of the larger electron-electron system taking
into account the interpretation of the negative energy levels and the identicity of the
particles.

3.7 Calculation of the Anomalous Magnetic Moment (g - 2)

We show now that our basic interaction action also contains besides Lamb shift,
spontaneous emission and vacuum polarization also the anomalous magnetic moment
in the same single expression. We shall also introduce at this occasion the more general
four-dimensional energy-momentum Fourier expansion instead of the energy Fourier
expansion (22) which was appropriate for the fixed external field problem.
The interaction action is given by

H~nt = -2
e
2
Jdxdyjll(X)D(x - y)jll(Y) (54)

vVe expand the fields as four dimensional Fourier integrals

(55)

J -
and insert it into the action
e2 1 e-ik(x-y) -
Hljnt = -2 (27r)4 dxdydkdpdqdrds¢(phll¢(p) k2 + iE ¢(rhll¢(s)
x ei(p-q)x+(r-s)y
HISTORY OF THE ELECTRON 135

which can be written as

Wint=_e2(271/jdPdqdrdSjll(p,q)(
2 r - s
~2'
+ZE
jll(r,s)o(p-q+r-s) (56)

where )I'(p, q) stand for the double Fourier transform

(57)

The o-function arises from the x, y and k-integrations. We separate the action into
two terms to satisfy the o-function

(i) p=q, hencer=s

(58)
(ii) p=s, henceq=r

Again as before the first corresponds to vacuum polarization; the second term contains
Lamb shift and spontaneous emission as real and imaginary parts of the energy shift
llE. It also contains the anomalous magnetic moment as the coefficient of the magnetic
part of the Lamb shift for any external field. For the calculation of (g - 2) it is thus not
necessary to solve a problem with an external magnetic field or to solve any external
problem for that matter.
The second term with (ii) can be written as

Here we recognize the c-number electron propagator-function Sex - y)

j dq1jJ(q);j)(q)e i (y-x)q == Sex - y) (60)

which satisfy the inhomogeneous wave equation

(61)

Let us also take the Fourier transform of S

Sex - y) = _1_ j dpe-ip(x- y ) S(p) (62)

(27r )4

Then the action (59) becomes

W5 ii )=_e 2 'jd dP.T.() ,IlS(phll .I.() (63)

mt 2z p 'I" P (p - P)2 + iE 'I" P
136 A. O.BARUT

It is related to the energy, more precisely to a mass shift by an overall o-function and
we can write
W(ii)
~E = --.illL =
J dp1f;(p)~M(p)lj;(p) (64)
(2iT )4
where we have introduced an effective mass matrix by

(65)

It remains now to evaluate the mass matrix tl11,1. First we expand the Green's
function or propagator in an external field as follows

where
(66)
It turns out that only the third term which is gauge invariant gives a nonvanishing con-
tribution to lowest order in 0': the terms containing AI' give vanishing contributions.
The mass operator becomes

(67)

The integrals can be performed giving 16

2'
~M(p) = ~_l_( -ie)-
2 (2iT)4 m
2110
dy(1 - y)a llll F
JlV

and finally
0' e 0' e (~ ~ .~ E~)
~M ( p ) = - - - a vF v = - - - a·B+w· (68)
2iT 2m Jl Jl 2iT 2m
Thus we recognize the anomalous magnetic moment to this order in front of the ii· B
term for any field as we have mentioned.
If we insert the mass operator into the energy shift formula (64) we can evaluate
it as the expectation value of the operator ii· B + ia . E. For relativistic Coulomb
problem for example the magnetic part of the Lamb shift can be analytically evaluated
exactly. 17
This way of calculating the anomalous magnetic moment also shows now how to
calculate higher order terms. vVe must take more terms in the Green's function expan-
sion (66). This may be much simpler than the diagrammatic method of perturbative
QED where there are already 891 Feynman diagrams in the order (0'/ iT)4 .
HISTORY OF THE ELECTRON 137

3.8 Covariant Analysis of Radiative Processes for Two-Body Systems

In this section we discuss how to treat radiative processes, like Lamb shift, etc.,
for a system like positronium or muonium beyond the naive reduced mass method.
As mentioned above the action formalism is more convenient than the equations of
motion.
We go back to our covariant 2-body action (37) and separate center of mass and
relative coordinates and momenta according to

1
r = Xl - X2 Xl = R+:t P = PI + P2
1 1 1
R=2(XI+X2) X2 = R - -r P = 2(PI - pz)
2
(69)
1 1
q=z-u z = Q +-q PI = -P+ P
2 2
1 1 1
Q=-(z+u) u = Q --q P2=-P-P
2 2 2

All quantities here are four-vectors. Then the action becomes

w= JdRdq~(R,r){ [1' (~P+p) -ml] [1' (~p-p) 01· n +1· n 0 -m2]

J
- ele2 D(r) - el + ~(r ~(Q,qhl-'
dQdq1/l 0 l ' nD (R - Q - q)) 0 l ' n<I>(Q,q)

- ~ JdQdq1.n01/lD(R-Q-~(r-q))~(Q,qh.n01/l<I>(Q,q)}<I>(R,r)
(70)
In the absence of a fixed external field the system is translationally invariant and the
generalization of the Fourier expansion (22) is the four dimensional Fourier transform
of the composite field <I>(R,r) which has actually one time variable, <I>(R,r.L), the
relative coordinates is a 3-vector r.L perpendicular to n.

<I>(R,r.L)= J d4p
(27r)4e
iPR
1/J(P,r.L)

We insert this expansion everywhere in our action and obtain

138 A,O,BARUT

-J J
TV - dRdr1. dPm - n ,r1.)e -iPnR{ [
dPn (27r)41j>(P
(27r)4 rl'P r1.,p e iP~R
I ' +.erel (.)]

1
- Z
J dP dP
(27r)4 (27r)4 dQdHdk d
[e 2 e-ik[R-Q+~(r.L-q.Ll]
k2
- '
,I' 0,' n1j>(Pr, q1.)e-zPrQ~(1'
0,' ne iP,Q1j>(P., H)
-ik[R-Q_l(r.L-q.Ll]
+ e; e
2
k 22 " n 0,1'¢(Pr, H)e- iPrQ " n 0'l'e iP,Q

X V'(PS) q1.] }1j>(Pm, r 1.)

(71)
where .ere! is the Lagrangian of the relative motion and is given by

and
1
r I' = Zhl' 0,' n +,' n 0,1') (73)

The result of performing the Rand Q-integrations, letting

kl' =,1' 0,' n - , ' n 0,1' (74)

)s

W = J (27r)4 -
dPn dPmdr 1.¢(Pn, r 1.) { [r I'PI' + kl'pl' - e) e2
m)l @;' n - m2l' n @ I - ~;I' 'i9 ~fl' ]

X 8(Pn - Pm) - -1
2
JdP(
-
27r
r
)4 dk dr 1. dq1.8(Pn - Pm - k)8(Pr - p. - k) [e)2 e-',1 kl".l.-q.l. )
dP'--
(27r)
4

X ~(I' @;, n;j;(Pr , q1.hl' @;, n + e~eitk(r.L -q.L);, n 'i9;I';j;(P" q1.h ' n @;I'l¢(ps. 1]1. }

¢(Pm,T1.)
(75)
Introducing the form factors
HISTORY OF THE ELECTRON 139

and (76)

for the two particles, we can write the action in the compact form

where

Note that kltnp = o.

Now we can perform the kG-integration, and without loss of generality set n
(1000), and obtain

#
(79)
where P stands for the principal value of the integral and means a summation over
discrete states and an integration over the continuum states. As in the case of the
Coulomb problem and (g - 2)-calculation, we separate the two terms corresponding
to
(a)n=m, hence'l'=s
140 A. O.BARUT

and
(b) n = s, hence m = r
and dictated by the 8-functions and obtain finally

(SO)
We recognize again the following terms:
(i) Term containing 8( k) + 8( -k). The contribution of this term to the dk-integral
vanishes.
(ii) The term Pl/k: This term corresponds to vacuum polarization.
(iii) The term with (i7rjk)[8(w nm - k) + 8(w nm + k)]. This term gives the spontaneous
emission or absorption from level n to m or vice versa.
(iv) The term with PA (wn~-k - wn~+k) gives the Lamb shift.
These are our formulas for the radiative processes of the two-fermion system 18. In
the limit they go over to the fixed center Coulomb problem on the one hand, and for
free particles to perturbative QED results.
For identical particles and particle-antiparticle system like positronium we have
to anti symmetrize our currents as discussed in Sec. VI. Thus the mutual interaction
action has two terms. The first is the usual direct interaction term

J -
(Sl)
(1) (2)
= _e 2 dxdY1h(x, y)J1l D(x - y)JIl<P(x, y)

corresponding to the potential

2(1) 1 (2)
V = -e ,..,/t - Til (S2)
r
The second term is

(S3)
HISTORY OF THE ELECTRON 141

where Po and p~ are the initial and final state energies and E is total conserved center
of mass energy of the whole system. In the positronium the relative momentum is
approximately zero so that we can set

PJl ~ (m, 0)

and the action becomes

(84)

Now we show that this term gives correctly the annihilation contribution to the hyper-
fine splitting in the n = 1 state of positronium, for example. The effective potential
above (84), when inserted into our wave equations gives an energy shift only for the
lewIs j = 1 ~ 0 and for j - 1 = 1 ~ o.

ma 4
bE(j = e= 0) ~ -3
2n
and
ma 4
bE(j - 1 = e= 0) ~ 4n 3

The difference is the annihilation contribution in the hyperfinesplitting

4
vJ::E HIS ( anm·h·l·
1 atlOn ) ~ -
3ma
-3- (85)
12n
To this order it agrees with perturbative QED. It is however obtained here in first
quantized QED with selffields.

3.9 Further Results

There are still discrepancies between theory and experiment in almost all tests of
QED. The Table III summarizes all measured levels in positronium, muonium and
Hydrogen, positronium lifetimes, the anomalous magnetic moments ae and aJl' and
some theoretical values in parenthesis. In reviewing some of these discrepancies, vV-
Lichten 18 writes "It seems likely that the problem lies in the difficulty of QED cal-
culations which have not been carried out to a high order enough, perhaps a totally
new type of calculation is needed". The self field approach to QED provides a new
type of calculation. It's important to have a complementary or alternate method to
perturbative QED, for a theory is tested not only against experiment but also against
other theories in order to clarify the basic assumptions and concepts, specially in view
of recent results that perturbative QED might be inconsistent or a trivial theory. Self-
field QED modjfies our notion of the quantized radiation field and the interpretation
of quantum theory. The emphasis is shifted from the field to the source of radiation,
an electronic charge distribution which objectively and deterministically evolves as a
142 A. O. BARUT

TABLE ill. BOUND STATE TESTS OF QED

POSITRONIUM

t 4 - 8619.6(2.7) MHz
(8625.2 - theory)
n=2 --,,.--_ _ 13001.3(3.9) MHz
( 13010.9 -theory)
18504.1(10) MHz
( 18496.1 -theory)

'I
, - 1233607218.9 ± 10 MHz
2430A (l 233 607 202 - theory)
- 5.1 eV

3S 03389.1 (0.7) MHz

t'-(~ 03.399.1
1
n=1 - theory)
IS
o
t 3S1~ 3Y=7.0514Il s-1
(7.0383 - theory) a e = 1 159652 188.4 (4.3) x 10 -12
t ISo~ 2Y = 7.994 ns- I (1 159652 192 (l08» - theory)
(7.9866 - theory)
MUONIUM (AND H)
.------.t-- F=2
2p _ _ _ 74 MHz
312 ---,..---1 ,
L-----1~ F=1
n=2 25 10.900 MHz
1/2

~187MHZ
F=O
(H: 1057.845(9) MHz)
10.1 ey" (1057.875 -theory)
-1221 A
(H: 2 446 062 413.70(41) MHz)

F=1 4463.30288 (16) MHz

n=1 2S112 - L - - - l ~63.303.6 - theory)
'--_I-~ F=O '\
(H: 14204057517667 (9) MHz)
~ = 11659110 (110) x 10- 10 (1 420 402 308 - theory)
(1 1659203 (20) - theory)
HISTORY OF THE ELECTRON 143

classical field and produces a selffield which acts back on the charge itself. Quan-
tized properties of the light reflect the discrete frequencies of the oscillating charge
distribution.
The two-body relativistic equation discussed in Sections V and VII gives us a pos-
sibility to make improved calculations for positronium and muonium, in particular.
In positronium, the experiments seem to be more accurate than the theory and the
perturbative calculations remain incomplete 19 . Considerable analytical work has been
done on the study of the two-body equation (39)ff: separation of radial and angular
parts and further reduction of the radial equations 2o • It turns out that the two-body
equation, when the electromagnetic potentials are kept to order a 4 , is exactly soluble
with an energy spectrum

(86)

with 1.1 = ml + m2, ~m = ml - m2, generalizing the Dirac spectrum. We have

treated the remaining potentials of order a 5 and higher as perturbations. But having
tested the equation in this way, one can now make direct nonperturbative numerical
calculations. The treatment of the negative energy states and the covariance of the
equation has also been discussed 22 according to the methods outlined in Section VI.
We give here some of the results zo .
1) For parapositronium, eq. (86), is exact including terms of the order a 4 since normal
and anomalous magnetic moment terms do not contribute to this order. It gives

ma Z ma 4 11 ma 4
Epara ps = 2m _ - - _ +- - - + 0(a 6 ) (87)
4n z 2n (2j + 1)
3 64 n4

2) Introducing the anomalous magnetic moments aI, az, which are in the selfenergy
term, as a Pauli-coupling we obtain the ground-state hyperfine splitting

with ( = mdmz. Numerically this gives 1420.348 MHz for H, and 4.463.060
MHz for muonium, compared to the experim~ntal values 1420.405752 and 4.463302,
respectively.
3) Positronium hyperfinesplitting including the annihilation term, eq. (85), gives

This "Lambshift" term - 2",. (¥+ Cn2)ma 4 has to be added perturbatively, but we
hope to calculate these terms and more eventually numerically.
144 A. O.BARUT

4) Positronium (n = 2,n = 1) splitting, including annihilation and anomalous mag-

netic moment contributions

3 2 0'2 Ry 35
b..E21 = -Ry - 0.4680980' Ry - - - - .
8 27r 96

5) Positronioum fine structure

1272 72
b..E(2 51 - 2 P2) = --0' Ry + -0' Ry - - 0 ' Ry + 0(0' 5 )
33
12 48 480
normal magnetic
(recoil ) (annihilation) ( moment )

6) H or muonium (n = 2, n = 1) splitting

where AI = m1 + m2, (= mdm2' f.L = m;w'2.

For other details and applications of self-field QED we refer to the literature listed
in the Appendix.
HISTORY OF THE ELECTRON 145

References to Chapter 3.
1. See e.g. W. Thirring, Principles of Quantumelectrodynamics (Academic Press, N.Y.
1958).
J.D. Bjorken and S.D. Drell, Quantum Fields (McGraw Hill, N.Y. 1965).
R.P. Feynman, Quantum electrodynamics (Benjamin, N.Y. 1961).

2. A.O. Barut, in Quantumelectrodynamics, ed. P. Urban, Supp!. Acta Physica Aus-

triaca, Vol. 2 (Springer, Berlin 1965); and The Theory of the Scattering Matrix,
(Macmillan, N.Y. 1962), Ch. 13.

3. G. Scharf, Finite Quantumelectrodynamics, Springer 1989, and These Proceedings.

4. See e.g. C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw Hill, N.Y.
1964).

5. A.O. Barut and LH. Duru, Phys. Reports, 172, 1 (1989).

6. J. Schwinger, Particles, Sources and Fields, Vol. II (Addison Wesley, Reading, MA

1973).

7. H.A. Lorentz, The Theory of Electrons, (Dover, N.Y. 1952).

8. A.O. Barut, Phys. Rev., DI0, 3335 (1974).

9. }... O. Barut and N. final, Phys. Rev., A40, 5404 (1989).

10. A. Lees, Philos. Mag., 28, 385 (1939).

11. D.A.M. Dirac, Pmc. Roy. Soc., (London), A268, 57 (1962).

12. 11..0. Barut and Y. Salamin, Phys. Rev., A37, 2284 (1988); A39, (1990).

13. 11..0. Barut and Y. Salamin, Zeits. f. Physik D, (in press).

14. 11..0. Bamt and N. Unal, A Regularized Analytic Evaluation of Vacuum polarization
in Coulomb field, Phys. Rev. D, 41, 3822 (1990).

15. A.O. Barut, J. Kraus, Y. Salamin and N. Unal (to be published).

16. For more details on (g - 2) calculations see A.O. Barut and J.P. Dowling, Zeits. f.
Naturf., 44a, 1051 (1989) and A.O. Barut, J.P. Dowling and J.F. van Huele, Phys.
Rev., A38, 4405 (1988).

17. For more details see A.O. Barut and N. Unal, Complete QED ofthe electron-positron
system and positronium (to be published).

18. W. Lichten, in The Hydrogen Atom, edit. by G.F. Bassani et al (Springer-1989), p.

39.
146 A.O.BARUT

19. A. Rich et aI, in "Frontiers of Qnantnmelectrodynamics and Qllantllmoptics ", edit.

by A.O. Barut (Plenum Press, 1991).

20. A.O. Barut and N. Unal, J. Math. Phys., 27,3055 (1986), Physica, 142A, 457 and
488 (1987).

21. A.O. Barut, Physica Scripta, 36, 493 (1987), and in "Constraint Theory and Rela-
ti'vistic Dynamics", L. Lusanna et ai, edit" (World Scientific, 1987), p, 122,
HISTORY OF THE ELECTRON 147

Appendix: References on Selffield QED

1. Nonperturbative QED: The Lamb shift, A.O. Barut and J. Kraus, Found. of Physics,
13, 189 (1983).

2. QED based on selfenergy, A.O.Barut and J.F. van Ruele, Phys. Rev. A, 32, 3887
(1985)

3. QED based on selfenergy vs. quantization of fields: Illustration by a simple model.

A.O. Barut, Phys. Rev. A, 34, 3502 (1986)

4. An exactly soluble relativistic quantum two fermion problem. A.O. Barut and N.
Unal, J. Math. Phys., 27, 3055 (1986)

5. A new approach to bound state QED. I. Theory, Physica Scripta, 142A, 457 (1987),
II. Spectra of positronium, muonium and Hydrogen. A.O. Barut and N.Unal, Phys-
ica, 142A, 488 (1987)

6. An approach to finite non-perturbative QED. A.O. Barut in "Proc. 2nd Intern.

Symposium on Foundations of Quantum Mech.", Phys. Soc. of Japan, 1986, p. 323.

7. On the treatment of Moller and Breit potentials and the covariant 2-body equation
for positronium and muonium, A.O. Barut, Physica Scripta, 36,493 (1987)

8. On the covariance of two-fermion equation, A.O. Barut, in "Constraint theory and

Relativistic dynamics", (1. Lusanna et ai, edit.) World Scientific, 1987; p. 122

9. Formulation of nonperturbative QED as a nonlinear first quantized classical field

theory, A.O. Barut in "Differential Geometric Methods in Theoretical Physics", (R.
Doebner et al, edit's), World Scientific 1987; p. 51

10. QED based on selfenergy: Spontaneous emission in cavities, A.O. Barut and J.P.
Dowling, Phys. Rev. A, 36, 649 (1987)
11. QED based on self energy: The Lamb shift and long-range Casimir-Polder forces
near boundaries, A.O. Barut and J.P. Dowling, Phys. Rev. A, 36,2550 (1987)

12. QED based on selfenergy, A.O. Barut, Physica Scripta, T21, 18 (1988)

13. Relativistic spontaneous emission, A.O. Barut and Y. Salamin, Phys. Rev. A, 37,
2284 (1988)

14. QED based on selfenergy: A nonrelativistic calculation of (g - 2), A.O. Barut, J.P.
Dowling and J.F. van Ruele, Phys. Rev. A, 38,4405 (1988)

15. QED based on selfenergy: Cavity dependent contributions of (g - 2), A.O. Barut
and J.P. Dowling, Phys. Rev. A, 38, 2796 (1989)
148 A.O.BARUT

16. QED based on self fields, A Relativistic calculation of (9 - 2), A.O. Barut and J.P.
Dowling, Zeits. f. Naturf, 44A, 1051 (1989)

17. Path integral formulation of QED from classical particle trajectories, A.O. Barut
and I.H. Duru, Phys. Reports, 172, 1-32 (1989)

18. Probleme relativiste a deux corps en eIectyrodynamique quanti que, Heelv. Phys.
Acta, 62,436 (1989)

19. Selffield QED: The two-level atom, A.O. Barut and J.P. Dowling, Phys. Rev. A, 41,
2284 (1990)

20. QED based on selffields: On the origin of thermal radiation detected by an acceler-
ated observer, A.O. Barut and J.P. Dowling, Phys. Rev. A, 41, 2227 (1990)

21. Relatiyistic Spontaneous emission in Heisenberg representation. A.O. Barut and Y.

Salamin, Z. f. Physik D, (in press)

22. Relativistic 251 / 2 -> 151 / 2 + I decay rates of H-like atoms for all Z, A.O. Barut
and Y. Salamin, Phys. Rev. A, (in press)

23. QED-The unfinished business, A.O. Barut in "Proc. III. Conf. Math. Physics"
(World Scientific, 1990) (F. Hussain, ed.) p. 493

24. A regularized analytic evaluation of vacuum polarization in Coulomb field, A.O.

Barllt and Unal, Phys. Rev. D, 41, 3822 (1990)

25. The Foundations of Self-field Quantumelectrodynamics, A.O. Barut in "New Fron-

tiers of Quantumelectrodynamics and Quantumoptics", (edit. by A.O. Barut), Plenum
Press, NY 1990

26. QED Based on Selffields: Cavity effects, J.P. Dowling, ibid,

27. Fundamental Symmetries of Quantumelectrodynamics in Symmetry in Science III,

Plenum Press 1989 (ed. B. Gruber), p. 3-13

28. QED in Non-Simply Connected Regions, A. O. Barut and I. H. Duru, Quantum

Optics

29. Contribution of the individual discrete levels to the Lamb shift in hydrogenic atoms,
A. O. Barut, B. Blaive and R. Boudet J. Phys. B.

30. Interpretation of selffield QED, A. O. Barut and J. P. Dowling Phys. Rev. A, (in
press)

31. Is Second Quantization Necessary? in Quantum Theory and the Structure of Space
and Time, Vol. 6 (edit. by 1. Castell and C. F. vm Weizsiicker), Hanser Verlag,
Munchen 1986; p. 83-90.
THE EXPLICIT NONLINEARITY OF QUANTUM ELECTRODYNAMICS

W.T. Grandy, Jr.

Department of Physics and Astronomy
University of Wyoming
Laramie, Wyoming 82071 USA

ABSTRACT. The underlying presumptions and principal features of major semiclassical theories
of radiation are examined critically. There are significant physical and mathematical difficulties
leading to the conclusion that, in their present state of development, these theories are incapable
of competing with QED either qualitatively or quantitatively. Some possible means of salvaging
the semiclassical approach are suggested, the motivation being to seek a more physically satisfying
picture than that provided by quantum field theory.

For over fifty years conventional wisdom has held that the only correct description of the
electromagnetic interaction of matter is with the quantized radiation field. Yet, for almost
as long numerous writers have considered this view not entirely acceptable, at best, and the
attempt to explain the data both quantitatively and qualitatively without field quantization
remains an active topic of interest. This article constitutes a review and analysis of some
contemporary efforts to construct a viable theory of leptons interacting with the classical
electromagnetic field (generically referred to as 'semiclassical' theories).
It is not unreasonable to ask the question immediately, Why study this approach?
Briefly, to help in answering this, we recall that the fundamental expression of electro-
dynamics stems from the linearity of Ma.xwell's equations, which suggests that the total
electromagnetic field is the sum of any external field A~xt acting on a source, plus the field
of the source itself, A~elf. Thus, in the covariant gauge 01' A~elf = 0, the system of electron
and fields, say, is described by the coupled Dirac and Maxwell equations:

(,1'11"1' - mc)1jJ(x) = ~,I' A~elf(x)1jJ(x), (1a)

DA~elf(x) = 47re1,b(xhl'1jJ(x) , (1b)

with 11"1' == (iliol' - ~A~xt). These highly nonlinear equations have proved extraordinar-
ily resistant to closed-form solution, and only quantum field theory (QFT) has succeeded
in solving them to the high order of accuracy required by experiment. This is achieved
by replacing the classical fields with operator fields, introducing the associated commuta-
tion relations, and adopting the notions of a physical vacuum and all-pervasive radiation
field. The result, in practice, is a perturbative theory commonly referred to as QED~with
or without the Latin connotation~which in its agreement with observation is the most
successful physical theory known.
149
D. Hestenes and A. Weingartslw/er (eds.), The Electron, 149-164.
© 1991 Kluwer Academic Publishers.
150 W. T. GRANDY, Jr.

But this quantitative success bears a qualitative price in the form of what some would
consider bizarre physical features. Primary among these is the interpretation of the vacuum
as a vibrant fluctuating reality which is a literal source and sink of real particles and fields.
Vacuum fluctuations lead necessarily to the notion of virtual states, and all the radiative
effects previously associated with the self-interaction of an electron with its own field are
now attributed to the fluctuating vacuum field. As Jaynes (1973, 1978, 1990) has noted
frequently, however, one can be forgiven some skepticism about the reality of vacuum
fluctuations upon examining actual numerical values: the energy density associated with
the Lamb shift would produce a Poynting vector about three times the total power output of
the sun, and a gravitational field disrupting the entire solar system! Although operator fields
provide a mathematical means for describing creation and annihilation, there is not yet any
understanding of a physical mechanism for these processes. In addition, the mathematical
description itself breaks down in the form of various infinities and, for example, acausal
behavior of the Feynman propagator, which is also highly singular on the lightcone. These
difficulties have been 'resolved' in part by the mathematically-questionable, but enormously
clever techniques of the renormalization program (e.g., Itzykson and Zuber, 1980).
Although several authors have suggested that radiative effects should be described by a
mixture of vacuum fluctuations and radiation reaction associated with the classical self-field
(e.g., Milonni, 1984, and references therein; Davies and Burkitt, 1980), we shall here focus
entirely on the question of field quantization itself. Moreover, there is very little argument
for the need for field quantization in scattering processes per se, so that the issues reside
principally with radiative processes such as the Lamb shift, spontaneous emission, and
anomalous magnetic moments, to which we restrict the following discussion. The generic
semiclassical approach to quantum electrodynamics thus begins with a formal integration
of Eq.(1b) to obtain coupled nonlinear differential and integral equations:

(2a)

AI'(x) = 4:e J
D(x - y)j,,[y; A(y)] d4 y

= 47l'e J
D(x - y)1jj(yhl'1j;(y)d 4 y, (2b)

up to a solution of the homogeneous equation, which is here taken as zero. (From now
on AI' will refer to the self-field.) It is just prior to this step that QFT and semiclassical
theories part company (e.g., Kiilh~n, 1958).
Among the general features of these equations yet to be decided are the boundary
conditions on the Green function D. If AI' is to be associated with a classical field, then
one would certainly expect it to be retarded (unless one were particularly attracted to
the Wheeler-Feynman absorber theory). Below, however, we shall see that arguments
are sometimes made for Feynman-Stiickelberg boundary conditions, despite the difficulties
already noted regarding the 50-called causal propagator D F . We remark here only that DF
implies that AI' is complex, and thus ol'jl' =I O. This is not a difficulty for QFT, because
the self-field does not appear-the vacuum field itself is not observable, only its effects.
A second feature of Eqs.(2) of some importance is that they are never' uncoupled.
That is, AI' can be zero only if jl'[Y; 0] == 0 (sometimes called Hammerstein's theorem in
THE EXPLICIT NONLINEARITY OF QUANTUM ELECTRODYNAMICS 151

the theory of nonlinear integral equations), but that is never the case here. It is therefore
difficult to know just what kind of approximation it is to set the left-hand side of Eq.(2a)
to zero and obtain stationary-state solutions for the Coulomb problem, say. On the one
hand it is obvious that there can be no such thing as exactly stationary states, while on the
other the approximation is evidently rather accurate in practice. In the absence of external
fields it is a theorem that no plane-wave solutions for 'IjJ exist (e.g., Das and Kay, 1989),
but there are unique global solutions to the initial-value problem described by Eqs.(2) for
t ~ 0 when 7r I' -> PI' (Flato, et aI, 1987).
The general Maxwell-Dirac problem is exceptionally complicated and remains poorly
understood. The quantum field-theoretic viewpoint leads to solution of the equations of
motion through a tortuous string of arguments often defying common sense, and the per-
turbative result constitutes an effective linearization. If the nonlinearity is a fundamental
feature of the system, then the excellent agreement with experiment suggests that the non·
linearity must be subsumed in the vacuum. The meaning of 'understanding' varies greatly
among personalities and contexts, but on one definition QFT can be considered superb in
that regard.
By way of contrast, the semiclassical approach described by Eqs.(2) alone constitutes
a first-quantized classical field theory for both 'IjJ and AI'" It is not necessary at this point
to interpret 'IjJ in other than the conventional way, as a probability amplitude, but the ma-
jor semiclassical developments of recent years tend to follow Schrodinger's interpretation
(Schrodinger, 192fl,1927). The wavefunction is interpreted literally as the matter field, in
which el'IjJ12 constitutes the actual charge distribution, and it is deemed sensible to focus
on a single particl«~ and the single event. A thoughtful and detailed discussion of objections
to this interpretation, and their possible resolutions, has been provided by Dorling (1987).
Because the issue actually goes to the heart of the so-called Copenhagen interpretation of
quantum mechanics itself, space limitations prohibit much further discourse here. Never-
theless, the interpretation is central to the approaches discussed below, and we shall have
a few more relevant comments later.

1. Semiclassical Theory I
It is important to emphasize that Eq.(2b) does not exhibit a solution for AI" but rather is a
nonlinear integral equation for the self-field. As is customary in treating coupled equations,
one can combine Eqs.(2) into a single integro·differential equation for 'IjJ:

(3a)

Although AI' no longer appears here explicitly, it is by no means eliminated, and the validity
of this procedure in the nonlinear case generally depends on what is done next with Eq.(3a).
At best the right-hand side is a time-dependent perturbation. Note that one could derive
this expression by including an additional term in the Dirac action:
152 W. T. GRANDY, Jr.

where the second line is written as such to facilitate later comment.

Equation (3a) has been studied to modest extent by a number of authors, including
Finkelstein (1949), Lloyd (1950), Kaempffer (1955), and Finkelstein, et al (1956). To
examine it in somewhat more detail we first rewrite it in a noncovariant Schriidinger form:

i1ifJ(!f;(x) = (Ho + Hself)1j!(X) , ( 4a)

with

Ho == co . 7r + eA oxt + --/mc 2 , (4b)

Hself == 4u 2 JD(x - y)"ijj(yhl'1j!(y) d4 y ·l,l'· (4c)

Because Hself is a time-dependent addition to the Dirac Hamiltonian, and never vanishes,
it appears that there are no precisely-stationary solutions, except for the ground state.
The nonlinearity of the equations of motion (4) prohibits superposition of solutions,
but 1j!( x) can still be expanded in terms of a complete set. Surely the Hilbert space remains
realized by L2 and is spanned by the set {'Pe} satisfying HO'Pf(a:) = fe'Pe(a:). It will be
convenient to define 'frequencies' We == felh, Wij == Wi - Wj, in terms of the stationary-state
energies fC. Hence,

1j!(x) == '[ae(t)e-iWd'Pe(a:) , (5)

c
and the continuous portion of the spectrum is included implicitly. Substitution into Eqs.(4)
yields the equations of motion for the coefficients:

(6a)

where

Hjc(t) == 41l'e 2 c L Joo

mIn -co
Kj£,mn(t - t'la;"(t')a,,(t')eiw,"n t' dt' . (6b)

The entire problem has thus been reduced to an integro-differential equation in a single
variable. Although the kernel KjC,mn is completely known, its precise form will depend
sensitively on the boundary conditions chosen for the Green function D(x - y).
Equations (6) had been studied years ago by Lanyi (1973), who derived them from
considerations of energy-momentum conservation with kernel defined by the radiation Green
function Dr (half the difference between retarded and advanced solutions). Through a
series of expansions and approximations, including the dipole approximation, he obtained
a reduced set of nonlinear equations for a two-level atom that are roughly equivalent to those
of the neoclassical theory to be discussed presently. Owing to his use of the homogeneous
Green function, his results were restricted to the problem of pure spontaneous emission.
Equations (6), however, are exact within the standard paradigm of first-quantized quantum
mechanics, for there are as yet no approximations, nor has there been a choice made for D.
THE EXPLICIT NONLINEARITY OF QUANTUM ELECTRODYNAMICS 153

But boundary conditions must be specified eventually, so let us first consider the
retarded Green function DR = Dp + !Dn in terms of principal-value and radiation Green
functions:

(7)

( )=_{+1,
where
Zo >0 (8)
E Zo -1, Zo <0 .
If Feynman boundary conditions are chosen instead we obtain the causal Green function
DF = Dp + !D 1 , where Dl is obtained from Dr by the replacement sin -+ cos. The
utility of writing the Green functions in this way will become apparent presently, but here
we simply note that the splitting has the same form as the classical decomposition into
velocity and acceleration fields.
Let us now return to Eqs.(6) and introduce the convenient notation

(9)

so that

ihbj(t) = €jbj(t) +4u2 c L be(t) L [JO J(je,mn(t - t')b;"(t')bn(t') dt' .

l mln- oo
(10)

When the self-field term is neglected one regains the expected stationary-state solution,

(11)

The next approximation is obtained by substitution of this solution into the integral in
Eq.(10), presumably corresponding to the one-photon approximation of QFT. In making
such an approximation, of course, we have linearized the equations, and it is not clear at
this point what violence has therefore been done to the physical description. The O( a)
form of Eq.(10) is now

ihbj(t) ~ Ejbj(t) + 41l'e 2 c L be(t)b;"(O)bn{O) JOO J(je,mn(t - t')eiwmnt' dt' .

.e,m,n -00

(12)

Define 'transition currents'

JI'(k)ji == J eik ,x <pj(x)jl'<.pe(x)d3 x

= J eik .x JI'(x)je d3x , (13)

154 w. T. GRANDY, Jr.

and recall the Green functions discussed in connection with Eq.(7). A short calculation
yields for the integral in Eq.(12)

where

The =f signs correspond to retarded and Feynman boundary conditions, respectively; indeed,
these are the only differences arising from the choice of Green function. Moreover, both
the b-functions and the sign differences come entirely from the homogeneous solutions to
Maxwell's equations, and the principal value is related to Dp only.
One now retains only the term £ = j in Eq.(12), in a sort of Wigner-Weisskopf approx-
imation, and extracts the j-terms from the remaining sums by discarding 'counter-rotating'
terms. The resulting approximate equation of motion is readily integrated to obtain

(15)

With E j == Ej + 6.Ej , we have

(16)

where the prime on the sums indicates that n of j, and the coefficients be(O) are to be
determined from initial conditions. Thus, except for very short times, the energy shift is
time dependent, or chirped. The first two (real) terms on the right-hand side of Eq.(16) have
the interpretation of vacuum-polarization and self-energy contributions, respectively, and
thus might be thought to constitute the Lamb shift. The imaginary part of E j corresponds
to spontaneous emission and, unfortunately, spontaneous absorption, which is not observed.
(These identifications are actually not that simple to make, and we shall discuss them
further below.) Owing to the primes on the sums in Eq.(16), the initial state cannot be
THE EXPLICIT NONLINEARITY OF QUANTUM ELECTRODYNAMICS 155

exactly an eigenstate of Ho, in agreement with the earlier observation that there are no
truly stationary states.
Equations (6) contain as a very special case the neoclassical theory developed by Jaynes
and his students (Crisp and Jaynes, 1969; Stroud and Jaynes, 1970), though the approxi-
mations leading to Eqs.(15) and (16) are not part of that theory. Rather, Jaynes restricted
his studies to a two-level atom and dipole interaction, and for this model was able to main-
tain the basic nonlinearity in the subsequent equations of motion. For pure spontaneous
emission the radiated energy is proportional to tanh ,6(t-to), where ,6 is twice the Einstein
A-coefficient, and the line shape has a sech 2 -structure. This, of course, is a signature of
soliton behavior and is the kind of pulse one might hope for in an unquantized, non-photon
theory, and for long times one regains the familiar exponential decay of QED. The latter
was only a presumption of the original theory of Weisskopf and Wigner (1930) at any rate,
and has remained such. One would naturally expect a more complicated structure in a
more detailed theory.
As mentioned earlier, similar results were obtained by Lanyi (1973), although that
work was restricted to pure spontaneous emission owing to the absence of the Dp-portion
of the Green function. No such restriction applies to the neoclassical theory, and a 'dy-
namical' Lamb shift containing the chirp emerges. Unfortunately, there is good evidence
that the chirp does not exist (Citron, et ai, 1977), thereby casting a serious cloud over the
neoclassical theory and the result of Eq.(16). On the one hand it might be expected that
the full nonlinear theory applied beyond two-level atoms might possess solutions in closer
correspondence to the observed physical behavior. Support for this expectation can be
found in the semiclassical calculations of Mahanty (1974), who was able to reproduce most
of Power's results (Power, 1966), including the Bethe logarithm. On the other hand, it is
not difficult to prove rigorously from the full nonlinear equations (6) that the first term on
the right-hand side of Eq.(16) is the exact static contribution to the energy shift-simply
presume that E j in Eq.(15) is independent of the time and substitute into Eqs.(6). The
outstanding question is whether or not the time dependence predicted in this approach is
experimentally determinable.

2. Semiclassical Theory II
Earlier we noted that Eq.(3a) can be derived from an action principle, and this has been
the point of departure for Barut and collaborators in developing an alternate semiclassical
approach (e.g., Barut and Salamin, 1988; see, also, Barut's article in these Proceedings).
With h = c = I, the total action follows from Eqs.(3):

(17)

Rather than carry out variations on W, Barut focuses directly on radiative corrections
to bound-state energies. Although the self-field remains implicit, the time evolution of
processes of interest is eliminated completely by working entirely within the action. Thus,
156 W. T. GRANDY, Jr.

one is already committed to small corrections to stationary-state quantities and any possible
time dependence of the energy shifts is suppressed.
The next and most far-reaching presumption in the development is that the state vector
iii can be expanded in a temporal Fourier series:

iIi(x) = t1/Jn(x)e-iEnXo, (IS)

where the coefficients 1/Jn remain to be determined but are taken to form a complete set,
and the notation now includes explicitly all portions of the Dirac spectrum. It is difficult
to understand this ansatz under any interpretation of quantum theory, for clearly iii is not
normalizable. Barut asserts that iii is simply an unnormalizable solution to the equations
of motion, and that 1/Jn represents an actual quantum-mechanical wavefunction. This seems
rather disingenuous, however, for if iii has no physical meaning neither does W. In light of
this view, a secondary feature of Eq.(IS) which is puzzling is the subsequent interpretation
of En as a system energy level, for which no arguments are presented.
It is useful to suspend these objections momentarily and investigate further the con-
sequences of the ansatz (1S). As a next step substitute the expansion (1S) into the action
and carry out the ko, Yo, and Xo integrations (in that order). The Feynman propagator
D F is chosen on grounds that this is a necessary choice to treat correctly both particle and
antiparticles in the theory. Why this criterion is applied to the 'photon' propagator in the
context of an unquantized electromagnetic field is not clear-below it will be seen that the
choice is effectively irrelevant. Substitution from Eq.(1S) results in four generalized sums
over states in WI, and the time integrations reduce these to two by introducing a factor
orEn - Em + Ee - E s ), a a-function. Subsequently the quantities {En} are found to be
complex (see below), so that at this point the meaning of the a-function is a bit obscure.
In addition, this is actually treated as a Kronecker-a in collapsing the sums, and arguments
are made that the only choices implied are n = m, r = s, and n = s, r = m. It is not
clear, for example, why En = i(Em + E s), Er = i(Em + Es) are not equally valid choices.
Bialynicki-Birula (19S6) has also objected to this kind of 'voodoo' mathematics.
The final developmental step is to establish an iterative scheme, in which the first
approximation consists of replacements 1/Jn --> 1/J~O), En --> E~O) + !:lEn, in terms of the
unperturbed solutions valid when self-fields are omitted. These are Coulomb wavefunctions
and energies in the present case of interest. After some algebra one finds the O( a) radiative
energy shift to be
(19)
where the first two terms on the right-hand side are real and comprise the vacuum polar-
ization,
 THE EXPLICIT NONLINEARITY OF QUANTUM ELECfRODYNAMICS 157

/
---- "

(b)
Fig.!. Feynman diagrams contributing in leading order to the Lamb shift: (a) vacuum polariza-
tion, and (b) electron self-energy.

contributions to the Lamb shift, the third (imaginary) term describes spontaneous emission,

x {i1l"[.5(Es - En + k) ± o(Es - En - k)]}, (22)

and JI"'(:Z:)ns is defined in Eq.(13). The ± sign in this last result exhibits the only difference
in choosing Feynman or retarded boundary conditions, respectively. As found earlier, the
real energy shift is independent of this choice. Note carefully that the sum-integral in each
expression is to range over all discrete and continuum states for both positive and negative
energies, although in the present model the negative-energy states are just the continuum
states of the positron.
One now attempts to interpret these expressions as done in the preceding section, which
is facilitated by reference to the perturbative results of conventional QED. The leading-order
contributions to the Lamb shift correspond to the Feynman diagrams of Figure 1, referring
to vacuum polarization and electron self-energy. In our notation the original evaluation by
French and Weisskopf (1949) yields ~En = ~E~a) + ~E~b), where

(23a)

(23b)

and Jlj,(k )ns is defined in Eq.(13). The quantilty Os = ±1, depending on whether s denotes
a positive- or negative-energy state. Although these hear a superficial resemblance to
Eqs.(20) and (21), their content and origins are rather different.
The first difference is that Eq.(23a) specifically excludes positive-energy states. In
QFT it arises conceptually as a modification of the photon propagator, rather than directly
as part of the energy shift, but perhaps neither view should be viewed as written in stone.
Secondly, the principal values in Eq.(21) are both summed over both positive- and negative-
energy states, and do not correspond to the two values taken by Os in Eq.(23b). These
remarks imply extra terms in Eqs.(20) and (21) over and above those of Eqs.(23), and the
158 W. T. GRANDY, Jr.

latter are known to yield close agreement with experiment. It would be remarkable if these
extra terms made a negligible contribution to Re(LlEn).
In the nonrelativistic limit Eq.(21) has been shown by Barut and Van Huele (1985) to
reduce to the low-energy result of Bethe (1947):

with
(24b)

In this limit there are no differences, because Eqs.(21) and (23b) reduce to the same thing.
But eventually this result must be matched to the corresponding high-energy expression
at the cutoff implied in Eq.(24a), and we have seen that this will not result in continued
agreement. Indeed, it is rather surprising that one obtains Eq.(24a) correctly.
Equation (22) has been studied at length by Barut and Salamin (1988), in which
they note that the 6-functions are mutually exclusive-the first implies En > E., and the
second vice-versa. The first has the obvious interpretation of spontaneous emission from
an excited state 'l/Jn·to a set {Es} of lower levels, and clearly explains why the ground state
is stable. But the second o-function predicts spontaneous absorption, which apparently
is not observed. This extra 'absorption' term seems to be a difficulty with semiclassical
theories in general (e.g., Davies and Burkitt, 1980), one not shared by QFT. Barut and
Salamin make the dubious claim that one can simply select the 'correct' term, and so
retain only the first. In that case the ± sign is irrelevant, and the spontaneous decay is also
independent of boundary conditions on the Green function D. Barut argues elsewhere in
these Proceedings that omission of the absorption term follows from energy conservation.
The term is there in the mathematics, however, and energy conservation has little to do
with its presence or absence. By way of analogy, there is nothing in energy conservation
forbidding an automobile from spontaneously cooling itself and jumping to the top of the
nearest building-it is rendered highly improbable by an additional selection rule, called
the second law of thermodynamics. Nor is the difficulty avoided by moving the basic level
of discussion from the action to the equations of motion themselves, for both o-fundions
still appear (e.g., Barut, 1988).
With the notation exp( -iEnt) = exp( iE~O)t) exp( -!r nt), the decay rate becomes

rn = -2Im(En ) = -e 2 tf f
s<n
d3 x d3 yJI'(re)ns JI'(Y)sn

(25)

The requisite integrals have been carried out and for various scenarios agree well with
some other theoretical calculations and some experimental results-the latter, apparently,
are sparse. If the dipole approximation is made and one identifies an electron velocity as
TIlE EXPLICIT NONLINEARITY OF QUANTUM ELECTRODYNAMICS 159

v = ca, then introduction of the Heisenberg (~quations of motion and a sum over 'photon'
polarization vectors yields a more familiar approximate form:

rn~ ~atw~sITnsI2, (26)

s<n

where W ns == En - Es. These matrix elements still contain relativistic wavefunctions, so

that the summation over the electron spin states will yield a factor of 2, thereby providing
the correct factor of 4/3 and Einstein's A-coefficient.
An additionally puzzling feature of these calculations is that the negative-continuum
states appear to be ignored in Eq.(25), though they satisfy the indicated inequality. They
are included in the conventional expressions of QFT, and the latter yield reasonably accu-
rate results (e.g., Goldman and Drake, 1981) . It is therefore a bit surprising that Eq.(25)
might produce some numerical agreement with other work.
Finally, the expression (20) has been evaluated to leading order in (Za) for s-states by
Barut and Unal (1990), in which they 'almost' reproduce the known results of QED (see,
e.g., Grandy, 1991). It should be noted also that this formalism has been applied to other
problems, including the Casimir-Polder effects (Barut and Dowling, 1987), and 'derivation'
of the electron anomalous magnetic moment (Barut and Dowling, 1989). We shall examine
this latter calculation in some detail in the Appendix.

3. Summary and Conclusions

Unquestionably, semiclassical calculations avoiding field quantization are capable of indi-
cating the presence of all radiative effects of interest. Classical-mechanical versions of the
Lamb shift have been illustrated more than once by Jaynes (1978, 1990), and we have seen
above the appearance of spontaneous emission in the neoclassical theory. What we now
call vacuum polarization was first derived semi classically by Uehling (1935), Schwinger, et
ai, (1978) derived the Casimir-Polder effects without field quantization, and an anomalous
magnetic moment for the electron of a/27r can be found by appropriate choice of relativistic
cutoff parameter either classically (Grandy and Aghazadeh, 1982) or semiclassically (Barut
and Dowling, 1988). But, as Jaynes (1978) has observed, "The mere fact of getting the
right numerical magnitude of ~WLamb cannot be claimed as a valid 'derivation' of the Lamb
shift if we do not get also the correct qualitative behavior." While applauding this standard,
in extending it to others one might wish also to broaden it by including an insistence on
mathematically unimpeachable procedures.
The neoclassical theory has not been applied extensively, and has focused primarily
on two-level atoms (though see Mahanty, 1974). It does, however, possess the considerable
merit of retaining the essential nonlinearity of the problem, and one consequence of this is a
line shape which is possibly a more realistic picture of the emitted radiation. Schrodinger's
interpretation of the wavefunction mayor may not prove central to semiclassical theories,
but there are more glaring obstacles to further progress that seem intrinsic to both the
neoclassical theory and its parent model of Section A. A first such obstacle is the appearance
of 'absorption' terms in both real and imaginary parts of the energy shift-their emergence
in Barut's approach as well suggests they are intrinsic. A second is the chirp in the Lamb
shift, for whose absence in the data there is good evidence. No ready resolution of these
questions appears to be in sight.
160 W. T. GRANDY, Jr.

Aside from the mathematical objections raised above, Barut's approach linearizes the
theory as a first approximation, thereby losing the essential feature which may be thought
to enable an avoidance of second quantization. Of course, this is also a shortcoming of
the semiclassical calculation in Section A (but not of the two-level neoclassical theory). In
addition, there is still an infinite renormalization term which one would have thought might
at least be finite when Coulomb wavefunctions are used for the unperturbed quantities.
There does remain the fact that both of these major semiclassical developments make
some contact with other theoretical calculations, as well as with some of the data. As
Khalatnikov (1989) reminds us, though, " ... no coincidence of theory with experiment
can justify logical gaps in the theorist's work." It is the supposed logical difficulties with
QFT, after all, that have motivated the work under consideration here, and there is no
point to falling into the same trap. There is qualitative encouragement, nevertheless, that
the basic semiclassical approach to the radiation field is sound. Indeed, devotees of QFT
themselves have shown a recent inclination to shift in subtle ways the specifics of the issue.
For example, Mandel (1986), and others (Kimble and Mandel, 1975; W6dkiewicz, 1980)
have argued that the radiative effects do not provide good tests of QFT as opposed to
semiclassical theories. Rather, it is argued that the quantum nature of the electromagnetic
field is really manifest only in photon correlation experiments. Although a number of
these appear capable of interpretation in terms of the classical field and its first-quantized
sources, no serious calculations along these lines are known to the author. Moreover, in
such experiments the issues of quantized fields and the interpretation of quantum mechanics
itself, in connection with Bell's theorem, are deeply intertwined. No doubt the advocates
of a semiclassical theory of radiation would settle for merely the radiative effects at this
time!
Jaynes has repeatedly emphasized that his motivation in constructing a neoclassical
theory was not so much the development of a specific replacement for QFT as it was to
uncover experiments which might suggest directions in which that development might pro-
ceed (Jaynes, 1973, 1978). Investigation of the Maxwell-Dirac theory by Barut and others,
however, has been more ambitious and has been aimed at providing just that direction. Al-
though both approaches seem to have run into serious difficulty, it can be argued that the
development simply has not proceeded far enough-principally because of mathematical
intractability. The truly essential feature of the Maxwell-Dirac equations is their non-
linearity, which has yet to be fully exploited. Owing to the complexity of the equations
in 3+1 dimensions, soliton-like solutions have not been found, but the possibility surely
exists. Such solutions would have enormous bearing on an understanding of the general
elementary-particle problem itself.
Unmentioned-at least explicitly-in semiclassical treatments to date is the desider-
atum of finding a physical mechanism for pair creation and annihilation. Quantum field
theory has eliminated the problem by fiat, but this procedure at least has the merit of
accommodating the phenomena. At the heart of any truly viable alternative to QFT must
be a natural description of the entire e+ e- spectrum, and in turn this suggests a need
to study the entire electromagnetic system as a single entity. The evidence for myopia in
looking at only individual pieces of the total e+ e-,-system has been before us for a long
time, but no one has yet seen how to put it together. Quite possibly it will be necessary
to include a vv-component as well. Earlier neutrino theories of light met insurmountable
obstacles, but perhaps only because the complete system was not being considered.
 THE EXPLICIT NONLINEARITY OF QUANTUM ELECTRODYNAMICS 161

In the author's view some variant of these generalizations is absolutely required if the
quest for a non-second-quantized theory of matter and radiation is to be sustained. At
a minimum, a first step might be to obtain solutions to the Maxwell-Dirac equations in
the absence of external fields and employ these as the zero-order input for investigation
of radiative corrections, rather than the intrinsically linear Coulomb solutions. Although
such solutions exist (Flato, et ai, 1987), finding them explicitly has not proved an easy
task. Nevertheless, we have noted earlier that, though there are no plane-wave solutions,
unique global solutions to the initial-value problem do exist. From the work of Mathieu
and Morris (1984) one can infer that there are no localized stationary solutions, even for
the Coulomb problem. In agreement with earlier discussion of the energy shifts, then, we
conjecture that there do exist localized, non stationary solutions. Once again, the major
issue will no doubt be the precise nature of the time dependence.

Appendix.
In a recent article Barut and Dowling (1989} claim to provide a derivation of the electron
anomalous magnetic moment. They consider the interaction term in the action, differing
by a factor -471" from our Eq.(3b), along with what appears to be a Fourier representation

(A-I)

This, however, is taken to be understood in the sense of Eq.(18) above, so that the Po-
integral may contain a sum over the discrete part of the spectrum, including spin. Sub-
stitution of Eq.(A-l) for each of the four functions occurring in the action, followed by
evaluation of three of the integrals, yields

The next step defies understanding, for the o-function is employed to evaluate two of
the integrals in Eq.(A-2). That is, the values s = p, r = q are selected as satisfying the
stated condition--which they do-and the continuum of other choices is ignored. Though
the procedure is clearly unacceptable mathematically, the claim is now made that the
physically interesting piece of the action with respect to 9 - 2 f. 0 is that corresponding to
these special values:

W(nt = ~2 j crp j d4x j d4yD(x-y)W(p)eip.x

x '"Y1'{j d4qlJl(q)W(q)e- i(x-y).q }'"YI'IJI(p)e- iPoy . (A-3)

Although we must consider a factor of 00 to have been omitted on the right-hand side, let
us follow the further development by Barut and Dowling anyway.
Denote the function within the braces in Eq.(A-3) as Sex - y), whose Fourier trans-
form is IJI (p) W(p), where IJI( x) is to satisfy the complete equations of motion obtained from
162 W. T. GRANDY. Jr.

Eqs.(17) above through variation with respect to W(x). The authors now claim that this
function S is the propagator, or Green function solution for the complete nonlinear equa-
tions, subject to the following boundary condition:

S(x-y) = O(YO-x o )l lJi(q)W(q)e-i(x-y).q d4 q

)l
(+)

-O(x O- yo lJi(q)W(q)e- i (x- y)·qd4 q. (A-4)

(-)
The notation (±) on the integrals indicates integration over positive- and negative-energy
states, so that the right-hand side is now independent of energies. In first approximation IJi
is replaced by the solution to the equations of motion for zero self-field, in which case S(x-y)
is asserted to become the familiar function satisfying (-y"7r" - m)S(x-y) = i8(x-y). One's
skepticism about the meaning attributed to S begins with the implication from its definition
that the general propagator is always a function of (x-y) alone. This is not generally true.
Moreover, the expression (A-4) is not at all equivalent to similar decompositions found in,
say, Bjorken and Drell (1964). Nevertheless, we follow the authors along this path and adopt
S(p) = (,"7r" -m )-1 as the leading-order approximation to the Fourier transform of S(x-y).
[The reader may find it challenging to try to demonstrate that lJi(p)W(p) = (J"p" - 1 mt
for free particles, let alone anything else.) They then rewrite Eq.(A-3) as

W'In!
=_ie 2
2
jd jd PW()
4
P
4 ,"S(p),,, 1Ji()
P (p - P)2 + if p

= j d pW(p) 8M(p) lJi(p) ,

4 (A-5)

which, after a change of integration variables, defines a mass shift

8M(p) == e2 _i_
2 (27r)4
j ,!tS(p+- .s)/" d s.
S2 tf
4
(A-6)

One notes that the interaction part of the action is proportional to the energy shift: Wint =
(27r )48E.
If the above procedure were correct this would be a remarkable result, for Eq.(A-6)
exhibits precisely the O(a) approximation to Schwinger's mass operator (e.g., Berestetskii,
et ai, 1982; p.477). But if one reviews the 'derivation' beginning with Eq.(3b) above, then
the implication is that the mass operator M (x, y) can be identified as the term in brackets
in the second line of Eq.(3b). This cannot be true even though Eq.(A-6) provides a first
approximation in momentum space. For one thing, the 'identification' in Eq.(3b) contains
no classical version of the vertex operator, which is absolutely essential in higher orders.
For another, when one actually works through the 'derivation' of Eq.(A-5) it is found that
a factor 8(0) has been omitted on the right-hand side. Thus, one can present anything one
pleases if it is multiplied by infinity! The observation by the authors that their expression
is equivalent to that found similarly by Babiker (1975) is simply irrelevant, for the latter
was derived in a mathematically consistent way employing field operators for the electron.
Appealing to Babiker's result is akin to merely cancelling the sixes in the fraction 64/16
and claiming the end to justify the means!
It would be very pleasing to see an unimpeachable derivation of the electron anomaly
that is independent of notions regarding the vacuum and field quantization. The work
under discussion, however, does not accomplish this.
THE EXPLICIT NONLINEARITY OF QUANTUM ELECTRODYNAMICS 163

REFERENCES

Babiker, M.: 1975, 'Source-Field Approach to Radiative Corrections and Semiclassical Radiation
Theory', Phys. Rev. A 12, 1911.
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Barut, A.O., and J.P. Dowling: 1987, 'Quantum Electrodynamics Based on Self-Energy, without
Second Quantization: The Lamb Shift and Long-Range Casimir-Polder van der Waals Forces
Near Boundaries', Phys. Rev. A 36, 2550.
Barut, A.O., and J.P. Dowling: 1989, 'QED Based on Self-Fields: A Relativistic Calculation of
9 - 2', Z. f. Naturf. 44a, 105.
Barut, A.O., and Y.I. Salamin: 1988, 'Relativistic Theory of Spontaneous Emission', Phys. Rev.
A 37,2284.
Barut, A.O., and N. Unal: 1990, 'Regularized Analytic Evaluation of Vacuum Polarization in a
Coulomb Field', Phys. Rev. D 41,3822.
Barut, A.O., and J.F. Van Huele: 1985, 'Quantum Electrodynamics Based on Self-Energy: Lamb
Shift and Spontaneous Emission without Field Quantization', Phys. Rev. A 32, 3187.
Berestetskii, V.B., E.M. Lifshitz, and L.P. Pitaevskii: 1982, Quantum Electrodynamics, Pergamon
Press, Oxford.
Bethe, H.A.: 1947, 'The Electromagnetic Shift of Energy Levels', Phys. Rev. 72,339.
Bialynicki-Birula,1.: 1986, 'Comment on "Quantum Electrodynamics Based on Self-Energy: Lamb
Shift and Spontaneous Emission without Field Quantization', Phys. Rev. A 34, 3500.
Bjorken, J.D., and S.D. Drell: 1964, Relativistic Quantum Mechanics, McGraw-Hill, New York.
Citron, M.L., H.R. Gray, C.W. Gabel, and C.R. Stroud: 1977, 'Experimental Study of Power
Broadening in a Two-Level Atom', Phys. Rev. A 16, 1507.
Crisp, M.D., and E.T. Jaynes: 1969, 'Radiative Effects in Semiclassical Theory', Phys. Rev. 179,
1253.
Das, A., and D. Kay: 1989, 'A Class of Exact Plane Wave Solutions of the Maxwell-Dirac Equa-
tions', J. Math. Phys. 30, 2280.Davies, B., and A.N. Burkitt: 1980, 'On the Relationship
between Quantum Random and Semiclassical Electrodynamics', Aust. J. Phys. 33, 671.
Dorling, J.: 1987, 'Schrodinger's Original Interpretation of the Schrodinger Equation: A Rescue
Attempt', in C.W. Kilmister (ed.), Schrodinger, Cambridge Univ. Press, Cambridge, p.16.
Finkelstein, R.J.: 1949, 'On the Quantization of a Unified Field Theory', Phys. Rev. 75, 1079.
Finkelstein, R., C. Fronsdal, and P. Kaus: 1956, 'Nonlinear Spinor Field', Phys. Rev. 103, 1571.
Flato, M., J. Simon, and E. Tafiin: 1987, 'On Global Solutions of the Maxwell-Dirac Equations',
Commun. Math. Phys. 112, 21.
French, J.B., and V.F. Weisskopf: 1949, 'The Electromagnetic Shift of Energy Levels', Phys. Rev.
75, 1240.
Goldman, S.P., and G.W.F. Drake: 1981, 'Relativistic Two-Photon Decay Rates of 2S1 / 2 Hydro-
genic Ions', Phys. Rev. A 24, 183.
Grandy, W.T., Jr.: 1991, Relativistic Quantum Mechanics of Leptons and Fields, Kluwer, Dor-
drecht.
Grandy, W.T., Jr., and A. Aghazadeh: 1982, 'Radiative Corrections for Extended Charged Particles
in Classical Electrodynamics', Ann. Phys. 142, 284.
Itzykson, C., and J .-B. Zuber: 1980, Quantum Field Theory, McGraw-Hill, New York.
Jaynes, E.T.: 1973, 'Survey of the Present Status of Neoclassical Radiation Theory', in L. Mandel
and E. Wolf (eds.), Coherence in Quantum Optics, Plenum, New York.
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Jaynes, E.T.: 1978, 'Electrodynamics Today', in L. Mandel and E. Wolf (eds.), Coberence in
Quantum Optics IV, Plenum, New York.
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and tbe Pbysics of Information, Addison-Wesley, Reading, MA.
Kaempffer, F.A.: 1955, 'Elementary Particles as Self-Maintained Excitations', Pbys. Rev. 99,1614.
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V, Teil 1, Springer-Verlag, Berlin.
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Spontaneous Emission as a Test of Quantum Electrodynamics', Pbys. Rev. Lett. 34, 1485.
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1835.
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Schwinger, J., L.L. DeRaad, Jr., and K.A. Milton: 1978, 'Casimir Effect in Dielectrics', Ann. Pbys.
115, 1.
Schriidinger, E.: 1926, 'Quantisierung als Eigenwertproblem', Ann. d. Pbys. 79, 361.
Schriidinger, E.: 1927, 'Energieaustauch nach der Wellenmechanik', Ann. d. Pbys. 83, 956.
Stroud, C.R., Jr., and E. T. Jaynes: 1970, 'Long-Term Solutions in Semiclassical Radiation Theory',
Pbys. Rev. A 1, 106.
Uehling, E.A.: 1935, 'Polarization Effects in the Positron Theory', Pbys. Rev. 48, 55.
Weisskopf, V., and E.P. Wigner: 1930, 'Linienbreite auf Grund der Diracschen Lichttheorie', Z.
Pbys. 63, 54.
W6dkiewicz, K.: 1980, 'Resonance Fluorescence and Spontaneous Emission as Tests of QED', in
A.D. Barut (ed.), Foundations of Radiation Tbeory and Quantum Electrodynamics, Plenum,
New York.
ON THE MATHEMATICAL PROCEDURES OF SELFFIELD
QUANTUMELECTRODYNAMICS

A. O. Barut
Department of Pbysics
University of Colorado
Boulder, CO 80309, USA

In this Note I clarify some of the mathematical developments and procedures of Self-
field QED which are different from those in the usual quantum field theory and has
caused some misunderstandings to those deeply attached or used to the techniques of
the latter. I think the critical remarks l are more on the way and the style we extract
observable quantities directly from the action rather than on the substance or the
philosophy of our approach which has its own different but complete interpretation.
By eliminating the vector potential AI' from the coupled Maxwell-Dirac equations
we found that the ,¢-field satisfies a nonlinear integro-differential equation 2 . Here
'¢ is a complex scalar field, not a probability amplitude. Then we look for possible
frequencies of such a field by expanding it into a Fourier series

~J(X) = ?!;1/;n(z)e- iEnt (1)

n

For some reasons there has been objections l to such an expansion, although this
is the proper way of treating any wave field, linear or nonlinear, which we do all
the time for time-dependent Schrodinger equation, for Navier-Stokes equation and
so on. Apparently they think of another expansion in perturbation theory where a
probabilistic, normalized wave function is expanded as

(2)

in which both1/; and 1/;n are normalized according to quantum mechanics, {1/;n} being
further an orthonormal set. We have no condition that our field (unfortunately labelled
by the same symbol1/;) in (1) should be normalized. The 1/;n and En in (1) have not
the same meaning as those in (2), again by custom the same symbols have been used,
regretable, as we realize. It is not clear at all that the series (1) diverges, as claimed,
because the unknown coeffiecients 1/;n satisfy coupled nonlinear equations, they are not
orthonormal, and these coupled equations provide us with the physical interpretation
165
D. Hestenes and A. Weingartshofer (eds.). The Electron, 165-169.
© 1991 Kluwer Academic Publishers.
166 A.O.BARUT

of'ljJn and En, the only quantities we need. At any rate as Heaviside put it :"This
series is divergent, therefore we shall be able to do something with it.". In contrast
the 'ljJn and En in (2) are known for a given problem. This is the big difference. Every
approach has its own interpretation and one should not confuse a language used in
one approach with the other.
Self-field QED has the logical structure of classical electrodynamics and the current
j It is the most important quantity. In both cases what we actually use, as developed
in an early version of the theory 3, is an expansion of the current j into its possible
frequencies. \Ve shall see that all observables are expressed in terms of the Fourier
coefficients of the current

(3)

Here Wnm are the observed frequencies of the atomic system. Schrodinger already
observed at the very beginning that one should be able to formulate atomic processes in
terms of the Fourier spectrum of matter and of radiation in space and time 4 . Inserting
(3) and the Fourier expansion of the Green's function D into the interaction action

and performing dxo, dyo, dko-integrations we obtain

TVint ""' J
dk V8(wnm
n1:rs
+wrs)j;m(k)DItV(wnm,k)j~S(-k) (4)

At this point we can also answer the critical queries about the 8-function in (4) which
in later work appeared as 8(En - Em + Er - Es). Now we could have developed the
whole theory in terms of the equations of motion without such a 8-function. But we
found that it is more direct and more interesting to calculate all the observables in a
unified form from the interaction action itself. One obtains so immediately the decay
rates, for example, instead of first finding the amplitudes and then squaring it to get
the rates. But the action being an integral over all space and time is infinite. However
this is a well known situation in S-matrix theory: one factorizes a factor 8(0) from
vVint to arrive at observables.
The next query concerns the ways the 8-function in (4) can be satisfied. In the set
of all frequencies {w nm } there are only two ways we can satisfy the 8-function: (a)
Wnm = a and w rs = 0, (b) Wnm = -w rs . We write then these two types of terms
separately and identify them with the contributions of vacuum polarization and Lamb
shift plus spontaneous emission, respectively. There are no other terms. In the form
8(En - Em + Er - Es) there seems to be at first more ways of satisfying the 8-function,
but these take one outside the set of frequencies{w nm }. One further query was what
happens to the argument of the 8-function when due to spontaneous emission the
energies get a complex shift (to which we shall come back immediately). The answer
is that these imaginary parts always cancel. At any rate the argument of 8-function
ON THE MATHEMATICAL PROCEDURES OF SELFFIELD QED 167

is always zero for observables as stated above. The summations go over all indices,
discrete or continuous, with their degeneracies, that satisfy the 8-function constraint.
We shall now see how this is done in the iterative solutions of the nonlinear equations.
Because the total action W vanishes when the equations of motion are inserted, we
obtain a condition to evaluate the energy shifts due to selffields iteratively. They are
given by

We may use the Fourier components of the Green's function in Coulomb gauge with
nonvanishing components
1
Doo = k2 ' DOk = DkO = 0
(6)

to obtain

Eint '" J dk [P'~tn(k)j8'( -k) + ~ jom(k)j;{'n( -k)

n,.m
(7)
+ pnm(k)
J
.mn(_k)
Jt
m
2
Wnm
_
k 2
k2
(8. _ k 2
Ji
kik t ) ]

In the last term we use the identity

w2
k2
- k2 =
1
2'
(w w )
~, - k + w + k - 2 (8)

and, because we have a sum over all nand m and the integrand is invariant under
the exchange n t-+ m, the two terms in (8) lead to the same contribution. Hence (7)
becomes

E int "" J [t/

dk o n(k)j8 s ( -k) + t;om(k)j[;'n( -k)
n,.m
(9)

This is the calorimetric interaction energy of all levels. We consider a fixed level nand
evaluate the energy shift of this level due to all others.In the last term of (9) because
of the pole of the integrand we use the standard formula

_1_ = p~_L _ i1C8(w _ k) (10)

w-k w·-k
168 A. o.BARUT

There is only one imaginary part to the energy shift. In some papers we wrote at this
stage two b- functions : b( w - k) + b(w + k), but we actually worked with one of them.
The first one gives the spontaneous decay from n to m, if m is a lower state, the second
from m to n, if n is lower. We combine these terms as explained above before we single
out a particular state n. This answers the remarks about "spontaneous absorption"l
interpretation of the second term. At any rate the causal Green's function has only
one imaginary part, and spontaneous absorption cannot occur by energy conservation.
Our formula for spontaneous emission,although it looks quite different and has been
obtained by an entirely different reasoning, has been shown recently to be exactly
equivalent, in the lowest iteration, to the QED formula 5 . Similarly, our formulas
for vacuum polarization and Lamb shift, although they look at first to be different,
are equivalent in lowest order of iteration to the formulas of QED, the first to the
expression used by vVichmann and Kro1l 6 , the second to the more recent and more
complete work of Mohr 7. Thus apparent looks should not be attributed to a difI"erence
in substance.
Finally two other remarks on the nature of Selffield-QED may help to clarify the
approach. The first is that it is not a "semiclassical" theory. "Semiclassical" usually
means that we keep both the matter and electromagnetic fields side by side, but
quantizing only the matter field and not the electromagnetic field, Here we have no
separate degrees of freedom for the electromagnetic field A,,; it has been eliminated.
But we have a nonlinear field theory for the matter field alone. Therefore we do not
think that the selffield-QED is an approximation to QED. We view it as an exact
theory of the Maxwell-Dirac system, and as far as we can see sofar in the lowest order
of iteration, it gives identical results to QED, but with a possibility that it may give
nonperturbative results and can be extrapolated to short distances. The question of
quantization of AI' does not arise. As we have often stressed, the quantized properties
of the field reflect the quantized properties of the source and there are no new degrees
of freedom in the field besides those of the source.
The second remark concerns the interpretation of negative energy states. Although,
as mentioned above, the Selffield-QED has the logical structure of classical electro-
dynamics, the source current is now the Dirac current which has a more complicated
frequency spectrum. Only the positive frequency solutions of the Dirac equation refer
to the electron, the negative frequency solutions have to be interpreted consistently,
even in first quantized theory, as the states of the antiparticle, the positron. However,
we often use in the calculations, with great advantage, the completeness relation which
involves both the positive and negative energy states. But then the correct physical
interpretation must be implemented. For example, in the calculation of vacuum polar-
ization contribution of the electron, we extend the contour of integration to negative
energy cut as well in order to use the Green's function of the Dirac equation, but for
the contribution of negative energy states we change e --+ -e, and then take one half
of the results. For the same reason we use the causal Green's function D(x - y) which
picks only one pole for tf > t, another for t' < t, thus indirectly controls the positive
end negative frequency states of the current source.
In conclusion, concepts, ideas and calculational techniques used in quantum theory
and QED should not be translated unchanged into another approach, the Self-field
QED. But with the correct interpretation the latter provides an efficient, unified and,
ON THE MATHEMATICAL PROCEDURES OF SELFFIELD QED 169

we think, a fully consistent framework for the whole field of radiative processes with
new directions of extrapolation.

References

1. T.Grandy, Jr.,These Proceedings. See also the earlier paper by 1. Bialynicky-

Birula, Phys. Rev. A 34, 3500( 1986) and reply to it, A.O.Barut, Phys. Rev.A 34,
3502 (1986)
2. A.O.Barut, These Proceedings, and references therein.
3. A.O.Barut, in Quantum Theory and Structure of Space and Time, Vo1.6 (edited
by L.Castell et al), C.Hanser Verlag, r-.1unchen 1986, p.83
4. E.Schrodinger, Die Naturwissenschaften, 17,326 (1929)
5. A.O.Barut and Y.Salamin, Phys. Rev. A (in press)
6. E.H.Wichmann and N.Kroll, Phys. Rev. 101, 83 (1956)
7. P.Mohr, Ann. of Physics, 88,26 (1974)
8. A.O.Barut and N.Unal, Phys. Rev. D 41, 3822 (1990)
NON-LINEAR GAUGE INVARIANT FIELD THEORIES OF THE ELECTRON AND
OTHER ELEMENTARY PARTICLES

F.I. COOPERSTOCK
Department of Physics and Astronomy
University of Victoria
P.O. Box 3055
Victoria, B.C. Canada V8W 3P6

ABSTRACT. We review the Einstein-Rosen program of building elementary particles in solitonic

structures in singularity-free non-linear gauge invariant field theories. The role of gravity via gen-
eral relativity is discussed. It is found that a zone of negative energy density surrounds the particle
core which is indicated to be much larger than 10-33 cm. A model that encompasses the electron,
muon and tau is found with particle sizes _10- 16 cm, within experimentallirnits. Spin and mag-
netic moment have the potential to be incorporated with fields of axial symmetry. The quarks can
also be modelled, but thus far, two additional coupling constants have been required. The new
approach of modelling the electron as a quantum soliton in Dirac-Maxwell theory is described.
Preliminary results indicate an emerging wave function with characteristic spread of the order
10- 16 cm.

1. Introduction
Field theory has been one of the crowning achievements of modem physics. It has described elec-
tromagnetism in Maxwell's equations, gravitation in Einstein's equations of general relativity and
in more recent times, it has merged with quantum theory. Its compelling elegance and logic have
led many researchers to believe that field theory holds the key to the fundamental description of
physical phenomena.
In the eyes of some of the greatest achievers in the history of physics such as Einstein, there was
a conviction that a proper field theory should be free of singularities. Moreover, the concept of
"particle" should not be separate from the field. From the well-known paper of Einstein and
Rosen [1]:

A complete field theory knows only fields and not the concepts of particle and motion.
For these must not exist independently of the field but are to be treated as part of it. On
the basis of the description of a particle without singularity one has the possibility of a
logically more satisfactory treatment of the combined problem: The problem of the field
and that of motion coincide.
171
D. Hestenes and A. Weingartshofer (eds.), The Electron. 171-181.
© 1991 Kluwer Academic Publishers.
172 F. J. COOPERSTOCK

Singularities signal the breakdown of physics and their avoidance has served as a useful con-
straint in the construction of physical theory. Another useful constraint has been that of gauge in-
variance. Indeed, in recent years, the concept of gauge invariance has come to play an increas-
ingly important role in the theories of fundamental interactions [2].
In terms of structure, nonlinearity has been a vital element in our efforts to model the complexity
of nature with mathematics. With regard to field theory, it is through the medium of nonlinearity
that structures emerge which we identify as fundamental particles such as the electron. The prob-
lem is to identify the ideal field theory (or theories) which describes the physical world. Its suc-
cess would be measured not only by its completeness and accuracy, but also by its capacity to pre-
dict new phenomena which are experimentally verifiable. These are the fundamentals of good
science. While esthetic appeal and elegance might be deemed desirable by most researchers in-
cluding ourselves, it is not our task to prejudge the workings of nature, which might follow a
course that is not in accord with our predilections or expectations. Of even less concern is confor-
mity to existing popular trends: progress in science is best served by an openness to new ideas.
It is my goal in this paper to describe the field theory of elementary particles which Rosen and I
[3] developed, as well as more recent wolk, both completed and in progress, which my collabora-
tors and I have pursued. This includes a more detailed treatment of the role of general relativity
[4], an attempt to encompass the quarks [5], spin and magnetic moment [6], and wolk in progress
to incorporate the Dirac theory in the formation of quantum solitons.
2. Historical Background
In Maxwell's theory of the electromagnetic field, we deal with the Lagrangian

(2.1)

where the Maxwell tensor, F IlV' is related to the four-vector potential All as
F",v = Av,u - A""v
(2.2)
and JIl is the current four-vector. The variational principle, or Lagrange's equations, yields the
source set of Maxwell's equations
Jt,v,v = -~f
(2.3)
In pure electromagnetic theory, elementary charged particles appear as point singularities. As we
discussed earlier, this is unsatisfactory as it signals the breakdown of physical theory.
Poincare had created finite non-singular "elementary" particles, but because of the Coulomb re-
pulsion of interacting elements of the finite structure, stresses had to be adjoined to maintain the
integrity of the particle structure. These "Poincare stresses" inject a phenomenological element to
the theory whereas a fundamental, unified structure is preferable, if attainable. Einstein showed
that with electromagnetic and gravitational fields alone, such a unified theory could not be real-
ized.
Gauge invariance provides the avenue by which new fields are added to build particles with
fields. It is well-known that a gauge transformation,
(2.4)
NON-LINEAR GAUGE INVARIANT FIELD THEORIES 173

where A. is a scalar function, retains the value of Fuv by virtue of the defining equation (2.2).
While Mie [7] developed a non-singular finite elementary particle by modifying Maxwell theory,
he ?id so with A!1 appearing explicitly in the Lagrangian. Thus, Mie's theory was not gauge in-
vanant.
Born and Infeld [8] created a new non-linear electrodynamic theory which, unlike Mie's theory,
had A appear in the Lagrangian only in the fonn of (2.2). Thus, their theory was gauge invariant
and it "id produce finite particle states. However, Rosen [9] showed later that this theory still con-
tained singularities.
Rosen noted that Mie's theory with A explicitly appearing in the Lagrangian did avert the sin-
gularity problem and he succeeded in JLtaining A while maintaining gauge invariance by intro-
ducing a new complex scalar field ",. This was actlteved by having", undergo a phase rotation

(2.5)
whenever ~u undergoes a gauge transfonnation (2.4). In this manner, Au is introduced explicitly
and gauge ihvariance is retained with derivatives of", entering into the Lagrangian only in the
fonn of a "gauge derivative",
(2.6)

Rosen [9] constructed his Lagrangian with the simplest scalar combinations, 'I"V and D!1'" D!1""
added to the free electromagnetic tenn,

(2.7)
with one non-trivial constant 0" introduced as shown. The particular choice of signs was the only
one which yielded particle solutions.
Variations with respect to A!1 and\ji yield the field equations

jPv, v = - ~, jl = ~ (iJif1Jl- 1JIif1Jl)

(2.8)
0v CDv1JI) - i£A"Dv1JI + o'lttp = 0 .
Note that the current source for the Maxwell equations derives from", coupled to A!1 itself via the
gauge derivative. The field equation for", is a modified Klein-Gordon equation.
Rosen found a continuum or non-singular, stationary, spherically symmetric particle solutions of
the fonn
1JI = (J(T) exp( - ifjJt) , jJ = constant, (2.9)
but the particle energies were all negative.
Years later, Finkelstein, Lelevier and Rudemlan [10] and Rosen, Rosenstock [11] examined a
purely scalar field Lagrangian with an additional quartic tenn, of the fonn

(2.10)
Because of the quartic tenn, the sign structure in (2.10) could be chosen differently from that in
174 F. I. COOPERSTOCK

(2.7) and still yield particle solutions. However, because of the new choice of signs, the particle
energies are positive. Moreover, only discrete particle solutions of the fonu
OCr) (r ...... 00) = Ae- ar Ir' a 2 = a2 - (J)2 > 0 (2.11)
are possible. These discrete states are readily plotted in the 9,r plane with the lowest energy
"ground state" having no nodes, the first excited state having one node, the second excited state
having two nodes, etc.
3. Lepton Modelling
Clearly, the charged versions of such states are realized by replacing iJp, by D and adding the
free electromagnetic part, -F "FIlV18n to the Lagrangian of (2. 10). This does indfed yield charged
quantized energy particle stites with positive mass. Numerical integration reveals that the particle
sizes which these states represent are of the order of 10-13 cm, the classical electron radius. How-
ever, in recent years, experiments have shown that the upper limit to the size of an electron is of the
order of 10- 16 cm, and hence this approach is inadequate.
Rosen and I [3] also considered a Lagrangian which couples the scalar 'I' of his original paper [9]
and the scalar (now called '1'1) of the quantized energy states [lO, 11) to electromagnetism:

0.1)

We found that with the proper choice of parameters, it is possible with this Lagrangian to find par-
ticle solutions whose sizes are even well below the experimental upper limit. At that point, we
endeavoured to realize our greater ambition, namely to model not only the electron but also the
other clearly ponderable lepton masses, the muon and the tau. However, in adjusting the parame-
ters to model the first excited state of '1'1 as the muon, the particle sizes were found to be of order
10-15 cm, which is beyond the experimental upper limit. Also, regardless of the coupling parame-
ters, it was not possible to adjust the second excited state to have the mass of the tau.
A successful model [3) was constructed by coupling the original [9] Rosen scalar, '1', with two
scalars '1'1' '1'2 of the quantized positive energy states [10,11] to electromagnetism:
L = - Fp.v[pv 18n - (rf'1/J)(Dp.1/J) + aw +
(3.2)

We found that we could choose parameters to fit the charge and the masses of e, Il and 't to within
0.06% of their experimentally determined values and the particle sizes which emerge are of the or
der of lO-16 cm.
An attractive element which is revealed in these models is that of confinement: as we consider
successively higher excitation states, the energies increase because the energy gains from the un-
coupled tenus in the energy density consistently outweigh the energy losses from the coupled nega-
tive binding energy tenus. However, because the latter increase in absolute value with successive
excitations, there is never a dissolution of the particle. This is attractive for two reasons: firstly, if
NON-LINEAR GAUGE INVARIANT FIELD THEORIES 175

these fields could become decoupled, a particle with negative energy, stemming from the original
Rosen scalar "', would exist independently, contrary to our experience. Secondly, this confine-
ment mechanism could conceivably be the protoype of thai which is responsible for confinement of
quarks in hadrons.
With regard to the question of stability, T.D. Lee and his collaborators [16] (see also [6]) consid-
ered a class of scalar field soliton solutions which included those generated by the Lagrangian of
(2.10). They found that the ground-stale solution is classicially stable but quantum-mechanically
metastable. The excited states were found to be unstable. Since this Lagrangian forms the base of
our particle models generated by scalar fields coupled to electromagnetism, it would be expected
thai the Lee et al stability results would hold for these as well. Moreover, the stability of the
ground stale and instability of the higher stales are what we seek in the modelling of elementary
particles.
4. The Role of Gravity
We recall that Einstein showed that electromagnetism and gravitation were not sufficient to model
elementary particles. With the weakness of gravitalion relative to the other interactions in nature,
there is a naIural tendency to dismiss gravitalion out of hand insofar as its capacity to influence
elementary particle structure. However, regardless of how small a mass may be, given sufficient
compactification, gravity can assume an important or even dominant role. This is readily seen from
the Reissner-Nordstrom metric

ds2 = (1- ~ + ~)dt2 - (1- ~ + ~) -1 dr2- r2tJ!il,

• r2 • r2
(4.1)

(where we now use geometrical units in which G=c=1 and all quantities are measured in centime-
ters). For m - r or e - r, gravity assumes vital proportions as deviations from Minkowski space
become important.
The coefficient of 2/r, namely m - e2/2r, in the goo component of the metric in (4.1) is the energy
which is localized within a sphere of coordinate radius r. The entire mass, m, is seen as r --t 00
and this is understandable because of the electromagnetic field energy e2/2r which is stored in the
field from a radius r to infinity. An observer aI radius r does not perceive this energy which lies at
radii beyond him.
The most interesting situation to consider is when r is sufficiently small to render
mer) == m- e2 / 2r (4.2)
negative. In this case, a neutral test particle at such an r value would be gravitationally repelled
rather than attracted. This phenomenon [12], often referred to as "Reissner-NordstrOm repulsion",
has hitherto been regarded as a curiosity of general relalivity, with no manifestation in the physical
world. However, when we consider the values of m and e for that most ubiquitous of particles, the
electron, namely 6.77 x 10-56 cm and 1.38 x 10-34 em respectively, we find that mer) becomes nega-
tive as we pass below r - 10-13 cm, the classical electron radius. If the electron were indeed of this
magnitude, this phenomenon would not be realized because the metric (4.1) would not be valid
within the particle itself. However, experiments tell us that the electron is no larger than 10- 16 cm,
and hence there is a zone from at least 10- 16 em to 10- 13 cm where Reissner-Nordstrom repulsion
would actually exist.
176 F. I. COOPERSTOCK

What are the important consequences of these results? Firstly, they establish that anti-gravity,
however limited, would appear to be a part of physics rather than merely science fiction. Secondly,
with the existence of a zone of negative energy density, a key condition of the well-known
Hawking-Penrose singularity theorems [13] is removed, as is the inevitability of the onset of singu-
larities in nature.
There are various points that are to be discussed in this subject. To begin, it is important to em-
phasize that the phenomenon is very limited, confmed to the zone up to the classical electron ra-
dius. We are not saying that there are negative masses. Indeed, as one examines mer) for r > 10- 13
cm, one finds that 99% of the positive observed mass of the electron is already contained at r -
to-II cm. Although the positive m is what we perceive in experiments to the present day, it would
be most interesting if an experiment could be designed to detect the negative energy inner core.
Unfortunately, the Coulomb interaction overwhelms the gravitational interaction and hence charged
probes would not be expected to be viable in this regard.
There is also the question of the applicability of classical general relativity in this domain. Since
we are considering such small distance scales, in fact within a Compton wavelength, inevitably the
question of quantum effects arises. However, according to references contained in [13], the mani-
fold structure of space-time remains intact to scales of at least to- 15 cm. Also, the quantization of
gravitr is believed to be required with certainty only at the considerably more extreme Planck scale
of 10- 3 cm, and the phenomena under consideration are safely beyond this, in the zone where the
manifold structure is secure. Thus, quantum considerations, insofar as gravitation is concerned,
would not appear to be relevant.
Another issue is that of the appropriateness of the Reissner-Nordstrom metric (4.1) for the gravi-
tational deSCription of elementary particles such as the electron. The electron has both spin and
magnetic moment and hence the Kerr-Newman [14] metric, which describes a spinning charge,
would appear to be a more appropriate choice. This is particularly underlined by the fact that the
Kerr-Newman metric reveals a gyromagnetic ratio which agrees with that of the electron.
Unfortunately, energy localization is more difficult to rationalize for these rotational states as op-
posed to the purely static states with spherical symmetry such as (4.1). However, recently, Virb-
hadra [15] has succeeded in demonstrating that the integration of the energy up to coordinate radius
r within the Kerr-Newman field gives the same result to third order in the spin parameter, using two
different pseudotensors, that of Tolman and that of Landau-Lifshitz. For the distances of concern
in the present modelling of elementary particles, the Virbhadra expression is not adequate and S.
Richardson and I are currently attempting to extract the exact integrated energy within r. With this
expression in hand, we will be able to compare the onset of "Kerr-Newman repulsion" to that of
Reissner-Nordstrom repulsion.
Gravitation theory can tell us more about elementary particles. We recall that with sufficient
compactification, any body, however small its mass, can reveal significant or even dominant effects
of gravity. We have considered a variety of non-singular field theoretic models of elementary par-
ticles in which the compactification is important. The results were:

a) whenever gravity played a significant role, the charge-to-mass ratio elm was of order unity or
less, elm ~ 1.
b) whenever gravity played a dominant role, elm approached 1.

However, for the known elementary particles, elm is very far from unity. For example, elm -
1021 in the case of the electron. Thus, if the model results have general validity, and if these results
NON-LINEAR GAUGE INVARIANT FIELD THEORIES 177

are not significantly altered by the inclusion of spin, we may conclude that gravity is not a signifi-
cant factor in the structure of elementary particles, at least for those particles with which we are
familiar.
These results in tum suggest further information about the sizes of elementary particles. For con-
sider the metric of (4.1). With the known e and m of the electron, for example, the metric compo-
nents show major deviations from unity when r is of the order 10-33 cm. Hence, gravitation be-
comes a dominant force at this, the Planck radius. However, if gravity is dominant only for parti-
cles with elm - 1, as suggested by all of our models, then it would necessarily follow that the
known elementary particles are much larger than 1O-33cm. Thus, experiments provide an upper
limit of 10- 16 cm and theory suggests a lower limit much greater than 10-33 cm. The challenge for
the future is one of narrowing these widely separated limits.
5. Spin and Magnetic Moment
To this point, the essential focus has been on models without spin. There is support for such an
approach at the level of classical modelling from the point of view that intrinsic spin may be seen as
a strictly quantum-mechanical attribute and that Dirac theory, which describes the interaction of the
electron with other particles and fields, successfully incorporates both spin and magnetic moment.
We will consider the electron with Dirac theory in the following section. However, before doing
so, it is of interest to consider an alternative approach, namely one of extending the classical field
theory in an attempt to encompass spin and magnetic moment [6] at the level of classical field the-
ory.
We consider, for the sake of illustration, the simplest Lagrangian which yields charged positive
energy particle states:

(5.1)

This yields field equations

(5.2)

¥'V '" = _ 4n.f ,

If we now consider solutions which are only axially symmetric, i.e. functions of rand e, then it can
be shown that this requires at least an additional azimuthal component of the four-vector potential,
A~,i.e.,

(5.3)

The most general form of the scalar field 'V is

1/J = ~(r,(J) exp (imt + istp), s = 27tn, n = integer

(5.4)
The s has been introduced as shown to assure single-valuedness of 'V.
When the energy-momentum tensor components are constructed from the fields in the usual man-
ner, we find that the x and y components of angular momentum vanish identically, as expected and
178 F. l. COOPERSTOCK

the z component is

LZ = j(xro - yT")dV = - jToq,dV

j dV[tf~'rA'r
(5.5)
=- + ~'(JA'(J/,2 +

+ ~2 (co - E~)(S - EA)] .

The magnetic moment is constructed from the current which, again, yields identically vanishing x
and y components and

Thus, provided solutions exist which render non-vanishing integrals in (5.5) and (5.6), there is
scope for modelling both spin and magnetic moment in these classical field-theoretic structures.
Unfortunately, it is far more difficult to find axially symmetric solutions of (5.2) because they arc
partial differential equations of considerable complexity. It would be interesting to see if solutions
could be found and if so, whether the new degree of freedom could lead to a modelling of elemen-
tary particles with a simpler Lagrangian than that of (3.2).
Although the present emphasis is upon the leptons, in particular the electron, we conclude this sec-
tion by noting that we have had a partial success in modelling the quarks [5]. We were not able to
model the clearly massive particles of the three families: the leptons, e, J.l., "t and the quarks, up,
charm, top and down, strange, bottom with a single set of coupling constants. Two additional con-
stants were required to fit all of the masses and charges (of which the latter two families are frac-
tionally charged) of the three families of particles. However, as before in the case of the lepton
modelling, the theory and the solutions predict particle sizes which are now found to lie in the ex-
perimentally acceptable range 10- 18 to 10-17 cm.

6. Quantum Solitons
We now consider an alternative approach to the modelling of the electron, and possibly the other
leptons as well. While the earlier discussion indicated that non-spherically symmetric solutions
have the potential to build spin and magnetic moment into the particle, there is a more direct route.
We recall that Dirac theory [17, 18] builds spin and magnetic moment via the spinorial structure of
the theory and the theory is very successful in describing the fine structure of the energy spectrum
of hydrogen. All that remains are the minute corrections from quantum electrodynamics to explain
the Lamb shift. In hydrogen, the Dirac equation is solved by coupling the electron to the Coulomb
field of the proton. In this treatment, the particles are treated as points. We have now embarked on
the following generalization: we treat the electron as a Dirac soliton built by the coupling of its own
electromagnetic field with its wave function, the Dirac spinor \jI. This is an extension of the
Einstein-Rosen program to the quantum domain. In contrast to the problem of the hydrogen atom
where the electromagnetic field is imposed via the Coulomb potential, we now have the Coulomb
potential enter as the boundary condition. The coupled Dirac-Maxwell equations now determine the
NON-LINEAR GAUGE INVARIANT FIELD THEORIES 179

structure of both fields, subject to conditions of regularity.

The Dirac Lagrangian becomes locally gauge invariant by the inclusion of the four-vector poten-
tial with minimal coupling as

L = ilici{!)PiJp1/J - meW - I~FIlY[pv - eifrf1/JAp (6.1)

where f are expressed in terms of the perbaps more familiar ex and P matrices as

(6.2)

These satisfy

a; = ay = a; = p2 = 1 (6.3)
and <Xx'~'~'P all anticommute in pairs. For a central field, the Dirac equation can be separated
exactly in spherical coordinates [17,18]. We define

p, == ~I& - ill) / r • a, - a.r / r. k == ~ (0'. L + If) (6.4)

where '" '" 71,.. ""

L=rxQ. cr'=
tv
(6.5)
and cr represents the three Pauli matrices. From (6.3-5), it follows that
III'

(6.6)
and the Dirac Hamiltonian H is
H = c<XrPr + itlcarPkr-1 + pmc 2+ V (6.7)
whence k commutes with H and is a constant k2: has eigenvalues (j + ~)2, j = 1/2, 3/2, 5/2, ... and
hence
k=±I,±2 •.... (6.8)
We choose a representation in which H and k are diagonal and represented by E and k (numbers).
The angular and spin parts of'l' are fixed by 'l' being an eigenfunction of k.
We now consider the electron as described by 'IJI generated by the coupling to its own electromag-
netic field. Writing 'l' with spinor components f/r, F2/r, G/r, Gtr with the F's and G's all func-
tions of r. the field equations are expressed as

d2 + iG2 - [1f(1-J.) + ~] Fl =a
180 F. I. COOPERSTOCK

Fz - ~F2 + [:2 (1 +.1.) - r] G1 o (6.9)

]11 - ~Fl + [:2(1 +.1.) - r] G2 =0

2 4.1re2 ....2....2 2 2
" V = - -2-
r (1'1 + 1'2 + G 1 + G2 )

A. _ E/mc2 , V eAo

with limit conditions.

.i ~ c,
L 'I.Vr r + C2 ' Cl'C2 constants
"-+00
Preliminary results have indicated an emerging Dirac soliton with characteristic spread of the wave
function of the order 10-'6 cm. However, the numerical integration of (6.9) is delicate and much
wolk remains to be done. The non-linearity of this system of equations renders it much more diffi-
cult than the corresponding linear problem of the hydrogen atom.
NON-LINEAR GAUGE INVARIANT FIELD THEORIES 181

References
1. Einstein, A. and Rosen, N. (1935) Phys. Rev. 48,73.
2. Quigg, C. (1983) "Gauge Theories of the Strong, Weak and Electromagnetic Interactions"
(Benjamin, Reading, Mass).
3. Cooperstock, F.I. and Rosen, N. (1989) Int. J. of Theor. Phys. 28,423.
4. Bonnor, W.B. and Cooperstock, F.I. (1989) Phys. Lett. A. 139,442.
5. Sharman, P.H. and Cooperstock, F.I. (1990) Can. J. Phys. 68, 531.
6. Cooperstock, F.I. (1989) Foundations of Physics Lett. 2, 553.
7. Mie, G. (1912) Ann. derPhysik 37, 511; (1913),40, 1.
8. Born, M. and Infeld, L. (1934) Proc. Roy. Soc. (London) A144, 425; (1934) A147,522;
(1935) A150, 141. See also Hoffman, B. and Infeld, L. (1937) Phys. Rev. 51,765.
9. Rosen, N. (1939) Phys. Rev., 55, 94.
10. Finkelstein, R., LeLevier, R. and Rudennan, M. (1951) Phys. Rev. 83, 326.
11. Rosen, N. and Rosenstock, H.B. (1952) Phys. Rev. 85, 257.
12. Papapetrou, A. (1974) Lectures on General Relativity, D. Reidel, Dordecht, Holland.
13. Hawkins, S.W. and Ellis, G.F.R. (1973) The Large-Scale Structure of Space-Time, Cambridge
University Press, Cambridge.
14. Newman, E.T. et al (1965) J. Math. Phys. 6, 918.
15. Vimhadra, K.S. (1990) Phys. Rev. D. 41, 1086.
16. Freidberg, R., Lee, T.D. and Sirlin A. (1976) Phys. Rev. 13,2739.
17. Dirac, P.A.M. (1947) The Principles of Quantum Mechanics, 3rd ed., Oxford, New York.
18. Schiff, L.I. (1955) Quantum Mechanics, 2nd cd., McGraw-Hill, New Yorlc
HOW TO IDENTIFY AN ELECTRON IN AN EXTERNAL FIELD

A. Z. CAPRI
Theotretical Physics Institute
Department of Physics
University of Alberta
Edmonton, AB.
T6G 2J1
Canada

ABSTRACT. We show that a construction previously obtained by us leads to a Lorentz

invariant vacuum even in the presence of an external field. This allows an unambiguous
definition of an instantaneous one-particle state.

1. INTRODUCTION
The title may suggest that the problem I want to discuss is simply an academic problem.
After all, we all know what an electron is. It is a particle with the following properties:
mass m = 9.109 389 7(54) x 10-31 kg
charge e = 1.602 177 33(49) x 10-19 C
spin s = 1/2 li.
So why is there such a vast literature on this subject? (A good list of references to recent
work is to be foumd in the book by Fulling as well as in the somewhat older book by Birrel
and Davies reference[l].) What is the problem about identifying an electron in a given
external field? The answer is that there are many problems. For example, if the external
field varies in time then there is a non-zero probability for pair creation and one can no
longer be sure that the state one is looking at is a single-particle state. Furthermore, two
observers not at rest relative to each other will see different external fields. If one of these
observers sees an electron, does it follow that the other observers also sees the same thing,
namely one electron? Or at an even lower level, if one of them sees a vacuum (zero-
particle) state, what does the other observer see?
This problem is not simply theoretical, although the experimentalist can frequently
avoid these difficulties by having the external field vary so slowly in time that for practical
purposes it is constant. Nevertheless if two experimentalists zip past each other at some
considerable speed they encounter the same difficulty. Still, even if the problem is not of
immediacy to experimentalists it certainly involves many important questions of principle,
especially if the external field is a gravitational field. Here a solution of this pr50blem is
viewed as a first step toward quantizing gravity. What i am trying to point out is that the
vacuum for an external field problem is a complicated entity. In fact, there is at present no
concensus among theorists on how to define it.
The reason for all these difficulties is that an external field destroys the fundamental
kinematic symmetries all relativistic theorie:s are supposed to have, namely Lorentz and
183
D. Hestenes and A. Weingartshofer (eds.J, The Electron, 183-189.
© 1991 Kluwer Academic Publishers.
184 A.Z. CAPRI

translation invariance of the field equations. As already stated, the external field looks
different in every Lorentz frame.
In this paper I shall to discuss only smooth (differentiable) external fields that
vanish rapidly for very large (positive or negative) distances or times. This guarantees that
asymptotically we have free fields and avoids the necessity for technical discussions that,
although important, are not germane to the physical problem of interest.

2. WHAT IS THE PROBLEM WITH THE VACUUM?

In a quantum field theory with time-translation invariance it is always possible to separate
the field into positive and negative frequency parts in a straightforward causal manner; the
future does not influence the past. If the theory also has Lorentz invariance, this
decomposition is also Lorentz invariant; positive frequency is positive frequency in every
Lorentz frame. In these cases one writes:

q,(x) = q,(+)(x) + q,H(x) (2.1)

Here q,(±)(x) are defined as follows:

(2.2)

The vacuum state 10> is then defined as the state annihilated by the positive frequency part
of the field, that is:

q,(+) 10> = 0 (2.3)

To illustrate what can go wrong when we have a time-dependent external field, we consider
a Dirac field interacting with a time-dependent electromagnetic field. The field
(Heisenberg) equation now reads:

[-iyll(dl1 - ieA I1 ) + m] q,(x) = 0 . (2.4)

This equation may be rewritten as an integral equation which incorporates the boundary
condition that at very early times the field is a free fieIdwhich we call q,in(x).

(2.5)
Here SR is the retarded Green's function for the free Dirac equation. That is,

[-iylldl1 + m] SR(X - y) = S(x- y) (2.6)

and

(2.7)
HOW TO IDENTIFY AN ELECTRON IN AN EXTERNAL FIELD 185

The obvious thing to do now seems to try and separate <Il(x) into positive and
negative frequency parts as in equation (2.2). This yields:

(2.8)

One of the major difficulties is now displayed in this equation. The positive frequency part
S(+)R of the retarded Green's function SR no longer satisfies equation (2.7). Thus, even
if the the external vector potential AjL(x) vanishes for xO < T ,where T is some finite
time, we find that <Il(+}(x) depends on AjL(x) for all times, including times xO < T .
Thus, the present depends not only on the past but also on the future.
The problem just described was recognized quite some time ago. In fact, back in
1972 G. Labonte' and I found a way to evade this difficulty [2]. What we did not realize
then was just how good this method is. I shall first review our construction and then go on
to show that it leads to a definition of an instantaneous vacuum that is seen as a vacuum
state by all observers connected by a Lorentz transformation to the frame in which the
vacuum was originally defined. Thus, although the theory no longer has space-time
translation invariance nor Lorentz invariance, we can nevertheless obtain a Lorentz
invariant, instantaneous vacuum state. Furthermore, the instantaneous one-particle states
will also be seen as one-particle states by all Lorentz-equivalent observers. Thus, there is
no ambiguity as to what constitutes an instantaneous state of one electron. The
construction used reduces to what would be an obvious procedure for a very slowly
varying external field. A more primitive version of these results was presented by us in
reference [3].

3. THE VACUUM

We now list certain minimal properties that we believe any reasonable vacuum state
should possess.
The instantaneous vacuum state at time t say, I~> should satisfy:
a) lOt> is a state of zero particles, where particles are defined as the quanta of the
instantaneous Heisenberg field.
b) for t --> ± 00, the vacuum lOt> --> 10out, in > .
c) The vacuum lOt> may not depend on times xO> t. It may only depend on times xO ~ t.
This is simply a requirement of causality.
We now show how to accomplish this for a Dirac field interacting with an external
electromagnetic field. The Heisenberg equation for this field is:

(3.1)

To begin, we defme an auxiliary field <Ilt(x) by the following equation and initial condition:

[-iyjL(djL- ieAjL(t,x» + m] <Ilt(x) = O. (3.2)

with initial condition:

(3.3)
186 A. Z. CAPRI

This condition guarantees that at time t = t the auxiliary field qJt(x) coincides with the
Heisenberg field qJ(x). The field equation for ~l(X) has time translation invariance and
can be decomposed into positive and negative frequency parts in a causal manner according
to equation (2.2).

(3.4)

The reason for introducing the auxiliary field $t(x) is to introduce time-translation
invariance which is essential if causality (condition c ) is to be achieved. A direct
decomposition of the Heisenberg field (whose field equation is not time translation
invariant) leads, as we have already seen, to acausal behaviour.
It is now possible to define the vacuum lOt> as usual as the state annihilated by the
positive frequency part of the auxiliary field:

$t(+) (x) lOt> = 0 (3.5)

This is the vacuum state defined by Labonte' and me. It is the state of interest to us.
Incidentally, the construction makes it obvious that at time t = t the Heisenberg field and
the auxiliary field coincide. They also both satisfy the same equation at this instant and, as
we have shown, have the same instantaneous hamiltonian. So at t = t the auxiliary field
can replace the Heisenberg field. What I want to show next is that the vacuum state defmed
above is Lorentz invariant in spite of all the non-covariant things in the theory. To discuss
this Lorentz invariance requires a slight change of notation. But first I have to explain what
exactly is meant by Lorentz invariance of a theory with an external field.
By Lorentz invariance of the Heisenberg equation (3.1) we mean that this equation
remains form invariant under a proper orthochronous Lorentz transformation if the external
field AIl(x) is replaced by:

A'Il (x') = All vAv(x) (3.6)

where

(3.7)

This is easily verified to be the case if ~(x) is replaced by $'(x') where

$'(x') = S(A)$(x) . (3.8)

Here SeA) is the (1/2,0) ffi (0,1/2) (non-unitary) representation of the Lorentz
transformation A and we furthermore have,

(3.9)

The notation we use here is the same as in the book by Bjorken and Drell [4].
To examine the transformation properties of the auxiliary field qJl(X) we now
introduce some more notation. Let n be a time-like unit vector
HOW TO IDENTIFY AN ELECTRON IN AN EXTERNAL FIELD 187

(3.10)

We next defme a space-like surface 0' by:

0': n.x = t (3.11)

If the points x are restricted to lie on the spacelike surface 0' we write x/O'. Thus, if f(x) is
some field and its argument (field point) is restricted, to lie on 0' , we write f(x/O'). This
simply means that f(xO,x) is replaced by f«Hn.x)/nO,x)
Under the Lorentz transformation (3.7), the surface 0' is transformed into AO'
where

AO': n'.x' = t. (3.12)

The auxiliary field is now labelled with the surface 0' rather than t. Thus, the field
equations for the auxiliary field now read:

[-iyl1(ol1- ieAI1 (x/O')} + m] ~o(x) =0 . (3.13)

with initial data

~o(x/O') = ~(x/O') . (3.14)

We are finally ready to prove the Lorentz invariance of the vacuum corresponding to this
instantaneous field.

4. PROOF OF LORENTZ INV ARIANCE

Consider the c-number corresponding to the quantized theory defined above. The
extension back to a q-number (second quantized)theory is easy and will be indicated at the
end of this section.
The Heisenberg field ~ has associated with it a self-adjoint hamiltonian h(A). The
corresponding hilbert space is L2(R3) with the usual Dirac scalar product. Associated
with h(A) is a M011er operator U(A) which is unitary since the electromagnetic field A dies
out rapidly in time. This allows us to write the Heisenberg field ~ as

~(x) = U(A) ~in(x) (4.1)

Here we have already introduced the incoming asymptotic (free) field ~in .We also defme
the transformed electromagnetic field AA by

(4.2)
188 A.Z.CAPRI

and a corresponding hamiltonian Ah and Ml£'ller operator U(AA). These correspond to the
respective quantities in the Lorentz transformed frame. Furthermore we have the
transformation properties of the free asymptotic fields:

(~in)(x) =SeA) ~in (A-Ix) =(V(A)~in)(X) . (4.3)

Now because the transformed Ml£'ller operator is again unitary we have:

(4.4)

Thus, we have unitary evolution from the same asymptotic field ~in to the Lorentz
transformed Heisenberg field A~.
We next define the self-adjoint hamiltonian ho(A) obtained from h(A) by restricting
the electromagnetic field A to lie on the surface (J. The auxiliary field ~o evolves with the
dynamics specified by this hamiltonian by starting from the surface (J with the initial data
specified by equation (3.l4).

~(xI(J) = ~(xI(J) . (4.5)

Corresponding to this evolution we have again a unitary operator Wo(A) such that
the field ~o evolves from the surface (J according to this operator:

~o(x) =Wo(A) ~(x/(J) =(Wo(A) Uo(A) ~in) (x) (4.6)

Here we have also introduced the Ml£'ller operator Uo(A) that evolves the Heisenberg field
~(x) from the asymptotic incoming field ~in to the surface (J. It is important to notice that
we again have unitary evolution from ~in to ~o.
Finally, we define the hamiltonian Ah o for the auxiliary field corresponding to the
Lorentz transformed electromagnetic field. Associated with this we have a unitary operator
AWAo (AA) such that:

(4.7)
From equation (4.6) we find:

~in (x) = (Ut o(A) Wto(A) ~o) (x) (4.8)

Thus,
(4.9)
HOW TO IDENTIFY AN ELECTRON IN AN EXTERNAL FIELD 189

This shows that under a Lorentz transformation the fields $0 and A$o are
connected by a unitary transformation. It only remains to translate all of this into a q-
number theory. But this is easy.
Consider the quantized field cp smeared with a test function f. We write this as
cp (f). A unitary transformation of the test function f say Uf induces a unitary
transformation V on the field cp according to:

Vcp(f)vt = cp(Ut). (4.10)

Thus, we have unitary transformations between the auxiliary field as defined in one Lorentz
frame and a second Lorentz frame. This means that a vacuum in one Lorentz frame is also
a vacuum in any other Lorentz frame and a one-particle state in one frame is also a one-
particle state in all other Lorentz frames. This is as much as one can expect, although it is
amazing that one can get even this much.
A natural question that arises is, "What has happened to all the interesting
phenomena that the relativists have found, such as the background of thermal particles in an
accelerating frame?" Preliminary calculations with a model consisting of a real scalar field
coupled to a c-number source indicate that these effects are still there, but arise in a totally
different manner. In that model it is the fact that the source for the auxiliary field is static
that leads to interesting features for the auxiliary field $0 It is shifted from the Heisenberg
field by a time-dependent c-number as well as a time-independent c-number. This second
part acts like a zero energy mode and can be taken with the annihilation (positive frequency)
part of the field or with the creation (negative frequency) part. Another way to handle this
time-independent term is to introduce a representation of the auxiliary field $0 in which it
has a non-zero vacuum expectation value both in the asymptotic vacuum lOin> and the
instantaneous vacuum 100 >, In this case the auxiliary field $0 would no longer be
unitarily equivalent to the asymptotic field $in . These various possibilities remain to be
investigated.

5. REFERENCES

[1] Birrel N. D. and Davies P. C. W. (1982) Quantum Fields in Curved Space,

Cambridge University Press, Cambridge/
Fulling S. A. (1989) Aspects of Quantum Field Theory in Curved Space-Time,
Cambridge University Press, Cambridge.

[2] Labonte' G. and Capri A. Z. (1972) 'Vacuum for external-field problems', II Nuovo
Cimento, lOB, 583 - 591.

[3] Capri A. Z. , Kobayashi M. and Takahashi Y. (1990) 'Lorentz in variance of the

vacuum for external field problems', Class. Quantum Grav., 7, 933 - 938.

[4] Bjorken J. D. and Drell S. D. (1964) Relativistic Quantum Mechanics, McGraw-Hill,

New York.
THE ELECTRON AND THE DRESSED MOLECULE

A. D. BANDRAUK
Departement de chimie
Faculte des sciences
Universite de Sherbrooke
Sherbrooke, Quebec, J1K 2Rl, Canada

ABSTRACT. Interactions of molecules with intense laser fields produce

highly nonlinear effects mediated by the bonding electrons. Recent
experiments in nonlinear photoelectron spectroscopy of diatomics such
as H2 have shown the presence of new electronic states and thus new
boundsta tes of the molecular ions, which are manifestations of the
dressing of the diatomic ion by the large number of photons present at
high intensities as predicted by theory. In this chapter we will
expose the theory of dressed molecules and present numerical calcula-
tions by coupled equations of these new phenomena which occur whenever
intense laser fields interact with molecules.

1. INTRODUCTION

Electrons are the "gluons" in molecules which prevent nuclei from

exploding apart. Thus electrons provide the necessary bonding to
create stable multinuclear species called molecules. Great progress
in our understanding of the electronic structure of molecules has come
from the introduction of the molecular orbital concept by Mulliken in
the 1950's and 60's. Thus as in atoms, electrons in molecules occupy
orbitals which envelope the whole nuclear space, creating stable mole-
cular species if the molecular orbitals are bonding and unstable spe-
cies if these are antibonding [1].
The bonding characteristics of molecular orbitals can be inferred
from photoelectron spectroscopy [2]. Recent improvements in this
method has even led to determination of the electron momentum distri-
bution in these orbitals [3]. A concomitant structure which appears
often in the photo electron spectrum is the vibronic structure of the
remaining molecular ion after photoionization. This structure which
is created by the coupling of the ionized electron to the core of the
ion reveals the vibrational structure of the molecular ion and the
191
D. Hestenes and A. WeingartsluJ!er (eds.), The Electron, 191-217.
© 1991 Kluwer Academic Publishers.
192 A. D. BANDRAUK

degree of coupling between both electron and ion [4]. We conclude

therefore that the electron serves as an essential probe in under-
standing molecular structure.
The advent of intense lasers has revealed some singular aspects
of the nonlinear behaviour of atoms in intense laser fields [5-7].
Recently, similar nonlinear phenomena (e.g., above threshold ioniza-
tion, AT!), have been observed in molecules [8-11]. In particular,
experiments on the nonlinear photoionization of H2 have revealed that
the vibronic structure of the molecular ion is considerably altered
with respect to the free ion [9-10]. It is the goal of this chapter
to examine a theoretical model, the dressed molecule, which can help
us understand nonlinear molecule-laser interactions, which interac-
tions we reiterate are induced by multiphoton transitions (real and
virtual) of the electrons in the molecule.
One can classify the regime of coupling between the laser and the
molecular system according to the nature of the process they induce.
The first regime is that corresponding to low-intensity lasers which
couple weakly with the system. As a result, the excitation processes
are well described by leading order perturbation theory, such as Fer-
mi's Golden rule. For molecules, this leads to a Franck-Condon pictu-
re of electronic (radiative) transitions [12]. At intermediate to
high intensities, one encounters a domain in which multiphoton proces-
ses begin to take effect. This is signalled by nonlinear behaviour of
the transition probabilities as a function of intensity. In particu-
lar two or more states may be strongly coupled together as a result of
being near resonant. An example of this is the Rabi oscillations of a
two level atom [5-6] or an n-level molecule [13]. Another example
which this chapter discusses in detail, is the nonlinear interaction
between rovibrational manifolds of different electronic molecular
states induced by intense laser fields. Judging from atomic experien-
ce, [14], one can establish the uFper limit of the intensity I of this
regime at 10 12 W/cm 2 (terawattlcm ), since for I ~ 10 13 W/cm 2 , ioniza-
tion rates exceed dissociation rates for many molecules. Finally one
has the very hi§h intensity limit available with current superintense
lasers (I > 10 1 W/cm 2 ), where Rabi frequencies (w = de/h, d = tran-
R
sition moment, e = electric field) are comparable to the laser fre-
quency, and highly nonresonant transitions compete with resonant pro-
cesses. Thus in the case of the nonlinear photoelectron spectroscopy
of H2 mentioned above [8-11], the photoionized electron continues to
absorb photons creating ATI peaks with a vibronic structure which has
no relation to the vibrational structure of free H+. We will show in
2
the present chapter, that the H+ core is dressed by the intense field
2
and that the structure of the ATl peaks reflects the nonlinear inter-
action of the ion core with the laser while at the same time remaining
coupled to the dressed photo ionized electron.
In particular we will show that intense lasers can create dressed
adiabatic states which are degenerate with the excited field-molecule
diabatic states, as a result of a laser induced avoided crossing bet-
THE ELECTRON AND THE DRESSED MOLECULE 193

ween the ground bonding state (2L+) of H+ and the dissociative anti-
9 2
bonding state (lL+) of that ion. From a semiclassical analysis of the
u
problem [15-16), one can predict a stabilization of new dressed mole-
cular states. This stabilization stems from the molecule resonating
between the two bound states, adiabatic (unperturbed) and adiabatic
(perturbed) of the molecule. Such stabilization of electronic states
at high intensities is currently being discussed extensively in the
atomic case [17-18). In the molecular case, the nuclear degrees of
freedom offer the possibility of creating stable new electronic states
by the laser induced coherent superposition of bonding and antibonding
states of the free molecule. In the following, we will show the rea-
lization of this effect within a more realistic close-coupling calcu-
lation involving many electronic-field states, as befits such a highly
nonperturbative problem. We also point out that at high intensities,
where Rabi frequencies exceed rotational spacings, laser-induced
orientational effects or alignment are expected to predominate in the
angular distribution of photodissociation fragments [19).

2. ELECTROMAGNETIC FIELD-PHOTON STATES.

In describing the field-molecule states of a radiation-molecule

system, one encounters the dichotomy of a classical description in
terms of the classical electric field e(x,t) and the n photon particle
quantum states In>. Maxwell's equations allow one to express free
electric field plane waves of frequency wand wave vector k, (one
dimension) as,

e(x,t) = eo sin (kx - wt) (1)

with total energy in a region of dimension L,

1
(2)
2:

where P and Q are canonical variables satisfying the relation [20-22),

P = Q = (Ll 1/2 e sin wt (3)

o

Equation (2) illustrates the harmonic oscillator analogy for the

field. Through a ~urther definition of amplitudes aCt) and the com-
plex conjugate aCt) ,
-iwt
aCt) = (Zhw)-ll2 (<<112 + iP) ae (4)

one can formulate the real field as

194 A. D. BANDRAUK

I'; = 1';(-) + 1';(+)

.
(5)

(I';
(-)
) =-i
(2Lh~ll2 art) e
Ikx
(6)

This renders the classical Hamiltonian (2) of the electromagnetic

field in the form
•
E=hwaa (7)

Quantization of the classical Hamiltonian transforms the positive

(1';(+)) and negative (1';(-)) partl¥. of the classical field defined in
t~rms of the amplitudes a and a into quantum field operators a and
a. Thus for a cavity of length L we obtain the electric field opera-
tors

8(+)
- i ( --;
hw fl2 art) e
Ikx
(8)

8(-) i ( n: fl2 a+(t) e -Ikx (9)

~(-)
The operator I'; creates photons of frequency wand wave number k
while the operator I';l+) annihilates photons. Taken together as an
ordered product of operators they yield the quantum field Hamiltonian
operator

(10 )

where the expressions

aln> = n1/2 In-1> ( 1ll

define the effect of these operators in the quantum photon number

states. The expectation value of the Hamiltonian operator (10) in the
eigenstate In> of photon number In> is

(12)

Thus the average energy of the field equals the energy corresponding
to an exact number of photons for that state.

Since either part 1';(-) or 1';(+) of the classical field I';(x, t) is a

solution of Maxwell's equations, one could expect that the operators
(8-9) have eigenvalues corresponding to some aspect of the classical
field, especially in the limit of large photon numbers. Thus rather
than quantize the field energy, ~ich leads to number states and
eigenvalues of the energy operator H, one can also quantize the field
THE ELECTRON AND THE DRESSED MOLECULE 195

ltlj!elf and search for eigenvalues and states of the field operators
8 l -) .

Thus the quantization of the field which can be expressed by,

A+ I
80:>=8 C+) I0:> (13)

produces field eigenvalues 8 C±) provided we can find the eigenstates

of a, and a+, i.e.,

alo:> = 0:10:> (14)

For the harmonic oscillator, these eigenstates are well known and are
called the coherent states of oscillator defined by, [21], [23],

10:10 e- cI 0: 12 )/2
10:>=I: In> (15)
o (n! )112

Combining equations (8), (9), (13) and (14) gives the real classical
amplitude as
nw ) 112
_ i ( - (o:e 1kX _ 0:*e- 1kX )
2

= (2nw)1/2 10:1 sin (kx - wt) (16)

Comparing this amplitude with that obtained from Maxwell's equation

(1). shows that the classical field 8 has an amplitude related to the
eigenvalue of the coherent quantum oscillator state 10:>, equation
(15). This is the coherent wave of controlled phase capable of being
produced by an ideal laser, and for this reason is called the coherent
state of light.
We can now evaluate the energy of this state obtaining from (10)
and (13),
AC-) C+) 2
E = <0:18 8 10:> = nwlo:l = nw<N> (17)

Thus from equation (15) we have the result that a coherent classical
wave is expressible as a superposition of an infinite number of photon
states In>. The wave has an average number of photons <N> defined by
(17) and the probability p of finding the nth photon state is given
o
by the Poisson distribution,

2 10:1 20 _10:1 2
I<nlo:> = -- e (18)
n!

Thus in the ensuing discussion we shall use the number state represen-
196 A. D. BANDRAUK

tation of the electric field which interacts with molecules. For high
intensities, i.e. large photon numbers, the Poisson distribution (18)
peaks at the average photon number <N> defined by (17) so that this
will be taken as the most probable photon state for which absorptions
and emissions of photons will occur. This photon number representa-
tion of the electric field will enable us to define field molecular
states for the proper description of the dressed states of the molecu-
le-radiation system.

3. THEORETICAL METHOD - COUPLED EQUATIONS

We shall elaborate in the present section on the coupled equa-

tions in the field-molecular representation which leads to a proper
and accurate description of dressed molecular states at high intensi-
ties [24-25].
For the present, let us consider the general case of photodisso-
ciation of a simple diatomic. The Hamiltonian for the system may be
partitioned into four components, namely,

H = Hm + H
na
+ H + H
f mf
(19)

in which the molecular interactions are denoted by H , the Hamiltonian

m
of the Born-Oppenheimer approximation, and H , the nonadiabatic per-
na
turbation. The quantized radiation fields were defined in section 2
and are represented by the term

H
r
r w a+ a
k k k
(20)
k

in which the summation is over the frequencies wk and wave vector k of

the modes. The creation and annihilation operators (a+,
k
ak ) have been
defined in equation (11). Lastly the radiative int,.eraction between
the molecules and the fields is denoted by the term Hmr and takes the
form in the quantized field representation and dipole approximation
[5-6], [26],

~ ~

H d· g (21)
mf

in which e denotes the polarization vector, V is the volume of the

cavity, and d(k) designates the dipole moment of the molecule for the
kth transition. The effect of nonadiabaticity can be treated simul-
taneously and can play an important role as in the multiphoton infra-
red dissociation of ionic molecules [27].
A measure of the various interstate couplings involved will help
in understanding the dynamics. Radiative couplings can usually be
THE ELECTRON AND THE DRESSED MOLECULE 197

expressed as a Rabi frequency

-+ -+
w (cm- l
R
) = d· tUh = 1.17 x 10- 3 d (a.u.) I [W/cm2]112 (22)

8n
__ g2 (23)
o
c

where a.u. denotes atomic units, c is the velocity of light, the in-
tensity I is reported in watts/cm2 and g is the maximum field ampli-
o
tude. For a dipole transition moment d - 1 a.u., and an intensity I =
1011 W/cm2, one obtains a radiative interaction of - 400 cm- t . This
is to be compared with the nonradiative (nonadiabatic) interaction
between the covalent and ionic state of LiF, <1/J(LiF) IHelll/J(Li+F-» '"
600 cm- t , as an example [26] whereas the vibrational frequency of LiF
is w(LiF) = 300 cm -t. It is clear that at high intensities (I > 10 to
W/cm2 ), radiative interactions are nonperturbative and will compete
with the nonradiative interactions, hence influencing considerably the
photodissociation ratios of branching into various product excited
atomic states.
We will endeavour to show in this section that the model of the
dressed molecule and the Born-Oppenheimer approximation [1], [12],
lead to the determination of the dressed or field-molecule eigenstates
as solutions to coupled differential equations that describe the
nuclear motion in the presence of the laser field. Thus bound-
discrete, bound-continuum, radiative and nonradiative (nonadiabatic)
can all be treated simultaneously for any coupling strength, thus
allowing us to go beyond the usual perturbative treatments. Since we
shall be dealing with bound states as initial conditions, the presence
of dissociative (continuum) nuclear states presents a problem, which
is curcumvented through the use of a scattering formalism that encom-
passes all possibilities. Thus, by introducing the technique of arti-
ficial channels for entrance [28] and generalized to include exit
channels also [25], one can simultaneously treat bound and continuum
states. It is thus possible by the present method to calculate rigo-
rously transition amplitudes for any radiative or nonradiative inter-
action strength in the presence of bound and continuum states, thus
covering both perturbative (Fermi-Golden rule) and nonperturbative
regimes.
We rewrite the total Hamiltonian ()9) by separating the radiative
and nonradiatlve perturbation, Hand H ,
mf na

H = H
o
+ V H
o
= Hm + H
f
V H
mf
+ H
na
(24)

Thus H is the zeroth-order field-molecule Hamiltonian, and V is the

o
total, radiative and nonradiative interaction. We now try to express
the field molecule eigenstates of the total Hamiltonian in terms of
198 A. D. BANDRAUK

the eigenstates of H, which are therefore direct products of the

o
unperturbed (Born-Oppenheimer) molecular eigenstates of H, and the
m
unperturbed field eigenstates of Hf , equation (12). We can therefore
define the field-electronic states

le,n> = Ie> In> (25)

where e is a collective quantum number (symmetry, spin, etc.) for

molecular Born-Oppenheimer electronic states, and n is the photon
number defined in equations (~-10). We now look for solutions of the
total Schrodinger equation: HI~E> = EI~E> with the total wave func-
tion expanded in terms of the basic field-electronis states defined in
(25) ,
1
~ F (R) le,n> (26)
en
e,n

F (R)' s are appropriate nuclear radial functions propagating on the

en
potential of the photon-electronic state le,n>. By substituting into
the total Hamiltonian defined in equation (24), and premultiplying by
a particular state Ie, n>, one obtains the set of one-dimensional
second-order differential equations for F (R):
en

d2 2m
{dR 2
+
n2
[E - Ve (R) nnw] } Fen (R)

2m
~ V (R) F (R) (27)
en,e'n' e'n'
n2 e' n J

where m is the reduced mass of the molecule, V (R) is the field free
e
electronic potential of electronic state Ie> obtained from ab-initio
quantum chemical calculations or from spectroscopic measurements [11,
[12]. We treat here rotationless molecules, although in principle
both rotational quantum numbers (J,M) can be included rigorously [19],
[28] .
Equation (27) for the field-molecule problem can be more succinc-
tly expressed in matrix form as,

F"(R) + W(R) F(R) =0 (28)

where the diagonal energy matrix elements are

2m
W (R) [E - V (R) - nnw] (29)
en,en f12 e
THE ELECTRON AND THE DRESSED MOLECULE 199

The nondiagonal elements that describe the couplings, i.e.,

2m
W
en,e'n'
(R) =-
112 [ v"',
en,e n
(R) +yr ,
en,e ,n_1
+] (30)

are of two types: nonradiative (v'" = Hna ) and radiative (yr = H ).

mf
Since each electronic potential Y (R) appears in the diagonal matrix
e
elements (29), we are able to sum numerically over all bound vibratio-
nal and continuum (unbound) states of the same potential. Thus only
the electronic and photon states need be specified explicitly in any
numerical calculation. Finally the radiative coupling yr are nondia-
gonal in the photon quantum numbers reflecting annihilation (absorp-
tion, ~n = -1) or creation (emission, ~n = +1) of a photon, equations
(21) and (30). The nonradiative (nonadiabatic) couplings remain dia-
gonal in the photon number n since they do not involve the field.
All numerical calculations are performed using a Fox-Goodwin
method, which has proved to be very accurate for molecular problems
(errors are of sixth order in the integration step [29]). The asymp-
totic numerical radial functions are projected onto asymptotic field-
molecule states le,n> and are expressed as,

F (R) = r Fe'n' (R)

en en
e' n'

Fe'n' (R) = k- l12 {o 0 exp [- i(k R + 0)] (31)

en en ee' nn' e

- S exp [i( k R + o)}

en,e'n' e'

2m
(E - Y (R ) - nhw)
n2 e ttl

o is an elastic scattering phase factor, which is zero for neutral

dissociating products but needs to be modified for charged products
[27].
The coefficients S are defined as the scattering, S-matrix
en,e'n'e'n'
elements, and the function F (R) corresponds to the nuclear radial
en
functions of the molecule in the final state le,n> for initial states
Ie' ,n' >. In practice one usually projects the real numerical func-
tions onto real asymptotic states, i.e.,
Fe'n' (R) = k-1/2 [0 0 sin (k R+o) + R cos (kR+o)]. (32)
en e ee' nn' e en,e'n' e

This projection enables one to obtain, from the numerical procedure,

the R matrix, which is related to the S matrix by the expression [30]
200 A. D. BANDRAUK

5 = (1 - iR)-l (1 + iR) (33)

and thus one obtains the transition amplitude matrix T,

5 = 1 - 2 n: i T (34)

In the molecular problems we shall encounter, invariably the

initial state is a bound state, so that one encounters the problem of
bound-bound transitions, or one has to calculate the probability of
transition from initial bound states to final continuum photodissocia-
tion states. One method, such as encountered in the complex-
coordinate method [311, calculties linewidths r which are squares of
the transition amplitudes r = -- ITI2. We have shown previously [24-
fl
251, [281, that it is possible to obtain transition amplitudes direct-
ly from the coupled equations (28), i.e., one can transform all tansi-
tion amplitude problems, including bound-bound transitions, into a
scattering problem by introducing additional artificial channels,
continua, as entrance and exit channels. The introduction of such
artificial channels into the coupled equations (28) permits us to
exploit the various relations between transition matrices in order to
extract the relevant photophysical amplitudes. Thus using the follow-
ing relations between the total Green's function G and the transition
operator T [321,

T=V+VG T=V+TG V (35)

o 0

G=G +G TG G = (E-Hl- 1 G (E-H )-1 (36)

o o o o o

one can obtain an expression for the transition amplitude TCC1 between
an entrance channel IC1> and a real physical continuum (dissociative)
channel Ic>,

T = exp (i T) ) V GO T (37)
C1,C 1 Cl,D ° DC

GO is the zeroth order (field-molecule) Green's function of the

o
initial bound state 10>, T)1 is the elastic phase shift for scattering
on the artificial continuum potentials of IC1>, and V is the cou-
cl,a
pIing (weak) between the artificial channel and the bound state. The
numerical solutions of the coupled equations (27-28) including the
artificial channel IC1> coupled to the initial state with n photons
10,n> permits us to extract each photodissociation amplitude

T = T exp (- iT) ) (V GO)-l (38)

oc Cl,C 1 C1,D °
All quantities on the right hand side of equation (38) can be calcula-
THE ELECTRON AND THE DRESSED MOLECULE 201

ted numerically [24-25]. The above method applies provided the ini-
tial state 10> is only weakly perturbed during the multiphoton proces-
ses, so that the unperturbed Green" s function GO is adequate. This
o
will be the case if the initial state is coupled nonresonantly to
resonant processes, as will be shown to occur in the H2 case (next
section). All multiphoton resonant processes and nonadiabatic inter-
actions are calculated exactly in Toc ' allowing us to join the weak,
perturbative nff~ime (I < 10 10 W/cm,2) to the strong, nonperturbative
regime (l > 10 1 W/cm2 ).
The advantage of the method based on the artificial channel le1>,
equation (38), gives direct access to numerical values of the transi-
tion photodissociation amplitude T from the initial bound state 10>
DC
to the final continuum Ic> Transition amplitudes are essential quan-
tities to calculate photodissociation angular cross sections or dis-
tributions [19]. Thus writing field-molecule states as

Ij;A jJ jM 1
j > = r- F j (J,R)t/I j (q,R) IA jJ jM
j > 1m j ,n j >
(39)

where t/I corresponds to the electronic eigenfunction of energy V (R),

j j
q represents the ensemble of electronic and spin coordinates, R is the
internuclear separation. The rotational states of the molecule are
defined by the normalized symmetric top function IAJM> , where A is the
electronic angular momentum, J is the total angular momentum and M is
the z-component of J [11]. The differential dissociation cross
section from some initial state 10> = Ig A J M > to the final states
j 000
Ij> = Ij;A j J j Mj >, is then given by the expression

J
<T(J M ;a,q» ~ I r exp CiJ jlll2) (2J+1)1/2 D jA (a,q>)T 12 . (40)
o 0 Kj j oj

4. THE DRESSED MOLECULE.

Having established in the previous section the necessary forma-

lism to treat multiphoton transitions in diatomic molecules beyond
perturbation theory, we now expose in detail the method in order to
help interpret the recent experimental results of van Linden van den
Heuvell [9] and Bucksbaum [10] on the nonlinear photoelectron spectrum
of H2 which exhibits above threshold ionization peaks (ATl), i.e., the
ionized electron keeps absorbing photons in the vicinity of the mole-
cular ion core. Each ATl peak now reveals a vibronic structure, as
the receding electron remains coupled to the core via coulomb and
polarization forces. Furthermore, measurements by Bucksbaum et al.
[10] on the proton yield demonstrate unusual yield dependencies on the
intensity of the laser.
We limit ourselves in the present work to the experimental laser
202 A. D. BANDRAUK

wavelength A =
532 nm. As pointed out by van Linden van den Heuvell
[9], this wavelength allows one to reach the Bir+ state of H via a
u 2
fi ve photon nonresonant transi tion (see figure 1). A sixth photon
cou~les radiatively and resonantly the B state to the doubly excited
2~u electronic state, the so called F state which crosses in a diaba-

tic representation [33] the Rydberg type E electronic state [12]. In

an adiabatic representation, the EF curve forms a double well as does
the GK potential. These two states remain coupled by a nonadiabatic
(non Born-Oppenheimer) coupling. One can however adopt the equivalent
diabatic representation where now the diabatic GF and EK curves cross
(figure 1) and are coupled by a nonradiative nondiabatic coupling due
to the fact ~ that in this representation the molecular electronic
Hamiltonian H is not diagonal. A residual nondiabatic coupling
~ m ~

<EKIH IGF>, the term H in the Hamiltonian (24) is operative. In

el ~

fact the diabatic EK and GF electronic potentials were obtained by

deperturbing with a 2 x 2 unitary transformation the spectroscopic
adiabatic electronic states EF and GK (for details see [12], [33]).
This procedure yield a nondiabatic coupling H (R) which is used in
na
the coupled equations (28). It is to be emphasized once more that in
the coupled equations formalism, both nondiabatic (we now use this
term in a diabatic representation rather than nonadiabatic which
applies to an adiabatic regime) interstate coupling and radiative
couplings are equivalent from a formal view point. The numerical
procedure presented in the previous section allows for the rigorous
treatment of radiative and nonradiative transitions on an equal
footing, from the perturbative (weak interaction) limit to the nonper-
turbative (strong interaction) limit.
The sixth photon is thus resonant with the vibrational states of
the GF and EK diabatic electronic potentials which further interact
nondiabatically. A seventh photon now couples radiatively these last
states to the x+(2r+) ground electronic state of H+. In this process,
9 2
a free electron is now created so that the electronic transition
moment involves the Rydberg electrons of the E and G states and the
ionizing electron in H+, (assuming that the H+ core is nearly the same
2 2
for the E, G and X+ states (see figure 1)). We emphasize that the F
state, which is doubly excited cannot couple radiatively directly to
the X+ state; i. e., the electronic transition moment <2~2ItllSO' f >,
u 9 c
where f is the ionized electron wavefunction is rigorously zero since
c
radiative transitions, if one neglects electron correlation, involve
only one electron excitation [12]. We thus have the interesting case
that the B state couples radiatively strongly to the F state, which
then couples nonradiatively to the Rydberg E and G states. It is from
these two Rydberg states that the seventh photon of wavelength 532 nm
can now access resonantly the H+ molecule, leading to ATI when the
2
ionized electron keep absorbing further photons. This last process
THE ELECTRON AND THE DRESSED MOLECULE 203

200
190 53Z nrn

180
A+ (2);+)
170 u

EO
H+
150 2
140
130 x+ (2);+)
g
K HZ
120
~. 110
''U
EI: 100
ulU
~.
~:J
~o 90
~r.
B(l);+)
>t 80 u

70
60
50
X(l);+) H2
40 g

30
20
10
0
-10
0 01 0.4 0.6 0.8 t2 14 16
R (crnl/Z)

Figure 1. Ten photon transition at A = 532 nm leading to

ionization and dissociation.
204 A. D. BANDRAUK

leads to dressing of the electron and various theoretical methods have

been developped over the years to treat this problem [5-6), albeit for
atoms only so far.
What we wish to point out is that in the course of ATI, a purely
electronic process as a first approximation, photons will interact
further wi th the H+ core leading to a dressing of the H+ molecular
2 2
ion. Firstly, a nonresonant three fhoton transition induces direct
photodissociation from the bound X+( r+) state to the repulsive, dis-
9
sociative A+(2r +) of H+. This is seen in figure 1, the standard non-
u 2
perturbative vertical image of multiphoton transitions. The more
complete nonperturbative representation is that of figure 2 where we
now use the field-molecule states defined in the previous section,
equations (25-26), (i.e., the total wavefunction is linear superposi-
tion of products of photon and molecular states). Let us now explain
in detail the meaning of this new representation. The ground X(lr+)
9
state with (n+5) photons couples radiatively nonresonantly to the
B(lr+) state leaving only n photons after a five photon transition.
u
Since this transition is nonresonant, it will be weak and can be
treated perturbatively. The remaining transitions, being resonant,
are strong and must be treated nonperturbatively. Thus the B(n) state
is coupled radiatively to two sets of states: the GF (n-1) and GF
(n+1) field-molecule states. The first (n-1) state corresponds to
removal of one photon from the field and is thus ascribed to an ab-
sorption. The second (n+1) state is the result of a virtual photon
emission. We remind the reader that the quantized electric field,
equations (8), (21) is explicitly written as the sum of an annihila-
tion (a) and a creation (a+) photon operator. The first corresponds
to absorption and the second to emission of photons. We must emphasi-
ze that at 532 nm wavelength the B ~ F tansition is resonant for ab-
sorption. Thus the GF (n-1) state crosses resonantly the B(n) state
in the Franck-Condon region for that transition. The B(n) ~ GF(n+1)
transition is nonresonant and is therefore called a virtual transition
(this transition is responsible for the Lamb shift of electronic
states in vacuum [21-23)). In the field-molecule picture one sees
immediately, figure 2, that this transition is nonresonant. In fact
the GF(n+l) state is 2hw in energy above the resonant B ~ F transi-
tion. This point help us establish the validity of the rotating wave
approximation, RWA, which neglects all such virtual transitions [5-6).
This approximation is therefore valid only if the Rabi frequency, wR
(equation 22), the radiative coupling between the Band F state is
much less than the energy separation between the resonant and the
virtual transition, i.e.

w «2hw (41)
R

This is the main reason why in the X ~ B five photon transition, only
THE ELECTRON AND THE DRESSED MOLECULE 205

OO~~~~-----------------------------,
532 nm
70
B(n+2)
60
X\n)
50
EK(n+l)
40 X(n+5)

30

X+(n-2)

EK(n-l)

-10 B(n-2)

-20 X+ (n-4)

-30 EK(n-3)

-40 A+(n-5)

-so

o 02 0.4 0.6 0.8 L2 l4

Figure 2. Field-molecule representation of figure 1 including

photon numbers n.
206 A. D. BANDRAUK

the X(n+s) and B(n) field molecule states are used. The virtual cou-
pling between the X(n+s) and B(n') states, where n' > n can be safely
neglected since the photon absorptions are themselves nonresonant, and
are therefore very weak. In conclusion, every resonant n ~ n-l absor-
ption is accompanied by a virtual n ~ n+1 emission. This explains
therefore the doubling of all electronic states in figure 2.
We now continue to follo~ the photon paths. The GF states are
coupled nondiabatically (via H , 1.e., v'" equation (30)), to the
na en,e'n
EF states with the same photon number since this is a nonradiative
transition. Now the Rydberg E and G(n-l) states couple resonantly to
the X+(n-2) state and virtually to the X+(n) of H+. The X+(n-2) state
2
couples nonresonantly to the A+(n-3) and virtually to the A+(n-l)
state. The A+(n-3) state couples radiatively to X+(n-4) and X+(n-2).
The first transition corresponds to the nonresonant absorption of the
ninth photon shown in figure 2. The virtual transition A+(n-3) ~
X+(n-2) serves to dress the X+ electronic state, and is depicted in
figure 3. Thus the X+(n-2) and A+(n-3) field-molecule states cross at
an energy above the v = 4 vibration of the X+ ground state of H+. The
2
~
symmetric radiative coupling <X+I~IA+> g gives rise to both the ab-
sorption X+(n-2) ~ A+(n-3) and the emission A+(n-3) ~ X+(n-2) proces-
ses. Similar crossings occur in the other field-molecule states which
must be added until numerical convergence is achieved. We repeat,
this is due to the fact that the classical coherent electric field g
is a linear superposition of photon states n, equation (15). Finally
we have a transition from the X+(n-4) to the A+(n-s) state. This last
state corresponds to the photodissociation-ionization of H (X1I:+) to
2 q
H+ (A~+) after absorption of ten photons, or the three photon photo-
2 u
dissociation of H+.
2
The figure 2 represents the minimal number of field-molecule
states required for a proper treatment of the nonlinear photoelectron
spectrum of H2 • As the intensity increases, more and more of those
states must be included until numerical convergence is achieved. In
the weak field limit, one recovers of course the direct perturbative
pathway described by figure 1. In the strong field limit, many more
pathways are allowed due to the virtual photon creation processes
which are normally neglected in the RWA regime. Thus the complete
state count as exhibited in figure 2 allows us to bridge the weak and
strong field limits.
The field-molecule representation depicted in figure 2 leads us
to make the following quick predictions. Firstly, crossings of field-
molecular states involving a one photon resonant process become laser-
induced avoided crossings as one increases the field intensi ty I.
Thus the crossings of the states X+(n), A+(n-l); X+(n-2), A+(n-3);
X+(n-4), A+(n-s) all undergo an avoided crossing as shown in figl1re 3
for various laser intensities. The new field-molecule states are
obtained by diagonalizing the diabatic 2 x 2 Hamiltonian (in a first
THE ELECTRON AND THE DRESSED MOLECULE 207

resonant approximation)

[
V
11
(R) + flw
V12 (R) 1 (42)
V (R) V (R)
21 22

gIvIng two new adiabatic states, called the dressed states of the
field-molecule system:
V (R) + V (R)
11 22 1/2
± 112 [ (V (R) - V (R))2 + 4V2 (R) ] • (43)
11 22 12'
2

where V11 ' V are the diabatic (zero-field) molecular electronic

22
potentials (figure 1), V is the radiative coupling (Rabi frequency,
12
equation (22)). Similar laser-induced avoided crossings occur at the
intersections of the B(n), GF(n-l); B(n+2), GF(n+l); B(n-2), GF(n-3)
states. These radiative avoided crossings are further perturbed by
the nondiabatic interactions with the EK states. These laser induced
avoided crossings induce nonperturbative intensity dependent changes
in the electronic potential and concomitantly in the vibronic structu-
re of transitions. Such laser induced effects have been considered by
various authors [34-37, 15-16]. A detailed study of laser induced
resonances, i.e., the nonlinear radiative lifetimes of photodissocia-
ting molecular states such as shown in figure 3 has been undertaken by
Bandrauk et al. [15-16]. In particular, a semiclassical approach used
previously in the theory of predissociation of molecules has proven to
be very useful in predicting the existence of these new resonances.
This is in keeping with the remark made above that in the field-
molecule representation, nondiabatic (nonradiative) and radiative
interactions are formally equivalent and can be treated simultaneously
in a unified formalism. The scattering formalism expounded in the
previous section is of course the most convenient method to treat
bound and continuum states simultaneously in the presence of large
radiative and nonradiative interactions. The experimental observation
of a laser intensity dependent vibronic structure of H+ was confirmed
2
recenty [9-10] and was therefore the first report of the laser induced
avoided effect illustrated in figure 3.
Figure 2 further demonstrates that at the wavelength of 532 nm
five open channels appear, i.e. channels below the initial zero ener-
gy. These channels correspond to dissociation of Hand H+ into neu-
2 2
tral atoms and protons. Thus the A+(n-3), X+(n-4) and A+(n-5) chan-
nels will produce H(lS) and H+ species with kinetic energies corres-
pondin¥ to the difference in energy between the zero line (energy of v
= 0, X ~+ of H 1 (figure 2) and the asymptotic energies of each state.
q 2
Hence three protons of different kinetic energy are to be expected.
The lowest energy proton will emanate from the A+(n-31 channel as a
 208
A. D. BANDRAUK

60
532 nm
50
A+2[+(n-l)
u
40

30
+ 2 +
X 1: (n)
20 g

~. 10
'''0

-.
E C
II .,
0
1:d
>t: -10

-20

-30

-40

-50

-60
0.00 0.20 0.40 0.60 0.80 1.00
R (cml/2l

Figure 3. Laser induced avoided crossing between X+(n) and A+(n-1)

field-molecule states of H! at various field intensities I (W/cm 2 ):
O· 10 12 ; nnn
5 x 10 12 ; «< 10 13 ; (JOO 5 x 10 13 ;
'U 14 .10
result of the tunnelling of the vibrational states of X+(n-2l at the
initial zero energy. A second higher energy proton will be produced
by the X+(n-4) channel, which corresponds to the ninth photon process
in figure 2, or equivalently the two photon dissociation of H+ via the
2
nonresonant process X+(n-2) -7 A+(n-3) -7 X+(n-4l. Figure 2 tells us
immediately that this final channel, X+(n-4) is coupled radiatively to
A+(n-S), so that a laser-induced avoided crossing will occur between
these two channels at the energy 32000 cm- 1 below the initial zero
energy. Thus the yield of the first low energy proton from A+(n-3)
and the second and third higher energy protons from X+(n-4) and
A+(n-S) are all expected to be nonperturbatively influenced at high
intensities by the laser-induced avoided crossings illustrated in
figure 3. The high kinetic energy protons from A+(n-S) are the result
of the three-photon nonresonant transition of H+, the last three pho-
2
tons in figure 1. The field-molecule picture illustrates that this
transi tion will be strongly affected by the two photon transition
X+(n-2) -7 X+(n-4) at high intensities. This transition merits further
THE ELECTRON AND THE DRESSED MOLECULE 209

elaboration. As indicated above, this is a nopresonant two photon

dissociation of the electronic ground state of H2 via the nonresonant
repulsive A+(n-3) state. Thus the A+(n-3) serves as a virtual state,
i.e., the dissociation products remain on the X+ potential, but the
radiative transition is induced by the A+ repulsive potential which is
nonresonant, and is therefore inaccessible. This is clearly seen in
figure 2 where the A+(n-3) state is always well above the X+(n-4)
channel. Thus only resonant processes give rise to crossing poten-
tials, whereas nonresonant processes always have well separated poten-
tial surfaces. This two photon nonresonant transition from bound X+
nuclear states to continuum X+ nuclear states has been called by us
previously ATPD (above threshold phot.odissociationl and is the analo-
gue of ATI [27]. The conditions for such processes is large transi-
tion dipole moments as in ATI where the ion-electron system gives a
dipole moment equal to the distance r between the two. In ionic mole-
cules such as LiF, the dipole moment of the system is R, i.e. the
distance between the ionic moities Li+ and F-. Thus a linearly diver-
ging dipole moment arises, creating a very strong coupling with the
radiative field as in the ATI case. For the X+ ~ A+ transition, we
will show below that the transition dipole moment is R/2, i.e. half
the internuclear distance. Thus in all three cases, similar nonlinear
absorption phenomena occur because of the large dipole or transition
moments which give rise to very large radiative couplings as the
ionized or dissociated species separate.
As in the ATPD of H+ described above, ATPD of H also occurs in
2 2
the B state of H . Thus from figure 2 one has also an open channel
2
the B(n-2) channel accompanied by the EK,GF(n-3) state, which are
coupled radiatively with the B state also with an RI2 transition
moment. Thus neutral H(1S) + H(n=2) atoms are expected to be also
created due to ATPD in the neutral B, EF, GK states of H2. In the
next section we will try to render these predictions quantitative as a
result of the numerical efficiency of the coupled equations (27) which
enable one to include as many channels as are deemed necessary by
convergence criteria. Furthermore, as emphasized above b one can rea-
dily cover the weak field, perturbative regime 2 I - 10 1 W/cm 2 to the
high field nonperturbative limit I - 10 14 W/cm, and include simulta-
neously bound-bound, bound-continuum, radiative and nonradiative tran-
sitions in one unified formalism and numerical method.

S. RESULTS AND DISCUSSION

As discussed in the previous sections, collision theory allows

one to obtain, using the artificial channel le1>, transition amplitu-
des from initial bound states to final bound states or continua. In
our case, we shall be calculating the transition amplitudes from the v
= 0 vibrational state of the ground electronic state X1L+ of H to the
9 2
various channels that are open according to the field-molecule dia-
gram, figure 2, i.e. all the channels which are below the zero energy
210 A. D. BANDRAUK

line which corresponds to the initial energy.

The input in the coupled equations (27) are the ab-initio poten-
tials of H2 and H; published in the literature [38-39]. These were
interpolated over 3000 points over an internuclear distance of 0.4 to
33 a. u. (in figures 1-3, distance units appear as cm1/2 since all
energies are reported in cm- 1 . Thus in H , due to the factor of 2m/h 2
2
in the nuclear Schroedinger equation (27), 1 cm 1/2 = 11 a.u.). In the
case of the EF and GK states, since these are calculated adiabatically
[40], these well known double well potentials were deperturbed [12],
[33], in order to produce the crossing diabatic potentials GF and EK.
A gaussian nondiabatic nonradiative interaction V12 was found of the
form V (R) = 3023 x exp [- 38.2 (R - 0.29)2] to give the above cited
12
adiabatic potentials EF and GK when inserted in equation (43).
As to the radiative couplings, two equivalent gauges are possible
[5-6], [26], the electric-field (multipole) gauge or the radiation
field (Coulomb) gauge. The radiative
-c)
coupling in the first is et • ~
whereas in the last it is eX/mc • p. Both gauges will give identical
results if a complete set of states is used, since the two gauges are
related by a unitary transformation. The use of one gauge or another
thus depends on its convenience. In the present problem, the electro-
nic transitions B -c) F and X+ -c) A+ involve excitation to electronic
states which have the same asymptotic limits. In fact the B -c) F and
X+ -c) A+ transitions are both 1s~ -c) 2~ molecular orbital transitions
q u
[9]. For these, the transition moment is easily shown to be:
1s + 1s 1s - 1s
a b a b
~(R) < letl > eRI2 (44)
,ii -12
in the limit of nonoverlapping atomic orbitals 1s and 1s . Clearly
a b
in the asymptotic atomic limit, these moments become infinite,
implying very strong coupling of the molecule to be electromagnetic
field upon dissociation. Unfortunately this creates divergent radia-
tive couplings in the electric field gauge. It is therefore best to
use the Coulomb gauge, which as the result of the commutation relation
p/m = (i/h)[H,r], results in the following convergent radiative coupling:
(E (R) - E (R»
A I j
------------- R/Z (45)
hw

1 aX
since ~ = c at Thus as R -c) 00, E1(R) - Ej(R) -c) 0 and the expression
(45) converges to zero. The Coulomb gauge transition moment (45) was
therefore used in the radiative transition calculations for the tran-
sitions B 7 F and X+ 7 A+. The X -c) B five photon transition was simu-
THE ELECTRON AND THE DRESSED MOLECULE 211

lated by using an arbitrary weak effective one photon transition,

which is of no consequence since this transition is perturbative.
Finally the E and G to X+ transitions being unknown were also given
the arbitrary transition moment ~ = 1 cm- i . Other transition moments
such as B ~ E, G were taken from the literature [41].
Calcula tions were performed for the transit ion ampli tudes Toc'
equation (38) for the open channels below the initial zero energy,
corresponding to the v = 0 level of Xi~+ (see figure 2). Since the
9
seventh photon falls in energy just above the v =3 level of X+(2~+),
9
then the v = 0 to 3 vibrational levels of X+(n-2) all lie below this
zero energy line (figure 2). It is to be emphasized that this figure
corresponds to zero kinetic energy of the el~ctron. In actual fact,
in the photoionization process H2 ~ H2 + e, the electron acquires
considerable kinetic energy which is then analyzed, thus exhibiting
AT! peaks [9-10]. Since we are interested in the dressing of the
molecular ion H+, we can obtain the energies and photodissociation
2
widths of the vibrational levels of H+ in the presence of the laser
2
field by examining the resonance structure of T In fact, these
oc
resonances show up in numerical calculation of Toc when one scans as a
function of energy the levels of H+ in anyone open channel, from
2
A+(n-3), B(n-2). X+(n-4) to A+(n-S). One takes into account the elec-
tronic kinetic energy by shifting up all the H+ channels, i.e. X+ and
2
A+. As one calculates T as a function of this displacement of the
oc
H+ channels (we iterate this corresponds to calculating T as a func-
2 oc
tion of electron kinetic energy), one finds that resonances appear
corresponding to v = 3, 2, 1, 0 successively from low to high electron
kinetic energy. This is as expected, since at low electron kinetic
energy, the molecular ion remains in its high vibrational states,
whereas at high electron kinetic energy, only low vibrational excita-
tion can remain by conservation of total energy. We illustrate in
fi~re 4 the two resonances attributed to v = 2 and v = 0 levels of
X+('1;+) of H+ at the intensity I = 10 13 W/cm2 . These resonances (nor-
9 2
malized to unity by dividing by the maximum value of the resonance)
were obtained from Toc where Ic> = IA+(n-S». They show a Lorentzian
behaviour for v = 0 and non-Lorentz ian behaviour at v = 2. Thus for
the first case, a linewidth can be readily obtained from the half-
width at the middle of the resonance curve, whereas in the v = 2 case,
the average half-widths at the middle was used as a first order appro-
ximation. We emphasize that the same resonances appear in all the
open channels, thus lending us to conclude that the energy shifts and
widths correspond to all possible photophysical processes included in
the calculation as illustrated in the field-molecule representation
(figure 2) which goes well beyond the perturbative description depic-
212 A. D. BANDRAUK

v-a A+(n-S) INT-1E+13

0.8

0.6

0.7

0 ...

O.S

0.4 ~
0.:>

1:JeillM 13884 131S04 '138M 13844 13"154 13884 13704 13724 13744 137M
SHFTlcm-t1

V-2 A+(n-S) INT-1E+13

0.0

0.8

0.7

0 ...

O.S

0.4

0':>

0.2

-
0.1

0
0242 8842 8842 10042 10242 10+42 101M2 10842 11042 11242
SHFT/om-ll

Figure 4. Laser induced resonances attributed as

v = 2, 0 levels of the dressed states, figure 3,
of H~ at I = 1013 W/cm 2 .
mE ELECTRON AND mE DRESSED MOLECULE 213

ted in figure 1.
We tabulate in the following table energies corresponding to
measured electron kinetic energies and widths of resonances for
various intensities as obtained from the numerical calculations des-
cribed above.

TABLE 1. Energies E (electron kinetic energy) and widths r of vibra-

tional levels v of H+ for various laser intensities I; ~ = 532 nm.
2

I v: 0 1 2 3 4 5 6 7
(W/cm2)

10 10 E(cm- 1 ) 7116 4926 2864 918

r(cm- 1) <10- 3
1011 E 7204 5018 2958 1016
r 0.001 0.01 0.005 0.006
10 12 E 8080 5914 3875 1961 177
r 3.8 2.0 4.3 2.5 3.9
10 13 E 13630 11664 9796 7334 5658 3606 2960 1631
r 4 138 686 980 0.13 502 285 139

At 10 10 W/cm 2, the radiative interactions are weak so that the

spacings of the H+ levels are still that of the unperturbed molecule
2
and linewidths r are below 10-3 cm- 1 . Since the lifetime T = 5 X 10 12
s/r(cm- 1), one sees that the radiative (photodissociative) lifetimes
of H+ at this intensity are larger than a nanosecond (10- 9 s). At
2
12 2
10 W/cm , one observes now levels with picosecond (10- 12 s) lifeti-
mes in addition to the appearance of an extra level. This is the
effect of the radiative interaction which pushes the dressed adiabatic
potential V_CR) down in energy, equation (43). The large energy
shifts are also the result of the laser induced avoided crossing. At
10 13 W/cm 2, more levels are seen to appear wi th lifetimes ranging from
a femtosecond (10- 15 s) to a picosecond. Such displacements of the
energy levels of H+ have been observed [9-10) and have been attributed
2
to the laser-induced avoided crossing illustrated in figure 3.
We also mentioned the fact that proton yields have been measured
and show unusual energy dependences as a function of intensity. In
particular it has been noticed by Bucksbaum et al. (10) that at low
intensities, high kinetic energy protons predominate, whereas at
higher intensities, lower kinetic energy protons predominate. Our
calculations show similar behaviour for the proton yield. Thus in the
next table we show the maximum intensities of the v = 0 and v = 2
resonances as a function of intensity for the two channels X+(n-4) and
A+(n-5L The first channel corresponds to two photon ATPD of H+ dis-
2
cussed above and the last, corresponds to the three photon dissocia-
214 A. D. BANDRAUK

tion of H+.
2

TABLE 2. Maximum intensities of resonances as a function of

laser intensi ty I.

Level: v 0 v 2
Channels: X+(n-4) A+(n-5) X+(n-4) A+(n-5)

I=10 t1 Wfcm2 4xl0- 6 4xl0- s 8x10- 4 9x10- 3

1=10 1~fcm 2 14 9 3.5 0.2

I=101~fcm2 10 3 1 65 10- 1

One sees clearly from the above table, that at low intensities the
A+(n-5) channel, i.e. the three-photon dissociation channel, is more
important than ATPD, the X+(n-4) channel. As the intensity increases
beyond 10 12 Wfcm 2 , this trend is reversed. This can be easily ratio-
nalized in terms of the field-molecule representation. Thus at low
intensities, the radiative interaction between the X+(n-4) and A+(n-5)
channel at the crossings of these is ineffectual and cannot influence
the high kinetic energy protons in the A+(n-5) channel. From the
Landau-Zener probability for a nondiabatic transitions one can predict
that high velocity particles at a crossing remain in fact diabatic,
i.e., they remain on the same curve (33). As the intensity increases,
the avoided laser-induced crossing predicted from equation (43) crea-
tes an upper potential V+(R) dissociating asymptotically to the
X+(n-4) state. Hence a turnover can be expected eventually in the
distribution of dissociating products with lower kinetic energy, i.e.
X+(n-4) products, predominating as a result of the laser-induced
avoided crossing of H+. This phenomenon was observed in (10) and was
2
presented as a clear evidence of such an avoided crossing. This laser
induced effect creating the new adiabatic potential V+ (R) can also
create new stable vibrational states of V+ (R) as we have predicted
previously [15-16). These have yet not been observed experimentally.
They should occur in the higher ATl peaks where the electron with high
kinetic energy can leave the molecule in the upper potential V+ (R)
instead of the lower V_ (R) potential which gives the laser induced
resonances enumerated in table 1.
In conclusion, we see that intense lasers modify considerably
electronic potentials, altering the vibrational energies and creating
laser-induced resonances. An important concept which helps clarify
these nonlinear phenomena is the laser-induced avoided crossing, figu-
re 3, in concert with the dressed molecule picture, figure 2. Laser-
induced resonances and laser-induced avoided crossings are the result
TIlE ELECTRON AND TIlE DRESSED MOLECULE 215

of intense electron electromagnetic field interactions. The electron

thus mediates the field-molecule interaction and also serves as a
probe for the properties of the dressed molecule.

Acknowledgments - We thank E. Constant, J.M. Gauthier and D. Marchand

in the preparation of this manuscript. We also wish to acknowledge
support from the Natural Sciences and Engineering Research Council of
Canada.

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5002-5008.
SCATTERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM

C. JUNG
Fachbereich Physik
Universitat Bremen
2800 Bremen
Germany

ABSTRACT. We study the classical scattering of an electrically charged

point particle off the Morse potential under the additional influence of
an oscillating electromagnetic field. We find scattering chaos, i.e. a
scattering function, which is discontinuous on a fractal subset of its
domain. This behavior is explained by the homoclinicjheteroclinic
structure of unstable periodic orbits. In the phase space a localized
chaotic invariant set A ( chaotic saddle ) is created, whose invariant
manifolds reach out into the asymptotic region and influence the generic
scattering trajectories. The transition probability to various final
energies shows a complicated pattern of singularities.

1. INTRODUCTION

One of the major themes of this conference is the scattering of electrons

under the additional influence of a strong electromagnetic field, so
called free-free-transitions. Such processes have been observed
experimentally for many years [1-7J. 1be theoretical description of these
experiments has been given quantum mechanically within a low frequency
approximation [8-llJ.
In the meantime an interesting new development in scattering theory
has emerged in the form of scattering chaos. So far this phenomenon has
been investigated for time independent systems mainly, i.e. for systems
without external fields ( see the reviews [12,13 J, however, for a
preliminary demonstration of chaos in field modified Coulomb scattering
see [14J ). It is well known that an external oscillating field can drive
bound states into chaos and so we expect, that such a field can drive
scattering processes into chaos as well. In this article let us try to
merge the ideas of free-free-transitions and of scattering chaos. Prior
to the investigation of a realistic but complicated system it might be
appropriate to investigate a simple model system first in order to
demonstrate the basic effects and to tryout appropriate methods. A good
model for demonstration is the scattering of an electrically charged
particle off a local I-dimensional Morse potential under the simultaneous
influence of a harmonically oscillati.ng external field. In periodically
driven Morse systems bound chaos has already been found [15,16J. And so
219
D. Hestenes and A. Weingartshofer (eds.), The Electron, 219-238.
© 1991 Kluwer Academic Publishers.
220 C.JUNG

we expect it to be a good candidate for scattering chaos too. To explain

the chaos in this system we essentially use ideas and methods which we
have used before in the treatment of scattering chaos in time independent
systems [17-20]. We modify them in the right way to make them fit to the
time dependent case.
So far chaos is a well defined concept in classical mechanics only.
The concept of quantum chaos has not yet been defined in a general and
satisfactory way. Therefore, in this article the classical side of
scattering chaos will be explained in detail and only in the concluding
section a few remarks on quantum scattering chaos will be made. In this
article we use the methods and the terminology which have become familiar
in nonlinear dynamics and in the mathematically oriented literature on
classical mechanics. Because of a lack of space it is not possible to
explain these issues in detail and we will use these expressions and
methods without comment. The reader, who is not familiar with nonlinear
dynamics, is referred to the textbooks [21-24].

2. SINGULARITIES IN THE SCATTERING FUNCTIONS

The system is given by the Hamiltonian function

H(p,q,t) - (p-A'cos (<a>t)) 2/2+exp (-2q) -2exp (-q) (1)

q is the l-dimensional position coordinate, p is the canonically conjugate

momentum, t is the time. wand A, the frequency and the amplitude of the
external field are free parameters.
The asymptotic limit of scattering is the limit q~.
The asymptotic Hamiltonian is
(2)
Ho (p,q,t) = (p-A'cos (<a>t)) 2/2

We introduce the reduced phase

(3)
tp=<a>t-q<a>/p- (A'sin(<a>t)) /p

p and ~ are conserved under the motion generated by Ho. Therefore we can
label asymptotes uniquely by giving p and ~. The set of incoming
asymptotes, denoted by Asin, has p<O. The set of outgoing asymptotes,
denoted by Asout, has p>O.
To compute a scattering trajectory, we take values of Pin and ~in
where Pin<O, take some large value of q and compute the corresponding
initial time tin mod 2~/w by inverting (3) and integrate the Hamiltonian
equations of motion generated by H. We stop the trajectory at some
terminal time tout at which q is back in the outgoing asymptotic region
while p>O. The outgoing asymptote is labelled by a pair of values for Pout
and ~out. It turns out to be instructive to know the time delay of the
scattering process, which we define by
SCATTERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 221

(4)

Now let us make a numerical experiment : We fix values for A and w, we

choose A=Q. 4, w=l. 4 for reasons which will become clear later. As an
example we fix the value of Pin at Pin=-l. Q, scan <Pin and plot Pout and Dt
as function of <Pin in Fig.l.
For people not familiar with scattering chaos the result may be
surprising : There are intervals of continuity of Pout and complicated
looking regions in between. In the complicated regions the time delay
becomes large. Fig. 2 gives a magnification of one of the unresolved
complicated regions of Fig.l. In this magnification new substructures
become visible. Fig.3 gives a further magnification in which more
substructures become evident. We could continue this process and keep on
magnifying complicated looking regions and would always find new smaller
intervals of continuity interrupted by complicated looking regions which
contain still smaller intervals of continuity e.t.c. ad infinitum.
This set of nested structures and substructures on all scales is
typical for fractal structures. The set of accumulation points of
boundaries of intervals of continuity defines a Cantor set along the <P-
axis. Such fractal clusters of discont:Lnuities of the scattering functions
have been found in all chaotic scattering systems encountered so far.
Therefore, the occurence of this phenomenon has been accepted as defining
criterion for classical scattering chaos.

3. TRAJECTORIES IN PHASE SPACE

The first step to explain the behavior displayed in Figs.I-3 consists in

looking at phase space trajectories coming from various intervals of
continuity. Some examples are shown in Figs.4a-e. In part a <pin=2.0 is
chosen, which belongs to the long interval of continuity outside the
chaotic region. This trajectory comes in, turns around and goes out again
directly. In part b we see a trajectory whose <Pin comes from the smaller
interval around <Pin=3.9 which can be identified clearly in Fig.l. This
trajectory comes in, turns around, but does not have enough energy to
escape directly. So it returns and comes in a second time. At the second
trial it has enough energy to escape. In parts c,d,e we see trajectories,
whose <Pin comes from still smaller intervals belonging to a deeper level
of the fractal hierarchy. They make several turns inside the potential
interior.
Figs.I-3 show that smaller intervals of continuity contain
trajectories making more revolutions inside the potential interior and
leading to longer time delays. In addition, Figs.I-3 indicate that Pout
goes to zero, if <Pin converges to the boundary of an interval. Supposing
that the external field is switched off adiabatically at very large
distance, then also the kinetic energy of the outgoing particle converges
to zero, if <Pin converges to the boundary. In total we obtain the
following picture : Inside any interval of continuity the scattering
trajectories all have the same qualitative structure. At small distances
222 C.JUNG

2~~CS\iid--V1
0.0
50

I-,
Ir-
-10
'""\

0.0 PHASE 6.28

Fig.l: Pout ( upper frame) and Dt ( lower frame) as function of ~in for
Pin=-l.O and parameter values A=O.4, w=1.4.

2.0 I I\j{\l h~~~it _M/\~M I

~ N" '~I"U1V~ 'I~
0.0
100~-~~

-20~~~--~~~~~

2.95 PHASE 3.75

Fig.2: Magnification of Fig.l.

SCATIERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 223

2.0

0.0
150~--~~~~~~---'

-30
3.18 PHASE 3.26
Fig.3: Magnification of Fig.2.

D
Q Q Q
Fig.4: Some phase space trajectories for parameter values A=O.4, w=1.4.
Parts a-e show scattering trajectories with Pin=-l.O and incoming phases
~a=2.0, ~=3.8, ~c=3.55, ~d=3.46. ~e=3.l9. Part f shows the periodic orbit
"y. Frame boundaries: QE(-1,3), pE(-2,2).
224 C.JUNG

beyond the boundary the first part of the scattering trajectories looks
similar to the trajectories inside the interval, then however the energy
is not sufficient to escape in the same way and the particle returns to
the potential interior and makes a few additional turns.
The fractal structure of the intervals is constructed in a
hierarchical way : On the O-th level there is the longest interval I CO )
which contains rpin=O. It contains trajectories coming in, making one
single turn and going out again directly ( see Fig.4a ). There remains one
connected unresolved complement CCO) on the rpin-axis ( note: because rp is
an angle variable, we identify rp=-O with rp=21C ). On the 1st level we cut out
of the interior of CCO) the 2 most prominent intervals 11(1) and 12(1). As
Fig.l shows, they lie around rpin=2.8 and rptn=3.9. These two intervals
contain all the trajectories making one additional loop inside the
potential interior ( see Fig.4b ). There remains a complement C(1) which
consists of 3 connected components. On the 2nd level we cut out of the
interior of C(1) the intervals containing trajectories making two loops in
the potential interior ( see Fig.4c ). We continue this scheme
iteratively. In the limit, between the cut out intervals a Cantor set of
measure zero remains. It is the set of accumulation points of the
boundaries of intervals.
Figs.4a-e give the impression, that scattering trajectories can be
attracted by some localized orbits. In the next few sections we shall make
this idea more precise. We close this section on phase space trajectories
by showing in Fig.4f that localized trajectory which will turn out to be
the most simple one and at the same time the most important one. It is a
periodic orbit, whose time of revolution is exactly equal to the cycle
time Tc=21C/w of the external field. In the following this trajectory will
be denoted by "y.

4. POINCARE SECTION

The best way to get an overview over the periodic orbits consists in
studying the appropriate Poincare map M. In the case of periodically
driven systems the best type of Poincare map is the stroboscopic map in
the (p,q)-plane taken for times t=O mod 21C/W, where W is the frequency of
the external field. This map is constructed like this :
Take an initial point (Po,qo) at time t=O, construct the trajectory
through this point, follow it until the time t-T c • At this time the
trajectory has reached the point (P1,ql). Define
(5)
SCATIERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 225

Alternatively we express this in the following way : Let ~t be the flow

map corresponding to the Hamiltonian (1). Then M~To applied to the plane
t=O mod 2~!w in the 3-dimensional extended phase space.
Any trajectory gives an infinite sequence of points in the Poincare
plane by marking its positions in the (p,q)-plane at all times tn=n To,
nEZ. Periodic orbits in the (p,q)-plane, whose time of revolution is an
integer multiple of To, correspond to periodic points of M.
We shall compare the Poincare plots with the phase portrait of the
system in the undisturbed case of A=O, i.e. in the absence of the external
field. In Fig.S a few trajectories of the field free case are plotted in
the (p,q)-plane. The broken line marks the separatrix to energy E=O.
Trajectories outside the separatrix are scattering trajectories with E>O.
Trajectories inside the separatrix are closed bound trajectories with E<O.
The point (0,0) is a fixed point. The time of revolution for
trajectories close to this point is T=2~!~2, i.e. the rotation frequency
is Oo=~2. Going further away from the origin the time of revolution
increases monotonically ( the frequency 0 decreases ) and towards the
separatrix we find T--, (}+O. For A=O and any field frequency w the
Poincare map restricted to the inter:lor of the separatrix is a monotonic
twist map, where the winding number of each invariant line is given by
w=O!w.
For weak perturbations, i.e. for small values of A some invariant
lines with irrational winding numbers survive. The invariant lines of the
field free case having rational winding numbers break into secondary
structures around elliptic periodic points and small chaos strips along
the invariant manifolds of hyperbolic periodic points. This szenario is
standard for generically perturbed twist maps [25,26].
We expect the strongest coupling between field and particle for ~o
and for most of our model calculations w=1.4 is chosen. For w«lo, but
!w-Oo ! small, an invariant line of the unperturbed system lying close
to the origin is in 1:1 resonance with the external field. For very small
amplitude A this line breaks into one elliptic fixed point X-y and one
hyperbolic fixed point xh' If A is increased a little bit more (
approximately at AzO.0015 ) xh collides with x o, the elliptic fixed point
coming from the origin in the field free case. x h and Xo destroy each
other in a saddle-node bifurcation. Afterwards X-y becomes the organization
center of the complete structure of the Poincare section.
In Fig.6 the case of A=0.04 is displayed. A few initial points (
marked by crosses) have been chosen arbitrarily and the next few hundreds
of iterates of these initial points are plotted. Starting from the outside
of the structure many KAM lines are already broken and some parts of the
(p,q)-plane, which are bound in the field free case, merge with the
scattering region. With increasing value of A more and more KAM lines are
distroyed. Fig.? shows the situation for A=O.l. Note the different frames
of Fig.6 and Fig.?
X-y stays elliptic until approximately A=AozO.35, where it changes
to inverse hyperbolic. For ~0.4 no elliptic periodic points and no KAM
lines are found. Scattering trajectories can come close to all points of
the (p,q)-plane. We find one global chaotic region. However, unstable
periodic orbits and aperiodic localized orbits still exist and they play
an essential role in the scattering behavior of the system. The fixed
226 C. JUNG

2.0

-2.0
-1.0 Q 2.5
Fig.5: Phase portrait for the field free case A=O. The separatrix between
scattering trajectories and bound trajectories is shown as broken line.

2.0

...

-2.0
-1.0 Q 2.5
Fig.6: Poincare plot for A=O.04, w=l.4. A few hundred iterates of some
initial points ( marked by crosses) under the Poincare map M are plotted.
SCATTERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 227

1.0

-1.0
-0.5 Q 1.5
Fig.7: Poincare plot for A=O.l, w=l.4. A few hundred iterates of some
initial points ( marked by crosses) under the Poincare map M are plotted.

2.0

-2.0
-1.0 Q 15.0
Fig.8: The first few tendrils of the unstable manifold WU(1) of the fixed
point ~ for parameter values A=O.4, w=1.4.
228 C.JUNG

point ~ of the Poincare map corresponds to the periodic orbit 1 shown in

Fig.4f.

5. THE CHAOTIC SADDLE

For A=O. 4, w=1.4 the fixed point ~ in the Poincare plane is inverse
hyperbolic. Its eigenvalues are J.il",-2.09, J.iZ=l/J.il' Now we look at the
stable manifold WS (1) and the unstable manifold WU (1) of ~ in the
Poincare plane. We take a short segment S=(x1,X Z) of WU close to ~ such
that xZ=MZ(x 1) and plot a few iterates of S in Fig.8. In total, WU (1) has
an infinite number of tendrils reaching out to unlimited distances. And
each tendril is a fractal arrangement of an infinite number of lines,
going in and out an infinite number of times. In the figures we only can
show a few tendrils and only a few branches within each tendril by
plotting a finite number of iterates of a finite number of points within
the initial segment S ( we have chosen 2000 points for Figs.8 and 9 ).
For q~lO the Morse potential can not be distinguished from zero
within the numerical accuracy. In this outside region the Poincare map
acts quite simply :
(6)

This action of Mas in mind, the reader can imagine how the plot would be
supplemented under continued addition of further iterates of the points
already present. All points of Wu(')') correspond to trajectories which
converge towards 1 for t~-oo.
In the same way we take a short segment R=(X 3 ,x 4 ) of WS (1), such
that x 3=Mz (x 4 ), and plot a few backward iterates of R in Fig.9. All points
of W'(1) correspond to trajectories which converge towards 1 for t~. The
comments to Fig. 8 apply also to Fig. 9, we only have to reverse the
direction of time.
Fig.lO shows a smaller neighborhood of ~ in the Poincare plane and
a few branches of WS (1) and WU (1) in this region. We see intersections and
homoclinic tangencies between Ws and Wu . Most important: If there is any
homoclinic point, then all images and preimages of this point under Mare
homoclinic points too, an infinite number of homoclinic points exist. They
correspond to trajectories which converge towards 1 for t~oo as well as
for t~-oo. The existence of homoclinic points of M implies the existence
of an infinite number of periodic points of arbitrarily high period and
an overcountable number of aperiodic localized points of M. We call a
point x of the Poincare plane localized, i f the q-coordinate of Mn(x)
stays limited for n~ and for n~-oo.
The localized points form a chaotic invariant set 11.' which is
contained in the topological closure of the homoclinic points. Other
points, not belonging to 11.', escape to q~ for t~ or for t~-oo.
Therefore, the chaotic invariant set A' in the Poincare plane corresponds
to a chaotic invariant set II. in the extended phase space, a so called
chaotic saddle. II. has measure zero and is a fractal arrangement of an
overcountable number of localized trajectories.
SCATIERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 229

2.0

-2.0
-1.0 Q 15.0
Fig.9: The first few tendrils of the stable manifold WS(~) of the fixed
point ~ for parameter values A=O.4, w=1.4.

0.0
..:,: "
.... #

::
........ ... .
.. .... ,
..... ~....:... -............
... ...... .
".
,', .... "of. '. .,
., .'t'... ) ..".: ...•..
','~',:"""",* ~ :'-'"
-1.0
........ .,... ,,'." .

-0.5 Q 2.5
Fig.10: Homoclinic intersections between WU(~) and WS(~) in the
neighborhood of ~ for parameter values A=O.4, w=1.4.
230 C.JUNG

Now we are ready to explain the occurence of scattering chaos : The

unstable trajectories in A all have their stable and unstable manifolds
which reach out into the asymptotic region. Incoming scattering
trajectories which start exactly on a stable manifold of a localized
orbi t, will converge towards this orbit and never escape again. This
subset of incoming asymptotes, which lead to capture, has measure zero in
the set of all incoming asymptotes. Let us have a look, how this set of
exceptional initial conditions looks like in the (pin'~in)-plane.
In out model system at parameter values A-0.4, ~1.4 we have only
one global chaotic region in the Poincare plane. Therefore, the homoclinic
structure and the structure of the invariant manifolds of anyone
particular periodic point is representative of the corresponding
structures of any other periodic point. The topological closures of the
homoclinic structures of all periodic points coincide. We choose as a
representative the most simple periodic point, namely ~. We place 10000
points on W'(r) and follow the trajectories through these points backwards
in time to the incoming asymptotic region. In the (pin'~in)-plane we mark
the values reached by these 10000 trajectories. The result is shown in
Fig.n.
Comparison with Fig.l shows, that along the line Pin=-l.O the
intersections with W'(r) mark exactly the boundaries of chaotic clusters
in Fig.l. This is easy to understand: Scattering trajectories become
complicated, when they start close to W. of some localized orbit. And as
explained above, the accumulation points of W'(r) are representative of
W' of any localized orbit and therefore also of W' for the whole of A.
Genuine scattering trajectories have in and out asymptotes,
therefore they are never chaotic themselves. However, they can follow
localized chaotic orbits for a finite time and trace out finite segments
of chaotic motion. The closer they start to W' of any localized orbit, the
longer they stay in the neighborhood of A and the larger the time delay
Dt becomes. In the limit of large Dt the relative probability of finding
a time delay Dt is given by
(7)
P (Dt) -exp ( -k·Dt)

where k is the average repellation rate of A, it is a measure for the

instability of A. Fig.12 gives a numerical distribution of Dt for the
parameter values A=0.4, ~1.4. We have taken 10 6 trajectories all with
Pin=-l. 0 and ~in distributed evenly in the interval (0,211'). The Dt -axis has
been cut into boxes of length ~=0.05 and hits of the 10 6 trajectories into
the various boxes have been counted. Fig.12 displays the logarithm of the
counts versus the mean delay time of the boxes. From this plot we read off
the value k:::::6.8.
In total we can characterize the situation of scattering chaos like
this : The bundle of incoming scattering trajectories hits A. Only a
subset of measure zero gets stuck. The other trajectories fly through the
gaps of A and transport some kind of shadow image of A into the outgoing
asymptotic region, where we can perform measurements. From the results of
these observations we can reconstruct the fractal properties of A. With
SCATIERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 231

0.0
..- ...... ', .; ..: ... :. ....... ..
~ ~
; •••.:... ; ....... •• •• • ::.....:: J' . ,•• ,,'

-2.0
0.0 PHASE 6.28
Fig.ll: Transport of 10000 points of WS(~) into the asymptotic (Pin'~in)­
plane for parameter values A-0.4, w=1.4.

12.0 x - - - - - - - - - - - - - .
1- . - . . . . . . . . -

X
X
Xx
X
Xx
Xx
X
X
XXX
Xx
Xx
4.0-l---------------'')(~~~
0.0 DT 1.1
Fig.12: Distributions of time delays. The vertical axis gives In(P(Dt))
versus Dt on the horizontal axis. The parameter values are A=0.4, w=1.4.
232 C.JUNG

this picture in mind, we interpret scattering chaos as a version of

transient chaos [27,28].

6. TRANSITION PROBABILITIES

The most important measurable quantity in field modified scattering is the

transition probability to various final energies of the projectile.
Usually in an experiment Pin is fixed and the phase 'Pin is evenly
distributed in (0, 21r). A detector registers outgoing particles with a
particular value Pout of the final momentum or a particular value Eout of
the final energy. We imagine that the field is switched off adiabatically
in the asymptotic region, such that Eout~Pout 2 /2.
To the relative count rates a(pout) or a(Eout) all trajectories
contribute, which end with this particular value of Pout or Eout ' These
trajectories can be read off from the plot of the function Pout ('Pin)
plotted in Figs .1- 3. In Fig.l let us imagine a horizontal line at a
particular value of Pout and register all intersections of this line with
the curve Pout('Pin)' These intersections mark the initial phases 'Pj of
those trajectories which contribute to a(pout) or a(E out )' In the case of
chaotic scattering there is an infinite number of them. Each one gives the
weight c j to the count rate a(E out ), where

c·-1 (E) - II-dEout (8)

-(qI·)
J dqlill J

The total transition probability to find Eout is given by

(9)

Fig.13 shows the numerical example of a(E out ) for the parameter values
A~O. 4, w=1.4 and the incoming momentum Pin=-l. O. 10 6 trajectories have
been started with 'Pin evenly distributed in the interval (0,271"). The
outgoing energy interval (0,2.5) has been cut into 2500 boxes of length
8E=10-3 and hits of the 10 6 trajectories into the various boxes have been
counted. In Fig.13 the vertical axis gives the logarithm of the count
rates versus the final energy on the horizontal axis.
The curve Pout('Pin) displayed in Figs.1-3 has an infinite number of
relative extrema, at least one maximum for each interval of continuity.
Accordingly, a(E out ) has an infinite number of singularities of the
qualitative shape c'CE-E j )-1/2 for E<E j and 0 for E>E j . In Fig.13 only a
finite number of these singularities is well resolved. The maxima from the
various intervals of continuity lie at different values of Ej . On the
energy axis they are arranged in a fractal pattern which is a projection
of the fractal structure of the chaotic set A in the extended phase space
( compare the corresponding discussion of angle singularities in the
differential cross section given in [19] ). Because of the limited
accuracy of any measurement of aCE out ) only a few highest levels of the
fractal structure can be resolved.
Here is a brief outline of the explanation, why the pattern of these
SCATIERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 233

singularities reflects the fractal structure of A : First imagine the

transport of WU (l) into the As out plane. It gives a picture in complete
analogy to the transport of WS ( 1) into the As in plane shown in Fig .11.
Just imagine Fig.ll to be turned upside down. Again WU (l) is
representative for WU of the whole of A. WU (l) in As out has an infinite
number of branches. Each branch j is just one line starting at p=O, going
to a maximal value Pj of Pout and winding back to p-O. Remember : We
identify rp=0 with rp=2'lf, therefore As out is a cylinder. The set of all
branches of WU ( 1) in As out forms a fractal arrangement, reflecting the
fractal structure of A. Accordingly, also the set of Pj values forms a
Cantor set along the pout-axis. The set of outgoing asymptotes which are
reached by the trajectories starting inside one particular interval of
continuity forms a line in As out of qualitatively the same shape as one
branch of WU ( 1). The set of accumulation points of the images of all
intervals coincides with the accumulation points of WU (l). Accordingly,
the set of maximal pout-values of images of intervals define the same
Cantor set as the set of Pj-values coming from maximal values of branches
of WU (l).

7. LOW FREQUENCY LIMIT

So far we have considered the case w~. What happens for other field
frequencies? For ~o the external field is so fast, that the particle is
not able to follow. The external field is essentially averaged out to zero
and no chaotic effects induced by the field are left.
In the other extreme w<{(J no chaos occurs either. Let us look at this
low frequency limit closer. It is of some interest, since the experiments
[1-7) done so far have used infrared lasers whose frequency w is a lot
smaller than atomic energies. Also the theories [8-11) are based on the
low frequency limit.
The experiments have been done with incoming energies Ein which are
large compared to the photon energy and to the interaction energy with the
field. No trapping of incoming scattering trajectories occurs. In the
picture of Fig.l1 the incoming momentum is always far away from p-va1ues
which are reaches by WS of A. Therefore, the time which the projectile
spends inside the interaction region of the local potential is very small
compared to Te. Then it is not necessary to take into account the
variation of the field phase during the projectile-target interaction
process. Outside this interaction region the canonical momentum p is a
conserved quantity anyway, in the outside region field induced transitions
to other momenta never occur. The field phase at which the projectile-
target scattering event takes place is just the reduced phase ~ given in
(3) for the incoming asymptote. Formally we define the phase l/J for any
point (p,q,t) of the extended phase space by
(10)
1/1=lim [Ult-q (t) ·Ul/p(t) -A·sin(Ult) /p(t) 1
234 C.JUNG

2.0-r------------:---,

-2.0+------------L..---;
0.0 ENERGY 2.5
Fig.13: Energy transfer cross section for Pin=-l.O and parameter values
A=O.4, w=1.4. The vertical axis gives In(a(E out )) versus Eout on the
horizontal axis.

-
~ ---- --------------------------------
-

----------------------------~
~

~ ---- --------------------------------
- ~

Fig.14: Phase portrait in the low frequency limit. Part a shows the case
of cos ~O. Part b shows the case of A·cos ~A. The q-axis is marked by
a broken line.
SCATIERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 235

where the limit t~-oo is taken along the actual trajectory through the
point (p,q,t). ~ is an approximate invariant of the motion in the limit
of small w. The scattering event is governed by the Hamiltonian
(11)
H= (p-A·cos (IJI) )2/2+V(q)

In Fig;l4 the phase portrait of (11) in the (p,q) plane is shown for 2
values of ~. Part a displays a line H-const. for the case of ~~/2 mod 2~,
which coincides with the field free case. Here the phase portrait curves
connect incoming asymptotes with a given value of Pin<O with the outgoing
asymptote with momentum Pout=-Pin. This is the usual picture for elastic
scattering. Part b gives the case of A·cOS(~)=A~O. The line p-O is marked
by a broken line in the plots.
Comparison of a and b shows that in the case of cos ~ ~O the plot
looks as if the plot of the field free case would have been shifted by
A=A cos ~. In this case such values of Pin and Pout are connected by a
trajectory, for which (pout-A)=-(Pin-A). The difference in energy between
the incoming and the outgoing asymptote is
2 2
4E. P out_p iII_- 2 A.p. +2A.2 (12)
22 m

Turned the other way around : It is possible, to connect the incoming

momentum Pin with a particular value of Pout' if IIPinl-IPoutll<2A. The
appropriate initial reduced phase is the one fulfilling

A-A·cos (IJI) _ 1P00001-lPinl (13 )

2

The probability to find an energy transfer of t>E is equal to the

probability to find an appropriate value of A or~. If incoming phases are
distributed evenly in (O,2~), then this probability to find
~ arccos (A/A) is given by

(14)

Compare similar considerations in [29). The scattering of the projectile

off the target is the same as elastic scattering with incoming momentum
p'in-Pin-A into outgoing momentum p' out=Pout-A. To obtain a cross section we
compare ingoing and outgoing fluxes. Therefore in the inelastic case a
momentum ratio is included. In total we obtain
(15)
236 C. JUNG

where 0 is the cross section with field and 0el is the corresponding cross
section without field.
In the 3-dimensional case we have a momentum shift only in the
direction of the polarization of the field. A and A become vectors
pointing in the direction of the polarization. The quantum counterpart of
these transitions only exist for ~E=Nhw,NEZ and the counterpart of (14)
is given by the square of a Bessel function ( see discussion in [29)
(16)

Inserting (16) into (15) explains the result obtained in [8).

Because of the existence of the adiabatic invariant (10) in the
extended phase space, the problem has become completely integrable and
there is no room left for chaos generated by the interaction with the
external field.

8. FINAL REMARKS

In this article some ideas on chaotic scattering coming out of the

workshop of nonlinear dynamics have been applied to the harmonically
driven Morse system. In the numerical computations the case of Pin=-l.O
has been demonstrated. What happens for other values of Pin? Fig.ll gives
an overview :
There is a critical value Pc",-1.8, such that for PinE(pc,O) the
stable manifolds of II will be crossed, when cp is scanned, and chaos
occurs. For Pin<Pc the incoming asymptotes do not meet WS of II.
Accordingly, the trajectories do never come close to II and the scattering
process is not chaotic. The scattering function Pout(CPin) will be smooth
and Dt('Pin) will be smooth and always finite.
So far we have looked at the case of A=0.4. For increasing A the
invariant manifolds of II spread over a larger range of p-values and
chaotic scattering occurs for a larger interval of incoming p-values, i. e.
Pc becomes smaller. In addition, the eigenvalue ~1 of 1 becomes smaller
( its absolute value increases) and II becomes more unstable. For
A<A c"'0.35, cP becomes stable and large scale KAM tori exist, which are
sticky for incoming scattering trajectories. A small amount of scattering
trajectories can stay inside the potential well for a longer time than
what is predicted by the exponential law (7). Trajectories inside KAM tori
are bound in the potential well for all times in past and future. The
field is not strong enough to dissociate these states. If 1 is stable and
lies inside KAM tori, then the invariant manifolds of II can not be
represented by the invariant manifolds of 1, which do not exist at all in
this case. However, the invariant manifolds of some other unstable
periodic orbit, which lies outside of all KAM tori, do the job. For A
becoming small, the invariant manifolds of II occupy only small strips in
the phase space and in the asymptotic region they concentrate in a region
near p=O, whose width shrinks to zero in the limit A~O. Otherwise the
general szenario is similar to the case of A=0.4 always.
Our whole discussion has been given wi thin classical dynamics.
However, scattering investigations are an important source of information
in the microworld, where quantum effects are essential. The question of
SCATIERING CHAOS IN THE HARMONICALLY DRIVEN MORSE SYSTEM 237

chaos in quantum dynamics is a very delicate one, because chaos in the

classical form can never occur in quantum dynamics because of the
uncertaincy. The complicated fractal structures which are characteristic
of classical chaos, are washed out in the quantum world. If we look at
sufficiently small scales, everything becomes smooth in quantum mechanics
( for the general problems of quantum chaos see the review [30]).
The best we can do, is to treat classically chaotic systems with
semiclassical methods and to look for ways in which features of classical
chaos have influence on semiclassical quantities. So far not much has been
done in the field of quantum scattering chaos and scattering in the
quantum version of system (1) has not yet been investigated. Therefore we
only mention a few ideas, which have emerged in the semiclassical
treatment of some other chaotic scattering systems. And we suggest that
things in system (1) behave in analogy.
The semiclassical scattering amplitude is a sum of terms from the
various contributing classical trajectories, which occur in the sum in
(9). Accordingly, the semiclassical cross section contains interference
oscillations from cross terms between all terms in the amplitude.
A first idea is, to look for chaotic effects in the statistical
properties of the fluctuations of the amplitude and the cross section. It
has been found, that the quantum cross section of a classically chaotic
scattering system has fluctuation properties of the kind we expect for a
random matrix system [13,31,32]. Random matrix systems are believed to be
generic quantum systems in the same way as chaotic systems are generic
classical systems. So the connection between classically chaotic
scattering systems and quantum random matrix behavior fits into the
general picture.
A second line of thought is to consider the wild fluctuations of the
quantum S-matrix as coming from very many poles and to express these poles
semiclassically by the unstable periodic orbits of the classical system
[33]. This idea runs in parallel to the expression of the poles of the
Greens function of bound systems by unstable periodic orbits [34].
A further idea is to Fourier transform the quantum cross section and
to interpret the complicated and fractally clustered arrangement of
oscillation frequencies as a washed out image of the fractal structure of
the classical chaotic saddle A [35,36]. Of course, this idea only makes
sense, if the system is already in the semiclassical limit, i.e. if the
wavelength of the incoming projectile is very small compared to the size
of the target. Then very many angular momentum values contribute to the
scattering cross section and in the inelastic case very many different
final states can be reached from the initial state. In the case of free-
free transitions this means, that the photon energy is very small compared
to the typical energy change of the electron. Then the discreteness of the
possible final energies is not important. In this case we can try to find
clusters of singularities in the cross section as a function of final
energy in analogy to the classical result of Fig.13.

9. REFERENCES

1. Weingartshofer A., Holmes J., Caudle G., Clarke E. and Kruger H.

238 C. JUNG

(1977) Phys. Rev. Lett. 39, 269

2. Weingartshofer A., Clarke E., Holmes J. and Jung C. (1979) Phys. Rev.
A 19, 2371
3. Weingartshofer A., Holmes J., Sabbagh J. and Chin S. (1983)
J. Phys. B: At. Mol. Phys. 16, 1805
4. Wallbank B., Connors V., Holmes J. and Weingartshofer A. (1987)
J. Phys. B: At. Mol. Phys. 20, L833
5. Wallbank B., Holmes J. and Weingartshofer A. (1987) J. Phys. B:
At. Mol. Phys. 20, 6121
6. Andrick D. and Langhans L. (1976) J. Phys. B: At. Mol. Phys. 9, L459
7. Andrick D. and Langhans L. (1978) J. Phys. B: At. Mol. Phys. 11,2355
8. Kroll N. and Watson K. (1973) Phys. Rev. A 8, 804
9. Kruger H. and Jung C. (1978) Phys. Rev. A 17, 1706
10. Mittleman M. (1979) Phys. Rev. A 19, 134
11. Rosenberg L. (1979) Phys. Rev. A 20, 275
12. Eckhardt B. (1988) Physica D 33, 89
13. Smi1ansky U. (1990) Course X of the Les Houches Session LII, ed.
Giannoni M., Voros A. and Zinn-Justin J., Elsevier, New York
14. Wiesenfe1d L. (1990) Phys. Lett. A 144, 467
15. Goggin M. and Mi10nni P. (1988) Phys. Rev. A 37, 796
16. Heaggy J. and Yuan J. (1990) Phys. Rev. A 41, 571
17. Jung C. and Scholz H-J. (1987) J. Phys. A: Math. Gen. 20, 3607
18. Jung C. and Scholz H-J. (1988) J. Phys. A: Math. Gen. 21, 2301
19. Jung C. and Pott S. (1989) J. Phys. A: Math. Gen. 22, 2925
20. Jung C. and Richter P. (1990) J. Phys. A: Math. Gen. 23, 2847
21. Arnold V. (1978) Mathematical Methods of Classical Mechanics,
Springer, New York
22. Abraham R. and Marsden J. (1978) Foundations of Mechanics,
2nd ed., Benjamin Cummings, Reading
23. Lichtenberg A. and Lieberman M. (1983) Regular and Stochastic
Motion, Springer, New York
24. Guckenheimer J. and Holmes P. (1983) Nonlinear Oscillations, Dynamical
Systems and Bifurcations of Vector Fields, Springer, New York
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27. Tel T. (1989) J. Phys. A: Math. Gen. 22, L691
28. Kovacs Z. and Tel T. (1990) Phys. Rev. Lett. 64, 1617
29. Friedland L. (1979) J. Phys. B: At. Mol. Phys. 12, 409
30. Eckhardt B. (1988) Phys. Reports 163, 205
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32. B1Umel R. and Smi1ansky U. (1990) Phys. Rev. Lett. 64, 241
33. Cvitanovic P. and Eckhardt B. (1989) Phys. Rev. Lett. 63, 823
34. Gutzwiller M. (1971) J. Math. Phys. 12, 343
35. Jung C. (1990) J. Phys. A: Math. Gen. 23, 1217
36. Jung C. and Pott S. (1990) J. Phys. A: Math. Gen. 23, in print
EXPERIMENTS WITH SINGLE ELECTRONS*

Robert S. Van Dyck, Jr., Paul B. Schwinberg, and Hans G. Dehmelt

Department of Physics, FM-15
University of Washington
Seattle, Washington 98195
USA

ABSTRACT: A basic description of past geonium experiments is given and the modifi-
cations which allow positron geonium to be formed is described. The use of compensated
Penning traps produces a harmonic axial frequency which has a resolution of 10 ppb. By
using synchronous detection and a magnetic bottle for coupling, the magnetic resonances
become observable. Stability of the radial position in this magnetic bottle is provided by
motional (magnetron) sideband cooling. The corresponding magnetic line shapes are pri-
marily determined by the Brownian statistical (axial) motion through this bottle. Finally
the beat-note between the nearly degenerate cyclotron and spin precession frequencies de-
fines the anomaly resonance and its value can be determined to ~l ppb statistical precision
by fitting to the noise-modulated Brownian lineshape. Present accuracy has produced the
most precise determination of the Q -1 (QED) and the positron/electron g-factor compari-
son. Results of new measurements using a phosphor bronze trap are also described which
show consistency with previous results using all-molybdenum traps.

1. Introduction

The importance of conducting experiments on this particle (or its anti-particle)

resides in our intrinsic interest in understanding the relatively simple system we
call "the electron". One of the earliest opportunities in recent history at which it
became clear that the electron's structur~ was not well understood, was provided
by the first accurate measurement [1] in 1947 of the hfs in hydrogen and deuterium.
A puzzling 0.2% discrepancy occured between the measured and predicted values of
this frequency interval. The discrepancy was ultimately traced to the electron's g-
factor (the ratio of its magnetic moment and its intrinsic angular momentum) where
the prevailing theory by Dirac had predicted this g-factor to be 2.000,000,000,000.
The interest to understand the electron's intrinsic structure (in addition to the
desire to explain such energy differences as the Lamb shift in hydrogen [2]) helped

* Supported by a grant from the National Science Foundation.

239
D. Hestenes and A. Weingartshofer (eds.), The Electron, 239-293.
© 1991 Kluwer Academic Publishers.
240 R. S. VAN DYCK, Jr. ET AL.

to inspire the development of the theory of quantum electrodynamics (see Sec. 7 for
a very precise g-factor prediction).
The first experiment specifically designed to measure the free electron's g-factor
was conducted by Louisell, Pidd, and Crane [3] in 1953, but their accuracy of 1% did
not provide an experimental value for the anomalous part "ae" of the free electron's
g-factor defined by
9 = 2(1 + a e ) . (1)
Inspired by this pioneering effort, Dehmelt devised a simple experiment in 1958 in
which electrons are stored in the field of a positive-ion cloud diffusing slowly in a
dense inert gas. This experiment yielded the first direct measurement [4] of the
anomaly (with a 3% accuracy). When he substituted the quadrupole trap, similar
to that shown in Fig. 1, for the optical pumping cell, the illustrious history of
the "Penning trap" was initiated at the University of Washington. Much of this
early history can be found in a number of previous reviews with various depths of
treatment [5-12] and at least one popular review [13] of the experiment is available.
The work initiated by Dehmelt led to the first non-destructive detection in 1968
of electron clouds in the Penning trap at very low pressures « 10- 11 Torr), cooled
by axial coupling to a resonant tuned circuit [14,15]. Then, noting the obvious
limitation on space charge, Dehmelt and colleagues leaped to the logical conclusion
that the cloud must be reduced to its irreducible limit, i.e. a single electron, ca-
pable of being continuously observed for several days (and months). In 1973, the
successfully observed [16] "monoelectron oscillator" made it attractive to complete
the development of the second critical aspect of the geonium experiment, that is, the
axial-frequency-shift detector [17], also referred to as the "continuous Stern Gerlach
effect" [18) (see Sec. 3). This initial fervor of activity culminated in 1977 with the
first ever high precision g-factor measurement on a single free electron [19].
The significance of the present series of experiments, using single charged parti-
cles, lies in our ability to measure some of its fundamental properties to a very high
precision in an environment which is relatively free of unwanted or unaccountable
perturbations. In addition, we have demonstrated the ability to isolate both the
electron or the positron for an indefinite period of time, making it quite sensitive
to mild rf stimulation and detection (subject of Sec. 3). For these experiments, the
particle is embedded in a very strong magnetic field which causes it to rotate in a
circular orbit within a plane perpendicular to the magnetic field; thus its magnetic
moment and charge are made to execute precessional and cycloidal motions in this
field (with respective frequencies w s and We that are nearly degenerate). The device
which we call the Penning trap is used to constrain motion along the magnetic field
direction; however, it itself alters the frequencies of these radial motions due to the
 EXPERIMENTS WITH SINGLE ELECTRONS 241

symmetry
aXIs z
(0)

vo
o

( b)

Figure 1. Electric trapping potentials defined by the Penning trap. (a) Electrode surfaces
(two endcaps and one ring) are placed upon a given set of equipotentials, amongst the
family that satisfies Laplace's equation, by choosing a particular Ro and Zo. The z-axis
is the rotational axis of symmetry. (h) An axial frequency is obtained by choosing the
appropriate bias potential Va for the ring relative to the endcaps (from Ref. 7).

electric trapping fields. Because these fields produce the first order perturbations
to the observable magnetic frequencies, the trapping device will be described first
(see Sec. 2). Because the position of the particle now becomes important due to
the specific detection mechanism discussed in Sec. 3, the method used to center
the particle is treated next in Sec. 4. The two magnetic modes (cyclotron and
precession) are then treated in the following two sections respectively. In all these
modes, we discuss the relevant perturbations to the motions in their respective
sections. Finally, in Sec. 7, the results of our measurements are presented with a
242 R. S. VAN DYCK. Jr. ET AL.

signal voltage
x -band (axial resonance)

tuned circuit
}
(60-70 MHz)

trapping
electrodes
(hyperbolic)

"'--1I'""""II/V','VV'" ax i a I d rive
(v z )

superconducting solenoid

Figure 2. Schematic of the geonium apparatus. The appropriately biased hyperbolic

endcaps and ring electrodes trap the charge axially while coupling the driven harmonic
motion to an external LC circuit tuned to the driven axial frequency. Radial trapping
of the charge is produced by the strong magnetic field obtained from a superconducting
solenoid. Also shown are the microwave multiplier diode and nickel ring for magnetic
coupling to the axial resonance (from Ref. 6).

particular emphasis on the new phosphor bronze trap which was constructed specif-
ically to investigate possible systematics encountered in these experiments.

2. Trapping Device

2.1. BASIC DESCRIPTION

The Penning trap is shown schematically in both Figs. 1 and 2 as a three electrode
trap with cylindrical symmetry. The magnetic field, Eo, (obtained from a stable
superconducting solenoid) is applied along this symmetry axis and is designated
by convention as the z-axis. Using cylindrical coordinates r, z, these electrodes are
machined along hyperboloids of revolution (relative to the z-axis) given by:
z2 = Z~ + r 2 /2 (endeaps )
(2)
r2 = R~ + 2z2 (ring)
where 2Zo and 2Ro represent the minimum endeap separation and the minimum
EXPERIMENTS WITH SINGLE ELECTRONS 243

ring diameter respectively. A potential Va is applied to the ring electrode relative

to the grounded endcaps by means of the standard cells shown in Fig. 2; as a result,
the following harmonic potential (shown in Fig. l(b)) is established:
r2 - 2z2
V(r, z) = Va 4d 2 (3)

where d is the characteristic trap dimension defined by 4d2 = 2Z~ + R~. This volt-
age distribution then produces the harmonic restoring force that binds the charge
axially, assuming the correct sign of the potential has been chosen.
Immersing the entire apparatus in liquid helium produces a low background gas
environment in these devices. This low ambient temperature also improves signal-
to-noise (S / N) of the detector such that single charged particles can be readily
observed. The endcap opposite the LC circuit is used to drive the axial motion
which then induces an image current into this external circuit, tuned to the particle's
axial frequency. The resulting voltage across the tuned circuit is amplified by the
following preamplifier. The magnetic bottle, shown schematically as a nickel wire
in Fig. 2 is used to couple the total magnetic moment to the axial resonance. The
microwave power which drives the observed cyclotron motion is obtained from an
X -band source that is multiplied up to the appropriate frequency by a Schottky
diode placed inside the vacuum envelop between an endcap and the ring.
The motion of the trapped electron is described for an ideal Penning trap by the
following set of equations:
2 eVO
Wz = ·-d
me
2 (4a)

eB a
We =-- (4b)
me c
(4c)
The first frequency in Eq. 4 (for w z ) is associated with the axial harmonic oscillation
of the trapped particle and is the subject of Sec. 3. The second is its unperturbed
cyclotron frequency (in the absence of the Penning trap). The last equation rep-
resents the radial motion; in the ideal case, w_ == 8e is the frequency of the slow
magnetron motion (the subject of Sec. 4) and W+ == w~ is the perturbed cyclotron
frequency (the subject of Sec. 5). The composite motion is then a superposition of
all three normal mode frequencies, as shown in Fig. 3.
For the ideal trap, it follows that

(5)
244 R. S. VAN DYCK, Jr. ET AL.

Three Modes of Motion

Figure 3. Illustration of the motion of a charge isolated within a Penning trap. The
magnetron motion is characterized by an orbit radius (~ 0.014 mm) which is at least
1000 times larger than the typical cyclotron radius. Driven axial amplitudes are also
typically less than the magnetron radius since they are usually less than the rms-thermal
amplitude (~0.018 mm) (adapted from Ref. 13).

However, in real traps, we find that the observed magnetron frequency, W m , is

slightly larger than be due to trap imperfections and a non-zero angle between Eo
and z.But, since Wm <i( W z <i( w~, we can ignore the difference between be and wm ·
The last of the important frequencies is associated with the spin magnetic mo-
ment of the electron. It is purely quantum mechanical in nature and is given by

(6)

where J-lB is the Bohr magneton (/-LB == eh/2m e c). Producing a strong enough
oscillating magnetic field at the very high frequency Ws (vs ~ 140 GHz) turns out
to be somewhat difficult. Using the near degeneracy of Ws and w~, it is far more
convenient to excite the difference frequency, w~ = Ws - w~, which is referred to as
the anomaly frequency and is the subject of Sec. 6. Thus, in a Penning trap, the
anomaly is obtained from

(7)

which turns out to be quite accurate and insensitive to all leading perturbations
such as angle-tilt and electrode eccentricity 120], as well the anharmonicity of the
trapping field.
EXPERIMENTS WITH SINGLE ELECTRONS 24S

Figure 4. Cross-section of the double trap apparatus used in the electron-positron com-
parison experiment. An OFHC copper pin base with non-magnetic feedthrus is sealed to
an all-metal beryllium copper envelope via a compressed indium O-ring. The lower trap is
a well compensated Penning trap used for precision measurements; the upper trap is used
only for the storage of positrons which can be transferred into the lower "experiment" trap
when needed (from Ref. 6).
246 R. S. VAN DYCK, Jr. ET AL.

2.2. A VACUUM TUBE

The present apparatus containing two traps in series is shown in Fig. 4. Initially,
both traps consisted of all-molybdenum electrodes, but recently the lower "exper-
iment" trap was rebuilt using copper and phosphor-bronze. The upper "storage"
trap was designed specifically to trap positrons and will be described in a following
subsection. A field emission point (FEP) is shown in the endcap closest to the
pin-base and is negatively biased relative to all endcaps in order to produce an
ionizing electron beam along the symmetry axis of each trap. The outer vacuum
envelope is made of beryllium copper with threaded rings (of the same material)
that are used to apply pressure to the indium melted into each end of the vacuum
tube. The pin bases are made of OFHC copper and each contains several cryo-
genic ceramic- to-constantan feedthrus. The upper pin base (not shown) is used for
mounting a sputter-ion pump whereas the lower base locates the experiment trap
at the solenoid's field center.

2.3. COMPENSATION FEATURE

One of the main features of our modified Penning traps is the use of compensation
electrodes [21] of the form shown in Fig. 4. These "guard" rings protrude into the
truncation region between endcap and ring electrodes. (Later versions were also split
on one side for "anomaly" excitation.) Several methods are available to tune up the
trap. The most common method uses the amplitude asymmetry versus frequency
as shown in Fig. 5. The procedure involves symmetrizing the axial resonance using
increasingly stronger drives. Until the 6th-order perturbation enters, symmetry
should occur at the same guard setting. A second method involves minimizing the
noise in the correction signal of the axial-frequency-shift detector described in the
Sec. 3.3. lipon varying the compensation potential, a de shift in the well depth is
also observed by means of this shift detector. The size of this effect was observed
to depend exponentially on how close the compensation electrodes are placed in
the truncation region relative to trap center and the size of the gap at truncation.
In addition, the ability to tune the trap was observed to scale with this shift in
well depth for symmetric Penning traps. Later, these observations were verified [22]
theoretically by Gabrielse using a numerical relaxation program which computes the
exact field in the real Penning trap. For Fig. 5, the normalized 4th-order coefficient,
C 4 , of the anharmonic potential is negative when the resonance pulls to the low
frequency side, thus indicating that the guard potential was set too low.
EXPERIMENTS WITH SINGLE ELECTRONS 247

r20Hz1
c
c::
Cl
.iii

c
')(
c
c::
Q)
>
.;:
'0

endcop drive frequency - -

Figure 5. Typical case for an anharmonic axial resonance swept down in frequency. When
the normalized coefficient, C 4 , of the 4th-order term in the potential is negative (i.e. the
guard potential is set too low), the resonance is pulled to low frequency side. Note that
10 dB less drive nearly restores the completely symmetrical line shape (from Ref. 12).

2.4. ADAPTATIONS FOR POSITRONS

Since the basic geonium apparatus can accept "+" or "-" charges simply by ap-
plying the potential of appropriate sign, the full sensitivity of the experiment is
available for positrons as well as for electrons with all the systematics that are
discussed in the following sections being the same for both. Thus, the principle
challenge for this variation of the geonium experiment is the trapping of a single
positron which was first demonstrated [23,24] by Schwinberg in 1979. The basic
mechanism for losing some of the positron's energy, such that it might fall into a
potential well, is by radiation damping of the axial motion due to the resistance of
the LC circuit tuned to W z . For the radial motion, synchrotron radiation quickly
damps the 50-100 ke V initial energy in the cyclotron motion within a few seconds
down to the thermal range.
Since the requirements imposed on the trapping electrodes for high precision
measurements are inconsistent with those imposed in order to achieve an adequate
capture rate, it was decided to have two traps, one of which would be optimized
248 R. S. VAN DYCK, Jr. ET AL.

Storage Trap Experiment Trap

S::~
probe V
o

Figure 6. Schematic of the double-trap configuration. The storage trap contains a sealed
22Na positron emitter located at 3R o/4. The sideband excitation (SBE) probes are used
in the radial centering process for off-axis-loaded positrons and position stabilizing in the
experiment trap. Each signal endcap is tuned to the axial frequency via an external
inductor for detection of axial motion, rf driven from an opposite endcap (from Ref. 47).

to capture the positrons. This double-trap configuration is shown schematically

in Fig. 6. The source of the positrons consists of a 2.6 yr half-life sodium-22 salt
in a sealed container and is visible in Fig. 4, located inside the largest endcap
(farthest from the pin-base). In order to fit both a field emission point (FEP)
and the source within the same endcap at the same radius, as well as to improve
the trapping rate, each is mounted at 3R o/4 ~ 3.55 mm from the central axis.
The FEP is required in order to optimize the centering process using an adequate
number of easily obtainable charges. The source hole through this endcap is 1.2 mm
in diameter and the line, parallel to the trap axis, which passes through this source
hole is referred to as the "load line". The positrons of interest (with 50-100 keY in
the radial motion and a cyclotron orbit smaller than the entrance hole) will travel
down this load line guided by the axial magnetic field, reflect at the opposite endcap,
reflect again at the source cap because the magnetron motion has rotated it away
from the load line, and finally lose enough energy after one magnetron period to be
permanently trapped. Only a very small fraction of those emitted satisfy all these
criteria, typically ~ 10- 9 . Due to the extremely anharmonic nature of this trap,
positrons loaded at 3R o/4 can not easily be made visible unless there are more than
~ 50 in the cloud. The cloud must first be cent.ered by a strong axial-magnetron
sideband drive (see Sec. 4) in order to observe and estimate its size or possibly
transfer part of it into the experiment trap.
The actual transfer is accomplished by pulsing the two adjacent endcaps (storage
trap signal cap and experiment trap drive cap) to the common ring potential for a
EXPERIMENTS WITH SINGLE ELECTRONS 249

ambient =4.2 oK 4
T = 1.7sec

3 en
0 c:
c: ~
01 ~

en

-
en
0
0
a.
)(
0 0
c:
2 ~

Q.l Q.l
> .c
~ E
~ ::J
c:

OL---~4----~8----~----~--~20
o
time (minutes)

Figure 7. Positron ejection record. Positrons transferred from storage to the experiment
trap are detected via strong off-resonance drives (with fixed amplitude and frequency).
Then, intense rf pulses at W z +w m are applied to the SBE probe in order to systematically
eject one charge at a time until only one remains (from Ref. 45 and 47).

few microseconds. Typical transfer efficiencies between 25 and 50% have been ob-
served; however with pulses much longer than 10 JiS, the transfer tends to be very
inefficient, possibly due to radial drifting during the passage between traps. Once
positrons have been identified to be in the experiment trap, their exact number is
estimated (see Sec. 3) and the excess beyond one are systematically ejected (see
Fig. 7) by using intense rf pulses, also at the sideband cooling frequency. The
rf amplitude of the ejection! cooling pulse is carefully adjusted such that at least
10 consecutive pulses are required in order tei eject one positron from the cloud.
Note however that empirically the probability for being thrown out increases as the
number drops since the individual charges have larger amplitudes as the number
decreases (i.e. center-of-mass amplitude scales inversely with the number trapped
250 R. S. VAN DYCK, Jr. ET AL.

c L

Figure 8. Input detection circuit with equivalent £c representing the trapped electron.
The trap's net capacity, C, is tuned out with an external inductor, L, yielding a large
u;,
parallel resistor, R, with equivalent noise generator, presumably at 4 K. The preamp
is represented as ideal except for the equivalent series noise generator depicted by u;,
which is presumably not at 4 K (adapted from Ref. 25).

for fixed drives). Once a single positron is isolated, the drive signal is reduced by
orders of magnitude in order to resolve the narrow (4-6 Hz) axial resonance, typical
of a well-compensated trap.

3. Axial Resonance

3.1. OBSERVATION OF TRAPPED CHARGE

As indicated in Fig. 2, the trapped particle is driven on one endcap and detected on
the other cap. A fairly straight forward calculation shows that driving the trapped
electron is equivalent to exciting a series £c tuned circuit [25]. For a single electron
with mass me and charge e, the series inductance becomes

£1 = 4meZg (8)
(C l e)2
where C 1 is a dimensionless constant of order unity which represents the finite ge-
ometry of the electrodes. Note that in this expression, C 1 e represents the effective
charge on the electron at trap center. For an infinite parallel plate arrangement,
C I = 1, but for a Penning trap, C I < 1 because the effective charge is reduced
by the attraction of some lines of force emanating from the electron onto the ring
electrode (and is theoretically predicted [26] to be 0.8). The electron oscillator in
the Penning trap is therefore replaced by the equivalent capacitor Cl in series with
equivalent inductance £1 such that w;= (£1 CJ)-l As shown in Fig. 8, the £c series
EXPERIMENTS WITH SINGLE ELECTRONS 251

TABLE 1. Summary of experimental parameters for the last two traps.

molybdenum phosphor
symbol description trap bronze trap
wz/27f axial frequency 64.0 MHz 72.7 MHz
wm/27f magnetron frequency 14.5 kHz 18.5 kHz
w~/27f cyclotron frequency 141 GHz 1424 GHz
w~/27f anomaly frequency 164 MHz 165 MHz
Iz/27f axial linewidth 6 Hz 6 Hz
6/27r magnetic bottle step-size 1.3 Hz ±1.8 Hz
Bo magnetic field 50.5 kG 50.7 kG
B2 magnetic bottle 155(4) G/cm 2 ±254(10) G/cm 2
Vo ring-endcap potential 10.2 volts 10.2 volts
2Zo minimum endcap separation 6.70 mm 6.29 mm
2Ro minimum ring diameter 9.47 mm 7.57 mm
Rm typical magnetron radius ~ 0.0014 cm ,...., 0.0014 cm
Rc cyclotron radius for n = 0 114 A 114 A
T.e. trap constant 20.0 MHz/V 1 / 2 22.6 MHz/V 1 / 2
Q tuned circuit quality factor '" 1000 ,...., 900
C1 E-field coupling constant 0.78(4) ,...., 0.76
£1 single lepton inductance 4500 H 3600 H
C tuned circuit capacitance 15 pf 15 pf
R tuned circuit resistance 170 kfl 140 kfl
q effective anharmonic coefficient '" 5 X 10- 5 '" 1 X 10- 4
B~Z~/Bo ii-field inhomogeneity 3.4 X 10- 4 ±4.9 X 10- 4
eVo/2m ec 2 well depth/rest energy 1.0 X 10- 5 1.0 X 10- 5
kTz thermal energy, at Tz = 4.2 K 3.6 X 10- 4 eV 3.6 X 10- 4 eV
hw~ quantum of cyclotron energy 5.8 X 10- 4 eV 5.9 X 10- 4 eV
hw z quantum of axial energy 2.6 X 10- 7 eV 3.0 X 10- 7 eV
hW m quantum of magnetron energy 6 X 10- 11 eV 8 X 10- 11 eV
-eVoR!./2R~ typical magnetron energy '" -5 X 10- 5 eV '" -5 X 10- 5 eV

circuit is effectively connected in parallel with the net trap capacitance, C, which
is then put in parallel with an external inductor L such that the LC combination
is also resonant at W z . This external parallel tuned circuit then provides the large
series R that damps the axial motion.
Now, when the electron is subjected to an axial rf electric field, the driven motion
will induce image currents in the signal endcap or an equivalent current [27]
i = (C}e)i (9)
2Zo
will flow through the £c series combination. To be observed, we drop this current
across the large resistor R whose value is determined by the Q of the external tuned
circuit. As summarized in Table 1 for the recent molybdenum trap, some typical
252 R. S. VAN DYCK, Jr. ET AL.

parameters are C ~ 15pf, liz '" 64 MHz, Q ~ 1000, and the equivalent parallel
resistance is R = Q /wzC ~ 170kO. The steady-state axial motion, driven by an
electric field at frequency w with amplitude Erf, has an axial amplitude given by
(eErf/m e )
z = ------~~~~--~~ (10)
{(w; - w2)2 + 1;,lw2} 1/2
where Erf = C I Vrf/2Zo for a given rf voltage applied to the drive endcap only and
Iz,1 represents the single electron's damping coefficient for the harmonic motion at
W z . From Eq. 10, it follows that the relative driven on-resonance amplitude is Zd
Zd
Zo
= (CI2VoVrf) ( Wz
Iz,1
) ;::: Vrr/2.5/L V (11 )

where typically Vrf 10 n V. From the expression for total energy of a harmonic
Zn, the relative energy for a driven electron in a Penning trap is:
"v

oscillator (mew;

W k (Zd)2 ( 2Vrf )2 (12)

eVo = Zo = CleRw z

It should be noted that the formal definition of Q z of the axial resonance (Q z ==

wz/iz) is 27r times the average energy stored (.ei 2 ) divided by the energy lost/cycle
(power dissipated x period = i 2 R/ w z ), from which it also follows that the ideal
axiallinewidth (FWHM) is IZ = Rlf. For a 27r(6 Hz) wide line in the experiment
trap shown in Fig. 4, this relation predicts fl 4,500 H for a single electron.
"v

The equivalent current on resonance (w = w z ) can be found from Eq. 10 by

substituting i = wz into Eq. 9:
. (Cle)(eErr) ~~f
z- -- (13)
- 2/zmeZo - R
where the expression for fl has been used. For a moderate drive applied to the
endcaps which is effectively -160 dB down from a 13 dBm (or 1 volt rms) drive
source, one finds a typical current of ~ 10- 13 amps for a single electron with ~ 6 Hz
axiallinewidth. This current also corresponds to a drive energy that is about equal
to kTz when Tz = 4 K. As a result, the on-resonance signal amplitude is about 17
n V in this case. If ~lIdet is the detection bandwidth, then the signal voltage can be
compared to the rms noise voltage associated with a resistance R given by

(14)

which, for a 170 kO resistor in liquid helium, yields 6 n V /...;-Ih. As one can see in
Fig. 8, if the trap is ideal and the fc circuit is tuned to make it resonate at the parallel
tuned circuit frequency, the series circuit would totally short out the resistor Rand
EXPERIMENTS WITH SINGLE ELECTRONS 253

o
C
0'
Vl

.2
x
o
c
(])
.?
....
o

II - 59, 382, 500 Hz

Figure 9. Axial resonance signals at ::::: 60 MHz. The signal-to-noise ratio of this::::: 8 Hz
wide line corresponds to a frequency resolution of 10 ppb. Both absorption and dispersion
modes are shown with the latter mode appropriate for the frequency shift detection scheme
employed in these geonium experiments (from Ref. 5).

its equivalent noise generator u;.The remaining noise in the experiment would
then come from the series equivalent input noise generator u~ associated with the
imperfect preamplifier. However, the anharmonic part of the real trapping potential
keeps the finite temperature electron from totally shorting the trap. Experimentally,
we see the noise reduce by about 50% when on resonance relative to the off-resonance
noise. An example of the quadrature components of this axial resonance is shown
in Fig. 9. The resolution in this case is ~ 1 x 10- 8 or about 0.5 Hz out of 60 MHz,
limited by the voltage fluctuations in the standard cells shown in Fig. 2.

3.2. ISOLATING THE SINGLE ELECTRON

When the cloud contains N electrons, the center of mass contains NC1e effective
charge and moves with velocity zcrn, yielding (from Eq. 9) a total current
NC1ez cm
(15)
2Zo
However, inspection of Eq. 13 shows that on resonance and under ideal conditions,
the observable voltage signal is simply Vrj , the value of the rf potential applied to the
drive endcap only. There is no number dependence in this case. Instead, we detune
254 R. S. VAN DYCK, Jr. ET AL.

5
T am b=77.4°K
-0 T = 1.7 sec
c:

-e
<Il
01 4C:
'en
Q)
u u
Q)
c: G)
o
§ 3 ....
o
....
Q)
lis
.0
o
.;;: 2 E
:J
o c:
c:
Q)
.~
....
"0

o
time-
Figure 10. Identification of a single electron. A strong off-resonance signal (with fixed
frequency and amplitude) is applied continuously to the drive endcap of the trap. Then,
the FEP is turned on for'" 10 sec with,...., 0.1 nA of current until a nonzero signal is
observed. The trap is occasionally dumped and reloaded until the smallest quantized
signal is observed (from Ref. 5).

our drive oscillator by an amount Ow such that w. ~ Ow ~ ",1, Again taking

i = wz and evaluating Eq. 10 for w = w. ± ow, we find that the amplitude of the
velocity is
.
Zcrn 0
(ff) _- eErf (16)
2m.ow
which is the same for a single or N electrons. From Eq. 15, it follows that

v"ig(off) = N[ VrfR] (17)

2l 1 ow
for the off-resonance signal. If the rf amplitude and the detuning are held constant,
the signal voltage is proportional to the number of charge quanta in the trap.
To load this charge, a current of ,...., 0.1 nA is emitted from the field emission
electrode for 10-20 seconds. After each load, the detection system is switched
on and the signal observed. Then, the trap is dumped by reversing the sign of
the trapping potential. The experiment is repeated and the results are shown in
Fig. 10. Note that there is a minimum voltage step, of which all others are inte-
ger multiples. After the minimum voltage step is achieved, the drive is reduced to
the nominal on-resonance power and the frequency is swept through the resonance,
EXPERIMENTS WITH SINGLE ELECTRONS 255

m = -1/2 m =+ 112

4--- 3 5.2

3--- 2 3.9
8 liZ
(Hz)
2--- 2.6

1.3

a
Bo =50.5 kG To =42°K
. 8l1z = {1+2n+2ml a.65Hz

Figure 11. Lowest Rabi-Landau levels for a geonium atom. The axial frequency (shown
in the right-hand scale) corresponds to the coupling via the fixed magnetic bottle field for
the last molybdenum trap. The lowest state (n == 0) which is occupied by the electron or
positron 80-90% of the time differs by 1.3 Hz depending on the exact spin state. This is
the signature used to indicate that a spin has flipped (from Ref. 6).

obtaining the result shown in Fig. 9. We have also verified that at multiples of
this minimum voltage step, the axial resonance linewidth varies as multiples of the
Rill. The amplitude signal (which is the Lorentzian line shape) is used to monitor
the lock signal (see Sec. 3.3.) which is itself derived from the quadrature component
(or dispersion shape also shown in Fig. 9).

3.3. MAGNETIC COUPLING

The ultimate goal of this experiment is to observe the magnetic resonances, w~ and
Ws (or w:) but observation must come indirectly through the purely electrostatic
axial resonance. In order to provide a weak coupling to this frequency, a small
magnetic bottle is produced from a ring of nickel (as used in the early compensated
Penning traps) or from a superconducting loop of NbTi, either of which is placed
in the midplane of the ring electrode. The latter is the secondary of a transformer
whose current can be externally adjusted to produce what we refer to as our "vari-
able bottle" [28]. If B2 represents the quadratic gradient, the net bottle field can
be described in cylindrical coordinates, r, z, by
(18)
256 R. S. VAN DYCK, Jr. ET AL.

I-<_-----.,omp. >----1

electron
equivalent
error
signal

correction voltage
(corresponds to
frequency shifts)

Figure 12. A schematic representation of the superheterodyne system used to detect the
electron's motion. The upper half is the standard axial drive and synchronous detection
whereas the lower half generates the frequency shift information. The feedback signal is
generated by mixing the phase-shifted drive synthesizer with the amplified electron signal
to produce an error voltage. This error is integrated and fed back as a correction voltage
to the ring bias circuitry, thereby producing a frequency lock (from Ref. 5).

where Bo is the resultant constant-field term.

The coupling to the axial frequency follows from the orientational potential en-
ergy associated with the total magnetic moment ji of the electron relative to the total
field B: W m = - ji . B. This term can now be added to the electrostatic potential
energy, - e F( r, z), to yield the resultant axial frequency [7]. When combined with
the total magnetic moment for the electron, we obtain the quantum-mechanically
correct expression (with 9 ;:;:: 2):

Wz = Wz 0
,
+ (n + m + -21 + Wm)
-q b
w~
(19)

where n, m and q are cyclotron, spin and magnetron quantum numbers respectively
and b = 2I-£BB2/(meWz) ~ 27r(1.3 Hz) for electrons in the last molybdenum trap
shown in Fig. 4, before the variable-bottle trap. Figure 11 shows the Landau level
scheme for this "geonium atom" and the associated shifts in the axial frequency.
In order to observe these shifts, the axial motion is incorporated into a feed-
back loop as illustrated in Fig. 12. In this apparatus, the dispersion-shaped axial
resonance (shown in Fig. 9) is used as an error signal applied to an integrator, whose
EXPERIMENTS WITH SINGLE ELECTRONS 257

output fine-tunes the total ring voltage. If something causes W z to shift relative to
the synthesizer output, then a nonzero error is integrated until W z again coincides
with the synthesizer's reference frequency and the error returns to zero. The output
of the integrator is the correction signal which will now reflect any shift in the axial
frequency due to all sources (i.e. noisy voltage, anharmonic shifts with change in
stored energy, tilt in the trap axis, and in particular, cyclotron and/or spin quantum
jumps via the magnetic bottle).

3.4. OTHER AXIAL FREQUENCY SHIFTS

The largest perturbation is associated with the non-zero angle, 8, between the mag-
netic field and the axis of symmetry for the trap. The effect of such a perturbation
is to weakly couple all the normal mode frequencies producing small, but observable
shifts. In particular, when 8 « 1, Brown and Gabrielse have predicted [20] that the
observable axial frequency Wz is

(20)

(In principle, maximizing wz will remove the angle 8.) Typically, the relative shift
in the axial frequency is -38 2 /4 ~ 10- 5 for () ~ 0.0035 radians (or "'-' 0.2°). For
small fluctuations, 88, in the average residual angle e,
it follows from Eq. 20 that

(21)

Because the observed stability of W z due to all causes approaches 1 x 10- 8 , the
average fluctuations must be less than 0.1 % of e.
There are also a few small perturbations associated with the non-uniformity
of the magnetic field, the non-quadratic terms in the electric potential, and the
relativistic corrections. Their effects on all the normal modes have been studied by
Brown and Gabrielse. From their analysis [8], one can show that the most significant
corrections to W z are:

B •~Z02 Il7n 3C4' IF" k B'2 Z20 \'

e·o) TF

13:: . d'o + 2 . ~Vo + 13:: -c-

(' "
6C4
VV.q
me;; eVa (22)
wz -

where vt'n, IV k , and IFq represent the actual energies associated with the cyclotron,
axial, and magnetron motions respectively. The first term is the effective bottle
shift associated with cyclotron excitation, where B~ = B2 - (elO/2meC2)(Bo/Zg) is
reduced by the relativistic mass shift. The second term is the effective anharmonic
shift ofthe axial frequency with axial excitation, with reduced coefficient: C~ = C4 -
258 R. S. VAN DYCK, Jr. ET AL.

e Vo / 4m e c2 . In Table 1, we list the relevant dimensionless ratios for our experiment:

B~Z5/Bo, C~, and eVo/2m e c2 j each is less than'" 5 x 10- 4 . For the energies, one
can obtain Wk from Eq. 12, Wq from -(eVo /2)(Rm/ Ro)2 and Wn from (n + ~ )nw~
with n < 10. Using some parameters from Table 1, we find that all energy ratios
should be less than 5 x 10- 4 , yielding the upper limit on the relative shifts in W z ,
due to these perturbations, at 3 x 10- 7 .
Finally however, there is one large shift in axial frequency which is used for
calibration purposes. When very strong anomaly drives are applied to one endcap or
to the split guard rings (see Sec. 6), the axial frequency is found to shift proportional
to the drive power. For anomaly power applied to the drive endcap only, as in the
very earliest geonium experiments, the shift was a direct measure of the amplitude
of the electric field used as an off-resonance axial drive. When applied to the guard
rings on the other hand, the circulating current in a second LC circuit, tuned to w~,
develops a differential voltage across the loop inductance of the guards which is then
capacitively coupled over to the endcaps. Since the drive cap typically has a very
low impedance, unlike the signal cap, the rf drive is again asymmetrically applied.
The shift in axial frequency itself is due to the production of a pseudo-potential
associated with the rf trapping field as routinely observed in Paul traps 29].

4. Magnetron Motion

4.1. METASTABLE RESONANCE

The next normal mode which is relatively easy to observe is the slow guiding-center
motion in the radial plane at the magnetron frequency, w m . The primary interest
in determining this frequency is that it allows us to determine the angle between
jj and i. This is quite important in the double trap configuration shown in Fig. 4
because positrons can not transfer into the experiment trap (see Sec. 2.4.) if the
angle is much greater than 0.25°. Actually, this motion should properly be called an
anti-resonance since it has a radial potential hill, not a binding potential well that
would naturally restore any displacement from equilibrium. The motion, however,
is indeed metastable since the decay out of the trap by all dissipative forces is very
slow, as experimentally verified to be » 10 5 sec.
In order to observe this frequency, an additional drive at w' is applied to a guard
probe that protrudes slightly into the gap between endcap and main ring electrode
in order to produce the sideband excitation (SBE). This drive field is not uniform
at the center of the trap, and will contain an axial component of the form E' = Ayi
where coordinate axes are preferentially chosen such that the probe lies in the yz-
plane. Thus if A = Ao sin(w z + wm)t and y = R'rrt sinwmt for the instantaneous
EXPERIMENTS WITH SINGLE ELECTRONS 259

position of the electron, then, the sideband drive becomes:

if' = [Ao sin(w z + wm)tJlRm sinwmt]z

(23)
AoRm cos [wzt l'z - --2-
= --2- AoRm cos [( W z + 2w m )t 1z.
'

Thus, the first term appears to be a constant axial drive, similar to the one used to
lock up the axial resonance. Note that the strength of the drive field is proportional
to Rm. Now suppose this auxiliary drive is not exactly resonant, i.e. w' = W z +
Wm + 6; then, a sharp resonant "perturbation feature" exists 6 in frequency away

from W z . This feature includes a beating of the free magnetron motion with the
driven motion and when this SBE drive is swept, such that b --; 0, the beat note
is faithfully reproduced in the correction signal shown in Fig. 13 (for the phosphor
bronze trap), assuming the loop time constant is not too long. By sweeping both
directions, the onset of perturbation-lock allows a determination of I/m to better
than 0.01 Hz or 1 ppm.

4.2. CHANGING THE MAGNETRON ORBIT

An uncontrolled parameter could exist when electrons are loaded into the trap,
due to the random initial magnetron orbit radius. Since the amount of radial
wander with time out of the trap depends on the initial orbit radius and since
the magnetic field is purposely made inhomogeneous (see Sec. 3.3.), the electron's
cyclotron frequency would vary in time in some non-linear way. Therefore, by
necessity, this means shrinking the magnetron orbit to as close to zero as possible.
The nature of the centering process is best visualized by considering conserva-
tion of energy. "\,\'hen a photon of energy, h(w z + w m ), is absorbed from the SBE
probe drive, hw m is absorbed by the magnetron motion while the remaining part,
hw z , is harmlessly added to the axial motion which quickly damps away because
of the strongly coupled tuned circuit, held at some fixed temperature T z . Now,
the magnetron energy is strongly dominated by the negative radial potential hill
given by -mew;R~/4, compared to the very small kinetic energy -mew~R~/2.
Thus, adding a positive amount of energy hW m to this negative hill will reduce the
magnetron radius, Rm. This we designate as a "cooling" or "centering" drive, in
260 R. S. VAN DYCK, Jr. ET AL.

-
.....
.c
II>

o
><
o

o QI Q2 Q3 0.4 Q5
lIrf - 69,512,088.0 Hz

Figure 13. The W z - Wm cooling resonance using SBE probes in the phosphor bronze
trap. The magnetron sideband of the axial resonance is observed as a pulling of the locked
axial resonance and a subsequent beating between the excited magnetron motion and the
applied magnetron drive. Such resonances allow Wm to be determined to 1 ppm and the
magnetron radius to be reduced because of the absorbed energy hw m .

contrast to the application of a sideband at W z .-- Wm which drives the electron

radially out of the trap. This technique [5] is also referred to as "motional sideband
cooling" .
In principle, cooling will continue to occur by application of this upper sideband
drive until the occupation quantum numbers (k for axial, q for magnetron) in the
two separate motions become equal [8]' i.e. q = k. In terms of the thermodynamic
temperature Tz of the axial motion, the average axial energy can be written hWz(k+
~ ) = k B T z · If the driven energy, used to lock the axial frequency, is much less than
kBTz, then Tz is essentially that of the thermal reservoir produced by the strongly
coupled tuned circuit and the preamplifier. The minimum magnetron energy is
determined by the condition:

-m ew z2 R 2m /4 = -hwm(q 1 Iq=k
+ :2) (24)
EXPERIMENTS WITH SINGLE ELECTRONS 261

and the resulting theoretical minimum orbit, Rmin, can then be written

1/2
R . _ ( 2ckBTz )
mm - ewzBo (25)

which explicitly shows the dependence on magnetic field. It is also insensitive to

well depth varying as the inverse 4th root. For typical conditions listed in Table 1,
Rmin '" 6 X 10- 5 cm. However, we are unable to obtain this theoretical limit due to
some unknown heating mechanism since we observe [5] '" 20 times this value. One
possible cause could be the finite angle between jj and z which could couple the
modes, thus allowing the axial drive to be a direct source of residual heating.

4.3. FREQUENCY SHIFTS

As it happens, the magnetron frequency is very sensitive to the angle of tilt, 8,

which occurs between jj and z as well as another parameter, €, which is associated
with the asymmetry of the electrodes (or simply their average ellipticity). From the
1982 analysis of Brown and Gabrielse [20], the observed magnetron frequency wm
can be approximated by the following formula (for f « 1 and 8 « 1):

(26)

where 8. is defined in terms of the observed axial and cyclotron frequencies: 8. =

w; !2w c .The condition € « 0.01 is expected to hold for these high precision Penning
traps (by construction) and thus may be ignored in most cases in comparison to
the effect of the tilt angle 8. All large compensated Penning traps used in electron
g-2 work have always had wm > 8. which is consistent with the dominance of 8.
Again, there are a few minor perturbations associated with the non-uniformity
of the magnetic field, the non-quadratic terms in the electric potential, and the
relativistic corrections. From the 1986 analysis by Brown and Gabrielse [8], we find
that the most significant of these corrections to Wm are:

~Wm = {2B2Z~} W~n + {6C 4 _ B2Z~ } W k + {6C4 _ 2B2 Zg} W; (27)

Wm Bo eh Bo e\1o Bo eva

where all (neglected) relativistic terms are less significant by at least (wz!w~)2 <
3 X 10- 7 . Each term in this shift is at least an order of magnitude smaller than
the one obtained from Eq. 26 and is also below our level of sensitivity except for
excitation energies> 0.01 eV.
262 R. S. VAN DYCK, Jr. ET AL.

1.3 ± 0.1 Hz / Landau level

I
N
.,IOppbi-- 6 ...
Q)
.D
....
'+- 5 E
:J
.c C

'">. 4 E
:J
u
c c
Q) 3 0
:J :J
CJ CJ
...
--o
Q)
C

....0...
0

)(
o

o 2 4 6 8 10
lIRF -141, 338,780 kHz

Figure 14. Single electron cyclotron resonance for the 1.3 Hz bottle in the last molybdenum
trap. The characteristic magnetic line shape has an exponential tail that reflects the
Boltzmann distribution of axial states coupled via the z2 term in the magnetic field. The
sharp edge feature corresponds to the field at the trap center and the solid line represents
a fit to the data with a width corresponding to aSK axial temperature (from Ref. 6).

5. Cyclotron Resonance

5.1. LINE SHAPE

The shape of the cyclotron resonance is almost exclusively determined by the mag-
netic bottle and the thermal Boltzmann distribution of axial states. In addition, the
relativistic pulling effect has some influence on the line shape, causing the resonance
to appear somewhat broader than expected [30]. It is beneficial, when possible, not
to have the axial locking drive applied if the true cyclotron line shape is to be deter-
mined. This follows from the z2 term in the magnetic bottle which will contribute
a cross term of the form Zth Zd that will also broaden the resonance (Zth is the
thermal amplitude of the axial motion). Necessarily then, the detection and excita-
tion for this resonance are alternated and the observed shifts in W z signal the onset
of the cyclotron resonance, assuming that the frequency is swept up in magnitude
for a positive magnetic bottle. Figure 14 shows a typical cyclotron resonance taken
with this scheme for the fixed positive bottle in the last molybdenum trap shown in
Fig. 4. A large negative bottle would tend to yield a resonance which is nearly the
mirror image of this positive bottle line shape, except for the effect of the relativistic
EXPERIMENTS WITH SINGLE ELEcrRONS 263

mass shift on the leading edge.

The main requirement of the alternating scheme is that the alternating rate not
be too slow compared to liTe where Te is the classical cyclotron decay time predicted
by radiation damping. The alternating scheme allows time for axial energy to decay
away by injecting a delay ;:c 100 ms before onset of cyclotron excitation. In addition,
the axial damping time (Tz '" 50 ms) needs to be short compared to Tc in order
that some signal is registered in the detection part of the cycle. It is worth noting
however, that if the axial locking drive is scaled down significantly such that lock
is just barely possible, a continuous cyclotron resonance can be swept out whose
low-frequency edge agrees very well with the alternating low-frequency edge (shown
in Fig. 14), but without quite as good resolution. Alternating conditions were quite
favorable in the previous molybdenum traps since Tc was measured to be '" 1 sec
[6]; in this case, it was possible to achieve a 1-ppb short-term resolution of the field
at the center of the trap. However, in the case of the phosphor bronze trap, we
now find that Tc "-' 0.1 sec and, therefore, we must use the low-axial drive method
for deducing w~; because of the lower S / N of this method, resolution was about a
factor of 2 worse.
The high frequency tail of the cyclotron resonance shown in Fig. 14 represents
the Boltzmann distribution of axial states, and the lie linewidth corresponds to
A ':= eE2kBTz := kBTz Ii (28)
oWe 2 2 to ..
mecw z z
TLW

where Ii is defined for Eq. 19. If this expression is used to experimentally determine
the axial temperature, we often find that Tz '" 10 K. This could be due to the
equilibration to an amplifier whose temperature is greater than ambient, but some
of the discrepancy is believed to be due to the relativistic pulling effect [30]. Here,
the onset of excitation of the edge for positive magnetic bottles immediately pulls
the true resonance edge to a lower frequency, thus artificially reducing the apparent
peak response. Since this reduction will be more significant (in terms of absolute
signal) near the peak than on the tail of the resonance, the shift has the effect of
broadening the lie resonance linewidth.

5.2. PERTURBATIONS TO THE CYCLOTRON FREQUENCY

The largest shifts in w~ are due to the non-uniformity of the magnetic field and the
relativistic mass shift. Again using the 1986 analysis of Brown and Gabrielse [8], it
can be shown that
6w~:= {-eVa} T~-n + {E~Zg} Wk + {2E2Zg} Wq (29)
w~ m e c2 eVa Eo eVa Eo eVo
264 R. S. VAN DYCK, Jr. ET AL.

where the conspicuous absence of the C 4 term occurs because its contribution is
reduced by the factor (wz!w~)2 < 3 X 10- 7 . Again using some of the parameters
listed in Table 1, this equation becomes

-Aw~
w'
= {-0.2 Wn + 3.4 W k + 6.8W}
q x 10 -5 (30)
c

where all energies are measured in e V. The first term represents the relativistic
correction and corresponds to '" 1.2 ppb per integer change in the cyclotron quantum
number. For a positive bottle, swept up in frequency, it should not affect the edge
resolution but only the height of the excitation, unless the noise pedestal of the 14th
harmonic of the X-band source from the Schottky diode heats up the cyclotron
temperature and allows the resonance to be prematurely pulled into excitation.
Normally, the average occupation level in a 50 kG magnetic field is 0.2 at 4 K.
The second term is responsible for the shape of the cyclotron resonance and was
the subject of the last section. The last term generates the most concern for us
in our g-2 experiment since a change in the metastable magnetron orbit yields a
corresponding change in w~ and WS' It illustrates the importance of cooling the
magnetron motion. Assume that Rm has a typical value [5] of '" 0.0014 cm; then
Wq '" -5 X 10- 5 eV. Thus, a -3 ppb relative shift occurs in the cyclotron frequency
which must be stable over time if a 1 ppb accuracy is to be maintained. Ideally, if
one could reach the cooling limit, the relative shift could be kept below 0.02 ppb.
The use of the variable bottle represents our first attempt to reduce or eliminate
the effect of the magnetic bottle in these measurements.
Another possible source of shifts in w~ that was considered in earlier sections was
associated with the angle of tilt () between jj and z, as well as possible ellipticity t
in the trapping electrodes. In terms of 8e defined in Sec. 4.3., the observed cyclotron
frequency w~ can be written in the form

(31)

which differs from Eq. 5 only by the discrepancy between measured and calculated
magnetron frequencies, scaled by the ratio of magnetron and cyclotron frequencies.
The relative shift is given by

(32)

where the quantity (wz!w~)4 :0:; 10- 13 for parameters chosen for our experiments
EXPERIMENTS WITH SINGLE ELECTRONS 265

(see Table I) and the last factor is usually no larger than 10- 4 . Thus, this shift is
totally negligible for precision in the foreseeable future.
However, a much more serious problem arises for the phosphor bronze trap which
contains the variable bottle. This latter feature consists of a closed superconducting
loop embedded in the midplane of the ring electrode. Since total magnetic flux
through this loop must be fixed for all time (i.e. B cos 8 = constant), a change in the
direction of the axis of the loop corresponds to a change in area of the loop normal
to the magnetic field. Currents are then established in the superconducting wire
which permanently alters the magnetic field at trap center according to 8B I B =
li88 where li ~ 88 now represents the average residual angle between the axis of
the superconducting loop (which should be the z-axis by construction) and the
magnetic field. From Eq. 21, variations in the cyclotron frequency can be related
to fluctuations in the axial frequency due to variations in 8:

( 8w~) 2(8wz) (33)

w~ loop = 3 -z;; fJ'

The limitation that the observed axial stability is ~ I-Hz out of 72 MHz for all causes
in the variable-bottle trap leads to 9 x 10- 9 for the upper limit to fluctuations in
the magnetic field. As it happens, we observe on the order of 10 ppb wander in
the field for this trap. The impulses to the magnetic axis are believed to be due to
the movement of a hydraulic elevator which is about 10 meters from the magnet.
We observe", 1 mG field changes at the distance of 10 meters from this elevator
as it moves from ground to the fourth floor of the physics building (as required for
day-time classes). Clearly, the use of the variable bottle puts a severe constraint
on the allowed variations in 8 or else requires that li be reduced by an order of
magnitude (which is indeed possible).

5.3. CAVITY EFFECTS

One observation that was particularly suspicious in our early work was the determi-
nation that the optimum alternation rate for observing cyclotron resonances in the
molybdenum trap shown in Fig. 4 (at 50.5 kG) was about 0.5 Hz (i.e. 1 second on
and 1 second off). It was apparent to one of us (H. G.D.) that the exci tation to higher
cyclotron quantum states was not decaying as quickly as expected from the classical
free-space damping time of 0.1 sec at this field. This took the form of a warning
from Dehmelt [9] as early as 1981 to guard against shifts in w~ "due to accidental
resonances with standing wave or 'cavity' modes" inside the trap structure.
266 R. S. VAN DYCK. Jr. ET AL.

5.05 Tesla
1/8/84

o
:;::::
:£ 0.2
Q)
>

O.I O
'-------'----------1-----.-L----J
Q5 1.0 1.5
time delay (sec)

Figure 15. Cyclotron damping time for a single electron in a microwave cavity (i.e. last
molybdenum trap). By delaying onset of axial detection drive in the normal alterna-
tion scheme, the initial resonance amplitude is found to decay exponentially with a time
constant which is ten times longer than predicted for free space radiation (from Ref. 6).

To observe the inhibition of spontaneous emission, the initial amplitude of the

cyclotron edge is measured for various fixed time delays after turning off microwave
power and turning on the axial detection drive. The response time for the axial
motion was less than 50 ms and thus did not affect these measurements. The
relative initial amplitude is then plotted on a semi-log graph versus the fixed delay
time as shown in Fig. 15 in order to determine [6] that the classical decay time
was 1.0 ± 0.1 s, or about 10 times longer than Tc predicted by free space radiation
damping. It was quite fortuitous that these early experiments exhibited such long
decay times. It meant that much less microwaye power was required to see a sharp
Zrms = 0 frequency edge, and therefore put less demand on the strength of the
carrier relative to the multiplied-up noise pedestal of the Klystron used in the
microwave frequency chain. This type of inhibition in a Penning trap was first
directly measured and reported [31] by Gabrielse and Dehmelt for a trap which
did not have a magnetic bottle, but did have a much more open structure. As a
result, they found a decay time Tc which ranged from 86 to 347 ms for Eo near
60 kG. In the phosphor bronze trap, some effort was made to reduce the Q of the
microwave cavity and subsequent measurements of Tc [32] indicated that the trap
had a cyclotron decay time in excellent agreement with the classical free-space value
at '" 50 kG.
EXPERIMENTS WITH SINGLE ELECTRONS 267

The presence of shifts in the cyclotron frequency could only be detected by chang-
ing the magnetic field but even with this change, the true unshifted frequency would
be difficult to deduce without some theoretical model as a guide. Since the g-factor
is related to the ratio of spin precession frequency divided by the cyclotron fre-
quency, a shift of several parts in 10 12 in We without a corresponding shift in W.
would produce a comparable systematic error in the measured g-factor (and 10 3
times larger shift in a e ). Since present experimental and theoretical precision for
the electron g-factor is on the order of several parts in 10 12 , this systematic effect
becomes very important and represents the principle motivation for our repeating
the single electron g-factor measurements in the new phosphor bronze trap with a
somewhat more open cavity structure.
The first complete theoretical treatment [33] of the effect of a microwave cavity
on the cyclotron motion of the trapped electron was based on a cylindrical model
which was chosen at the time because of the ease of theoretical treatment that the
geometry afforded. It predicted that for a cavity with a Q ~ 1000, shifts in the
cyclotron frequency could exceed 70 parts in 10 12 . However, a closer look [34] at
the model suggested that for this same Q ~ 1000 with Te(cavity) ~ 10Tc(vacuum),
the maximum and probable shifts would be on the order of 8 and 4 parts in 10 12
respectively. The latter uncertainty thus represents the limitation to the accuracy
of the last precision measurement [35] of the electron's g-factor inside a fixed-bottle,
molybdenum trap. To see if a Q '" 1000 was reasonable, an experimental technique
was devised to directly observe the cavity modes [36], based on the well-established
bolometric technique [14,25] for clouds of several hundred electrons located at the
center of the Penning trap. The results of such studies using the molybdenum
trap shown in Fig. 4 was an indication that this trap has a lower than expected Q
(probably less than 500).
It should also be noted that this work had stimulated further theoretical interest
in pursuing a numerical calculation of the mode structure in an ideal Penning trap
[37]. Unfortunately, subtle variations in slits and holes in the structure as well as
deviations from the perfect family of hyperboloids do not allow for any reasonable
agreement between the observed and the predicted mode structure for this particular
g-2 Penning trap. Future efforts to account for the mode structure will either
concentrate on using very low Q cavities that can not exhibit significant frequency
pulling or else on measuring the few nearest modes on each side of our operating
frequency in hopes of predicting [34,37,38] the true cyclotron frequency. The latter
approach requires much more experimental effort, but does have the advantage
of putting less dependence on an absolutely clean microwave source and may be
necessary for the future relativistic mass shift method of detecting spin flips.
268 R. S. VAN DYCK, Jr. ET AL.

In addition, there is also a more subtle electrostatic perturbation due to the

intrinsic constraint of a real charged particle surrounded entirely by conducting
surfaces. The simple model of a point charge located inside a grounded sphere
of radius a has been used to predict [39] the electrostatic shifts due to the image
charge located outside the sphere. Using the experimental condition that Wz is held
constant by the feedback loop (described in Sec. 3.4.), the observed shift in w~,o for
a single electron at the center of the trap becomes

We ,1 = w~,o - 3~/2we,1 (34)

where w~ 0 represents the cyclotron frequency in the limit of zero trapped electrons
and ~ ='e2/mea3. Since it was experimentally determined that a is approximately
Ro in a 3 times smaller quadring Penning trap [40], we predict that for a single
electron in a g-2 trap that

( ~W~)
-/
2
ec ~ 5
= 3m3 B2 ~ X
10-15
. (35)
We image RO 0

This is indeed negligible and may be ignored for all the g-2 experiments.

5.4. OBSERVED SHIFTS IN PHOSPHOR BRONZE TRAP

5.4.1. Variation with Microwave Power. An example of a shift in w~ with microwave

power was published earlier [35] and appeared to vary quadratically. The source of
this shift was a mystery whose explanation is still not available, though we know
much more about it now. In particular, with the implementation of the variable
bottle in the phosphor bronze trap, we have found that this shift depends on the
applied bottle strength. In Fig. 16, we have plotted this shift versus relative X -band
power (with arbitrary reference) for the two extremes of the variable bottle. The
relative power levels are obtained by calibrating the rectified de current through
the multiplier diode versus a known amount of microwave attenuation. The zero
shift limit near 4.3 mW corresponds approximately to the apparent zero for the
14th harmonic, below which the cyclotron resonance abruptly disappears (i.e. the
n = 1 level is not excited). The difference between this shift and that which was
reported earlier could be due to either the different multiplier diode used or the
lack of direct X-band power radiating onto the electron since a high pass filter is
used in the phosphor bronze trap. A direct determination of B~ is possible from
the magnitude of the single quantum step associated with the spin flip according
to Eq. 19. For the positive bottle extreme, B~ = 245(10) G/cm 2 and for the neg-
ative extreme, B~ = -262(10}G/cm 2 . These values would correspond to an intrinsic
EXPERIMENTS WITH SINGLE ELECTRONS 269

1500
(0)
1200

900
N

--
:I:
600 l -9
2xlO
.s:::.
i
(/) 300
>.
u
cQ)
O~-L~-L~~~~~-L~~~~~~

-e
:::l
CT O,,-n~rT'-,,-''-,,-''-rT'-''-'''
~

-
c -300

o
!:i-600
>.
u
-900

-1200

-1500
4 6 8 10 12 14
cyclotron drive power in mW (or b. reference)

Figure 16. Residual systematic shift of the cyclotron frequency versus applied X-band
microwave power for a single electron in the phosphor bronze trap. The solid curves
represents a linear fit to these data with the constraint that the "zero" occurs at fixed
power level which can no longer excite the resonance.

bottle of -8.5(14) G/cm 2 , which is consistent with -6 G/cm 2 predicted by the

relativistic mass correction. However, the data does not show symmetry about zero
effective magnetic bottle. The slope for maximum negative bottle is about 40%
larger than the magnitude of the slope for the maximum positive bottle; but if the
shift is proportional to the applied B2 quadratic gradient, there should be complete
symmetry since only the sign of the current in the magnetic bottle was reversed.
Another characteristic of this shift is that it appears to be associated only with
the measurement of the cyclotron edge and to not be present in anomaly resonances
if the simultaneously-applied microwave power is non-resonant. Part of the motiva-
tion for using such non-resonant drives was to eliminate this type of shift by causing
the same shift to occur in the anomaly resonance (which it did not do in this case).
270 R. S. VAN DYCK, Jr. ET AL.

~ 800 r-r--...-r-r-,-,..-;r-r-,-...,-,,.--,-...,-,"--'-T"'l'"
N

-
I
Bo = 50.7kG
!"!:: 600 82 =+ 245(10) G/cm 2
~
<JI
>.
u
C
Q) 400
::J
0'

•
Q)
.....
~

§ 200
~

'0
u
>.
u
20 40 60 80 100
oxio I drive power in fJ- W (orb. reference)

Figure 17. Shift of cyclotron frequency versus simultaneously applied weak axial drives
for a single electron in the phosphor bronze trap. Adequate data was not available for the
maximum negative bottle, but the sign of the shift is consistent with the sign of B 2 .

In our discussion of Eq. 29, we also did not include a term proportional to the
cyclotron energy which depended on the magnetic bottle strength, because it was
many orders of magnitude smaller than the relativistic term as well as the size of
the shifts observed. Upon reviewing other terms in this theoretical shift equation,
only the magnetron energy term is consistent with both the dependence on B2 and
the sign of the shift (recall Wg is negative and observed cyclotron shifts are indeed
negative for positive bottles). The size of the required magnetron orbit which would
give such shifts is also quite believable:

(36)

Thus assuming that Rm,o is close to the observed cooling limit of 0.0014 cm, then
a 1-ppb shift in the cyclotron frequency would corresponds to a 10% increase in
magnetron radius for the maximum bottle strength used.

5.4.2. Variation with Axial DriVE Power. On occasion, we find that it is necessary
(or convenient) to measure J)~ simultaneously with axial drive applied, i.e. without
alternating excitation and detection periods. Subsequently, we observe shifts in the
cyclotron frequency proportional to the power of the axial drive. Figure 17 illus-
trates the typical shifts associated with axial drive, measured in the new phosphor
bronze trap. The (arbitrary) reference power, at unity, is taken to be the same for
both runs that were combined to yield the graph. We have also observed that the
cyclotron shift with axial power reverses sign when B2 is reversed. Thus, the shift
EXPERIMENTS WITH SINGLE ELECTRONS 271

..c
0.30
0.
x +155 (15) G/cm 2 (I)
>.
g25 • +245(10) " (2)
Q)
::::l • -262(10) " (3)
0-
Q) • +105(8) " (4)
.:: 20

-
c::
e
0 15
U
>.
U

-
E
c:: 10
.....
I/)
5 (4)

-e
0
c::
0
u
..... 5 15 20 25 30 35 40
fractiona I shift in axial frequency (ppm)

Figure 18. Shift of cyclotron frequency versus simultaneously applied anomaly drives for
a single electron in the phosphor bronze trap. The strength of that drive is measured as a
fractional shift in the axial frequency according to its effect on well-depth for an rf-trapping
field. From the available data, these positive shifts appear to vary quadratically with B 2 .

is expected to vary linearly with B2 and should be described by the second term in
Eq. 29. This equation predicts that a shift of 1 ppb should correspond to an axial
energy of 0.00003 eV. From Eq. 12, this energy corresponds to VrJ = 4 nV which
agrees quite well with our estimate for the single endcap drive actually applied.

5.4.3. "Variation with Anomaly Drive Power. The one shift which we can find no
plausible source is the shift in cyclotron frequency associated with applied anomaly
drive. In the past, we attributed this shift to an off-resonant axial drive whose
non-zero average position in the magnetic bottle caused a true shift in the effective
magnetic field. As a field shift, this effect was expected to affect w~ in the same way
and thus not affect the determination of tIe to first order. Figure 18 summarizes all
available data on this effect at Bo = 50 kG. The data for the old molybdenum trap
(i.e. B2 = 155(15) G/cm 2 ) have been properly scaled to aCCOU'lt for the different
axial and anomaly frequencies relative to those of the phosphor bronze trap. Note
that this older data does not agree with any data from the newer trap, presumably
because of different capacitive effects in the two traps. The curious observation is
that the shift is always positive, independent of the sign of B2 and, from the three
values of the bottle shown for the phosphor bronze trap, the shift appears to vary
272 R. S. VAN DYCK. Jr. ET AL.

quadratically with the magnitude of B 2 . Clearly, this shift is not characterized by

the second term in Eq. 29.
These observations also rule out many other possible shift mechanisms. For
instance, the relativistic mass shift is even smaller than the B2 contribution and it
has the wrong sign. The anharmonic potential shifts are still smaller yet and have no
dependence on B 2 . Even a relativistic ve
x ET magnetic field has been investigated,
but is likewise very small. Measurements of Vm during application of anomaly power
do not find absolute shifts in the magnetron frequency comparable to those in v~.
Obviously, more data is needed in order to discover the true nature of this effect.
Also we must continue to test a. versus anomaly power in all future experiments to
see if any part of the shift in w~ does not appear in w~ for a. determinations.

6. Anomaly Resonance

6.1. CLEAN "BEAT-NOTE"

The anomaly resonance is actually a beat-note between the precession rate w. of

the magnetic moment about Eo and the cyclotron rotation at We ;::: Ws about the
same axis. This beating was dramatically illustrated by the work on electrons and
positrons completed in 1971 by Rich et ai. [41], and the work on positive and
negative muons completed in 1977 by the CERN collaboration [42J. Excitation of
the resonance is produced by a simultaneous two photon process, in which the spin
is flipped and the cyclotron state changes by one unit (see Fig. 11). This yields
an advantage of 3 orders of magnitude in precision for g-factor measurements over
any method which independently measures Ws and We since major perturbations are
reduced accordingly by the simultaneous transitions.
To give a particular example, the relative shifts in Ws due to the bottle field are
(but not including terms of order w~/w~2 [8]):

6w s
Ws
= +{ B2Z~}
Bo
W!: +{2B2Z~} Wq .
Bo
e~To eVo
(37)

As with the cyclotron frequency, terms associated with the anharmonic potential
do not appear at this order of precision. Upon comparing this equation with cor-
responding terms for the relative cyclotron perturbation, given in Eq. 29, the dif-
ference frequency, Ws - w~, then has the same functional form shown in Eq. 37 but
now for 6w~/w~.
The shift in w~ associated with the relativistic mass increase has been considered
in our previous review [7] and there we obtained

WaI = Wa '0 (
1- WI:)
---2
2mec
(38)
EXPERIMENTS WITH SINGLE ELECfRONS 273

B{-+----~-----+----~----~~~Z

Z =0 axial motion

~
p
cyclotron orbit

Figure 19. Mechanism for inducing spin flips by the electron's axial motion in the magnetic
bottle. From the electron's frame of reference, a magnetic field is seen to be rotating at
w~. This field is then modulated by the axial motion at w~, thus yielding sidebands at
w~ ± w~ where Ws = w~ + w~ (from Refs. 5 and 18).

where a large 1-ppb dependence on cyclotron energy is now absent. Thus, if Wk is

a coherent axial energy comparable to kTz , then, ~w~/w~ ~ -0.35 ppb.

6.2. AXIAL EXCITATION TECHNIQUE

An added benefit of using a magnetic bottle for detection is that it can be used
fortuitously to generate the appropriate perpendicular spin flipping field, read-
ily associated with standard NMR experiments. The technique is based on axial
amplitude modulation, at the anomaly frequency,.)f the non-homogeneous mag-
netic field seen by the electron, whose component in the radial plane has the
form given by Eq. 18: br = -B2TZ. Figure 19 illustrates the motion through
this bottle at the two extremes of the axial motion, where for this discussion
the magnetron motion can be safely ignored. From the frame of reference of
the electron, going around its cyclotron orbit, it sees a magnetic field rotating
about the z-axis, which points in opposite directions at the two extremes of the
motion as shown. Thus, from this rotating frame, the radial magnetic field is
274 R. S. VAN DYCK. Jr. ET AL.

given by:

(39)

where
br (l) = ~B2ReZIl cos(w~ + w~)t
(40)
br (2) = -~B2ReZa.cos(w~ -w~)t.
The cyclotron radius, R e , in a 50.5 kG field and for the n = 0 ground state cor-
responds to ;::: 114 A. The axial amplitude Za., produced by the electric field Err
applied to one endcap, is given by Eq. 10 when far off resonance:

Z" (41 )
Zo

For our typical conditions (see Table 1), this gives Za ~ 0.003 em for a 1 volt rf
drive yielding an amplitude B2RcZa = O.531-rrf (inJLG) assuming v~f is in volts. The
radial component br (1) corresponds to the appropriate perpendicular spin flipping
field at frequency w~ + w~ = We + Wa = W s , whereas the second component will
correspond to an insignificant non-resonant perturbation. From standard NMR
theory, the br (1) component will then produce spin flips with a Rabi frequency

(42)

which is ~ 27r(O.7 Hz) for a 1 volt rf endcap drive for the last molybdenum trap
shown in Fig. 4. The average time between flips is then given [7] by

( 43)

which, on resonance, is about 6 sec for a 3 Hz wide anomaly line.

Unfortunately, applying a volt of anomaly power would also produce a significant
amount of liquid helium evaporation from a 50n terminated drive line. Such time
varying heat loads then upsets the equilibrium of the axial frequency lock loop and
reduces the effective SjN. However, a reduction in anomaly power would then
require much longer data-taking periods. This problem was alleviated somewhat by
applying a simultaneous microwave drive in the tail of the cyclotron resonance in
order to heat the cyclotron motion into higher excited states on the average with
correspondingly larger values of Rc and therefore larger br ( 1).
EXPERIMENTS WITH SINGLE ELECfRONS 275

6.3. GUARD-RING EXCITATION TECHNIQUE

Every major systematic shift encountered so far appears to be related to the strength
of the detection magnetic bottle. To get around these shifts, the bottle must be
reduced which means some other mechanism is needed to generate the required
spin-flip field. As a first step, a direct current loop is made possible by splitting the
guard rings on one side only. The injection of current is arranged to yield currents
flowing in opposite directions for each guard (i.e making a pair of counter-rotating
current loops) and is therefore expected to produce a constant radial magnetic
gradient (near trap center) of the form [6]

Bl :::::: 1.3Irffs Gauss/cm (44)

where Irf is in amps and fs is the rf electrode shielding factor. The measured spin
flipping rate indicates that fs ~ 0.1. A tuned circuit attached to the guard rings
also enhances the production of rf current. The radial field then becomes:

( 45)

One immediately notes the similarity to the two components described in Eq. 39,
in which the amplitude B2RcZ .. is replaced by Bl Re. If Rc is again associated with
n = 0, this amplitude becomes;::; 0.74Ir rf.(in JLG), assuming Irf is in amps.

m =+1/2

m = -112

m =+1/2

m= -112

o 2 4 6 8 10 12 14 16 18
time (minutes)
Figure 20. Example of spin-flip data. The alternation between detection drive and anomaly
excitation is much slower than for cyclotron excitation in order to minimize the power in
the anomaly field. The excitation frequency is kept constant for each 12-20 alternation
cycles while spin-flips are being detected in the correction signal. Note the increased
rate of flipping near the peak of the resonance shown in (a) compared to the rate on the
exponential tail shown in (b) (from Ref. 6).
276 R. S. VAN DYCK, Jr. ET AL.

6.4. RESONANCE LINE SHAPE

The excitation cycle for anomaly resonances involves turning off the axial drive at
Wz, and turning on the anomaly drive at w~ with the microwave drive also applied,
but tuned into the tail of the exponential line shape of the cyclotron resonance.
The subsequent detection cycle then involves turning off both anomaly and cy-
clotron drives while the W z drive registers the floor of the lock-loop correction signal
Examples of the type of anomaly data obtainable with this method are shown in
Fig. 20. The anomaly drive was kept fixed in frequency for some specified num-
ber of excitation/ detection cycles and then the number of successful flips is plotted
versus the applied drive frequency. An example of such early resonances is shown
in Fig. 21(a) for run 109. The dotted line is added only to aid in recognizing the
resonance shape. Before a theoretical line shape was available with which to fit
this data, an estimate had to be made as to where the leading edge would be for
Zrms = 0, typically taken at one-third of peak height. The lack of a sharp edge is
due to the noise modulation (or statistical fluctuation) of the anomaly resonance by
the axial motion within the inhomogeneous magnetic field. The estimation process
produced a serious "apparent" systematic shift of the anomaly with anomaly power.
This shift is related to the tendency for this type of resonance to saturate (or to be
power broadened). A maximum spin flipping rate is 0.5 (with equal probability of
being left in spin-up or spin-down state). As more anomaly power is applied, the
resonance edge is broadened and the "apparent" anomaly frequency (and therefore
a e ) is artificially shifted to a slightly lower frequency. However, the extrapolation
to zero power should be consistent with our presently accepted value of a e .
The basis for this unusual line shape is found in Eq. 18 for which the magnetic
field depends weakly on the instantaneous value of z2. As described in Sec. 5.1.,
the magnetic width, which represents the 1/ e linewidth, will be proportional to the
average axial energy, and will reflect the Boltzmann distribution of axial states at
some temperat ure Tz:

(46)

In other words, the weak magnetic coupling to the axial frequency also causes the
random thermal axial fluctuations to appear in all magnetic resonances. This effect
was accurately described [43] by Brown in 1984 by writing the stochastic average
of the axial motion as a functional integral which transcribes it into a soluable
one-dimensional, quantum-mechanical barrier penetration problem. The resulting
line shape profile x( w) as a function of frequency is given for very weak rf drives by
 EXPERIMENTS WITH SINGLE ELECTRONS 277

8
I
Zla (a)
.-
I
"
",
6

4 If/ ,,
,,
2
f/ "1-,
- --V
C\I
"-
(/)
....
0- 0

-
C
0-
(/)
S
0
....
Q)
( b)
.D 6
E
::J
C

o
40 45 50 55 60 65 70
IIrf - 163,910,300 Hz

Figure 21. Early anomaly resonance (run 109 taken in the last molybdenum trap) showing
strong saturation on the peak. These early runs, obtained by plotting the number of
flips observed out of a fixed number of tries vs anomaly frequency, were found to show a
systematic shift towards lower v~(Zrms = 0) with increasing applied power. As shown in
(a), this arose from the artificial choice of determining the resonance edge at ~ 1 /3rd of
peak height. As the new line shape fitting shows in (b), the correct Zrms = 0 is near the
top of the peak (adapted from Ref. 6).
278 R. S. VAN DYCK, Jr. ET AL.

4 { ,',z 00 (,' _,z)2n(,' + ,.)-2n }

X(w) = -Re
7r (,' + ,.)2 ~ (n + ~ h' -,z/2 + ,e/2 - i(w - wo) (47)

where Re denotes the real part, " == (,~ + 4i,z~W~)l!2, ,z is the corresponding
axiallinewidth, and ,e is the natural cyclotron damping linewidth in the absence of
the magnetic bottle, However, moderately hard anomaly drives are typically used
such that near the peak, the probability for a spin flip approaches ~ as indicated
earlier. If ie represents the period of time that the excitation is on, then the net
probability for observing a spin flip is given by

(48)

where the strength of the drive is contained in the Rabi frequency WRa. Note
that when WRa is sufficiently large, r --t ~ (i.e. saturated). If run 109, shown in
Fig. 21(a), is fitted to Eq. 48, one sees in Fig. 21(b) an excellent example of this
saturation effect where the Zrms = 0 position is near the peak instead. Using fitted
data, the apparent systematic shift of a e with anomaly power totally disappears.
Recent work in the new variable-bottle phosphor- bronze trap has required that we
modify our excitation method due to the 10 times shorter cyclotron decay time (now
comparable to the axial damping time). The result of the 10 times less dwell time
in the n = 1 state is a strong asymmetry between spin-up and spin-down transition
rates. As a result, we now prepare each spin-flip event by first forcing the electron
into the spin-up state. This is accomplished by strong (saturating) anomaly drives
with some cyclotron assisting to further enhance the transition probability. How-
ever, once prepared in the up-state, the anomaly power is reduced to low nominal
levels, corresponding to (rf-trapping) axial shifts of 200-300 Hz, and the cyclotron-
assist is turned off. Figure 22 shows two examples of resonances obtained with
this method at the extremes of the variable magnetic bottle. Figure 22( a) shows
the anomaly shape for the maximum negative bottle case when ,./27'1 ~ 1.0 Hz.
The solid curve represents the theoretical fit which yields such fitted parameters as
!::.w a /27r ~ -3.4 Hz and w a /27r = 164,992,769.36 Hz. Figure 22(b) then gives the
usual anomaly resonance associated with the maximum positive bottle, for which
,./27'1 ~ 0.5 Hz. Again, the solid curve is a fit which yields !::!.w a /27r ~ 3.9 Hz
and Wa /27'1 = 164,791,544.80 Hz. With this particular excitation method, the fits
as shown in Fig. 22 are obtained by letting the theoretical number of observations
per data point be twice the actual experimental number. This is done in order to
use the same fitting program and at the same time to avoid saturation at ~ since
saturation now occurs at unity.
 EXPERIMENTS WITH SINGLE ELECTRONS 279

12 (a) B2 = 245(10) G/e

10 Yz /2rr = 0.5 Hz
Yc /2rr = 1.6 Hz
8

~ 2

- 0
c
.0. 50 55 60 65 70
III

'0 lIrf - 164,791,800 Hz

....
~ 12
§ (b)
c 10
B2 =-262(10) G/em 2
8 Yz/2rr= 1.0 Hz
6 ~/2rr = 1.6 Hz
6.11~ =-4.0Hz
4

o
55 60 65 70 75
IIrf - 164,992,700 Hz

Figure 22. Anomaly data taken at the two extremes of the variable bottle in the phos-
phor bronze trap. The natural cyclotron linewidth, Ie = 27r(1.6 Hz), is characterized by
the free-space damping at Te = 0.10 sec. The axiallinewidth, IZ, was reduced in each
case by shifting the axial resonance off the tuned circuit during the anomaly excitation,
whereas .6.l/~ represents the fitted bottle width for the actual anomaly resonance. The
line shape now clearly shows the expected exponential tail, but with the low-frequency
edge still smeared by the axial noise modulation. Error bars are derived from the bino-
mial distribution, and are computed with the least-squares-fitted curve as the true parent
distribution.
280 R. S. VAN DYCK, Jf. ET AL.

7. Results, Conclusions, and Goals

7.1. THE g-FACTOR RESULTS

From the time of our first demonstration in 1976 [19] that these geonium experi-
ments could indeed determine the g-factor of the free electron to very high precision
(i.e almost 20 times better than any previous method at this first demonstration),
there have been four major experimentation periods in which significant improve-
ments have been made. The first of these is the work of 1979 [9,44,45] which is
concerned with the measurement of the g-factor at three different magnetic fields,
using our first molybdenum compensated Penning trap in a glass envelope:

a e (1979) = 1159652200(40) x 10- 12 . (49)

The use of three different fields was done at the time as a very simple expedience for
finding an accurate g-factor since systematics were expected to depend on relative
field homogeneity. The next period that ended in 1984 [6] noted the use of a new
(molybdenum) double-trap configuration (shown in Fig. 4) and several improve-
ments in technique, most notably the use of guard-ring drive and line shape fitting
to eliminate the major systematic error. The work of the next to the last period
in 1987 [35] used the same double trap to uncovered the presence of systematic
dependencies on power, the prediction of the cavity shift and yielded the precision
comparison of the electron with the positron. Finally, the work that ended in 1989
[46] utilized our new phosphor bronze trap which features a low Q cavity in order to
test for cavity shifts. It also features the variable magnetic bottle which allows us
to compare g-factors with different values for B 2 . Thus, another very real division
of our geonium experiments can be characterized by the use of either a traditional
(asymptotically symmetric) molybdenum Penning trap with a fixed magnetic bottle
in which RV2ZJ = l.0 or the special (nearly orthogonal) phosphor bronze trap with
a variable bottle in which R~ !2Zg = 0.72. A true orthogonal trap [22] would have
R~/2Z(~ ~ 0.674 and is characterized by the non-interaction of guard compensation
potential and the absolute well depth, Vo.

7.1.1. Traditional Molybdenum traps. 1\1 uch of this early work concentrated pri-
marily on developing hardware and terhlllques, and a little on systematics. For
example, the development of a newall-metal vacuum envelope became necessary
because of the unreliability of the glass envelopes. The introduction of the cyclotron-
assist for the spin-flip cycle allowed us to use less anomaly power and thus reduce
the disruption to both the magnetic field stability and the detection S! N from the
increased liquid helium evaporation. For the same reason, we also introduced the
EXPERIMENTS WITH SINGLE ELECTRONS 281

~ 194~-.~.-,,-.-..-.-,,-.,-.-ro-r-r'-,,-.~
Q
X
0192
o
Q
~ 190
lO _j___ __ __j-(V~l~ ~88(9)
~ 188
o 2xIO 9
q 186
o
I
1
cWI84~~-L~~~~-L-L~~~~-L-L~~~~-L~
- 0 400 800 1200 1600 2000 2400
axia I shi ft due to anomaly drive (Hz)

Figure 23. Summary of 1983-84 anomaly runs using guard-ring excitation and line shape
fitting. The relative power of the anomaly drive is again measured as an axial frequency
shift due to the rf effect. The residual anomaly-power dependence is < 1-ppb over the
range of drives used, but there was an undetected (at that time) -4 ppb error due to the
residual systematic shift of the cyclotron frequency (from Ref. 6).

guard-ring excitation for producing the observable spin flips. To narrow the anomaly
resonances, we established an alternating detection/excitation scheme which also
eliminates the perturbation of the detection axial drive on the anomaly frequency.
By far the most significant improvement in technique was the utilization of the
line shape fitting program to predict the "true" location of the Zrms = 0 anomaly
edge. By taking power broadening into account, the fitting process successfully
eliminated the systematic shift in past data that had an adequate amount of the
line shape with which to fit to the theoretical shape. This effect was discussed in
Sec. 6.4. The apparent systematic which was reported earlier [44,45,47] is clearly
not present in the runs shown in Fig. 23 which comprises the 1984 data.
Another improvement in technique was the addition of a small shift in trapping
potential during the w~ and w~ excitation phases. This shift is done to reduce
the axial damping linewidth, /z. The Brownian-motion line shape theory [43] has
confirmed that the anomaly resonance will have a sharper [9] Zrms = 0 edge if this
axiallinewidth is reduced well below the magnetic resonance width.
At this point, our precision had made it possible to observe a residual shift of the
cyclotron frequency with cyclotron drive power (as described in Sec. 5.4.). After
correcting the cyclotron resonances for this effect, the 1987 data is displayed in a
plot of 6a e versus microwave power (in Fig. 24(b)) which is directly related to the
rectified current through the multiplier diode and similarly versus rf anomaly power
282 R. S. VANDYCK, Jr. ET AL.

192
(0) • e+
• e-
190 T

N
-g
188

186
tI f t
x
g 184
0
C\J 182
lC) 4
1 5 6 7 8
U)
anomaly power in ppm of axial shift
ai
!f?
(5 192
0 (b)

If
ciI 190
Q)

-3 188

186

184

182·L-L-L-L-L-L-L-L-L-L-L-~~~~~J-J-~
320 340 360 380 400
cyclotron power in microamps of rectified rf drive

Figure 24. Summary of most recent molybdenum trap data used in the electron/positron
comparison. In (a), the data is plotted vs anomaly power measured in ppm of the axial
shift when anomaly power is applied. In (b), the same data is plotted vs cyclotron power
measured in microamps of rectified rf drive. Any residual systematic. shift is below the
I-ppb level of uncertainty (data obtained from Ref. 35).

(in Fig. 24( a)) which is related directly to the axial frequency shift produced by
the anomaly drive as discussed in Sec. 3.4. Because the same microwave diode,
microwave generator, and trap were also used for the 1984 data, we are therefore
justified in correcting each run of the earlier data series for the shift in cyclotron
frequency associated with the strength of the microwave field. The data shown in
Fig. 23 contain this correction. As a result,

a e(1984)corr. = 1,159,652,189(4) x 10- 12 (50)

which is 4 ppb lower than the published result [6]. The uncertainty was limited by
apparent magnetic field fluctuations (from background gas collisions inside a leaky
vacuum tube), but was at least an order of magnitude less than for the 1979 data.
EXPERIMENTS WITH SINGLE ELECTRONS 283

The statistical error was about half of the reported uncertainty.

Finally, our improved precision has brought considerable theoretical interest in
possible cavity shifts due to the mode-pulling effects. Using the model of a cylin-
drical cavity as developed by Brown et al. (1985) [33], and the constraint that the
classical decay time is inhibited by a factor of 10 (shown in Fig. 15) in the neighbor-
hood of mode-structure (believed to exist for a comparable sized cylindrical cavity
at 50 kG), it was estimated that a probable shift in a e of no more than 4 x 10- 12
would exist; the following g-factor anomaly was thus quoted [35]:

a e (1987) = 1159652188.4(4.3) x 10- 12 (51)

where a statistical error of 0.62 x 10- 12 in the weighted average of all electron runs
was combined in quadrature with our estimate of l.3 x 10- 12 for the uncertainty in
the residual microwave power shift and the 4 x 10- 12 potential cavity-mode shift
discussed above. Also mentioned at the beginning of this section, similar runs were
taken at this same time using positrons instead. These runs are included in Fig. 24
and give essentially the same result within their uncertainties [35]:

(52)

where the common cavity-shift estimate is not included in the final uncertainty.

Phosphor Bronze Trap. This most recent work features a trap which is about
10% smaller than our previous traps, has a phosphor bronze ring electrode and
compensation guards, and OFHC copper endcaps. The structure was purposely
left more open than the traditional molybdenum traps in order to spoil the Q of
the microwave cavity. At high field, it indeed appears to be low since the classical
decay time is measured to be the free space value in the immediate vicinity (0.2%)
of 50.7 kG. However, at 36 kG, the trap looks less open because of the now longer
wavelength. Thus, over a 2% range, the cyclotron decay time was measured to vary
from 2 to 4.5 times the free space value. Again using the cylindrical model as our
guide, we estimate that the cavity Q for this trap could be on the order of 100. In
fact, according to Brown et al. [37]' cavity shifts in hyperbolic cavities are expected
to produce shifts only half as large as those of comparable cylindrical cavities. Thus,
this trap should produce results relatively free of cavity shifts in comparison with
earlier molybdenum traps, assuming that one can make these measurements at fields
where the cyclotron decay time is near the free space value.
The primary implication of a 10 times shorter classical decay time is that less
uninterrupted dwell time exists in the excited Landau levels (see Fig. 11). As a
result, much more anomaly power is needed to flip the spin from the required n = 1
level, if the spin is initially in the down state where it predominantly resides in the
284 R. S. VAN DYCK, Jr. ET AL.

TABLE 2. Summary of runs taken in the phosphor bronze trap

using the variable-bottle feature. Power shifts associated with ax-
ial, cyclotron, and anomaly drives are summarized as a net shift in
the third column and corrected anomalies have statistical errors in-
cluded. All cyclotron calibrations are made relative to "spin-down"
state.

Magnetic Relative [(Corrected a e )-

bottle a e shift 1 159652000]
Run (G/cm 2 ) (ppb) X10- 12
161 +245(10) -0.7(7) 182.0{2.8)
162 +245{1O) -0.7(7) 189.0(2.6)
164 +245(10) +0.5(9) 189.4(3.3)
165 +245(10) +0.7(8) 191.3(2.3)
168 -262(10) +2.0(8) 183.7(2.4)
169 -262(10) +1.0(8) 182.8(2.7)
170 -262(10) +1.8(9) 177.7{2.3)
171 +245(10) +6.9(9) 181.2{3.6)
172 +245(10) + 1.5(8) 189.0{l.9)
173 +245(10) +2.4(7) 187.9{2.0)
175 -262(10) -3.4(8) 188.4(2.2)
176 -262(10) -l.5(7) 182.5{2.3)
177 -262(10) +2.9(8) 184.1 (l.8)
179 -262(10) +2.3(8) 188.6(2.6)

n = 0 level. The observed detrimental effect of high anomaly power is reduced

axial 5/ N due to the boiling of liquid helium. Thus, we first prepare the electron
by forcing it into the spin-up state as described in Sec. 6.4. As indicated, when
analyzing this data, we report to the fitting program twice as many attempts to flip
the spin as the actual number used. The program expects a saturation to occur with
a relative probability of 0.5 for the spin to flip on each try. However, when we start
with the spin-up state, then we can approach a probability near unity for flipping
to the spin-down state for moderately hard drives because of the low probability of
leaving the lower energy state. Clearly, this simple approximation fails in the limit
of very high power since again, the probability of flipping the spin approaches 0.5
irrespective of initial state. Thus, we have been very careful to keep well away from
even the apparent saturation at unity.
The second implication of a 10 times shorter classical decay time is associated
with observing the cyclotron resonance line shape as described in Sec. 5.1. Our
axial 5/ N is not adequate for the fast alternating sequence required (~ 5 Hz) for
taking the cyclotron resonance in comparison with the optimal rate of 0.5 Hz used
in the old molybdenum traps when Tc ~ 1 sec. In addition, higher excited Landau
levels now decay away even faster, thus requiring much more microwave power, even
EXPERIMENTS WITH SINGLE ELECTRONS 285

when using continuous axial detection. This requirement put a severe constraint
on the microwave source, which had to be greatly improved in order to get a clean
source that did not artificially broaden the resonances.
Because of the use of continuous detection, we are now required to correct the
cyclotron frequency for the measured shifts due to axial drive as shown in Fig. 17
in addition to those due to the applied microwave field as shown in Fig. 16. For the
same reason, we must also add the shifts shown in Fig. 18 due to the anomaly power
which is applied during the observation of the anomaly resonance, since again axial
S / N was not adequate to allow for the simultaneous application of off-resonance
anomaly power during the observation of the cyclotron resonance. Table 2 lists this
series of runs taken in the phosphor bronze trap, with half taken at each extreme
of the variable bottle. The net corrections are also listed, and only in one case is
this correction not comparable or less than the final uncertainty in the measured
anomaly. In Fig. 25, we show the anomaly plotted versus Wk, We, and W" for these
runs, after corrections have been applied.
Finally, as indicated in Sec. 5.2., a magnetic field wander of ~ 10 ppb is observed
in this trap which contains a closed superconducting loop. As a result, the anomalies
measured in the runs listed in Table 2 are spread out in a somewhat non-statistical
distribution between two limits, about 12 ppb apart. This effect is clearly evident in
Fig. 26 which is a histogram of all runs taken in the phosphor bronze trap at the two
limits of the variable magnetic bottle. There is a possible residual B2 dependence in
these results since the average of the runs using B2 = 245G / cm 2 exceeds the average
of the runs using B2 = -262 G/cm 2 by about 3.1 x 10- 12 . Therefore, because of
the nature of this distribution, it seems only prudent to report the simple average
with the given standard deviation as an estimate for the uncertainty of the mean:

a e (1990) = 1159652185.5(4.0) x 10- 12 . (53)

As evident from Table 2, the uncertainty in Eq. 53 is about twice the typical sta-
tistical errors of the individual runs (which is most of the error listed in the fourth
column). This result clearly agrees with the measurements shown in Eq. 49-52.
Because of the possibly 10 times lower cavity Q, we are justified in not including
any uncertainty associated with unknown cavity shifts at this level of confidence.

7.2. "a e " AND QED

The intrinsic interest in this very precise measurement of the electron's g-factor
resides in the ability of theorists to calculate the same quantity to roughly the
 286 R. S. VANDYCK, Jr. ET AL.

195

II
• 245 G/cm2
·-262 II

N
-0
x
(a)

I 2 3
I n
4 5 6
anomaly power in ppm of axial shift
7

80
C\J 190
I{) (b)
<D
O"i
I{) 185

-
0
0
0
180
1
0
CII 2 4 6 8 10
cyclotron power In mW (arb. reference)

20 40 60 80 100
axial drive power in fLW (arb. reference)

Figure 25. Summary of the 14 runs obtained using the phosphor bronze trap at the two
extremes of the variable bottle. In each case, the computed anomaly is plotted versus
the amount of (a) anomaly, (b) microwave, and (c) axial drive power applied during the
excitation part of each cycle. At the given statistical level of uncertainty, no residual shift
remaIns.
EXPERIMENTS WITH SINGLE ELECTRONS 287

-c: 3 I I I I I I I I
Q)

E
Q)
~ overage = 85.5 (4.0)
-
:;)

~ 2 -
Q)
E - +
0.0-
o
>-
u - -
c:
Q)
:;) - + + - - - - +- + + +
c-
Q)

.t: 0 I I II il I I I II
76 78 80 82 84 86 88 90 92
[ a (e-)- 0.00I,159,652,100]x 10 12

Figure 26. Histogram of the 14 runs obtained using the phosphor bronze trap. The
"+" or "-" sign represents the sign of the variable bottle used for each indicated run.
The distribution is believed to be a representation of the wandering magnetic field which
occasional would make relatively fast fast jumps to a new field value between cyclotron
calibrations.

same accuracy via the powerful theoretical technique of Quantum Electrodynamics

(QED) which is based on the processes of virtual emission and absorption of photons
and the polarization of the vacuum by electron/positron pairs. These radiative
corrections can be arranged into an infinite power series of the quantity oIrr where 0:
is the fine-structure constant, and C i are the corresponding coefficients of the power
series. This particular lepton system is also fortunately blessed with understandable
and calculable effects due to the strong and weak interactions such that they do not
limit the present accuracy of the g-factor prediction.
The leading term has one Feynman diagram due to a single virtual photon ex-
change and its coefficient C] was first shown [48] by Schwinger to be exactly 0.5. The
second term allows for two virtual photon exchanges or a single vacuum polarization
bubble containing an e+ / e- pair, yielding a total of 7 Fynman diagrams. The C 2
coefficient in front of this term is known analytically to be -0.328,478,965, ... (most
recently predicted [49] by Sommerfeld and Petermann). The next order of complex-
ity contains 72 Fynman diagrams and the corresponding C3 coefficient has been
evaluated in part analytically [50] and in part numerically [51]: C 3 = 1.175,62(56).
As if this was not difficult enough, Kinoshita undertook the mammoth job of eval-
uating [52] the 891 Feynman diagrams associated with the 8th order term, yielding
C 4 = -1.434(138). The remaining contributions due to the other-lepton loops,
hadronic terms, and the electroweak effect have all been collected [53] into the
term: 8a~ = 4.46 x 10-]2.
288 R. S. VAN DYCK, Jr. ET AL.

In order to make connection between theory and experiment, one needs to have
an accurate measurement of the fine-structure constant, a. Unfortunately, the
tremendous improvement in precision of the g-2 experiment and theory have both
exceeded the present improvements in the accuracy of a. At present, the most
accurate independent measure of a is by the quantum Hall effect [54]:

a-I(QHE) = 137.0359979(32) (54)

which can be compared to the fine-structure constant computed from the ac Joseph-
son value of 2e/ h combined with the most recent low field determination [55] of the
gyromagnetic ratio r~ (for protons in a water sample):

a-l(acJ&r~) = 137.0359840(51). (55)

These two values do not even agree within their combined uncertainties. Using the
first of these (because it is more accurate), the theoretical value for a e [52] is

a( theory) = 1 159652 140(28) x 10- 12 (56)

where the error is almost exclusively due to the measurement error of a. Because
of this limitation, it is preferable to determine a from the g-2 experiment [33] and
QED theory [52]:

a -I (QED) = 137.03599222(94) (57)

where the uncertainty is still primarily determined by the theoretical error. How-
ever, it is anticipated that one day there will be comparisons of all three of these
major ingredients [a, ae(expt.), and ae(theory)] at the one part per billion level
of precision. This will provide a far more stringent test of QED than is presently
possible.

7.3. POTENTIAL FUTURE EFFORTS

The best method for handling the cavity problem for the future has not yet been
determined. It has been suggested by Brown, et al ~37] and Dehmelt [34,38] that the
characteristics of several neighboring modes can be investigated and the theoretical
model can then be used to predict the region in frequency space which is free of
shifts. Such regions do exist, but considerable effort may be required to find them.
However, it is always preferable, if a choice exists, to design the experiment such
that all regions of space are relatively shift free. More than likely, both low cavity
EXPERIMENTS WITH SINGLE ELECTRONS 289

Q traps will be used and the nearest modes will be studied by actually measuring
g-2 at the maximum shift positions.
The next task will be the improvement of axial sensitivity in order to allow still
smaller axial frequency shifts to be observed, with the intent of reducing the size
of the magnetic bottle. The clear evidence is that all major systematic effects,
other than cavity shifts, depend directly on the strength of this bottle in some
way. A more stable voltage source (possibly from a Josephson junction array)
would translate immediately into improved S / N, since voltage fluctuations in the
standard cells are believed to produce the present 10 ppb resolution. A higher Q
tuned circuit would also generate a larger signal for the same oscillation amplitude
in the trap.
However, since the real goal is to achieve an almost perfectly uniform magnetic
field (with some means of detecting magnetic moments), two possible methods
have been suggested for which this condition is approached by at least an order
of magnitude. One method involves utilizing the existing relativistic bottle in some
way. The primary impediment to this approach is axial sensitivity which presently
requires 10-20 times larger bottles just to see a spin flip. Several elegant methods
have been proposed [30,56-59] which might enhance the effect in order to yield the
required S / N, but no technique has yet demonstrated that it will work. The second
method involves another application of the variable magnetic bottle. Here, the
B 2 -term can be modulated [60] at some slow rate and a PSD can be used on the
axial-frequency-shift detector to pick out the corresponding modulated magnetic
moment (see Eq. 19). This technique has been attempted, but has so far achieved
only partial success. In all cases, the primary objective is improvement in precision
(though improvement in accuracy is also desirable) when the test to be performed is
an ultra high precision comparison of electron and positron g-factors. Systematics
such as cavity shifts or B2-related shifts will be common to both. Thus, a statistical
improvement is all that is required for the next improved comparison of matter and
antimatter.

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29. McLachlan, N.W. (1947) Theory and Applications of Mathieu Functions, Oxford
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Comparison of Electron and Positron 9 Factors', Phys. Rev. Lett. 59,26-29.
36. Van Dyck, Jr., R., Moore, F., Farnham, D., Schwinberg, P., and Dehmelt, H. (1987)
'Microwave-Cavity Modes directly Observed in a Penning Trap', Phys. Rev. A (Brief
Reports) 36, 3455-3456.
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Penning Trap Microwave Cavity', Phys. Rev. A 37, 4163-4171.
38. Dehmelt, H. (1987) 'Single Atomic Particle at Rest in Free Space: Shift-Free Sup-
pression of the Natural Line Width?', in W. Persson and S. Svanberg (eds.), Laser
Spectroscopy VIII, Springer- Verlag, New York, pp. 39-42.
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ber Dependency in the Compensated Penning Trap', Phys. Rev. A. 40, 6308-6313.
40. Van Dyck, Jr., R., Schwinberg, P., and Bailey, S. (1980) 'High Resolution Penning Trap
as a Precision Mass-Ratio Spectrometer', in J.A. Nolen, Jr. and W. Benenson (eds.),
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A iomic Masses and Fundamental Constants 6, Plenum, New York, pp. 173-182.
41. Wesley, J.C. and Rich, A. (1971) 'High-Field Electron g-2 Measurement', Phys. Rev.
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the 9 Factor of the Free Positron', Phys. Rev. A 5,38-49.
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Anomalous Magnetic Moment and Electric Dipole Moment of the Muon, and a Direct
Test of Relativistic Time Dilation', Nue. Phys. B 150, 1-75.
43. Brown, L.S. (1984) 'Geonium Lineshape', Ann. Phys. (NY) 159,62-98; also Brown,
L.S. (1984) 'Line Shape for a Precise Measurement of the Electron's Magnetic Moment',
Phys. Rev. Lett. 52,2013-2015.
44. Van Dyck, Jr., R.S., Schwinberg, P.B., and Dehmelt, H.G. (1979) 'Progress of the
Electron Spin Anomaly Experiment', Bull. Am. Phys. Soc. 24, 758.
45. Schwinberg, P.B., Van Dyck, Jr, R.S., and Dehmelt, H.G. (1984) 'Preliminary Compar-
ison of the Positron and Electron Spin Anomalies', in B.N. Taylor and W.D. Phillips
(eds.), Precision Measurement and Fundamental Constants II, Nat!. Bur. Stand.
(U.S.) Spec. Pub!. 617, pp. 215-218.
46. Van Dyck, Jr., R.S., Schwinberg, P.B., and Dehmelt, H.G. (1990) 'Consistency of the
Electron g-factor in a Penning Trap', submitted to Book of Abstracts for the Twelfth
lnt. Conf. Atomic Physics (ICAP-12), and this publication.
47. Schwinberg, P.B., Van Dyck, Jr, R.S., and Dehmelt, H.G. (1981) 'New Comparison of
the Positron and Electron g Factors', Phys. Rev. Lett. 47,1679-1682.
48. Schwinger, J. (1948) 'On Quantum-Electrodynamics and the Magnetic Moment of the
Electron', Phys. Rev. 73,416-417.
49. Sommerfield, C. (1957) 'Magnetic Dipole Moment of the Electron', Phys. Rev. 107,
328-329; Petermann, A. (1957) 'Fourth Order Magnetic Moment ofthe Electron', Helv.
Phys. Acta 30, 407-408.
50. Levine, M.J., Park, H.Y., and Roskies, R.Z. (1982) 'High-Precision Evaluation of Con-
tributions to g-2 of the Electron in Sixth Order', Phys. Rev. D 25,2205-2207.
51. Cvitanovic, P. and Kinoshita, T. (1974) 'Sixth-Order Magnetic Moment of the Elec-
tron', Phys. Rev. DID, 4007-4031; also Kinoshita, T. and Lindquist, W.B. (1977)
'Improving the Theoretical Prediction of the Electron Anomalous Magnetic Moment',
Cornell preprint CLNS-374.
52. Kinoshita, T. and Lindquist, W.B. (1990) 'Eighth-order Magnetic Moment of the Elec-
tron V. Diagrams Containing No Vacuum-polarization Loop', Phys. Rev. D 42, 636-
655; see also Kinoshita, T. (1988) 'Fine-Structure Constant Derived from Quantum
Electrodynamics', Metrologia 25, 233-237; and Kinoshita, T. (1989) 'Accuracy of the
Fine-Structure Constant', IEEE Trans. lnstrum. Meas. 38,172-174.
53. See, for instance, Kinoshita, T. (1978) 'What Can One Learn from Very Accurate
!Yleasurements of the Lepton Magnetic Moments?', in B. Kursunoglu, A. Permutter,
and L.F. Scott (eds.), New Frontiers in High Energy Physics" Plenum Publishing,
New York, pp. 127-143; also Kinoshita, T. and Lindquist, W.B. (1981) 'Eighth-Order
Anomalous Magnetic Moment of the Electron', Phys. Rev. Lett. 47,1573-1576.
54. Cage, M.E., et al. (1989) 'NBS Determination of the Fine-Structure Constant, and
of the Quantized Hall Resistance and Josephson Frequency to Voltage Quotient in Sl
Units', IEEE Trans. Instrum. Meas. 38,284·289.
55. Williams, E.R., et al. (1989) 'A Low Field Determination of the Proton Gyromagnetic
EXPERIMENTS WITH SINGLE ELECTRONS 293

Ratio in Water', IEEE Trans. Instrum. Meas. 38,233-237.

56. Dehmelt, H., Van Dyck, R., and Schwinberg, P. (1979) 'Proposal for Detection of
Geonium Spectra via Radial Displacement', Bull. Am. Phys. Soc. 24,49l.
57. Dehmelt, H., Van Dyck, R., Schwinberg, P., and Gabrielse, G. (1979) 'Proposal to
Detect Spin Flips in Geonium via Linked Axial Excitation', Bull. Am. Phys. Soc. 24,
675.
58. Dehmelt, H. and Gabrielse, G. (1981) 'Faster, Simpler Schemes to Distinguish n = 0,1
in Geonium', Bull. Am. Phys. Soc. 26, 797; (1984) 'Comb Excitation Scheme for
Resolving the Cyclotron Spectrum of Geonium', Bull. Am. Phys. Soc. 29,44; (1984)
'Quasi-Thermal, Multi-Step Excitation Scheme for Geonium Cyclotron Spectroscopy',
Bull. Am. Phys. Soc. 29,926.
59. Gabrielse, G., Dehmelt, H., and Kells, W. (1985) 'Observation of Relativistic, Bistable
Hysteresis in the Cyclotron Motion of a Single Electron', Phys. Rev. Lett. 54, 537-
539.
60. Schwinberg, P.B. and Van Dyck, Jr., R.S. (1981) 'Geonium Spectroscopy Using a
Modulated Magnetic Bottle', Bull. Am. Phys. Soc. 26, 598.
EXPERIMENTS ON THE INTERACTION OF INTENSE FIELDS WITH ELECTRONS

A. Weingartshofer
Laser-Electron Interactions Laboratory
Department of Physics
St. Francis Xavier University
Antigonish, Nova Scotia
Canada B2G lCO

ABSTRACT. This is an attempt to present an overview of currently

performed experiments that investigate some fundamental aspects of the
interaction of electrons with intense electromagnetic fields (laser and
microwave). The electrons are free or in a continuum state of the atom
or ion.

1. THE ROLE OF A GOOD EXPERIMENT

At the moment, we have the situation where the art of experimentation in

intense laser-atomic physics - which provides much information on laser-
electron interactions is ahead of theoretical and computational
methods. Many laboratories have not only provided some astounding new
observations but, more important, they have also presented them to the
physics community in a quantitative form. The results are impressive but
the systems that have been investigated are complex and in order to
explain them, theorists have to make approximations and find it difficult
to go beyond qualitative agreement with experimental data. To try to
develop a theoretical model that can incorporate "all" the subtle aspects
of multiphoton ionization and also the quantitative observations of the
generation of "multiple harmonics", is surely a challenge.
Experimentalists are not only discovering new phenomena but they are
also exploring them in quantitative terms and as a function of
fundamental parameters like intensity or polarization. One of their
major problems is to provide absolute values of laser intensities,
ionization cross sections or reaction rates. They are essential to
differentiate among the theories that have been proposed to explain the
same phenomena. The laser laboratories at C.E.N. de Saclay, France are
making great efforts in this direction [1].
Theoretical approaches may succeed in reproducing quali tati vely,
most of the observed features and, in principle, this would be sufficient
if the theory can also establish their dependence on specific atomic
structure or single out the possible situations that involve primarily an
explanation in terms of "free electron dynamics in intense lasers", i.e.
"the simpleman's theory".
295
D. Hestenes and A. Weingartshofer (eds.), The Electron, 295-310.
© 1991 Kluwer Academic Publishers.
296 A. WEINGARTSHOFER

Experiments in multiphoton ionization (MPI) and above-threshold

ionization (ATl) are very complex phenomena because of the large number
of parameters that can enter the process. Of less complexity are
experiments in simultaneous electron-mul tiphoton excitation of atoms
(SEMPE) because we have control over important parameters that determine
these off-energy shell processes and where ionization can be avoided. Of
still lesser complexity is the "elastic" version of the previous
experiments: elastically scattered electrons in the presence of an
intense field, also known as laser-induced free-free (FF) transitions.
They all are important in applied problems.
From the point of view of theory, they all are complex. There is
also a positive aspect if one considers that there is a common link
between them that can be explored to increase our understanding of
intense laser-atomic physics. Processes that can be controlled
experimentally can provide a testing ground for theoretical
approximations and new ideas before they are applied somewhere else, i.e.
the approximations and ideas can now be modified or extended with
confidence. It is an obligation of the experimentalist to devote time
and effort in designing experiments to test the limitation of widely used
approximations and establish their regime of application. We have
pursued this fundamental philosophy in our Laser-Electron Interactions
Laboratory.

2. GENERAL OVERVIEW

In dealing with laser-electron interactions it is convenient to

distinguish five basic experimental situations that describe the state of
the electron with respect to an atom and the intense radiation (laser or
microwave) .

a) a free electron: since free electrons cannot absorb photons, for

the interaction to take place effectively a "laser-stimulated
mechanism" must be invoked which is known in the literature as
"stimulated Thomson scattering" (elastic) and "stimulated Compton
scattering" (inelastic). This gives rise to light-induced forces
that change electron energies and momenta and are referred to as
"ponderomotive" effects. Multiples of photons can only be exchanged
with the radiation field in the presence of a third body, i.e. a
potential, and therefore can only occur in the case of a bound
electron or an electron colliding with an atom, i.e. a scattering
process, as described below.
b) a bound electron: bound in the ground state of the atom or "quasi-
free", bound in a high Rydberg state close to the ionization limit.
The electron can also form a negative ion, i.e W which has an
electron affinity of -0.754 eV. Current experiments of interest
investigate the ionization of atoms when illuminated with intense
electromagnetic radiation (laser or microwave). The ion yield as a
function of field intensity can be examined but principally they
rely on the interpretation of the photoelectron spectra of the
ejected electrons recorded with an electron spectrometer. The
EXPERIMENTS ON LASER-ELECTRON INTERACTIONS 297

experiments fall into four categories: mul tiphoton ionization (MPI),

above -threshold ionization (ATl), mul tiphoton detachment (MPD) , and
tunnelling if the intensity reaches a critical value.
c) electrons scatterin& from an atom in the presence of an intense
laser field are referred to as laser-induced free-free (FF)
transi tions, although the initial and final electron scattering
states are NOT completely free states, only asymptotically free
states (even though the time of collision may only last 10-15 s). It
is also possible to experiment with scattering resonances which
occur at well-defined electron impact energies, and the electron
attaches itself to the target atom forming a temporary negative ion,
nevertheless a truly bound state that may live 10' times longer
than the duration of a normal collision.
d) electron-multiphoton-atom a simultaneous interaction, carried out
in a controlled three-beam experiment.
e) electrons produced in new sources; i.e. Penning electrons.

Free electrons in (a) originate in experiments with bound electrons

described in (b). As a matter of fact, they playa crucial role in the
overall processes of MPI, MPD and the more intriguing phenomenon of ATI
where the electron absorbs a much greater number of photons than
necessary for the atom to be ionized. ATI experiments have greatly
increased our understanding of the interaction of intense fields with
free electrons. Intense laser atom physics relies on these electrons for
information on the behaviour of atomic structure when exposed to high
intensities (1014 W/cm2 ) . Recent developments of femtosecond laser pulses
has provided a tool to investigate these properties [2). See also
article by A. Bandrauk in this Book. The field has made remarkable
progress and has reached a very interesting state but there are also many
challenges, especially in the theoretical interpretation of the
complexity of processes that occur. Since there is a necessity to
understand laser-matter interaction (for applied reasons) it has become
one of the most active areas of research and is well-documented [3,4).
The more elementary laser-electron interaction represents one part of the
overall program and the intention in this presentation is to discuss some
relevant aspects of this interaction and establish some common links with
the more complex processes. First, we want to summarize some general
notions.
Experiments with microwave fields occupy a very unique place in this
area because of their special techniques and they also represent the
limit of the low-frequency regime. The contribution of Tom F. Gallagher
to this book illustrates all this in a magnificent way in an article by
the title "Ionization in Linearly and Circularly Polarized Microwave
fields". Microwave ionization of Rydberg atoms can investigate high
order mul tiphoton ATI (_10 5 microwave photons) processes and show in which
way they are fundamentally different from ATI at optical frequencies.
It was mentioned above that intense laser-atom physics relies on the
ejected electrons for information of atomic structure. Another important
source of information is the emitted radiation in an MPI process which
contains very high harmonics of the laser field. This radiation is now
being examined quantitatively and systematically in C.E.N. de Saclay,
298 A. WEINGARTSHOFER

France. The essential aspects of this phenomenon is reviewed in this

book in an article by the title: "Multiple Harmonic Generation in Rare
Gases at High Laser Intensity" prepared by Anne L'Huillier, Louis-Andre
Lompre and Gerard Mainfray.
Two very recent areas are now demonstrating their worth as new
powerful tools to probe both, laser-electron interactions as well as
particular aspects of intense laser-atom physics. Although multiphoton
processes are being observed, all the complexities of MPI are avoided by
working at low laser intensities, 10· W/cm'. These new areas are
described in this book by Barry Wallbank in "Absorption and Emission of
Radiation During Electron Excitation of Atoms" and by Harald Morgner in
"Penning Ionization in Intense Laser Fields". This area also illustrates
some possible mechanisms that can lead to laser-controlled chemistry.
The process of multiphoton ionization of an atom under the influence
of an intense oscillating electromagnetic fields is a fundamental
question. Leo Moorman addresses the problem from a new point of view in
an article entitled "Microwave Ionization of H Atoms: Experiments in
Classical and Quantal Dynamics", Moorman discusses the theory of
microwave ionization based on experiments performed in the laboratory at
Stony Brook, NY, which reveals that the process may be divided into
different regions of distinct behavior depending on the precise
experimental situation. Some regions may be described using classical
atomic dynamics while others require quantum atomic dynamics. The author
discusses the level of accuracy of this division. Microwave ionization
of atoms has become one of the experimental and theoretical prototypes
for studying quantum chaos. It is of interest to observe that also in
the last paper of the theory section, Christof Jung addresses a similar
problem by proposing a theory to merge the ideas of free-free transitions
and scattering chaos.
Finally in the theory section of this book, Andre Bandrauk discusses
the theory of intense laser interaction with molecules in a very timely
article by the title "The Electron and the Dressed Molecule". Molecules
present a system of higher complexity and therefore is the restricted
domain of only a small group of investigators. His predicted laser-
induced effects have been observed experimentally and the results
illustrate magnificently that this area is important for fundamental as
well as applied reasons.
Our current understanding of MPI and ATI is, that, somehow we can
think of it as a four stage event. First, the atom is deformed by a
dynamical Stark shift. Two, the deformed, or dressed, atom absorbs
photons and forms an ion and an electron in a continuum state in the
combined fields of the laser and ion (long-range potential). Three,
laser-induced electron transitions between continua may take place and
even laser-stimulated recombination. Four and final stage, the electron
is released and it has to escape from the laser field as a free electron
and ultimately find its way to the detector.
When does the electron come free, and when can we assume free
electron laser interaction? There is no clear answer to this, but, at
this point, one distinguishes two aspects: the "quiver" and the "drift"
motion of the electron. The former is referred to as the
"ponderomotive" aspect of the laser-electron interaction and this is now
EXPERIMENTS ON LASER·ELECTRON INTERACTIONS 299

fairly well understood, although some confusion existed in the past [5].
The "drift" aspect of this interaction which can impart large amounts of
kinetic energy to the ejected electron, as has been demonstrated in
recent experiments [6,7], has raised some interesting questions. These
are typical ATI experiments where the energy comes from the absorption of
photons in excess of the minimum number required for ionization.
However, this raises the question: where does the momentum come from
since photons carry very little momentum [8]? Some further reference to
this will be made in Section 4.
The oft-made analogy between electron scattering and the ejection of
an electron during the ionization process is a very valuable concept that
has been explored repeatedly, e.g. Collisions and Half-Collisions with
Lasers is the title of a workshop organized by N. K. Rahman and
C. Guidotti [9]. It has been pointed out [8] that the analogy is far
from complete. In the low frequency regime one essential difference is
that, although both the collision duration and atomic orbital period are
short compared to the cycle time, in a scattering process the electron
passes by the core only once, while in an ionization process the electron
orbits the core many times. Rosenberg [10] has suggested that
multiphoton ionization (MPI) may be thought of as the second half of an
induced resonance reaction. It is not difficult to realize that there is
a link between microwave excitation and ionization of highly excited
Rydberg atoms and "resonant electron scattering" in a laser field via the
formation of a temporary bound state. In this context it is illuminating
to refer the reader to page 192 of Bandrauk's article on the physics of
the mul tiphoton ionization of the H,-molecule, where he describes the
photoionized electron as continuing to absorb photons that create above-
threshold ionization peaks which reflect the vibronic structure of the H;-
ion "dressed by the intense field".
It is appropriate at this point to recall one important result that
we have learned in scattering experiments in the presence of a radiation
field [11,12]. The measurements clearly demonstrate the transfer of
multiples of the photon energy of a CO 2 - laser into the kinetic energy
of electrons while being scattered on an atomic target. This is
basically a confirmation of the Kroll and Watson equation [13] which
plays a fundamental role in laser-electron interactions and has occupied
the minds of many theorists and presents challenges to the
experimentalists. Further reference to this type of quantified "inverse
bremsstrahlung" or laser-induced free-free (FF) transitions of electrons
will be made in Sections 3 and 4.
Negative ions represent a unique system for the investigation of
MPl and ATl. They have usually one bound state and electron correlations
playa primary role in determining binding energies. Negative hydrogen
is a typical example, the ion has only one bound state (the ground state)
and, in contrast to ionization of atoms, the detached electron [14,15]
experiences a short-range (l/r') potential as it leaves the ion rather
then the usual long-range Coulomb potential seen in MPl and ATl. The
four- stage event of the process is considerably simplified. Experiments
of MPD of electrons from negative hydrogen, H-, have recently been
performed (16] where a beam of ions (produced externally) was intersected
by a focused CO. TEA laser. We want to remark that negative ions also
300 A. WEINGARTSHOFER

play an important role in electron scattering, i. e. electronically

excited "Feshbach resonances" that form a system of bound "temporary
negative ions" [17). These resonant states have also been investigated
in the presence of a strong radiation field [18,19). The negative ions
are produced (internally) and since the photon energy (hw) is much larger
than the resonance width (8), the field changes sign many times during
the resonance lifetime so that decay by tunnelling is not very likely.
Further reference to this will be made in Sect. 4.

3. LASER-STIMULATED PROCESSES

Once the electron is free (see stage four above) it can no longer absorb
photons but it will continue exchanging momentum and energy with the
laser field in a very significant way. This can only be explained in the
photon language of quantum electrodynamics, i.e. high-intensity coherent
light scatters from a free electron in a stimulated manner known as
"stimulated" Thomson scattering (elastic: stimulated scattering among
different momentum eigenmodes) or "stimulated" Compton scattering
(inelastic: stimulated exchange of photons between different frequency
modes) [20,21,22).
These theoretical notions originated with the early development of
high intensity laser beams and were summarized by Kibble in 1966 [22) in
a paper with the title "Mutual Refraction of Electrons and Photons" where
the complete analogy between the processes of refraction of light by
electrons and of electrons by light is emphasized. Kibble goes on to
show that the important effects in most cases of practical interest (huge
number of photons) can all be understood in purely classical terms.
Solving Newton's equation for an electron subjected to a Lorentz force
that is turned on during the time scale of one radiation cycle will set
the electron into a "quiver" motion. Thomson scattering arises because
the electron is accelerated by the external electric field, and thereby
absorbs and reradiates energy [21). Dealing with the fine details of the
interaction of the electron (mass m and charge ~) with the spatially
varying ~ and ~ fields of the radiation field (i.e. the focused laser) is
referred to by the word "ponderomotive" and we speak of ponderomotive
forces (i.e. a field gradient force) and a ponderomotive potential, which
is the time-averaged kinetic energy of the oscillating electron

Wp

where Eo is the electric field amplitude, I the intensity in W/cm2 ,w and

~ the angular frequency and wavelength of the radiation. This energy
depends inversely on the square of the laser frequency and therefore
ponderomoti ve effects are very important for infrared and microwave
radiation.
Experiments suggested in the paper of Kibble [22) have been
attempted [23), but it is only very recently that confirmation of these
effects has been carried out in a series of very elegant experiments
EXPERIMENTS ON LASER-ELECTRON INTERACTIONS 301

conducted at AT&T Bell Laboratories [24,25,26]. These results clarified

the role of the ejected free electrons in MPI and ATI processes and have
been of fundamental assistance in t:he interpretation of the observed
photoelectron energy spectra.
Ponderomotive forces are dramatic examples of the predominance of
stimulated processes in free electron-laser interactions. In this
regard, it is illuminating to consider another example that was mentioned
before, FF transitions or electron scattering from an atom under the
influence of an external field [12]. During the scattering event, the
electron is accelerated and, on the basis of classical electrodynamics,
an accelerated charged particle will emit electromagnetic radiation, i. e.
bremsstrahlung. For an electron with an incident energy, E1 = 10 eV, the
"spontaneous" emission of a photon with a perceptible energy has only
negligible probability. However, if the electron scattering event takes
place inside a laser field, then the induced analog of bremsstrahlung can
occur and we will observe a laser "stimulated" phenomenon where
absorption as well as emission will occur with equal probability.
Notice, however, that in contrast to stimulated Thomson scattering,
during the electron-atom collision, multiples of laser photons (hw) are
exchanged i. e. the scattered electrons can only increase or decrease
their kinetic energy in units of laser photons, and this is now observed
routinely [11,27,28].
It is of interest to remark that the emitted photons in FF
transitions have not been investigated. This project is in the planning
stage. It has been predicted [29] that there is a high probability that
these emitted photons contain a high proportion of multiple frequencies
of the driving wave frequency, i..e. the most elementary example
imaginable of "mu1ti.p1e harmonic generation".
Mu1ti.p1e harmonic generation in the MPI of the rare gases at high
laser intensity has been investigated in C.E.N. Sac1ay, France and some
exciting experimental results and their interpretation are discussed in
this book in an article prepared by Anne L'Hui11ier, Louis-Andre Lompre
and Gerard Mainfray. The phenomenon has been known for over 20 years
[30] but it is only recently that it is being investigated systematically
and quantitatively.
Both laser-stimulated Thomson scattering and FF transitions take
place in intense radiation fields where a very large number of coherent
photons are being scattered and therefore we are, in principle, dealing
with a classical radiation field. FF transitions become a semi-classical
problem because the interaction of the electron beam with the target atom
has to be treated quantum mechanically [11,12]. On the other hand,
stimulated Thomson scattering assumes a totally classical physical
picture as was described above, i.e. a classical electron in an
oscillating electromagnetic field [21,22].

4. INTERESTING SITUATIONS THAT RAISE SOME QUESTIONS

When an atom is ionized, the free electron is born with an amount of

ponderomotive (i.e. quiver) energy Wp and its motion is the superposition
of a "drift" motion and a "quiver" motion, which as it travels through
302 A. WEINGARTSHOFER

the beam waist (finite size, 20 ~m) is reconverted into translational

motion by the field gradient force (ponderomotive force), thereby leaving
the electron energy unchanged, owing to the conservative nature of the
potential. This reconversion of energy was predicted 25 years ago by
Kibble [22] and has recently been observed experimentally by Bucksbaum et
al [24,26] who demonstrated in a very elegant way the equivalence of the
electron's leaving the focus of the field to leaving the region of a
potential energy Wp.
This reconversion cannot be consummated in the case of ultrashort
pulses that terminate before the electrons have time to traverse the
dimensions of the focused pulse. A substantial fraction of the quiver
energy cannot be recovered and the energy is effectively transferred back
to the laser field. Two recent experiments on ATI processes were
performed under similar conditions by Gallagher [6] and Corkum et al [7]
with circularly polarized radiation in the long wavelength limit: CO 2 -
laser and microwaves. The results were remarkable, the electrons picked
up an unusually high drift momentum, while it was just remarked above
that for ultrashort pulses the energy is returned to the wave and does
not contribute to ATI so, where does the drift motion come from?
The interpretation of Gallagher [6] and Corkum et al [7] is that
these are free electron phenomena and the observations are primarily the
result of the interaction of a newly freed electron with the laser field.
The electrons are removed from the primary influence of the atom almost
instantaneously by tunnelling or dc field ionization (fraction of cycle
of the radiation) and therefore are released into a well defined phase of
the electric field. The quantitative discussion is based on classical
physics. The fact remains, however, that these observations bring a new
insight to ATI of a laser-electron interaction that will generate much
discussion in the physics community.
Two theorists, Cooke and Shakeshaft [8,14] have written a very
interesting article on these observations which they describe as a
"puzzle" with the following words "the electron is released into the
field with a small mechanical momentum, and so one may wonder how the
electron can later acquire a large momentum ..... while the radiation
field provides the photoelectron's energy in a photoionization experiment
it cannot provide the photoelectron's momentum". They propose a non-
radiati ve transfer mechanism that underlines the ideas of electron
scattering in their theory. The drift energy originates in the
absorption of photons followed by an electron-core scattering that is
elastic so the drift momentum really originates in the electrostatic
force which initially binds the electron to the atomic core, they call it
a "soft scattering of the electron from the atomic core to which it is
bound". The theory is illustrated with very vivid physical images and it
will surely stimulate many minds to try to improve on the model.
However, a new theoretical interpretation is not excluded.
Although the analogy between scattering and ionization in a low-
frequency field is far from complete, it has been used very successfully
as a basis for a theory that explains several of the characteristic
structures observed in ATI spectra. This is clearly illustrated in an
article by Kupersztych [31] with the title "Inverse Half-Bremsstrhlung in
MPI of Atoms in Intense Light Beams". In the first part of the problem
EXPERIMENTS ON LASER-ELECTRON INTERACfIONS 303

Kupersztych takes basically the same approach as Gallagher [6] and Corkum
et al [7], i.e. the electron is removed from the ground state into the
continuum in a time shorter than an optical period and then comes
strongly under the influence of the intense electromagnetic field but
still acted on by the long-range Coulomb field of the ion. The electron
can absorb photons and momentum while receding from the ion. The ionized
atom and the collision centre are the same and, therefore, the name half-
collision and the similarity with free-free (FF) transitions as a
mechanism to absorb additional photons. The only difference is that in
FF transitions we scatter electrons from a neutral atom while in the ATI
process the scattering is from an ion (long-range Coulomb field). This
approach has the merit that it represents an attempt to investigate the
physical relationship that may exist between ATI and FF processes. Is
there a clear connection, and is it possible to detect the influence of
the long-range Coulomb field of the ion on FF processes? Mittleman [32]
has written a very comprehensive article on the Coulomb extension of the
Kroll and Watson theory [13] which shows special peculiarities that we
don't fully understand.
The FF process that has been discussed so far is basically an
elastic collision (the target atom changes no internal energy) and the
duration of the collision (approximately hlE i , where Ei is the incident
energy of the electron) is much shorter than the laser period (l/w) in
the low-frequency regime. Therefore, photon absorption will occur
primarily immediately before or after the collision but is very unlikely
to occur during the collision proper. This is the essence of the low-
frequency approximation and the condition is that hIEi«l/w, or hw«E i ,
i. e. also the name "soft-photon" approximation. The consequences are
that the atom takes on the role of "passive" observer and simply supplies
a potential to consummate the FF process. This was first recognized by
Kroll and Watson [13] in their formulation of FF processes, i.e. the FF
cross section is simply the radiationless elastic cross section
multiplied by a factor that reflects only the properties of the laser and
the electron but not the atom. This result has been confirmed, so far,
by experiments but attempts are being pursued intensely to determine the
circumstances under which the breakdown of the "soft photon"
approximation can be observed [11,27,28,33].
One possible approach is to scatter electrons resonantly by making
Ei = Er , the resonance energy, so that the electron is captured
temporarily forming an electron-core scattering resonance with a narrow
width r and a lifetime h/r which is 10' times longer than hlE r , i.e. a
true bound state is formed in the presence of the radiation field. One
might expect some significant influence on the cross section of the FF
process under the new circumstances. The expected changes were examined
[19a,19b] by measuring the FF cross sections of the p-wave neon and argon
resonances around 16.1 and 11.1 eV for the incident electron energy.
Significant abnormalities were observed that cannot be predicted by the
soft -photon approximation [34]. These results are, of course, very
promising and are now being investigated in our laboratory at higher
laser intensities.
304 A. WEINGARTSHOFER

5. ELECTRON-MULTIPHOTON-ATOM SIMULTANEOUS EXCITATION

In FF processes the atom remains in the ground state and, therefore, we

are examining elastic collision processes. In contrast, in this series
of experiments the structure of the atom will be modified, first by the
laser field and then excitation under the simultaneous action of one
electron and several photons i.e. a three-body multiphoton collision
performed in a controlled three - beam experiment. The process can be
monitored by detecting the excited state directly (if a metastable state)
or indirectly by observing the emitted UV radiation, the energy and
angular distribution of the scattered electrons, or the ion yield if one
is probing the ionization threshold in order to determine the ionization
cross section as modified by the laser field or investigating laser-
stimulated electron-ion recombination.
So far, we have used a TEA CO 2 - laser with a peak power of 10· W/cm'
operated in the multimode optical configuration and linearly polarized.
These experiments present a new approach to examine both photon-electron
interactions and 1aser-electron-atom interactions. One may want to
investigate how electrons and photons can couple to perform a specific
function or explore atomic structure under the influence of an intense
field by monitoring e1ectron-multi-photon excitation processes, i.e. an
alternative way to understand intense laser-atom physics for a specific
system that is readily accessible. This condition poses limitations but
one has to remember that the investigation has the advantage of being
performed in a controlled manner and can produce quantitative
information, i.e. cross sections and their dependency on several
parameters like intensity, polarization, etc.
This technique is only at its beginning and so far it has been used
to measure cross sections for the following.

i) Simultaneous off-energy shell excitation of He 23 S by an

electron and one to four photons. Notice that the process
starts with a singlet state He liS and therefore we are also
observing an interesting electro exchange mechanism [35].

ii) Simultaneous electron-photon excitation of metastable He (2"S),

and the 23p states of neon and argon [36].

iii) Quanti tati ve experimental investigations of the effects of

laser intensity and polarisation on the formation of He 23 S by
simultaneous electron-photon excitation [37].

iv) Absorption and emission of radiation during electron excitation

of the 2"S and 2"p states of helium. We report electron spectra
resulting from the inelastic scattering of 4S-eV electrons from
He atoms in the presence of an intense CO 2 - laser [38].

One can foresee a wealth of new experiments of this nature as a

testing ground of laser-electron interactions and atomic physics in the
presence of intense lasers. As a way of illustration I would like to
make a few specific comments about the type of experiments described
EXPERIMENTS ON LASER-ELECTRON INTERACTIONS 305

under (i) above. Barry Wallbank will discuss the example of case (iv) in
an article in this book.

5.1 COMMENTS ON THE OFF-ENERGY SHELL EXCITATION He l'S ~ He 2'S

Geltman and Maquet [39] have examined these results and their calculated
cross sections are in good qualitative agreement with them. This is
interesting since they considered a simple extension of the Kroll and
Watson formula [13] for laser-assisted collisions which would demonstrate
that the "soft-photon" approximation 'has still some validity under these
circumstances. Further comments on this will be made below.
Trombetta [40] is intrigued by the electronic exchange that has been
observed in the excitation "singlet ~ triplet" state of helium and has
calculated cross sections for the electronic exchange in laser-assisted
elastic electron-hydrogen collisions. He has reached the remarkable
conclusion that the exchange mechanism may strongly dominate the
differential cross section. This, is of course, a very significant
result since this would be a clear example of the use of a laser to
control specific excitation mechanisms that may have practical
applications, i.e. laser-assisted chemistry.
In this inelastic process, all three pairs of collisional subsystems
violate energy conservation. We have, then, the possibility of
observing, for example, inelastic electron-atom differential cross-
sections or total excitation cross-sections at electron kinetic energies
which are off the "energy shell" from the point of view of the electron-
atom subsystem. Sundaram and Armstrong, Jr. [41] have taken the
viewpoint that ATI is a highly complex process and make an interesting
analysis of the participating mechanisms which involve both excited bound
states and continuum states, all invoke intermediate. off-ener~y shell
transi tions. It is our hope that in the future our experiments may
contribute to this analysis.
Finally, it is illuminating to interpret the results from the point
of view of the "simpleman's theory" [42] which in this case is really a
quali tati ve discussion of the Kroll and Watson [13] approximation
discussed in section 4. The basic idea, as illustrated in the diagram,
is that we are viewing an
"instantaneous" collision between
an electron (with velocity Va) and
an atom inside a laser field
(E ~ Eo cos wt). The electron
linearly will acquire an oscilatory motion
polarized (i.e. "quiver") and therefore a
laser quiver velocity Vq(t) ~ (eEoImw) E

...
sin wt. The duration of the
collision is "ultrafast" compared
to the period of the laser (2~/w),
and therefore the electron will
a 19 eV electron, V. collide with the atom at a given
time to with an instantaneous
velocity V-V. + Vq(t o) and a
306 A. WEINGARTSHOFER

kinetic energy ~mV~ = ~m [V. + Vq(to»)2 or ~ m [V: + Vq2 + 2V•. Vq(to)]. It

is easy to see that the middle term is negligible compared to the third
term which is the one of interest in this discussion. We wish to evaluate
the term 2V•. Vq(to) and demonstrate that under optimum conditions (i.e.
sin wto - 1) can acquire an energy equivalent to FOUR CO 2 - laser Photons,
so that the combined energy of the incident electron (19.34 eV) plus the
FOUR PHOTONS (4 x 0.120) add up to 19.820 eV or the excitation energy for
the transition He llS .... He 2'S.
And indeed, this can be demonstrated for the following experimental
conditions: The incident electron has an energy of 19.3 eV with a
corresponding velocity V. = 2.6 x 10· cm/s and we are using a pulsed TEA
CO 2 - laser of intensity I = 10· W/cm 2 and angular frequency w = 2 X 1014
S-l and a wavelength = 10.6 ~m. Assuming sin wt - 1, substitution will
result in

2 V. (eEo/mw) x 1 = 2 x 2.6 x 10· x 25.6 x .[lci'" x 10.6

= 4.5 X 10-1 ' ergs = 0.48 eV (FOUR PHOTONS)

The "simpleman' s theory" provides a physical picture consistent with

energy conservation involving FOUR-PHOTONS, but it takes more
sophistication to reproduce the "quantized " photon effects observed in
the experiment [37) which indeed are qualitatively reproduced in the
calculations of Geltman and Maquet [39] mentioned above.
It is difficult, however, to visualize a mechanism to explain the
TEN-PHOTON absorption/emission that has been measured in FF processes in
the backward electron scattering direction and the same laser intensity.
In this geometry one would expect a cancellation of effects.
Changing the plane of polarization of the laser reduces the effect,
also in accord with the "simpleman's theory", however when V. and E are
perpendicular to each other the cross section should totally ~ but
the experimental results show that this is not the case. We observe a
very significant signal that has not been explained.
The absorption of photons implies not only energy but linear and
angular momentum as well. The problem of linear momentum is not obvious
and was discussed in Sect. 4. Here we want to make reference to the
angular momentum. In the simultaneous electron-photon excitation
experiment, linearly polarized radiation was used, i.e. each photon can
change the angular momentum quantum number i by ± 1, on the other hand,
with circularly polarized radiation i must increase by one with the
absorption of each photon. This illustrates the potential of detailed
examination that these types of experiments can offer to explore laser-
electron interactions for future experiments.

6. NEW SOURCES OF ELECTRONS: THE PENNING ELECTRON

It was said in the introduction to this presentation that it is an

obligation of the experimentalist to design new experiments to explore
intense laser-atom physics and laser-electron interactions under new
conditions. So far we have considered experiments where the electron
EXPERIMENTS ON LASER-ELECfRON INTERACTIONS 307

originates in a MPI process or in a controlled electron beam. In

contrast, the ionization energy to produce Penning electrons is provided
from an excited rare gas atom in a collision process where the partners
simply have to come in "contact". It could be interpreted as
autoionization of this collision complex which is a very fast process (10-
15S ) . This represents new initial conditions which offer various
experimental possibilities. The important point that we want to make
here is that "field-free" Penning ionization is well understood and the
observed electron-energy spectra of many systems are very sensitive to
external perturbations. In principle, we have here a new system that can
provide information on both, laser-electron interactions or the
"perturbed atomic structure" by an intense laser field.
The Penning technique uses controlled two beam experiments and
examines the emitted Penning electrons according to their energy and
their angular distribution. To monitor the experimental parameters the
standard procedure is to compare the Penning electron spectra with those
produced by a conventional VUV photoionization source (He resonance line,
hw = 21.2175 eV), i.e. one photon. Two remarks have to be emphasized:
one, the experiments are performed under good controlled conditions
second, the VUV photon ionization source can also offer some interesting
experimental possibilities with intense lasers physics.
These "new idea" experiments are discussed in this book in an
article by Harald Morgner.

7. REFERENCES

1. Morellec, J., Normand, D. and Petite, G. (1982) Nonresonant

multiphoton ionization of atoms, Adv. At. Mol. Phys. 18, 97-164.
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understanding, Physics World, September, 47-50.
(b) Crance, M. (1990) Nonperturbative ac Stark shifts in hydrogen
atoms, J. Opt. Soc. Am. B7, 449-455.
3. (a) Cooke, W. E. and McIlrath, T. J., eds. (1987) Feature on
multielectron excitation in atoms, J. Opt. Soc. Am. B4, 705-862.
(b) Kulander, K. and L'Huillier, A. eds. (1990) Feature on high-
order processes in intense laser fields, J. Opt. Soc. Am. B7, 403-
685.
(c)Bandrauk, A. D. ed (1988) Atomic and molecular processes with
short intense laser pulses, Plenum Press.
4. (a) Mittleman, M. H. (1982) Introduction to the theory of laser-atom
interactions, Plenum Press, New York.
(b) Chin, S. L. and Lambropoulos, P. eds. (1984) Multiproton
ionization of Atoms, Academic Press.
(c) Faisal, F. H. M. (1987) Theory of Multiphoton Processes, Plenum
Press, New York.
(d) Agostini, P. and Petite, G. (1988) Photoelectric effect under
strong irradiation, Contemp. Phys. 1, 57-77.
5. Miloni, P. W. and Ackerhalt, J. R. (1989) Keldysh approximation, A',
and strong-field ionization, Phys. Rev. A 39, 1139-1148.
308 A. WEINGARTSHOFER

6. (a) Gallagher, T. F. (1988) Above-threshold ionization in low-

frequency limit, Phys. Rev. Lett. 61, 2304-07.
(b) Fu, P., Scholz, T. J., Hettema, J. M. and Gallagher, T. F.
(1990) Ionization of Rydberg atoms by a circularly polarized
Microwave field, Phys. Rev. Lett. 64, 511-14.
7. Corkum, P. B., Burnett, N. H. and BruneI F. (1989) Above-threshold
ionization in long-wavelength limit, Phys. Rev. Lett. 62, 1259-62.
8. Cooke, W. E. and Shake shaft , R. (1991) Nonradiative transfer of
momentum from an electromagnetic wave to a charged particle, Phys.
Rev. A 43, 251-57.
9. Raman, N. K. and Guidotti, C. eds. (1984) Collisions and half-
collisions with lasers, Harwood Academic Publishers.
10. Rosenberg, L. (1981) Intermediate- and strong-coupling
approximations for scattering in a laser field, Phys. Rev. A 23,
2283-92.
11. (a) Weingartshofer, A., Holmes, J. K., Caudle, G., Clarke, E. M. and
Kruger, H. (1977) Direct observation of multiphoton processes in
laser-induced free-free transitions, Phys. Rev. Lett. 39, 269-70.
(b) See also Weingartshofer, A. and Jung, C. (1984) Multiphoton
free-free transitions, Ref. 4b pp. 155-87.
12. Kruger, H. and Jung, Ch. (1978) Low-frequency approach to
multiphoton free-free transitions induced by realistic laser pulses,
Phys. Rev. A 17, 1706-12.
13. Kroll, N. M., and Watson, K. M. (1973) Charged-particle scattering
in the presence of a strong electromagnetic wave, Phys. Rev. A 8,
804-09.
14. Becker, '.I., Long, S. and McIver, J. K. (1990) Short-range potential
model for multiphoton detachment of the H- ion, Phys. Rev. A 42,
4416-19.
15. D6rr, M., Potvliege, R. M., Proulx, D. and Shakeshaft, R. (1990)
Multiphoton detachment of H- and the applicability of the Keldysh
approximation, Phys. Rev. A 42, 4138-50.
16. Smith, W. '.I., Tang, C. Y., Quick, C. R., Bryant, H. C., Harris, P.
G., Mohagheghi, A. H., Donahue, J. B., Reeder, R. A., Sharifian, H.,
Stewart, J. E., Toutouchi, H., Cohen, S., Altman, T. C., Risolve, D.
C. (1991) Spectra from multiphoton electron detachment of H-, J.
Opt. Soc. Am. B8, 17-21.
17. Schulz, G. J. (1973) Resonances in electron impact on atoms and
diatomic molecules, Rev. Mod. Phys. 45, 378-486.
18. (a) Andrick, D. and Langhans, L. (1978) Measurements of the free-
free cross section of electron-Ar scattering, J. Phys. B: At. Mol.
Phys. 11, 2355-60.
(b) Langhans, L. (1978) Resonance structure in the free-free cross
section of electron-Ar scattering, J. Phys. B: At. Mol. Phys. 11,
2361-66.
(c) Andrick, D. (1980) Free-free processes in electron-atom
scattering: Experiment, Electronic and Atomic Collisions, N. Oda
and K. Takayanagi, eds., North-Holland Publishing Co., 697-704.
19. (a) Andrick, D. and Bader, H. (1984) Resonance structures in the
cross section for free-free radiative transitions in e--He
scattering, J. Phys. B: At. Mol. Phys. 17, 4549-55.
EXPERIMENTS ON LASER-ELECTRON INTERACTIONS 309

(b) Bader, H. (1986) Resonance structures in the cross section for

free-free radiative transitions in e--Ne and e--Ar scattering, J.
Pys. B: At. Mol. Phys. 19, 2177-88.
20. Brown, L. S. and Kibble, T. W. B. (1964) Interaction of intense
laser beams with electrons, Phys. Rev. 133, A705-A7l9.
21. Frantz, L. M. (1965) Compton scattering of an intense photon beam,
Phys. Rev. 139, B l326-B 1336.
22. (a) Kibble, T. W. B. (1966) Mutual refraction of electrons and
photons, Phys. Rev. 150, 1060-69.
(b) Kibble, T. W. B. (1966) Refraction of electron beams by intense
electromagnetic waves, Phys. Rev. Lett. 16, 1054-56.
23. Bartell, L. S., Thompson, H. B. and Roskos, R. R. (1965) Phys. Rev.
Lett. 14, 851. See Ref. 22a pp. 1061 and 1065.
24. (a) Bucksbaum, P. H., Bashkansky, M. McIlrath, T. J. (1987)
Scattering of electrons by intense coherent light, Phys. Rev. Lett.
58, 349-52.
(b) Bucksbaum, P. H., Freeman, R. R., Bashkansky, M. and McIlrath,
T. J. (1987) Role of the ponderomotive potential in above-threshold
ionization, J. Opt. Soc. Am. B, 4, 760-64.
25. Bucksbaum, P. H., Schumacher, D. W. and Bashkansky, M. (1988) High-
intensity Kapitza-Dirac effect, Phys. Rev. Lett. 61, 1182-85.
26. (a) Bucksbaum, P. H. (1988) Above-threshold ionization and quantum
mechanics of wiggling electrons, Ref. 3c, pp. 145-55.
(b) Freeman, R. R., Bucksbaum, P. H. and McIlrath, T. J. (1988) The
ponderomotive potential of high intensity light and its role in
multiphoton ionization of atoms, IEEE J. Quant. Electr., 24, 1461-
69.
27. (a) Weingartshofer, A., Clarke, E. M., Holmes, J. K. and Jung, C.
(1979) Experiments on multiphoton free-free transitions, Phys. Rev.
A 19, 2371-76.
(b) Weingartshofer, A., Holmes, J. K., Sabbagh, J. and Chin, S. L.
(1983) Electron scattering in intense laser fields, J. Phys. B: At.
Mol. Phys. 16, 1805-17.
28. (a) Wallbank, B., Holmes, J. K., Weingartshofer, A. (1987)
Experimental differential cross sections for multiphoton free-free
transitions, J. Phys. B: AT. Mol. Phys. 20, 6121-38.
(b) Wallbank, B., Connors, V. W., Holmes, J. K., Weingartshofer, A.
(1987) Experimental differential cross sections for one-photon free-
free transitions. J. Phys. B: At. Mol. Phys. 20, L833-L838.
29. Kruger, H. (1990) Multiple harmonic generation by laser modulation
of Zitterbewegung: the electron - a highly nonlinear electromagnetic
medium, THE ELECTRON 1990 Workshop, August 1990.
30. Baldis, H. A., Pepin, H. and Grek, B. (1975) Third harmonic
generation from laser-produced plasma, Appl. Phys. Lett., 27, 291-
92.
31. Kupersztych, J. (1987) Inverse half-Bremsstrahlung in multiphoton
ionization of atoms in intense light beams, Europhys. Lett., 4, 23-
28.
32. Mittleman, M. H. (1988) Extended low-frequency approximation for
laser-modified electron scattering: Coulomb effects, Phys. Rev.
A 38, 82-92.
310 A. WEINGARTSHOFER

33. Jung, C. and Kruger, H. (1978) On the accuracy of soft-photon

approximations for resonance free-free transitions, Z. Physik A 287,
7-13.
34. Jung, C. and Taylor, H. S. (1981) Possibility of experimental
separation of resonances and cusps from background in electron
scattering, Phys. Rev. A 23, 1115-26.
35. Wallbank, B., Holmes, J. K., LeBlanc, L. and Weingartshofer, A.
(1988) Simultaneous off-shell excitation of He 23 S by electron and
one or more photons, Z. Phys. D - At. Mol. and Clusters, 10, 467-72.
36. Wallbank, B., Holmes, J. K. and Weingartshofer, A. (1989)
Simultaneous electron-photon excitation of metastable states of
rare-gas atoms, J. Phys. B: At. Mol. Opt. Phys. 22, L6l5-L6l9.
37. Wallbank, B., Holmes, J. K. and Weingartshofer, A. (1990)
Simultaneous electron-photon excitation of He 22S: an experimental
investigation of the effects of laser intensity and polarisation, J.
At. Phys. B: At. Mol. Phys. 23, 2997-3005.
38. Wallbank, B., Holmes, J. K. and Weingartshofer, A. (1989) Absorption
and emission of radiation during electron excitation of the 21S and
21p states of helium, Phys. Rev. A 40, 5461-63.
39. Geltman, S. and Maquet, A. (1989) Laser-assisted electron-impact
excitation in the soft-photon limit, J. Phys. B: AT. Mol. Opt. Phys.
22, L4l9-L425.
40. Trombetta, F. (1991) Exchange effects in laser-assisted elastic
electron-hydrogen scattering. To be published.
41. Sundaram, B. and Armstrong, Jr., L. (1990) Modeling strong-field
above-threshold ionization, J. Opt. Soc. Am. B7, 414-24.
42. Kupersztych, J. While I was presenting a seminar at C.E.N. de
Saclay, France, Dr. Kupersztych worked out the numerical problem and
after de seminar he handed me a piece of paper with the words "here
is a possible explanation of your experiment." I express my
appreciation to Dr. Kupersztych for the splendid suggestion.
IONIZATION IN LINEARLY AND CIRCULARLY POLARIZED MICROWAVE FIELDS

T. F. Gallagher
Department of Physics
University of Virginia
Charlottesville, VA 22901
U.S.A.

ABSTRACT. Ionization by microwave fields provides a useful bridge between processes

normally thought of as being photon and field driven. In addition, it is possible to see clearly
that some phenomena are simply free electron phenomena. Investigations of ionization by
circularly polarized microwaves and ATI are described to illustrate these points.

1. Introduction

Recent experiments with high intensity lasers have shown that in some cases multiphoton
processes can be though of as field driven processes rather than photon absorption.[I] This
notion is hardly surprising since higher order processes require lower frequency fields, and
at some point the static or quasi static limit is encountered.
With the goal of studying the evolution from photon to field driven processes we have been
investigating several aspects of microwave ionization and excitation. Here we discuss two
aspects of these studies, ionization of Rydberg Na atoms by circularly polarized microwave
fields and above threshold ionization (A TI) of Na and K Rydberg atoms in the low frequency
limit.

2. Microwave Ionization With Circular Polarization

In optical experiments it has been observed that substantially higher powers are required to
effect multiphoton ionization with circularly, rather than linearly, polarized light.[2) This
difference is attributed to the fact that there are more paths to ionization, and hence more
potential resonances using circularly than linearly polarized light. Since the difference in the
number of available paths increases with the number of photons absorbed, microwave
ionization of a Rydberg atom, requiring hundreds of photons could show striking differences.
With this in mind we have begun the experimental study of microwave ionization by
circularly polarized microwave fields.
In our experiment Na atoms in an atomic beam pass through a Fabry Perot microwave
cavity, where they are excited to a Rydberg state using two pulsed tunable dye lasers.p) The
lasers are tuned to the 3s-3p and 3p-Rydberg transitions at 5890A and -4140A respectively.
The atoms are excited to the Rydberg states in the presence of the circularly polarized field,
and we observe the ions which are produced after the microwaves have been turned off.
The experimental approach to producing the circularly polarized field is to use a Fabry
311
D. Hestenes and A. Weingartshofer (eds.). The Electron, 311-320.
© 1991 Kluwer Academic Publishers.
312 T. F. GALLAGHER

Perot cavity which supports two orthogonally linearly polarized 8.5 GHz fields 90° out of
phase. The major experimental problem is to minimize the interaction between these two
nominally degenerate cavity modes, as any interaction lifts the degeneracy. To minimize the
interaction we feed the cavity with the two orthogonal polarizations from orthogonally
polarized waveguides through irises in the Fabry Perot mirrors. In our 8.5 GHz cavity the two
modes are offset by 2 MHz, and they have Q's of 2,000 and 2,100, so the linewidths of the
modes are 4 MHz. Due to the fact that the Q's are not identical we can achieve circular
polarization in steady state, but not as the cavities modes are filled with the microwave field.
To ensure that we are using circularly polarized microwaves we excite the atoms to a Rydberg
state in the microwave field.
A much higher circularly polarized field is required to ionize the atoms than linearly
polarized field, as shown by Fig. 1, a plot of the threshold fields, where 50% ionization
occurs, for linear and circular polarization. As shown by Fig. I the circularly polarized
microwave field required for ionization follows an E = 1/16n 4 Iaw. This field is the same as
the static field required to ionize a Rydberg Na atom and much higher than the field required
for ionization by linearly polarized microwaves, E = 1/3n5 .[4]

n
30 40 50
10 3

•
5 ••
4

3
••
••
•• ••
2 E=_1_

E ••• 16n4

0 • ••
•• ••
...
10 2
G
•••
w • ••
5
4

3 •
• •• •
••
.
2 E=_1_
•
• ••
3n s

10'
3 4 5 106 2 3 4 5 107

n4

Figure I. Ionization threshold fields for linear (.) and circular (e) polarization as a function
of n when the atoms are excited in the microwave field.
IONIZATION IN LINEARLY AND CIRCULARLY POLARIZED MICROWAVE FIELDS 313

In addition there is a sharp dependence of the observed ionization signal on the

relative phase between the two polarizations in the cavity. This point is shown for n=46 in
Fig. 2.
If-(~

rl
(a) (b)
201-
"iii
t:
II
/
C>
(/j
t:
0

~
C
.2
Q)
>

e'"
I
;;:

"
~
01-
L---"--- ' _------L------.!,~~--ir_----L-L_J.-----'-----~
70 90 110 L 70 90 '10' 70 90 110
Relative Phase (clegrees)

Figure 2. Microwave ionization signal as a function of relative phase between the horizontal
and vertical fields when the atoms are excited to the energy of the 46d state and the
attentuation is set between the linear and circular polarization ionization thresholds.
Specifically, scans of the phase are shown for microwave fields of (a) 34.6 V/cm, (b) 53.5
V/cm, and (c) 61.5 V/cm.

When E is just below 1/16n4 , so that ionization by a circularly polarized field does not occur,
a very small phase deviation from the 90° phase shift required to produce circular polarization
leads to a sharp increase in the ionization as shown. As shown in Fig. 2 when the field
amplitude is lower the phase is not so sensitive, which is hardly surprising. Apparently a
small ellipticity in the polarization produces ionization, while pure circular polarization does
not.
Since the threshold field is the same as for a static field, it is tempting to simply call
the field quasistatic. However it is not. First the symmetry is different. In a static field
azimuthal symmetry is preserved and m is a good quantum number. In a circularly polarized
field states of l + m even and odd form two mutually exclusive sets of states. Second, the
microwave frequency of 8.5 GHz is not always small compared to the Lm separation and not
always small compared to the separation of the Stark states. For both these reasons the most
simple minded quasi static picture is inappropriate.
As is often the case, a variation of the simplest picture does provide a good
description. If we transform the problem to a frame rotating at the microwave frequency, the
microwave field is in fact static and cannot induce transitions. The transformation to the
rotating frame, often used to describe two level resonance experiments, is discussed Salwen.[5]
If we have a microwave field given by

E~ i E cos!.)t + Y E sin!.)t • (1)

and we transform the problem to a frame rotating about the z axis at angular frequency w, the
field becomes a static field in the x (or y) direction in the rotating frame. If we use as eigen
states in the laboratory frame the usual spherical nlm states, a rotation through angle 0 about
the z axis ~ransforms the wave function tP nlm to eim4> tPnlm. In the rotating frame 0 = wt and
tPnlm -+ e- 1ffiwt tPnlm' in the energy is shifted by -mw. Transforming the Na nlm wave
functions to the rotating frame only adds -mw to their energies. In principle, it is then a
simple matter to diagonalize the new Hamiltonian matrix containing a field in the x direction
to find the energy levels in the rotating frame. In practice, the fact that there are n2 /2 levels
314 T. F. GALLAGHER

for each principal quantum number complicates the problem. To develop an understanding
we have diagonalized the Na Hamiltonian matrix, for n = 5, 6, and 7;, in a frame rotating at
1500 GHz, so that the frequency is 2% of the n = 4-5 interval. A frequency of 8.5 GHz is
2% of the n = 25 to 26 interval. The result of the numerical calculation is shown in Fig. 3 for
*
e+ m even. There are two points to note. At zero field m 0 levels are displaced from their
normal energies, as shown clearly by the p m = ± I states. More important, when the Stark
manifolds of adjacent n overlap there are avoided crossings, just as in a static field. In the
limit of very low frequency the Stark manifolds overlap at E = 1/3n 5 , but the overlap occurs
at lower fields as the frequency is raised.

I
-8000
5s-j--_ __

500 1000 1500 2000

Field Amplitude (kV / em)

Figure 3. Schematic energy level diagram for Na near n=24 in a frame rotating at 102 GHz.
For clarity we have only shown s, p, and d states, which are assumed to have quantum defects
of 1.35,0 and 0 respectively. Their energies in the nonrotating frame are shown by the
arrows. Note that in zero field the p m=± I and d m=±O levels are displaced from the non
rotating energies of the p and d states by +102 and +204 GHz respectively. The lowest field
avoided crossing, between the m=±2 levels, is negligibly small, while those in fields larger
than 1/3n5 are substantial due to the admixture of the s state with its non zero quantum
defect. In fields greater than 1/16n4 the avoided crossings are with unstable states, and in this
region ionization occurs very rapidly. In the rotating frame the field is static, and there are
no transitions. Ionization occurs if atoms are excited above the classical field for ionization,
E=I/16n 4 .

We have also done the same calculation for H, and the levels cross, as in a static field.
Thus the avoided crossings in Na are due to the non coulomb core coupling not the
transformation to the rotating frame. In static fields 1/3n 5 < E < 1/16n 4 the core coupling,
is manifested in avoided level crossings, and in fields E > 1/ 16n4 the same coupling between
discrete states and the underlying Stark continuum of ionized red Stark states leads to
ionization. [6] In other words field ionization of Na at E > 1/16n4 is really autoionization.
In the rotating frame the situation is presumably the same. There are avoided crossings for
IONIZATION IN LINEARLY AND CIRCULARLY POLARIZED MICROWAVE FIELDS 315

E < 1/16n4, presumably due to the m = 0 and 1 parts of the wave function, and ionization
when E > 1/16n4. In other words we have almost come back to the original notion; ionization
by a circularly polarized microwave field is field ionization in the rotating frame.
It is interesting to consider the pronounced effect of ellipticity in the polarization in
terms of the above rotating frame description. When the polarization is elliptical, in the
rotating frame there is a field oscillating at 2w superimposed on the static field. If the static
field in the rotating frame exceeds 1/3n 5 , the atom is in the field regime in which there are
many level crossings, and even a very small additional oscillating field can drive transitions
to higher lying states via these level crossings.
The above description of ionization by a circularly polarized microwave field is clearly
a field description. At the other extreme is photoionization, which is also described as
auto ionization in the rotating frame. For example photo ionization of an m = 0 state to the m
= I continuum appears as auto ionization into the degenerate m = I continuum with a rate
proportional to the light intensity. This notion is easily extended to low order multiphoton
processes which occur via virtual intermediate states. An interesting, but outstanding question
is how to connect these pictures to the quasi static picture advanced above to describe
microwave ionization in a circularly polarized field.

3. Above Threshold Ionization

Above threshold ionization (A TI), the absorption of more photons than necessary for
multiphoton ionization was first observed by Agostini et al. in the energy spectrum of
electrons produced by multiphoton ionization of Xe by intense 1.06 /Lm light.[7] Shortly
thereafter Kruit et al observed that the lowest energy peaks in the electron spectrum were
suppressed as the intensity was raised.[8] This was attributed to the upward shift in the
ionization limit by the ponderomotive energy, Wp=e 2E2/4mw2, in the intense laser field.[9]
The ponderomotive energy is simply the average kinetic energy of a free electron of mass m
and charge e oscillating in a field E coswt. The fact that the ionization limit and therefore the
Rydberg states shift in this manner is not a property of intense light, but is generally true for
light which is of frequency low compared to the transitions available to the ground state, but
high compared to the frequencies of strong transitions of the Rydberg state in question. For
example the room temperature black body radiation induced shift of the ionization limit is
equal to the ponderomotive energy.[I 0] In the laser field the ionization limit is increased, so
more photons must be absorbed, the excess energy going into the oscillation of the
photoelectron. However when the photoelectron leaves the focus of the laser beam the
oscillatory ponderomotive energy is converted to translational kinetic energy. Thus the
ponderomotive energy does not alter the energies of the detected electrons, assuming of course
that the laser field remains present as the electrons leave the focus. However it does alter
which electrons are actually observed. No electrons can be observed with an energy less than
the ponderomotive energy.
Examination of optical ATI experiments using linear polarization indicates that there are
typically ATI electron peaks at energies as high as three or four times the ponderomotive
energy, implying that the ponderomotive energy can be used to crudely estimate the amount
of AT!. Since the ponderomotive energy scales as 1/w2 , it is large for even modest microwave
fields. The ponderomotive energy for 8 GHz microwaves is I eV for an intensity of
I 04W/cm 4, a power level that is easily produced. With this in mind we decided to see if
microwave A TI existed and, if so, if it was simply understandable.
316 T. F. GALLAGHER

The experimental approach is to analyze the electrons from Rydberg atoms ionized by
an 8.2 GHz microwave field.[11] The first experiments were carried out using the
arrangement shown in Fig. 4.

signal

_ _ _ _ retarding
voltage

f electrons.

YJ2
['cavity
atomic r- ----l1 laser
beam beams

septum microwave
power

Figure 4. Schematic digram of the apparatus for electron energy spectroscopy.

The microwave cavity is a rectangular cavity with a septum and operates on the TE 101 mode
at 8.2 GHz. The Na atomic and laser beams pass through holes in the sidewalls of the cavity,
so that a pencil shaped volume of excited atoms is created by the two dye laser beams which
excite the Na atoms from the ground state to a Rydberg state via the 3ps state. The energies
of electrons ejected through a hole in the top of the cavity are analyzed either by a retarding
potential or by time of flight. The Na atoms are excited are excited to the Rydberg states in
the presence of the microwave field. When atoms are excited to the energy of the 40s state,
electrons are observed at energies as high as lOeY when the computed ponderomotive energies
are -3eY, and the energy distribution is peaked at -3W p' On the other hand when atoms are
excited with the laser to the energy of the ISs state the electron energy is observed to peak
at Wp.
The above results represent the energy spectrum of the detected electrons, but are not
indicative of the energy spectrum of the electrons as ejected from the atoms, due to the
ponderomotive force. Ponderomotive forces were first shown to distort the angular
distributions of ATI electrons from laser multiphoton ionization by Freeman et. al,[i2] and
the effect is, if anything, worse in the microwave case. The ponderomotive force Fp is given
by

Fp = -VWp
2
__e_ VE 2 (2)
4mw 2

In a focused traveling wave laser beam F p is radially outward. In a standing wave beam there
IONIZATION IN LINEARLY AND CIRCULARLY POLARIZED MICROWAVE FIELDS 317

is also a variation in E2 along the beam propagation direction, leading to ponderomotive forces
in the propagation direction, the Kapitza Dirac effect.[13] If we define a rectangular
coordinate system with its origin at the center of the cavity, we can write the square of the
microwave field amplitude in our TE 101 cavity as

(3)

where a and b are the transverse dimensions of the cavity. In our cavity a ~ b. Inside the
cavity there is evidently ponderomotive force in the x-y plane, which is approximately
radially outward, since a ~ b, and increases approximately linearly with distance from the z
axis. There is evidently no z component to the ponderomotive force inside the cavity.
However at the exit hole in the top of the cavity the field amplitude changes from Eo to zero,
so there is a ponderomotive force in the z direction, and electrons gain energy Wp in the form
of motion in the z direction on leaving the cavity.
Electrons ejected from the atom in the +z direction with large kinetic energies, >W p'
are unaffected by the ponderomotive forces in the cavity, escape from the cavity, and reach
the detector. However those electrons ejected from the atom with low, -0, kinetic energies
are virtually certain to be deflected horizontally and will not leave the cavity and be detected.
Thus low energy electrons are suppressed in the detected electron spectrum. This distortion
is most dramatic for high n states, as has been shown by applying a small bias voltage to the
septum.[l4] As shown by Fig. 5, when a voltage is applied the electron spectrum at high n
is seen to consist of not only a peak at 3W p but a peak at Wp as well.
Energy (eV)
10 8 7 6 5 3

(a)

.
(b)

0
:>
(e)
D

<
m
0
co
(d)
'"
.
0
0

w
(e)

50 100 150
t{ns)

Figure 5. Electron time of flight spectra when Na atoms are excited to n=71 in the presence
of a 3.0kV/cm 8.173 GHz microwave field. The voltages on the septum are (a) OV, (b) -
2.00V, (c) -4.00V, (d) -6.00V, and (d) -8.00V. At higher magnitudes of the septum voltage
the later peak, due to the low energy electrons is quite evident. With small voltages the low
energy electrons are lost due to ponderomotive forces in the cavity. Note that each additional
negative two volts on the septum raises the electron energy by leV.
318 T. F. GALLAGHER

At low n, where virtually all the electrons have low energy the dominant effect is an overall
suppression of the signal. Taking into account the effect of the ponderomotive forces in the
cavity, it is apparent that as ejected from the atom the electron energy distributions for low
n and high n are respectively singly peaked, at WP' and doubly peaked at Wp and 3W .
To explain these electron spectra requires only a simple model, sometimes calfed the
"simpleman's theory", developed independently by van Linden van den Heuvell and
Muller,[l4] Corkum et al [15] and Gallagher [II]. Consider an electron liberated in a classical
field E=Eosinwt at time to with initial velocity v=O. At any later time t the velocity is given
by

(4)

and the time average velocity by

- eB
v = --cosc.>lo. (5)
mW

The average kinetic energy is given by

(6)
which is composed of two parts, an oscillation energy of Wp and a translational energy of 2
Wp cos 2wto. When the electron leaves the microwave field the energy of oscillation is
converted to directed translational motion, so the energy of Eq. (5) is the energy of the
electrons when detected.
When the atom is excited to a low n state, the microwave field is too small to directly
field ionize the initially excited state, and the atom makes transitions to higher n states until
field ionization is allowed, which is most likely to occur near the peak of the field, wto = 7r/2.
For this value of wto the electron is ejected from the atom with no translational energy, and
W ~ Wp' in accord with the observations. On the other hand, if a high n state is excited it can
be iOnIzed at nearly any phase of the microwave field. The excitation and ionization are
therefore simultaneous and computing the energy spectrum is simply a matter of computing
(dW /dtorl. Since the function cos 2wt o spends most of its time at its turning points, wto=O and
7r, the electron energy spectrum is peaked at Wp and 3WP' in agreement with observations.
If the atoms are excited to a Rydberg state and then exposed to a pulsed microwave
field we see electrons of energy approximately equal to the ponderomotive potential due to
the field necessary to ionize the atom. For low n states this energy can be quite high, but for
high n states it is vanishingly small.
The simpleman's theory agrees quite well with electron energies observed at
microwave and CO 2 ~ser frequencies. From Eqs. (4) and (5) it is apparent that there is phase
dependence in both v and W. The phase dependence has been observed explicitly by phase
locking a microwave oscillator to a mode locked laser which excites K atoms with 5ps laser
pulses.[l6] A phase shifter between the oscillator and the microwave cavity allows the
microwave phase, wt o' at which the atoms are excited to be varied at will. The electron
energies are analyzed by time of flight.
At low n there is no phase dependence, as expected. Irrespective of when ionization
occurs several microwave cycles are required to make transitions to higher n states and
ionization occurs near the peak field, resulting in electrons of energy - WP' For high n states,
which are readily ionized by the field, both v and W exhibit the predicted phase dependence.
IONIZAnON IN LINEARLY AND CIRCULARLY POLARIZED MICROWAVE FIELDS 319

For half the microwave cycle, no electrons are detected, indicating that the electrons are
ejected down, away from the detector. In addition the electron energy varies with phase, as
shown in Fig. 6.

Energy (eV)
16 14 12 10 9 S 7

(e)

unlocked

O'

30'

60' (j)to"'-It

~
C
=> 90'

.e
.e 1200

~ 150'
C>
iii
c
e 160'
U
~
W 210' rota'" 0

240 0

2700

~
300'

3300

360' 0010= It '

40 60 SO
t(ns)

Figure 6. Electron time of flight spectra as a function of phase when the microwaves are
phase-locked and the n-=42-46 d states are excited. The top trace is unlocked for comparison.
The microwave intensity is 33kWcm-2 (3.5kVcm- 1;Wp -=2.leV). There is an offset of 4eV on
energy scale due to septum voltage of -S.OV.

When wto-=O the electrons have energy 3Wp and when wto=±11"/2 the electrons have energy
W-=Wp, as predicted by Eq. 5. Electrons detected with energy - Wp have very low kinetic
energy in the microwave field, and irrespective of the direction in which the electrons are
ejected from the atom they are detected because of the bias field. On the other hand electrons
produced at wt O=±1T have kinetic energy 2Wp in the cavity but are directed downward, hit the
septum, and are not detected.
These experiments clearly validate the simpleman's theory. In addition they
demonstrate, that the ionization, as opposed to the excitation en route, occurs in a single cycle,
otherwise there would be no phase dependence in the ATI spectrum. Thus ATI in this
frequency range is fundamentally different from ATI at optical frequencies, where well
resolved ATI peaks are observed, indicating that ionization takes place over many cycles.

4. Conclusion

As shown by these experiments, at low frequencies multiphoton processes are well

320 T. F. GALLAGHER

described as field driven processes, and some of the effects, such as ATI are simply the
reflection of the response of a free electron to an oscillating field.

S. Acknowledgements

This work has been supported by the Air Force Office of Scientific Research under
grant AFOSR-90-0036.

6. References

1. Augst, S. D., Strickland, D., Meyerhofer, D. D., Chin, S. L., and Eberly, J. H. (1989)
'Tunneling Ionization of Noble Gases in a high intensity Laser Field', Phys. Rev. Lett.
63,2212-2215.
2. Lompre, L. A., Mainfray, G., Manus, C. and Thebault, J. (1990) 'Multiphoton
ionization of rare gases by a tunable-wavelength 30-psec laser pulse at 1.06 11m', Phys.
Rev. Ali, 1604-1612.
3. Fu, P. M., Scholz, T. J., Hettema, J. M., and Gallagher, T. F. (1990) 'Ionization of
Rydberg Atoms by a Circularly polarized microwave field', Phys. Rev. Lett. 64, 511-
514.
4. Freeman, R. R., McIlrath, T. J., Bucksbaum, P. H., and Bashkansky, M. (1986)
'Ponderomotive Effects on Angular Distributions of Photoelectrons', Phys. Rev. Lett.
57,3156-3159.
5. Salwen, H., (1985) 'Resonance Transitions in Molecular Beam Experiments. I.
General Theory of Transitions in a Rotating Magnetic Field.., Phys. Rev. 99, 1274-
1286.
6. Littman, M. G., Zimmerman, M. L., and Kleppner, D. (1976) 'Tunneling Rates for
Excited States of Sodium in a Static Electric' Field'" Phys. Rev. Lett. 37, 486-489.
7. Agostini, P., Fabre, F., Mainfray, G., Petite, G., and Rahman, N. K. (1979) 'Free-
Free Transitions Following Six-Photon Ionization of Xenon Atoms', Phys. Rev. Lett.
42, 1127-1150.
8. Kruit, P., Kimman, J., Muller, H. G., van der Wiel, M. J. (1983) 'Electron spectra
from multiphoton ionization of xenon at 1064,532, and 355 om" Phys. Rev. A 28, 243-
255.
9. Muller, H. G., Tip, A., and van der Wiel, M. J. (1983) 'Ponderomotive force and AC
Stark Shift in multiphoton ionization', J. Phys. B lQ, L679-L685.
10. Gallagher, T. F. and Cooke, W. E. (1979) Interaction of Black Body Radiation with
Atoms', Phys. Rev. Lett. 42, 835-838.
11. Gallagher, T. F. (1988) 'Above Threshold Ionization in the low frequency limit', Phys.
Rev. Lett. 61, 2304-2307.
12. Bucksbaum, P. H., Schumacher, D. W., and Bashkansky, M. (1988) 'High-Intensity
Kapitza-Diract Effect', Phys. Rev. Lett. 21, 1182-1185.
13. Gallagher, T. F. and Scholz, T. J. (1989) 'Above Threshold Ionization at 8 GHz', Phys.
Rev. A 40, 2762-2765.
14. van Linden van den Heuvell, H. B. and Muller, H. G. (1989) 'Limiting cases of excess
photon ionization', in S. J. Smith and P. L. Knight (eds.), Multiphoton Processes,
Cambrdige University Press, Cambridge, pp. 25-34. S. J. Smith, and P. L. Knight,
(eds.) (Cambridge Univ Press., Cambridge, pp. 25-35.
15. Corkum, P. B., Burnett, N. H., and Brunei, F. (1989) 'Above-Threshold Ionization in
the Long-Wavelength Limit', Phys. Rev. Lett.~, 1259-1262.
16. Tate, D. A., Papaioannou, and Gallagher, T. F. (1990) 'Phase sensitive above threshold
ionization of Rydberg atoms at 8 GHz', Phys. Rev. A 42, xxx.
MULTIPLE HARMONIC GENERATION IN RARE GASES AT HIGH LASER
INTENSITY

Anne L'HUILLIER, Louis-Andre LOMPRE and Gerard MAINFRA Y

Service de Physique des Atomes et des Surfaces
C.E.N. Sac/ay, 91191 Gifsur Yvette, FRANCE

ABSTRACT. We briefly review the main experimental results on high-<:lrder harmonic

generation in rare gases and their interpretation.

Introduction

Recent experiments show the production of very high harmonics of the laser field.
Up to the 17th harmonic of a KrF pump laser (248 nm) has been observed in neon
by the group at the university of Illinois at Chicago (McPherson et al.(1987)). In
Saclay, we have seen up the 33rd harmonic of a Nd-YAG laser (1064 nm) in a 10
Torr argon gas jet (Li et al.(1989)). The intensities of the harmonics drop fairly
quickly for the first harmonics, then exhibit a broad plateau and finally decreases
again sharply. The length and width of the plateau depend on the atomic medium
and on the laser intenSity.
Calculations of single-atom photoemission spectra going beyond
perturbation theory reproduce fairly well the experimental results (Kulander and
Shore (1989), Potvliege and Shakeshaft (1989), Bandarage et al. (1990), Eberlyet
al., 1989). This seems to imply that propagation effects play no role or rather that
they affect all the harmonics in the same way. However, a calculation of phase
matching in the weak field limit shows that phase matching should severely
degrade with the order. We present a calculation of harmonic generation in a rare
gas medium, involving the resolution of the paraxial propagation equation for a
non-perturbative polarization field. It reconciles experimental results and
321
D. Hestenes and A. Weingartshofer (eds.), The Electron, 321-332.
© 1991 Kluwer Academic Publishers.
322 A. L'HUILLIER ET AL.

calculations of single-atom emission spectra, since it shows that phase matching is

the same for all the harmonics of the plateau region.

1- Experimental mmlts with a. Nd-Y AG laser

An harmonic generation experiment consists in focusing an intense laser

radiation into a rather dense rare gas medium (a few Torr) and in analyzing along
the propagation axis the VUV light emitted during the interaction. In our case, we
use the fundamental frequency of a mode-locked Nd-YAG laser (40 ps pulse
width- 1064 nm wavelength), with a maximum energy of 1 GW at a 10 Hz
repetition rate. The laser pulse is focused by a 200 mm-focallength lens over a 18
;tm focal radius (confocal parameter b = 4 mm). The gaseous medium is provided
by a pulsed gas jet producing a well--collimated atomic beam (1 mm FWHM) with
a 15 Torr pressure at 0.5 mm from the nozzle of the jet. The VUV light (from 350
nm to 10 nm) is analyzed along the laser axis by using a grating monochromator
and detected by photomultipliers or a windowless electron multiplier at
wavelengths below 120 nm.
Fig.1 shows the number of harmonic photons obtained in xenon at a 15 torr
pressure at several laser intensities, 3xl0 13 W.cm-2, solid line, 1.3xl0 13 W.cm-2,
dashed line, 9xl0 12 W.cm-2, dot-dashed line, 7x10 12 W.cm-2, dotted line,
5xl0 12 W.cm-2, double-dot-dashed line (Lompre et al.,1990). We only observe
odd harmonics (we did not however look for second order harmonic generation).
This is to be expected for harmonic generation in an isotropic gaseous medium,
with inversion symmetry. At the lowest laser intensity, the conversion decreases
with the order. This is a rather intuitive result: the harmonic signal decreases as
the nonlinear order increases. However, at higher laser intensity, a plateau
appears. The length of the plateau increases with the intensity up to the point at
which the medium gets ionized (above 1.3xl0 13 W.cm-2). The highest harmonic
detected is the 21st, which means an energy of 25 eV, a wavelength of 50 nm. The
vertical scale is an order-{)f-magnitude estimate of the number of photons
produced at each laser shot. The efficiency for the plateau harmonics at 3xl0 13
W.cm-2 is about 10-8- 10-9. This represents a very high brightness, 10 17
• 2
phi As(mrad) .
MULTIPLE HARMONIC GENERATION AT HIGH LASER INTENSITY 323

VI
C
o
......
o
.c
a.

t...
QI
.0
E
::J
Z

9 11 13 15 17 19 21 23 25
Harmonic order
Figure 1 : Number of photons obtained in xenon at a 15 Torr pressure at several
intensities (see text). The incident wavelength is 1064 nm, the confocal parameter
estimated to 4 mm, the interaction length 1 mm.

Another way of looking at these results is to plot the number of photons as

a function of the laser intensity. Figure 2 shows the intensity dependence of the
5th and 17th harmonics. The signal increases first rather rapidly with the laser
intensity, before saturating when the medium is ionized. Ionization limits
harmonic generation because the main medium responsible for the conversion gets
depleted and also because phase matching conditions break down owing to the
presence of free electrons (Miyazaki and Kashiwagi, 1978, L'Huillier et al., 1990).
Below saturation, the qth harmonic signal varies with the laser intensity as rP with
p = q for the first orders and p < q for the highest orders. Table 1 summarizes the
effective orders of nonlinearity determined by a least-square fitting procedure for
the different harmonics. Some harmonics (in particular the 13th) present a more
complicated behavior than a simple power law, which might be due to the
influence of AC Stark shifted resonances.
324 A. L'HUILLIER ET AL.

10 7

Xe H17
Xe
10 8
HS
1'/
,-
,~
I i
,-•"
10 6 ~
~
f
:' :-
V1 "

z ~

I
0
0-
0
10 7 ! i
:
I
a..
u.... 10 5
0
0:::
"
-
"-'
al !
.
L
:::> 10 6 I

f
z
"

10~

i1 )

10 5 J
10 3
10 12 1013 10 14 10 12
LASER INTENSITY (Wcm- 2, LASER INTENSITY (Wcm- 2,

Figure 2 : 5th and 17th harmonics as a function of the laser intensity

Table 1: harmonic power laws below saturation

3 5 7 9 11 13 15 17 19 21
3.3±O.2 6±O.5 7.5±O.7 8.3±O.7 7. 7±O. 7 5.3±O.5 11.7±O.7 12.5±1 13±1 12.5±1

Similar results have been obtained with the other heavy rare gases (krypton
and argon). The distributions obtained in Xe, Kr, Ar at 3xl0 13 W.cm-2 are shown
in Fig. 3. The conversion efficiency decreases from Xe to Ar, which is not
surprising, since xenon is more polarizable than lighter rare gases. However, the
maximum order that can be observed increases from 21 in Xe, 29 in Kr to 33 in
Ar. The 33rd harmonic (32 nm, 38 eV) is the shortest wavelength radiation that
we were able to produce with our Nd-Y AG laser system (limited, however, to
MULTIPLE HARMONIC GENERATION AT HIGH LASER INTENSITY 325

about 20 mJ in 40 ps). Atoms with higher ionization energies have in general a

lower conversion efficiency but they can produce more harmonics. Moreover, they
can experience a higher laser intenSity without being ionized. The results shown in
Fig.3 have been obtained at an intensity of 3x10 13 W.cm-2, which is higher than
the saturation intensities for xenon and krypton. The Ar harmonic intensity
distribution presents an anomaly. The 13th harmonic is missing. This may be due
to breakdown of phase matching conditions or strong reabsorption in the vicinity
of (possibly AC Stark shifted) Rydberg states.

109~-------------------------------.

q
\
\
10 8 \
\
.\
\. \\
\\. \
III 107
c: \L""t\
2
0 \. \ q ,P'--a.,.....
-...
~ '0.,"'0..
a..
10 6 \ \
A
.....
~\ Kr
0
, ;' \
\
CI.I
'.\ \\ '1 '~ \
.c / \ b
/ I

\V/ 'r', r·.J

"
E . ."'-. \ .
::l
z .........
lO S .
'w-.....- "-
.". Ar
\
.~
10'
Xe Kr Ar \
\
103
3 7 11 15 19 23 27 31
Harmonic Order

Figure 3. Number of photons obtained in Ar, Kr and Xe at 3x10 13 W.cm-2, 1064

nm
326 A. L'HUILLIER ET AL.

11- Interpretation of the experimental results

1- Singlfr-atom response
The coherent spectrum emitted by a single atom is the square of the
Fourier transform of the dipole moment d( t) induced by the laser field. d( t) can be
obtained from the wavefunction of the atomic system, as the expectation value of
the dipole operator. This problem has recently received considerable attention.
Various approximations have been used, time-dependent methods (Kulander and
Shore, 1989, 1990, Eberly et al., 1989), Floquet theory (Potvliege and Shakeshaft,
1989), model calculations (Bandarage et al., 1990, Becker et al., 1990, Sundaram
and Milonni, 1990). Figure 4 presents results of calculations performed by
Kulander and coworkers in xenon (Krause et al., 1991). It shows the square of the
dipole moment as a function of the harmonic order at 3x 10 13 W.cm-2. Note the
close similarity between theoretical and experimental results in particular for the
length of the plateau and the position of the cut off at high frequency.

Xenon SPo A.=1064nm 3xl013 W/cm 2

10. 1 , I I
, , , , I
, I
, , I
, , , I I

~I
I
10' 3

10' 5
B
'iii
I':

....1:"
10. 7

'Eli
Q
....l
10' 9

10. 11

10. 13
0 5 10 15 20 25 30 35 40
Harmonic

Figure 4. Relative harmonic intensities as a function of the harmonic order q.

MULTIPLE HARMONIC GENERATION AT HIGH LASER INTENSITY 327

Most of non-perturbative calculations, even crude models such as a

two-level system (Sundaram and Milonni, 1990), are able to reproduce
qualitatively the experimental results, with the decrease in efficiency for the first
harmonics, the plateau and the cut off. This is probably a very general property of
a strongly driven nonlinear system. As a further illustration, we show in Fig.5 the
spectrum emitted by a classical anharmonic oscillator, obtained by solving the
II ,
following equation: x + Wo2x + rx + vx3 = Fcoswt. The parameters are chosen to
be Wo = lOw, r = 1, v = F = 500. Surprisingly, the main features (plateau and cut
off) observed in the experiments are reproduced by this simple model.

Figure 5. Spectrum
10'
emitted by an
anharmonic oscillator.

3 7 11 15 19 23
2- Many-atom response

The generation of the qth harmonic field is described by the propagation

equation

where kq is the wavevector of the qth harmonic field, w the laser frequency and ~
the nonlinear polarization field, equal to 2..,yd( qw), where ..,y denotes the atomic
density and d( qw) the qth harmonic component of the time-<iependent dipole
moment. Equivalently, this can be written as (Jackson, 1975)
328 A. L'HUILLIER ET AL.

with R= Ir-rll. The calculation of the harmonic profile in the far field is thus
reduced to a three dimensional integration over the nonlinear medium (L'Huillier
et al., 1990, 1991). The number of photons emitted at a given harmonic frequency
is obtained by integrating the temporal and spatial distribution as follows

Nq = 44w f r'l6'q(r' ,t) 12 dr'dt

The single-atom contribution is usually factorized out as Nq = Id(qw)1 2 IFql2,

where 1d( qw) 12 is the single atom contribution and 1F q 12 the phase matching
factor, reflecting the integration of the propagation equation. This factorization
can actually be done analytically for an incident Gaussian beam, in the weak field
limit. In a more general situation, in a strong field regime, one calculates
numerically the number of photons and then derives the phase matching factor.

..-
o
~
10-8

3 5 7 9 11 13 15 17 19 21 23 25
Harmonic Order

Figure 6. Phase matching factor 1F q 12 for different intensities and geometries (see
text)
MULTIPLE HARMONIC GENERATION AT HIGH LASER INTENSITY 329

Figure 6 shows the phase matching factor 1F q l:l as a function of the order q
both in the weak field limit (solid lines) and in a strong field regime (3xl0 13
W.cm-2, dashed lines). Two different geometries have been investigated : a
collimated geometry where the laser confocal parameter b (4 mm) is larger than
the interaction length L = 1 mm (squares in the figure) and a confocal geometry
where b (1 mm) is of the order of the interaction length (circles).The difference
between the two intensities is striking. In the weak field limit, 1F q 12 decreases
rapidly with the process order. In contrast, at 3xl013 W.cm-2, 1F q 12 stays
approximately constant, independent of the focusing geometry. Note the twelve
orders of magnitude difference between the weak and strong field regimes in the
tight focused geometry.
10 11 r - - - - - , r - - - r - - - r - - - - , r - - - - - r - - - , - - - , - - - - r - - , - - , - ,

10 9 ~ ~\

--0\
VI \
c'·· ,
.::
o
".\ \
'. \
..
' ""
o---o-~
..c '. \ \ ",'"
Cl. '. ". __ ' "

~ 10
7
Y.:\- "'"
~
.0
"\'" .
.-\ '\
\
'
E .~ ....... ~. \
~ ~,
°
\ .. \ ....... \ . -0 ....
105 !,_
.
....
".
0- \
.........• . \
~ ...,
,
\
\

\ .......... \
...• \ ~
103L-~_~_~~~_~_l~_~__L-~L-~~~
3 5 7 9 11 13 15 17 19 21 23 25
Harmonic order
Figure 7. Calculated number of photons for b = 4 mm; L = 1 mm, 15 Torr. Same
intensities as in Fig.I.

The numbers of photons Nq , calculated at the same laser intensities as in

Fig.l are shown in Fig.7. The results (L'Huillier et al., 1991) are in good
agreement with the experimental data, for the absolute value of number of photons
330 A. L'HUILLIER ET AL.

as well as for the general behavior of the harmonic intensity distribution as a

function of the incident field.
In the experiments reported in this paper, the pressure is rather small.
Consequently, the phase mismatch ~k= kq-qk1 = (nq-ndqw/c is close to zero. It
remains much smaller than the effective phase mismatch induced by focusing
which originates from the 7r phase slip across the focus of a Gaussian beam and
which is centered around -2q/b. Perfect phase matching is not realized. In a weak
field picture, increasing the nonlinear order q has about the same effect as
decreasing the confocal parameter b. High--order harmonics become more tightly
focused. The phase matching function evolves from a sinx/x type of function for
the lowest orders to a much more rapidly varying function for the highest orders,
such that the wings are completely damped (orders--of-magnitude lower than the
average of the sinx/x oscillations). This explains why, in the perturbative limit,
the phase matching factor decreases rapidly with the nonlinear order, simply
because the geometry becomes more and more of the tight focus type. In a strong
field regime, the dipole moment varies much less rapidly with the laser field
strength, for example as IE IP, where p is the effective order of nonlinearity.
Consequently, the high harmonics do not get tightly focused as in the weak field
limit. They are defocused, in some cases exhibiting rings. The magnitude of the
phase matching function in the wings of the distribution, i.e. away from the
maximum, stays constant, since it reflects the geometry of the interaction and the
variation of the dipole moment with the intensity. The phase matching factor does
not depend much on the nonlinear order as shown in Figs.6,7.

III- Experimental results in neon with alps Nd-Glass laser.

In conclusion, we show in Fig.S recent experimental results obtained in a 15

Torr neon vapor with alps Nd-Glass laser system developed in our laboratory.
The laser wavelength is 1053 nm, the pulse width 1.2 ps, the energy up to 1 J,
though only 25 mJ are used in the present experiment for avoiding damaging the
UV analysis grating. We use aim focal length in order to increase the conversion
efficiency (b 3 scaling, see Lompre et al., 1990). After a rapid decrease for the first
harmonics (up to the 13th harmonic) there is a very long plateau, extending up to
the 53th harmonic (62 eV, 20 nm). These preliminary results are encouraging
MULTIPLE HARMONIC GENERATION AT HIGH LASER INTENSITY 331

because one might expect to reach shorter wavelengths and higher efficiencies by
increasing the incident laser intensity.

Energy (eV)
9 10 20 30 40 50
10 r--r---'---'r---.---~--~~

Figure 8. Harmonic Ne
generation in neon III
c
at 1053 nm, 15 torr. ....00

-
.s=-
The laser intensity o...
is estimated to be 0
14 -2 '-
5xl0 W.cm ,the CII
..0
pulse width, 1.2 ps. E
::J
Z

5 9 13 17 21 25 29 3~ 37 41 45 49 53
Harmonic Order
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ABSORPTION AND EMISSION OF RADIATION DURING ELECTRON EXCITATION OF ATOMS

BARRY WALLBANK
Department of Physics
St. Francis Xavier University
Antigonish, Nova Scotia
Canada, B2G lCO

ABSTRACT. Recent progress in experiments studying the excitation of

atoms by electron impact in the presence of an intense carbon dioxide
laser is described. Possible future experiments in what is a new field
in atomic collision physics are also discussed.

1. INTRODUCTION

Atoms are normally excited into higher states from their ground states
through collisions with electrons or photons. Recently, however, it has
been demonstrated that excitation can also occur through the
"simultaneous" impact of an electron and one [1] or more [2] photons.
This excitation mechanism, first predicted by G6ppert-Mayer [3], may be
described by

where an electron of incident energy Ei is scattered from atom A in the

presence of a laser, photon energy ~w, and emerges with energy
Ei + II~W - En after exciting the atom to a higher state, excitation
energy Eo., and absorbing (11)0) or emitting (11<0) II laser photons. If the
atom were to remain in its ground state then (1) would describe the so-
called free-free transitions that have been studied for several years
[4].
Experimentally, the simultaneous electron-photon excitation (SEPE)
process described by (1) has been studied for the excitation of the 23 S
state of helium through the detection of the metastable atoms at electron
energies close to the threshold of excitation (Ei-E.x) in the field of a
CW CO 2 laser (1-10' W cm- 2 ) [1,5]. The much more interesting problem of
excitation involving more than one photon, requiring higher laser
intensities (1-10· W cm-') , has been examined by detecting the metastable
atoms produced through the excitation of helium (2'S and 2' S), neon (,P,)
and argon ('P,), again where Ei-Eex, using a pulsed, CO 2 TEA laser [2,6,7].
333
D. Hestenes and A. WeingartshoJer (eds.), The Electron, 333-339.
© 1991 Kluwer Academic Publishers.
334 B. WALLBANK

These data describe the total excitation cross-sections in the presence

of the laser.
If the scattered electron is detected in experiments examining (1)
then differential scattering cross-sections, which should be much more
sensitive tests of theory than total cross-sections, may be obtained.
For Ei-E~, there are technical difficulties in performing such
experiments but at higher incident energies it has been demonstrated that
differential scattering cross-sections are measurable [8). The
experimental conditions used to obtain these data (Ei -45 eV, 1_10· W cm- 2 ,
excitation to the 2'p and 2 ' S states of helium) may also be more tractable
theoretically.
In order to interpret the total cross-section experiments a simple
extension of the Kroll-Watson treatment [9), used to examine free-free
transi tions in the field of a low frequency laser, has been applied
[10,11) . These calculations were successful in reproducing the main
features of the data but are inappropriate for some experimental
conditions. The earliest calculations were primarily confined to cases
where the electrons have incident energies well above the threshold for
excitation of the hydrogen atom to the 2' S state from its ground state
[12 -15) . These calculations should be relevant to experiments where
scattered electrons were detected.
The experimental data and theoretical treatments mentioned above
will be briefly reviewed and possible future progress in this new field
of atomic collision physics will be discussed.

2. RESULTS AND DISCUSSION

2.1 Total Scattering Cross-Sections

Figure 1 presents data typical of those obtained in these types of

experiment for excitation of helium to the 23S state (Eex=19. 820 eV) by
the SEPE process. The change in metastable atom yield due to the
presence of a pulsed CO 2 laser, accumulated over 5000 laser pulses, is
shown as a function of incident electron energy around the excitation
energy. The change in yield can be seen to rise to a maximum around
Ei=Ee. before dropping to such an extent that a decrease in excitation
cross-section in the presence of the laser, compared to field free
excitation, is observed. Similar general behaviour has been observed for
exci tation of the He 23S, He 2' S, Ne 3P2 and Ar 'P2 metastable states
[2,6,7). Excitation of the He 23S state has been examined as a function
of two of the variables that the laser brings to the collision process,
the polarization and intensity of the laser [7). It has been observed
that the maximum changes in cross-section occur when the laser
polarization and the incident electron energy are parallel but with
significant changes (-1/3 of the maximum increase at Ei=Eex) still
occurring when the laser polarization is perpendicular to the incoming
electron direction.
The main features of these experimental data have been reproduced
using a simple Kroll-Watson Ansatz for He 23s [10) excitation and the
closely related Instantaneous Collision Approximation for He 2'S and 2' S
ELECfRON EXCITATION OF ATOMS IN LASER FIELDS 335

Ul
(])
.!:!l
:J
0..
...
(])
Ul
jll!
g it
800
0
0
!
0 f
tIl
•t
....
0
......
0 - - - -
1]
]i
>-
(])
:0
d -800
lii
0
a:; -1 +1 -2 +3
2: n
Incident electron energy (E"e: n'f1w)

Figure 1. The change in helium metastable yield due to the laser as a

function of incident electron energy in units of the photon energy with
respect to the 23 S excitation energy.

[11]. This would indicate that the atom plays a fairly passive role in
the process with the laser-electron interaction being most important in
determining the cross-section for the SEPE process. However, neither of
these treatments is applicable to the situation where the laser
polarization and incident electron energy are perpendicular.
Recently, the role of electronic exchange in laser-assisted elastic
electron-hydrogen collisions has been reconsidered [16]. It has been
predicted that the exchange, under certain circumstances, may largely
dominate the differential scattering cross-section, although, in the
corresponding field-free situation, it is only a small correction to the
direct amplitude. This is expected to occur in some angular regions when
only a few photons are exchanged but over the full angular range for
many-photon processes. The effect is enhanced when the collision energy
is low and should therefore occur at relatively weak laser fields for
such collision energies. These findings may be of particular importance
to the data discussed above which were obtained for pure exchange
transitions. Indeed, this interaction may be playing a dominant role in
these collisions and, therefore, this exciting, new aspect of electron-
atom collisions in a laser field is certainly worthy of further study.
336 B. WALLBANK

2.2 Differential Scattering Cross-Sections

The first theoretical treatments of SEPE considered the excitation of the

2' S state of hydrogen at high incident-electron energies (E 1»E •• ) [12,15].
One can consider SEPE as resulting from two processes: (i) the energy of
the electron is changed by the direct absorption or emission of one or
more photons while scattering from the atomic potential, and (ii) the
atom may be 'dressed' by the laser field with the electron scattering
from this 'prepared' atom.

2'P
I

.........
-'!
~

d
c
01
'i./i
"0
ClJ 5
U
::J
"0 0
.~
L
I -5
(J)
fj)
g -10
"0
(J)
N
d
E L
o +1 +2
0
Z Electron energy

Figure 2. The electron signal detected after scattering 45 eV electrons

through 12° in the energy region of excitation of the 2'p and 2' S states
of helium in the absence of the laser (upper) and the change in electron
signal recorded when the laser is present. The change in signal has been
normalized to the field-free signal at the excitation energy of either
the 2' p or 2' S state and the electron energy is displayed in units of
laser quanta with respect to the excited states.
ELECTRON EXCITATION OF ATOMS IN LASER FIELDS 337

The Kroll-Watson treatment ignores the latter process. However, at

high incident electron energies and small scattering angles, differences
in cross-section between absorption and emission of photons, occurring
through the interference, either constructive or destructive, of the
scattering amplitudes from mechanisms (i) and (ii) have been predicted
[15]. At small scattering angles the two amplitudes may be of similar
magnitude making interference effects particularly visible. The actual
occurrence of such marked differences between absorption and emission
depends on such parameters as the laser intensity and polarization, the
atom, the incident electron energy etc. and requires detailed calculation
of the scattering amplitudes.
The data shown in Figure 2 was obtained at a scattering angle of 15 0
in the region of excitation of the 21p and 21S states of helium. These
data are the recorded electron counts in the presence of the laser minus
those in its absence normalized to the field-free count rate at the
electron energy corresponding to the appropriate excitation, either 21p
or 2' S, as a function of scattered electron energy. A decrease in cross-
section at the excitation energy and an increase at ±lhw are clearly
seen. The data around the 2' S excitation show a greater statistical
uncertainty because of the much lower field-free excitation cross-
section. The peaks at + 1hw represent the change in excitation cross
section due to SEPE involving the absorption of one photon and those at
-lhw involving the emission of one photon. These data are reminiscent of
those published for elastic scattering of electrons from atoms [4] which
showed a depletion in signal at the elastic peak with this loss in signal
being distributed over the peaks due to the absorption and emission of
photons, to satisfy an overall sum rule. Such a sum rule appears to be
satisfied for the data presented in Figure 2. These experimental results
demonstrated that the differential cross-sections for SEPE are, in
principle, measurable for the 21p and 2' S states of helium. The
interested reader should consult Ref [8] for full experimental details.
In Figure 3, preliminary results are displayed for the differential
scattering cross-sections for SEPE to the 2' p state of helium involving
either the absorption or emission of one photon over the angular range
12 to 34 at an incident electron energy of 45 eV with the laser
0 0

polarization parallel to the incident electron direction and similar

laser intensities to that used for the data of Figure 2. There is some
indication in these data that at small scattering angles «20 0
the
)

cross-section for emission of one photon is somewhat higher than for

absorption. There also appears to be a feature at _28 where the
0

emission cross-section shows a dramatic increase while that for

absorption shows a similar size decrease. However, these differences are
almost within one standard deviation and therefore should be considered
with some scepticism. We are now involved in trying to improve the
uncertainties in the data to enable us to state definitively whether the
expected differences in cross-section for emission and absorption arising
from interferences of the scattering amplitudes are indeed observed under
the conditions of these experiments.
338 B.WALLBANK

, , • 1

10

o
o
o
.......
3
~
+1 ~o.. Emission
o..N
N 15

10

5f-! f t 1 f ! I f
1 I
10 20 30
Scattering angle

Figure 3. The change in electron signal due to the laser at scattered

electron energies corresponding to 1 laser photon above (absorption) and
1 laser photon below (emission) that of excitation of the 2ip state of
helium as a function of scattering angle. The incident electron energy
is 45 eV and the change in electron signal is normalised to the field-
free signal at the 2ip excitation energy.

3. FUTURE PROSPECTS

The investigation of collisions in laser fields is an exciting new field

of research that is of fundamental importance to collision physics as
well as to applied fields such as laser heating of plasmas and presents
almost unlimited possibilities for future experiments. The interesting,
new, theoretical ideas on the electron proposed at this meeting may
reveal even more possibilities since photon-electron interactions may be
one of the few ways open to examine some of these properties.
Our immediate goal at St. Francis Xavier University, however, is to
pursue experiments investigating ionizing collisions in a laser field.
The motivation behind such experiments, involving electrons, UV photons
and metastable atoms as projectiles, have already been discussed by
Professor Harald Morgner, University of Witten and the interested reader
is directed to his contribution for further details.
ELECTRON EXCITATION OF ATOMS IN LASER FIELDS 339

4. ACKNOWLEDGEMENTS

This work was performed with the help and collaboration of Mr. J. K.
Holmes and Professor A. Weingartshofer, and with the financial support of
the St. Francis Xavier University Council for Research and the Natural
Sciences and Engineering Research Council of Canada.

5. REFERENCES

1. Mason, N. J. and Newell, W. R., (1987), J. Phys. B, 20, L323.

2. Wallbank, B., Holmes, J. K., LeBlanc, L. and Weingartshofer, A.,
(1988), Z. Phys. D, 10, 467.
3. G6ppert-Mayer, M., (1935), Ann. der Physik, 2, 273.
4. Weingartshofer, A and Jung, C., (1984), 'Multiphoton Free-Free
Transitions', in S. L. Chin and P. Lambropoulos (eds), Multiphoton
Ionization of Atoms, Academic Press, New York, pp. 155-187.
5. Mason, N. J. and Newell, W. R., (1989), J. Phys. B, 22, 777.
6. Wallbank, B., Holmes, J. K. andWeingartshofer, A., (1989), J. Phys.
B., 22, L6l5.
7. Wall bank, B., Holmes, J. K. and Weingartshofer, A., (1990), J. Phys.
B, n, 2997.
8. Wallbank, B., Holmes, J. K. and Weingartshofer, A., (1989), Phys.
Rev. A. 40, 5461.
9. Kroll, N. and Watson, K., (1973), Phys. Rev. A, ~, 804.
10. Geltman, S. and Maquet, A., (1989), J. Phys. B. 22, L4l9.
11. Chichkov, B. N., (1990), J. Phys. B, 23, L333.
12. Rahman, N. K. and Faisal, F. H. M., (1976), J. Phys. B, 2, L275.
13. Rahman, N. K. and Faisal, F. H. M., (1978), J. Phys. B, II, 2003.
14. Jetzke, S., Faisal, F. H. M., Hippler, R. and Lutz, H. 0., (1984),
Z. Phys. A, 315, 271.
15. Schwier, H., Jetzke, S., Hippler, R. and Lutz, H. 0., (1987),
'Continuum Transitions in Mu1tiphoton Ionization and Electron
Scattering', in S. J. Smith and P. L. Knight (eds.), Multiphoton
Processes, Cambridge University Press, Cambridge, pp. 43-57.
16. Trombetta, F., (1991), 'Exchange Effects in Laser-assisted Elastic
Electron-Hydrogen Scattering' To be published.
PENNING IONIZATION IN INfENSE lASER FIElDS

HARAlD M)RGNER
Depl:J1't:mmt cf Science
University mtt.en/H!n:lecke
St.cx:lcurrHm-J 0
D-S81O mteen, F.R.C.

ABSTRACf. The es;ential fuatures cf Penning iaJi2atien are revie"\'led in mler to provide
the basis fa the dis::ussien cf laser m:dified Penning iaJi2aticn. E"IJeCted experimmtally
<:h<Ervable effects are des:ribed. The relaticnship to free. free transiticns is di~.

1. lDraIuctlat

The ~ cf this CCIltributien is to des:ribe e"lJeCted effects that an itlt.enle laser has
en the the spcnta.neaJs ~ cf Penning icnimtien

A* + B -~ A + B+ + e-

where A*denct.es an ex:ited rare gas atcm and B a target atcm a- nrlecule, usually in
its grwnd state, but net nec:e:mrily In A ncteVlathy fuatw-e cf Penning icnimtien as
cppc.:a:rl to ether cdlisicnal icni:zatien events is the fact that this pro::ess occurs even in
the Iimt cf 2El'O kinetic energy. Thus, Penning ionization can be vie"\'led as autdcnimtien
cf the cdlisien c:m:plex A*B.
The rmin emphasis cf this ccntributien is net a rig<ralS tlEa-etical fi::rrrulatiCll, but ra-
ther a dis::ussien cf how the laser m:rlificatien can be identified e"lJerlmmtally.
Several tlEa-etical sudies en laser- ~ a- laser- m:xiified Penning iaJi2atien have
been published over the last decade, e.g. [t] ,[2] ,[3] ,[4] . They all have in camxn
that they treat the field-rmtter interact.iat by the dipole approxirmticn. The presmI.
ccnsideraticns lead us to the ccnclu&en that the n:r::a iIqx:rtant. experinEntal features
are net accautted fa- in this way.
The paper is a-gani2£d as fdlaws: s:nien 2 is devct.ed to the preamt.atien cf air
knowledge en field free Penning iaJi2aticn. The nece:mry extensien cf the Haniltcn
cperatcr to the inccrpcratien cf the laser field and the qualification cf the tenn
'illt.enre' in the pre:e1t ccnte». is diocusred in the ne». s:nicn. The last lVlO s:nicns
341
D. Hestenes and A. WeingartshoJer (eds.!. The Electron, 341-351.
© 1991 Kluwer Academic Publishers.
342 H.MORGNER

deal with the qualitative des::riptlcn cf the ch.oErvable eftect.s an:i how field - mxlified
Penning ia1i2aticn can be vie~ as part cf a braider tcpic, I1IilIEly free-free transitiCllS
in the lrng range Gnlcni> field.

2. Fleld free I'emIlng iadaUCIl

We will ccncentrate wr dis:us.<icn cn the prctctype reacticn

(2)

since a large bcdy cf e~l an:i thecretical infcrtmticn is available fer this
syllteID. The tmin &:llI'Ce cf ~ infcnmticn is the ~Il-cs:x.py cf the enitted
elect.ral, i.e. the regiliraticn cf its kinetic energy an:i its directicn.

~I
A*+ B

Distance

tF
A+ B+
:.

Eo ~ax

Fig.1 Schermtical pctential curves fer initial (Ha*+Ar) and final state (He+Ar+).
Electrcns with tmldnum energy E max are enitted predominantly when the
nuclei are close to the classical turning point. The width of the electron
energy distribution (left pane)) reflects the variation of the energy sepa-
ration between the potential curves.

If the ia1i2atim takes plaoe at large internuclear ~tiaJS, i.e. with ess::ntially unper-
turbed atcni.c &ates, the energy cf the electrcn is given by

Eo= E [He*] - IP [Ar]

PENNING IONIZATION IN INTENSE LASER FIELDS 343

which is twghl.y SeV. H:Mever, sln::e iaJlmtim is enabled aJl.y as a cmsequence a

dale erx:cunter m:B. imizlng ew!JIls take place at srmU lnternuclear IEpIU'Iltims where
tlE erergy levels a beth at.c:Im are dls.urbed. Flgl sOOws how this leads 10 electnn
energies ether than aJl.y Eo. The electnn erErgy dUUibutim is rwgbly 100 raN bread.
Sitx:e tlE energy redulim a electnn spectroIEters is rwch better than tl1at, it is
palSible 10 distinguish experimmtally betl'leel1 electraJs tl1at ha~ different energies. This
has intereSlng calBequences with respect 10 the llEBSJreIl'Ent. cf tlE enissim directim.
It is palSible 10 deri~ angular distributims rx1 aJl.y as an average OYer aU eniu.ed
e1ectl'a1S, but als> dIffurentially ftr e1ectraJs rut cf a tinite energy interval within the
energy distribulim. Fig.2 (upper panels) shows the different angular distr.\butims cbtaiIEd
ftr different electnn erugies fran the sysem

(3)

Thes:! remIts shaw caMtdngly that electraJs with different. eJe"gies can be in diffe-
rent sates even thwgh they alginate fran ale aJJi the saJlE pnx;es.c!. If ale calculates
fran tlE Iabcratay tiJed franE back into the nrlecular tiled franE the differences
betv.een tlE angular distr.\butims becmE even m:re draSic [5], [6], [7] as shawn
in tig.2 ClCMel" panels). Acccrding 10 [8] it is helpful 10 diainguish three s.eps in tlE
prcx:essa Penning ialimtim:
1. tlE apprCBch a If and B, a typical half coI.lisim in tlE caIlext a nrlecular dyna-
nics
2 the electrCllic traruJitim, i.e. the switcll fran the bwnl calflguratim into ale with
ale electrm in tlE cmtirnwm
3. the nuclear miim a A+B +.
The rlgaws t.rea.tnEnt cf Itep 1 requires the ~ cf a natlcx:al aJJi energy depeOOent
pctential cperatcr which is familly c:aq>licated aJJi caq>U1atimally linE caJSJning.
H:Mever, in a recent study [9] it has been shawn tl1at m:B. Penning imizlng sySems
- in::luding reactim(2) - can be treated by lIBUIS a a local aJJi energy iJdepenlent
pctential withcut lcaing accuracy beym:1 experimmtal WlCeI1alnty. Fran the remlts a
ref. [9] it becatEs evident that this situatim is unchanged if tlE interactim with the
la.e- tield is added. General characteriaic reatures in Penning ialimtim exist. aJl.y ftr
Itep 2 as pcinted rut in ref. [8]. In the fi:ilowlng VIe list. the charact.erlaic prqJerties a
the electrCllic transition according to the discussion of ref. nO];
Verticality
Itep 2 is a vertical transitim in the IeJlBe that switclling fran initial 10 tinal
cmtirnwm calflguratim is fu.st ccnpa.red 10 nuclear miim. A satenEnt in the
literature U1] that the Fran::k- O:nicn approxiImtim dces net apply 10 Penning iali-
mtim is net in CCIltradictim with this. The reas::n being that in [11] the above
mmtiaJed famll natlcx:ality cf the pctential qJenita (with iIqlact m the calculatim
cf the initial sate wa~ functim a nuclear miioo) has been shifted 10 the transiticn
344 H.MORGNER

qJeratcr. The thus intrcrluced ncn- local transitim is fi:nmlly ncnvertical, r:f ca.\I're,
but the physical nEaning r:f verticality, Jl8JlEly the alxNe nentimed different tiDE
s::ales fir electrolic and nuclear m:ticn, is chvirusly net affected.
A ccrul!quenee ci verticality in the above sense is the conservation of total
spin during step 2: the time for the change of configuration is too short to
allow ~n ablt interactim r:f valence abitals to be effuctive. TItis &aterrent is
backed by Jl1.\Ch experiIJEntaI evidence [8],[10] . we ncte, hcMever, that ~n cm-
servatim does net hold thrwgh the whole process including &ep 1 and &ep 3.
Exchange mechanism for A" being a metastable
the ninimlm nuni>er ci electrms that tIllSt nnve during a Penning reactim like
(2) is tVlO. In principle the final &ate can be reachrl in tVlO pcssible ways
which are visuali2ed by 8lTOWS in the following. Fa the direct IJEChanilm v.e have

~e­

Illu s2sl + Ar<3 p6) (3a)

41

'I_ _~e

fE*(1::Qs) + Ar(3V') (3b)

( I

As lmg as the prqecti.le is in a IJEta&able &ate the pro:e~ is datinated by the

elChange IJEChanism A particular ruiking piece cf experiDEntal evidence has been
prerented by Ebding and Niehaus [12] . Trey had the prqecti.le He- mOling fa~
and the target Ar being at re~ in the labcratcry fra.Jre. Dcppler shift xreasure-
DEnts allCMed them to identifY the fust rmving Ii! atan as the &.'lI1'Ce r:f electrm
enissim. It is chvicus that the direct IJEChanilm ~d net give rire to a Dcppler
shift. We ncte here that this nny have &:lIE inpact mto the design cf an experi-
DEnt m la.rer- m:x:lified Penning icnizaticn: if the prerence cf the elect.ramgnetic
field shru1d enable the direct IJEChanilm this cwld easily be traced experiIJEntally
by Dcppler shift llEaslII'eIIEIlts. We will catE back to this cmsideratim in rectim
4 cf this paper.
The Independent Particle model
varicus pieces ci evidence prove that the tVlO electrms that are cmsidered to IIJJVe
during the Penning icnizatim pra;e~ do net elChange angular IIllIEntwn [10]. To
illuruate this further v.e write rut the tVlO electrm t.ranStim nntrix eleIJEnt as
 PENNING IONIZATION IN INTENSE LASER FIELDS 345

Region I Region II

o
j
01
~1~
or--.,----------,----"
o o~.-_--~-~-~-~~
90 180 o 90 180
Emission Angle Emission Angle

c:] Molecular o Molecular

Frame
~j 'm, / \

f\J
o 0~---,-~--9::-0.,---~-180
~
0"-----.,-----,...,....-"--,-_---,
o 90 180

Fig2 Angular distrtbutiens Region III

cf electrrns frcm
lE*(~ S)+Ar ~ o
lE+Ar++ e- for diffurent
electrcn energies. The
three energy regiens
carespcn::i to t.hae
indicated in fig. 3 . The
energy lncreares frcm
o 90
Emission Angle
180
regicn I to regicn ill.
The ICJY.eI' panels dis-
o
play the ccnvern.cn cf Molecular

/
the angular distrtbu- Frame
liens into the nrlecu-
lar fra.Jre.
Adcpted fran ref. [71 /

O~__~__---~~---~
o 90 180
346 H.MORGNER

<X(1hPHe1s (2HU(1,2) l<p Ar3p (2)'<p He2s (1» ( 4)

V'Iiue X dend.es the s::atterlng &.ate cf the en:itted elect.rm and the rubs::ripts label
the atcmic crbitals that the elect.raJs \\WId pcpulate if the cdlislrn partners V\ere
~ted adiabatically. The apprqlIiate transitirn cperatcrs U reed net be ~­
fied further in the presmt CCl1text. The above lJEIltirned rule has the crnrequence
that X (t) as '/\ell as <p Ar3p (2) Il1lSI. have sigrm-syn:JrlEf.ry with respect to the inter-
nuclear axis. A reccn:I interesting feature is the finding that the tVlO elect.raJs do
net eJ<t:hange m:m:mtum [10]. This rreans in particular that the irni2atirn pro::ess
is wbject to the reruictirn that the elect.rm labelled (1) in eq.(4) prererves its
in&.antanews nx:mmt.wn This has inpact rn the details cf the Dcppler shift.
llE9.SlI'e1IEIl Irel1tirned above [10].

3. Coupling of the electromagnetic field

Let ~l be the electrCl1ic HamltCl'lian cf the field free SYSem The electramgnetic field
is intraluced into the SchrOOinger equatirn in the &.andard way by adding the interac-
tirn cperatrr

(5)

V'Iiue A dend.es the vectcr pct.ent.ial in the three dinEnsirnal !pice. A shall des:ribe
linearly pclarized light with the pclarizatirn vectrr in z-directirn

A = Ao sin(wt-kr) = (O,O,Aoz) sin(wt-kr) (6)

Fer any tVlO elgen&.ates la>, I b> cf the field free HamItCl'lian Hel the transitirn IIRtrix
eieJre11t is given as

<blH int la ) = e(mc) -1 {<b lAP I aI ) + <b1A2 e/(2C)la)}

The first term which is linear in A can be rewritten as

TUn = Hmlh )A OZ < b Isin(wt-kr)V(r) z la > (8)

and the quadratic term yields

T d = (112) e(2c)-1 A 20 <blcos(2wt-2kr) la> (9)

qua z
PENNING IONIZATION IN INTENSE LASER FIELDS 347

~ \lie have Scipped <bla> sinre la>, Ib> are crnsidered <J't.hcgrnal. ~ electrcn
des;:rtbed by Ia>, I b> DJJYeS in tIE pd.en1ial VIr) which can be crnsidered as Crulcrri>
pctential V(r):Q/r fir wr plU'JlCl£S. With z: r Y 10 (e) this allows to trarufum eq'(8)
into

TUn = Um/h>A oz Q < blsin(wt-kr) '10 (e)la > (8a)

O::npuing (Ba) ani (9) shows that tIE brackets have tIE SIlllE crder cf rmgnitude.
Thus fer accessing tIE relative :Uength cf tIE quadratic tenn it suffices to CCIIplre tIE
respective prefacta's. ~ break even pdnt is detemined by

which ccrresp<nls to an intensity cf mre than 101Jwatt/cm2 fer tIE wavelength cf aC0 2
-Iarer. Such high intensities are tcially WlS.litable to 1tudy Penning iooi7aticn In this
intensity regirre IWItiphc.too iooi7atioo lEts in. This wwld IreaIl that He- ani Ar wwld
hardly have a chanre to collide bebe being iClli2ed by tIE larer field. Irx.leed, a plan-
ned experirrent 00 larer m:xlified Penning iooi7atioo is designed with an intensity below
109 Watti crJ [13]. As a cClla!quenre \lie cootent. wrrelves to c<nsider rnIy that part
cf tIE transitioo ~litude that is linear in tIE vectcr field A.
~ l£Ccn:l, likewise reaocnable, approxiImtioo crnsi1i.s in restricting tIE larer interactioo
<nto tIE nnt la:rely bamd electrcn cf eitlEr crnfigw-aticn, &*(1s2s)+Ar or & + Ar++e-.
In case cf tIE initial crnfigw-atioo this is tIE electrcn in tIE eldted 1i.ate that asynptctically
1i.arts as tIE 2s crbital cf heliwn In tIE final 1i.ate \lie have to crnsider tIE electr<n in
tIE crntinuum 1i.ate. C<nsequently \lie can treat tIE 1i.ates la>, I b> as me electrcn 1i.ates
with respect to tIE interactioo cpmltcr H as loog as \lie crnsider transitirns within eitlEr
cf tIE crnfigw-atirns (ct: cases 1 and 3 below).
We have to diwnguish three cases:
case 1: bcth 1i.ates 1 a> ,I b> beloog to tIE subspace cf Hel that e<ntains tIE eldted
(bamd) 1i.ates cf tIE helium atan. If rnIy tIE ICJ\l\ef lying Rydberg 1i.ates
are effuctively pcpulated than \lie are allCMed to rmke use cf tIE dipde ap-
prmdImticn ~ effect thus deocribed is tIE fcrrmtioo cf dre~ 1i.ates rf
He*{ts2s)+Ar
case 2: one state, say 1 a>, belongs to the subspace of the preceeding case \\here-
as 1 b> des::ribes an electr<n that DJJYeS in the field cf tIE nrlecular ioo
He Ar+. ~ linited spacial extensioo cf 1 a > again ju1tifies the dipde appro-
xiImtioo
case 3: bcth 1i.ates are part cf the subspace that containes the s:attering 1i.ates
cf &+Ar+. 1be loog range interactioo bet\\een this ioo and the electr<n is
a Crulcrri> pd.en1ial. Thus, the fcrIWlatioo (Ba) r:f the transitioo rmtrix
elemmt is applicable. Taking into acccunt that 1 a >, I b > are scattering
348 H. MORGNER

sates v.e lEe that the integraticn over spacial ccxrdinates <ices nd. em-
verge. The range that ccntributes to this integral exterds f<rmllly to in-
finity as leng as the lruer field is de&:rlbed as a plB.lE wave. It is obvious
that in this situaticn the exact shape of the laser beam profile must be
cf influence. Furtller, me may expect that this type of interaction with
the electromagnetic radiation is felt much more strongly than the other
cases. It is evident that the dipole approximation is necessarily invalid
for this matrix element.
Exirung literatw'e (1], [2] , [3], [4] is emmtlally ccn::erned with the cwpling acccr-
ding to care 2. Sa:rE authrs [1] do nd. specifY the final sate in tenns a eq.(3).
Other authcrs, e.g. re£ [3] sate e"Plicitly that they errploy the radiative IlEChanism
which can be visualired by eq.{3a) with the additicnal !'eIIRI'k that the transitirn is nd.
spcntanecus but driven by the external lruer field. We do net fdlow, hcMever, the
satement cf ref.[3] that the radiative IlEChanism shruld be active fa- He* 1S2s, 23 S.
Our reas:ning is that acccrding to (3a) the 2s eleclrai had to return to its gramd
sate which v.wld require a spin flip. The electramgnetic radiatirn cannd. do that directly
(Wlder circumtances that are ccnsidered to jurufY the electric dipole approxilIlltirn). An
indirect effuct via spin-crbit interactirn IJll.y be crnceivable in at.ar5 ether than helium
which is known to fdlow the LS-cwpling ocherre up to very high 1Illin quantum numbers.
Our own analysis cf cas:! 2 indicates that the external larer field can lead to beth fi-
nal sates de&:rlbed by the right sides a eq.{3a) and eq.{3b). &Mever, at intensities
ccnsidered here, i.e. below 10 9W / cn:t, the ccrrespcndi.ng transitirn IJll.trix elements are
IlJ.ICh srmller than there a the spcntanecus pra:ess un
Larer acticn acccrding to cas:! 1 can m:difY the character a the orbital of the eJdted e lec-
trcn. This can be e~ by the e~nsiaJ.:

I'll He2s )-1 L Cnl In!) (10)

with I nl) being Rydberg &.ates a heliwn This m:dificaticn lIlly be ncticeable in parti-
cular fir crntributirns with 1=1 which subrequently enable the spcntanecus direct pra;ess
(3a).
The nnt inprtant influence is to be e}q)eCted fran the IJll.trix element cf cas:! 3. It
ccrrespcnds to a free-free transitirn in the field a HaAr+ . The leng range character a
the l£Ar+ - e- interactirn shadd Imke this pra;ess even m:re i~nt than free-free
transitirns mediated by the shcrt. range pctential betv.een Ar and e-. The latter are
active at CD2 1ru:er intensities v.ell below 10 "w/ crrl- as shown by Weingartshcfer and
cdleagues [15],[16]. To alI' knowledge, the i~nce cf cas:! 3 has net been re-
~ befcre in the literatw'e.
PENNING IONIZATION IN INTENSE LASER FIELDS 349

4. Observable effects

The tmin e~ effuct cf the lae- field m Penning icrtimtlm is the change in the
electrm energy diSributicn As viruali~ s;:heImtlcally in fig.3 the energy spectrum
famllar &em the field free care is augmmted by peaks much are displaced &em the
aiginal peak by integer m.J1tiples cf the phctcn energy.
The net change in energy allows to judge the ninlmun crder to much the lae- has
interacted with the cdllsim syS.e:m. .Hc:Mever, the real crder cf interactim Imy be ID.JCh
higher than can be read cff the energy displacemm.t since m.J1tlples cf the phctcn
energy can be ab&:rbed as \'\ell as enitt.ed.
Thus, the inspectim cf the angular diSributim gives a ID.JCh mre detailed harrlle m
the pn:x::ess. The roe cf the angle bet~ the pdarlzatim vectcr and the internuclear
axis has been dis;:U5.'ai befcre by Dahler [1] fer care 2.
The experimmtal distinctim bet'\\eell the cares 1, 2 and 3 is net Wlique. Still, if the di-
rect pra:ess (3a) is ioolated via DewIer shift. llEaSllI'eIrellts as IIEIltimed at the em cf
l'ECtim 2., it is cbviaJS that Imtrix eletrents acccrding to care 1 (initial state) er acccr-
ding to care 2 mut have played a roe. Free-free transitims alene cannct switch the
enissim aigin &em lEllwn to Argcn

Field free

With Laser Field

Eo
~--..:j'll.J~W X tiw )( 1'1W---1

Fig.3 upper panel: field free electrm energy spectrum l~ panel: field - nnllfied
electrm energy spectrum
350 H. MORGNER

S. Synops1s with other free-free transitions

As diocllS!ai above the tmin etrect is to be expected fran the interactien cf the lal£f
field with the en:itted Penning electron. Thus laser modified Penning ionization can
provide intere:ting infcrrmtien en the free-free transitions of electrons in the
lcng range Crulcni> field. The charm cf this apprcach to free-free transiticns ccnsists in
the fuet that the starting ccnditicns cf the electrcnic state can be ccnudled to quite
s::rt"E detail and varied with ea.S'!: as dem::nstrated in s=ctien 2. different electrcn ener-

gies in the field free situatien lead to very different angular distributicns indicative
cf different states kf. fig2 and ref. [6], [7]). Different states in the field free care
have to be WKlern.arl as different starting ccnditions fir the laI£f driven free-free precess.
This type cf varying a pa.ra.rreter is OOtained alnm autamtically in a given eJqJerimmL
Another interesting experiment is the study of VUV photoionization in a laser
field. Again the laser is not the cause of the ionization, but interacts only with
the emitted electrcn. The lal£f intensity has to be kept in the sanE range as diocusred
fir Penning icni2aticn. Since the angular distributien cf phdoelectrcns withrut the lal£f is
very ~ll known ~ again have a situatien where the starting ccnditicns cf the electrcns
are known and different fran all ccnditicns encamt.ered in Penning icni2aticn. This type
cf e~ appears quite naturally as accc.mpanying Penning icni:zatien in a laI£f field
since in lal£f free experi1rents the ccnparis:n betv.een Penning icni:zatien and VUV photo-
icnization is a standard thing to do.
Of coorre, this diocussien shall nd. give the impressien that an eJqJerimmt in which
electrcns are ocatt.ered elf an ien in the presence cf a laI£f ~ superfluws. In cen-
trast, this arrangemmt VIWld result in again different starting ccnditicns cf the electrcn.
It is necessary to mmticn, hcMever, that this exper ment suffers fran t"'O shcrtccmngs
canpared to the experiIrents diocusred previaJS1y. One is eJqJerimmtal and is due to
the fact that ien beam; can be Imde cnly with very low nuniler densities ccn:pared to
neutral beam;. Thus, the camt rate VIWld be extreIrely low. The ether is cf t.hecretical
nature: whereas in Penning icni:zatien and pbctcicni:zatien the electrcn is set into a state
with a very linited nuniler cf partial 'Ml.ves the ocattering eJqJerimmt cannet avcid to
average over a large n1.lIIber cf angular m::JIEIlta. The que:tien is then either how
detailed infrrImtien can be drawn fran such an e~ cr how stringent a test cf a
t.hecretical result it cw1d provide.
In ccnclusien ~ prqx:re to ccn:iuct a careful study cf Penning icni2atirn and VUV
photdenizatien in an intense lal£f field, the PrqJer intensity being diocussed above.
Better understanding cf this field VIWld nd. cnIy be cf interest to basic ocience but at
the sanE tiIre be cf value in applicaticns like plasrm diagncSics and plasrm m:xlel-
ling.
PENNING IONIZATION IN INTENSE LASER FIELDS 351

Acknowledgment

'I1E auth<r wishes to eJq)l'e$ his gratitude to Prcf. A Weingart.sbcfur fir DJiking the
v.aicshcp en "The Electrm" a reality.

References

[1] Saha,HP. Dahler).s. ani Nie1sm,S.E. (1983) Phys.Rev. A2B, 1487-1502

[2] Saha,HP. ani Dahler).S. (1983) Phys.Rev. A2B, 2859-67
[3] Bellum).C ani Gecrge,T.F. (1979) J.OJem.Phys. 70, 5059-71
[4] La.m,K.S. ani Gecrge,T.F. (1985) Phys. Rev. A32, 1650-6
[5] Mrgner,H (1978) J.Phys.B11, 269-80 Hcffimnn, V. ani M::rgner,H (1979) J.Phys.B12,
2857-74
[6] Hm:zzEr,A M::rgner,H Rcth,K. and ZiImEr1llinn.G. (1985) xrvth ICPEAC, Bcd< cf
Abstracts, Palo Alto 1985, eds.MJ.Ccggiola, D.L. Hue::tis and RP.SaJDl
[7] He:rt2ner,A (1986) Diplcnthesis. Freiburg
[8] Mrgner,H (1981) CcmnAtcmMi.Phys. 11, 271-285
[9] Mcrgner,H (1990) OJem.Phys. 145, 239
[10] Mrgner,H (1988) G:nnlAtcmMi.Phys. 21, 195-215
[11] Gecrge,T.F. (1983) J.Phys.OJem. 87, 2799
[12] Niehaus, A. and Ebding. T.
ICPEAC. Book of Abstracts. eds. J.S. Risley and R. Geballe Wniv. Washington
Press. Seattle. 1975)
Ebding. T. (1976)
Ph. D. TheSiS, Freiburg
[13] Wallbank. 8.. Weingartshofer. A. and Morgner. H.
[14] Hertzner, A. and Morgner, H. 1990 to be published
[15] Weingartshofer. A., Holmes, J.K., Sabagh, J. and Chin, S.L.
(1983). J. Phys. B16, 1805-17
[16] Wallbank, B., Holmes, J.K. and Weingartshofer. A. (1967)
J. Phys. B20, 6121-38
MICROWAVE IONIZATION OF H ATOMS:
EXPERIMENTS IN CLASSICAL AND QUANTAL DYNAMICS

L. MOORMAN
Department of Physics
State University of New York
Stony Brook, NY 11794-3800
U.S.A.

ABSTRACT. Experiments on microwave ionization of hydrogen atoms at various

frequencies compared with theoretical calculations have shown that the problem
may be divided into different regions of distinct behavior depending on the pre-
cise experimental situation. Some regions may be described up to a certain level
of accuracy using classical atomic dynamics while others require quantum atomic
dynamics. At a more detailed level, the latest comparisons with theoretical calcula-
tions indicate that experimentally we can measure atomic quantal interferences in
an otherwise "classical region" and make links to classical atomic dynamics in an
otherwise "quantal region". In the latter case, "scars" promise to playa particularly
important role in the understanding of how quantal and classical dynamics merge
for a system whose classical dynamics is at least partly chaotic.

1. INTRODUCTION

This chapter deals with the recent advances in experiments and theory of microwave
ionization of hydrogen atoms. We will mainly discuss the experiments performed
in the laboratory at Stony Brook. Our theoretical discussion will focus on one
of the most recently proposed interpretations of strong local stabilities observed
in a region that, as far as the treatment of the atomic dynamics is concerned, is
of intermediate classical-quantal nature. The intertwining of quantal and classical
dynamics in this region and the fact that the latter has an irregular character makes
353
D. Hestenes and A. Weingartshofer (eds.), The Electron, 353-390.
© 1991 Kluwer Academic Publishers.
354 L.MOORMAN

this microscopic problem of the simplest atom placed in a microwave field extremely
rich and interesting.
To the experimentalist the hydrogen atom in a microwave field is a particularly
attractive system to study because it allows for careful control over the key pa-
rameters. Our experiments expose hydrogen atoms in a Rydberg state with initial
principal quantum number in the range from no = 24 to 90 to a carefully controlled
microwave field. Depending on the microwave frequency. w (atomic units are used
throughout au) in order to ionize a H atom for some no values as many as 300 photon
energies are necessary to bridge the energy gap to a final free-electron (continuum)
state.
Theoretically this system can be studied from two very different points of view
"Classical" theories regard both the atom and the field as classical quantities.
"Quantal" theories describe the atom quantum mechanically, but the field and its
coupling to the atom are still considered classically. This combined with the "irreg-
ularity" of the classical problem therefore represents a (time dependent) example
of a family of problems that may be grouped under the name "quantum chaos".
Classically. the non-linearity of the harmonic drive of the microwave field com-
bined with the Coulomb force may give the possibility for chaotic dynamics and
quantitative tests in which Liapunov exponents were calculated have proven this to
be so (1,2,3]. The chaotic dynamics in its turn provides the irregular trajectories
through which the atom may ionize, the effects of which were observed experimen-
tally (4]. For certain conditions - higher scaled frequencies - this is sometimes called
diffusive ionization and a theoretical prediction that in that case quantum mechan-
ics suppresses the classical diffusive ionization process (5] has also been verified by
at least two experimental groups (6.7]. The confirmation of this purely quantal ef-
fect in the hydrogen microwave ionization experiments initiated a lively discussion
among theorists as to what this quantal suppression of classical microwave ioniza-
tion originates from and whether such a behavior could successfully be described
by simplified, truncated theories or (classical) map approximations that may be
quantized in some way.
From the comparison of experimental results with theoretical calculations that
closely model the experimental conditions a picture has emerged in which the mi-
crowave ionization of hydrogen may be divided in different regions (8,10,9] for the
parameters that control the experiment. In some regions the experimental dat.a
compare very well to calculations based on classical atomic dynamics while in oth-
ers quantal atomic dynamical calculations are necessary. But, if one looks in more
detail (e.g. with higher resolution) one finds indications for quantized-atom inter-
ferences within the "classical" region, and links with classical atomic dynamics in
the "quantal" region. The proven accuracy (11,12] with which experiments can
be done and the wealth of information they have yielded so far make ionization
by microwaves a good probe system to study the transition of quantal to classical
dynamics, which some may conjecture to be the region where a correspondence prin-
MICROWAVB IONIZATION OF H ATOMS 355

ciple would be applicable. It is therefore appropriate to include these investigations

under the heading of Quantum Chaology [13,14].

2. THE HYDROGEN ATOM IN A MICROWAVE FIELD

The hydrogen atom is prepared in a Rydberg state with principal quantum number
in the range (24 < no < 90). The method by which this is done will be discussed
below. The free field energy spectrum of these states becomes denser with increasing
principal quantum number and their energies are approximately given by Eno =
-1!(2n5) (au). The external electro-magnetic field used is so strong that the atom
ionizes, or is brought on the verge of ionization. But because the ratio of the photon
energy to the free field energy splitting is of order unity or less, the atom has to
absorb at least the order of 10 2 times the energy of a single microwave photon in
order to be eventually ionized. Of course, although the final energy balance can
be made up this way, the process of ionization can not be considered in this way,
where the atom is decoupled from the field, because in the real experiment the atom
ionizes in the presence of the field.
The relevant parameters in microwave ionization of hydrogen, or single electron
systems in general, are the classically scaled frequency, n~w, and the classically
scaled amplitude, n~F. The scaled frequency is the ratio of the "external" fre-
quency w (au) to the "internal" Kepler frequency, l/n~(au). In terms of quantum
mechanical quantities this represents the ratio of the photon energy to the energy
splitting of free field n states at n = no. The scaled amplitude is the ratio of the
external electric field amplitude, F( au), to the mean internal field strength of the
nucleus - averaged over one classical Bohr orbit around the nucleus.
Under the large variety of experimental conditions investigated so far, we have
found not only quantum behavior but also clear indications of classical behavior in
the ionization process. Table 1 summarizes what the comparison of experimental
and theoretical investigations on the microwave ionization problem of hydrogen has
taught us. Experimental ionization curves were always obtained by measuring the
ionization signal as a function of field amplitude at a fixed frequency. These ion-
ization curves together with the results of theoretical calculations have indicated
up to now the existence of (at least) six regions the characteristics of which were
explained in Ref [8,9,10]. We emphasize that with respect to scaled frequency the
six regions are not disjoint; the boundaries overlap. The frequencies at which the
experiments were performed are also indicated in the second column of the table
because the quantum dynamics does not depend only upon classical scaled quan-
tities as is required by classical dynamics. The abbreviations under 'Exper. type'
are explained in the next section. The third column, labeled 'Exper. character.,'
listing the occurence of experimentally anomalies is based on a comparison of
 Region CLsc.freq. Freq. Exper. Exper. model Remarks
n~w (GHz) type character. characteristic Exp .... Th.
~
static "1 o ~' [19] I Tunneling through
° . static barrier 20
L__ t~~~i~-T ;S0.05 ;:;;-------. DIP- Absence of NMS in IThnneling through QM lower than classi-
I I ionization curves I slowly moving bar- cal threshold

'---n--i <:oj 7.58 NMS 3D/1D CL MC;

.,
NMS are QM that may
low ! 0.05 9.92 DIP+SA + changes of slope in QM S.eq solution;
lower or raise thresh-
frequency i to 0.3 11.9 ionization curves. QM adiabatic basis
7.58
~'~I . Classical scaling . 3D/lD Class.MCj
m 9.92 DIP+SA
",,01 of thresholds; Local I ID Q.approx. on
semi 11.9 SA
to 2 I stabilities in thresholds adiabatic states; ID
I
classical 26.4 SA I at = l/q
ng", S.eq.
36.0 DIE+SA
it I~ • I ~ 1 26.4 SA I Local structures over 3D/1D Class.Me;
LanSlhon I to 2 .36.0 DIE+SAi 50% of fl6F values. Is 1D QM S.eqj
I I ---1..!!1 there classical BC aling~( RCE basis

II V I
I I 1D S.eqj
I high I >2 26.4 SA Large struct.ures su-
I Delocalizationj
! frequency I rv 36.0 DIE+SA perimposed on rising Cantorus flux
I I hO ' "
l..... I~~';,.tion
I m I .-1 QRCE-basis ~ __ u_ --

QM perturbat.ion ... 0-- _o.o~ ".

> ino I theory

Table l:Six regions of Hydrogen micT07JJave ionization are distingui,hed by comparing Stony Brook r
experiments to several theoriu. This chapter contains more detailed information a,bout the regions II- V. 3:
Abbreviation, u,ed in the table are: lD(3D) = 1(.1) dimensional; CI=Clauical,' DIP. DIE: see ,ection ~
3; J.·fC=lvfonte Carlo; NltfS=nOll monotonic structures; QKM = quantized Kepler map; QkI = quantum ~
mechanical; QRCE-basis: basis of 'qua,i resonant compensated energy' sta,tes; sc.=lcaled; SA: lee section Z
j; S.eq = Schrodinger equation.
MICROWAVE IONIZATION OF H ATOMS 357

different experimental data only, i. e. no theoretical arguments, except scaling, are

used. The fourth column lists which types of theoretical models have been used.
The last column briefly notes general conclusions that follow from the comparison
between experiment and theory. In this paper we report on the regions II through
V. We will not discuss region I in which only preliminary measurements have been
made and for which the border with II in terms of scaled parameters is not well
investigated yet. Region VI is largely theoretical, combining results of calculations
with what is known from experimental strong field (pulsed laser) interactions with
atoms, which use frequencies far above the microwave region. Experiments were also
performed with two simultaneous microwave fields but these will not be discussed
in this contribution [15,16J.

3. EXPERIMENTAL METHOD

A very simplified picture of the apparatus is given in Fig 1 [17,18,21J. A fast beam
of hydrogen atoms (~ 15 keY) is produced by charge exchange of protons with
Xe gas. In two separate field regions with static fields F l , F3 two 12C1602 laser
transitions are made using a double resonance laser excitation technique. Typical
field values for Fi were about 30 kV fcm and F3 ~ 1 - 500 V fern. Since each laser
polarization is parallel to the direction of the static fields, only dm = 0 transitions
are driven. A typical scheme involves excitation of the extremal Stark state with
parabolic quantum numbers (no, nt, 1m!) = (7,0,0) to the state (10,0,0) with the
first laser, and from there to (no, 0,0) with the second laser. Spectroscopic scans
[17J made as a function of F3 showed that available laser lines and experimental
spectroscopic resolution allowed us to populate and resolve states with principal

CO2 laser Electron Static

beams deflection field
",. ionizer ,.-_ _-,
Atomic f plates
I
~ H(n*),p+ Detect
beam f
labeled
\ ~"~::'-""'()J ions
F2
Collimator
Cavity
MCP
electron
~
~
multiplier
Figure l.Schematic view of the apparatu6 uud in Stony Brook for the detection
of microwave ionization. Atomic beam particie6 enter from the left and are, after
preparation in highly excited 6tate6 no, ionized in the cavity due to the interaction
with a 6trong microwave field. For variou6 detection 6cheme6 lee text.
358 L.MOORMAN

1.2
W rl=0.13rnrn
0
~
I- ~'---===~~r 2=0.39
-.J
CL 0.8
r3=0.65
~
«
0.6
0
-.J
------- ----~~~\Iii'?------- ------:
,
_____________________ I

W I
0.4
LL I
I
I
I
() I
I
0.2 I
Il:: I
I
l- ,
,
I
I
e.)
~/-
L___ ----I
W
-.J
0
,,----
,, ,,------j,
, ,
, I
W ,
,
I

-0.2
-2 -1 o 2

Z (em)
Figure 2. Longitudinal (E:) and radial (Er) component8 (with regard to the direction
of the beam axi8) of the electric field amplitude in8ide the cavity and beam hole!.
The calculation by 'Superfish' wlvel Maxwell'! equation! for cylindrical geometrie!
for many radii. Three solution6 c/06e to the axis and normalized to the center of
the cavity are 6hown. Part of th e cavity contour, 8caled to the z axil, is !hown by
the dashed line. l! = 36.02 GHz for the TMo40 mode

quantum numbers in the range 24 :::; no :::; 90 for hydrogen [16,4].

Though a unique substate was produced in F 3 , it could not be maintained all
the way into the cavity. A nominally "zero field" region dominated by stray fields
before the atoms entered the cavity led to a statistical distribution of all substates
[4] of the given no-value, or classically an ensemble of orbits with a microcanonical
distribution of orbital planes and eccentricities (t :::; 1). Such an ensemble (or
distribution) will be called 3D. We will later refer to 1D models, in which the
MICROWAVE IONIZATION OF H ATOMS 359

0.5
,SUPERFISH (")
JO
~
N
0
=<
W
:E
~
r
-0.5

~ ENTRANCE/EXIT HOLE
BEAM RADIUS
-1
0 0.5 1.5 2

radius (em)
Figure 3. Longitudinal component (E.) of the electric field amplitude on the mid-
plane inside the lame cavity as Fig 2. The result of the calculation with 'Superfish'
is compared to an 'ideal cavity' solution. The latter, which has no beamholes, 16 a
Jo-Bessel/unction with the fourth node at the cavity wall/or thil TMo40 mode.

electron bounces back and forth on a line between the nucleus and the classical
turning point of the Coulomb potential, roughly corresponding to what would be
in quantum mechanics an extremal Stark state (22,23,24,25].) This 3D atom is the
initial state with which we perform actual experiments.
The beam of Rydberg atoms was narrowed by a collimator before entering a
circular cylindrical copper cavity with beam holes in the center of the two endcaps.
The cavity was resonantly driven in one of the modes TMopo with p=2, 3 or 4 such
that the atom on its way through the cavity experiences several hundred oscillations
of the field. Table 2 summarizes the atomic state properties, characteristics of the
360 L.MOORMAN

cavities and the field amplitude shapes for the different experiments done in our
laboratory.
The field amplitude in the cavity was specified as accurately as possible by
solving Maxwell's equations with a standard finite element computer program 'Su-
perfish' [26] which exploits the cylindrical symmetry of the cavity. The calculated
electric field amplitudes along the beam direction (E, ) seen by the atoms traversing
the cavity are given in Fig 2. The various curves represent the amplitude compo-
nents along straight line trajectories through the cavity at various distances (rj)
from the symmetry axis. The atoms on axis of the cavity see the largest maximal
field amplitude. A cross section of the cylindrical cavity, rescaled to the horizontal
axis, including correct wall thicknesses and hole sizes is indicated by the broken
lines in Fig 2.
We refer to the curves E. in Fig 2 which are normalized on the axis of the cavity
as the envelope function A(z). Seen from the rest frame of the atoms (in analogy
with pulsed laser experiments) this represents the pulse shape of the field, A( at),
where a-l = ~ is the characteristic timescale for the atom to travel the length of
the cavity. The turn on and off of the pulse therefore occur over a known number of
oscillations and its duration depends on the velocity of the beam. The field inside
the beamholes was well suppressed since the holes were of relatively small radius
and the walls were thick. The field amplitude was already reduced to 50% at the
inside of the cavity wall and virtually vanished outside the beamholes. As can be
seen in Fig. 2, the radial field amplitude ET became comparable to E, only inside
the holes, indicating that at that position the field was rotated 45° with respect to
the beam axis. However, such a large angle would be experienced only by those
particles at the extreme outside of the beam at r = 0.7 mm, and in addition only
there where the amplitude is reduced to less than 20%. It is important to note that
at the inside wall where the field amplitude is reduced to 50%, this rotation is less
than 25°. The field is to a good approximation longitudinal near the position where
the total field is maximal, i. e. where we expect ionization to be most probable.
We have done many tests on the accuracy of these calculations of the amplitude
envelope including in situ measurements by passing a small dielectric 'bead' through
the cavity and determining the local field amplitude from the resonance frequency
shift. These tests demonstrate that for the cylindrically symmetric modes used
in the experiments reported on here, the pulse shape is known very precisely (at
the percent level) and can be used reliably in numerical calculations of the atomic
dynamics.
With a collimator of typical radius 0.7 mm (see Table 2) for the high frequencies,
the total beam contained atoms that experienced different field amplitudes, F maz ,
ranging over 4 and 7% for the two modes used (~Fma.. ; Table 2). Fig. 3 shows
the field, E. (TMo4o) in the midplane of the cavity calculated by 'superfish' with
the beamholes included, compared to an analytic expression of an ideal mode for a
cavity without beam holes (a Jo Bessel function). The Fo values given throughout
MICROWAVE IONIZATION OF H ATOMS 361

in this paper will always be the geometric average of Fmaz over the cross section of
the beam. In some cases, not reported on here [11,12]' the beam was collimated
down until DoFmu was 0.2%, but a special calibration technique described in [27]
estimated the absolute accuracy in the value Fo generally to be 5%.
In response to the electric field, which is maximal on the axis and polarized
along the beam, atoms may be ionized depending on the field amplitude. To detect
the ionzation we employ three different methods with the following objectives:
A) Direct Ionization, Proton (DIP) detection method. In this method the protons
produced inside the cavity by the interaction with the microwaves are detected in
a particle multiplier downstream from the cavity.
B) Survived A tom (SA) detection method. The atoms that survive the microwave
field in a Rydberg state are ionized in a separate (longitudinal) static field ionizer
downstream from the cavity. Those ions are subsequently detected in the same
particle multiplier that is used in the previous method.
e) Direct Ionization, Electron (DIE) detection method. The electrons produced
in the cavity by the interaction with the microwaves are extracted from the beam
immediately after the cavity and then detected with a microchannel plate.
The names given to these methods are meant to indicate which object, electron,
proton, neutral atom, leaving the cavity we intended to detect. All three detection
methods made use of a label voltage on that part of the apparatus where the
charged particles of interest were produced. (For DIP the cavity body was held at
::::: 100 V, for DIE it was at ::::: -6 V and for SA the cavity body was grounded and
the static field ionizer was at about 150 V). This 'energy label' allowed for velocity
selection downstream from the cavity and the static field ionizer in a static field
lens system (not shown in Fig 1). In addition all measurements were performed by
phase sensitive detection on the first laser transition.
A notable side effect of the DIP method was that an atom existing in an ex-
tremely high Rydberg state after the cavity (say 10 times no) could be ionized by
the small field along its flight to the detector. This ion (or electron) would also
be energy labeled by the field in which it was formed and therefore pick up extra
velocity.
Another complication was that even a minute stray transverse component of the
magnetic field could produce a Lorentz force (or equivalently a 'motional' electric
force in the rest frame of the atom) strong enough to ionize extremely high Rydberg
states. The combination of these two ionization mechanisms sets the limits, or cutoff
values, n" indicated in the 4th line of Table 2. Notice the large values of n c , 160-
190, for the DIE experiment compared to the initial values no. In the experiments
reported on here the cutoff values were completely determined by effects outside the
microwave region and therefore the atom-microwave interaction dynamics inside the
cavity is completely free from static field interaction effects. This is different from
experiments done elsewhere [4,7], in which static fields were intentionally maintained
in the interaction region.
362 L.MOORMAN

H H H
type of experiment SA or DIP SA or DIP SA or DIE
no range 24-90 43-49 and 62-77 45-80
nc 90-95 90-95 86-92 or 160-UI(
I kinetic en"gy (keV) I 14.0 14.0 14.6
frequency (GHz) 9.9233 7.582 / 11.889 26.40 / 36.021
I nominal cavity mode Tl'vfo20 T M 020 / T M030 TAf030 / TM040
I cavity length (cm) 4.928 4.501 2.007
I cavity radius (cm) I 2.658 3.498 1.565
I collimator radius (cm) I 0.251 / 0.050 0.251 0.070
turn-on (5%-95%)( osc) 79 50/80 82
flat max. (95%-9S%)(osc) 231 200/300 334
turn-off (95%-5%)(osc) 95 50/80 82
calc. radial variation AFmaz of F 7% /0.2% 4%/10% I 4% /7%
absolute accuracy in Fo 5% 5% 5%
>::: U.14-U.44
n~w 0.02-1.1 0.5 - 2.8
and 0.43-0 ..').'1

Table 2:Experimental apparatus data for H microwave ionization and excitation

experiments reported in this contribution. The turn-on, -off, and fiat maximum de-
scribe in number of field oscillations the field pulse-shape that the atoms experienced
in their rest frame. The absolute accuracy for the microwave field amplitude on the
beam axis is based on an extensive calibration procedure [27}. Relative accuracy
within a given experimental curve, e.g., Fig. 5, is better than one or two percent,
but the field drops quadratically away from the beam axis by the amount shown in
the second-to-last line of the Table.

Direct ionization experiments were taken by slowly increasing the amplitude

Fo from a value where only a background signal was measured to a value where
ionization occurred. The signal was observed to increase and finally saturate to a
maximum, defining the 0 to 100% levels. This increasing signal, recorded in con-
junction with the microwave power, produces a curve whose shape is characteristic
of those of the DIE experiments as well. In contrast, the type of curve measured in
a survival experiment, SA, represents a decreasing signal from 100% down to 0%.
If one takes the inverse of the SA signal, a Q curve is obtained (Q=100%-SA). This
curve represents the amount of signal quenched by the microwaves and is similar in
appearance to that of DIP and DIE curves.
All experimental curves DIP, DIE and Q record the microwave field dependence
of the probability for hydrogen atoms to be excited to a final state with an energy
larger than that of a state with principal quantum number n" a cutoff which is much
larger than no in most cases. The contributing processes include true ionization, i. e.
excitation to the continuum by microwaves, as well as excitation to bound states
MICROWAVB IONIZATION OF H ATOMS 363

above nco The characteristic which distinguishes the three experimental modes is
the value of nco Another difference is that the SA method is sensitive to extreme
microwave de-excitation. Although its occurance is unlikely, (except in some special
cases) it can be diagnosed by varying the voltage in the static field ionizer which
determines the minimum n-state that will ionize. Provided that the value of the
static field voltage is kept high enough there should be no difference in behavior
among the detection methods. This was in fact experimentally verified.
Although complete curves were always recorded, a useful way to characterize
them is to extract the field amplitude at which 10% of the Rydberg atoms were
ionized which we will call the 10% electric field amplitude thre$hold (Fo(lD%)) or
simply the 10% threshold. Following this procedure, many comparisons carried out
for hydrogen at 9.92 GHz with no ranging from 32 to approximately 70, showed that
quench and ionization curves gave the same information - even if they contained
structure(s). When no approached n c , not surprisingly, differences became apparent.
Experiments done at 36 GHz also revealed differences between the DIP and DIE
methods. These will be discussed later.

4. CLASSICAL SCALING

In Fig 4A, five curves are shown representing measurements made on atoms in
three different initial states, no ;; 67, 48, and 43, and at three different frequencies
9.9, 26.4, and 36.0 GHz respectively. The electric field amplitudes necessary for
ionization seem to be very different for the three cases. Moreover, from a quantum
mechanical point of view one might not expect them to have much in common each
requires a very different number of photon energies to reach the free field continuum
( c. q. the binding energy of the initial states is 73, 53, and 47 nw). It may come as a
surprise then that by rescaling the electric field amplitude (F, expressed in atomic
units) with the fourth power of the initial principal quantum number (n~) these
curves become nearly congruent, see Fig 4B. A key point in the magical degeneracy
of these seemingly unrelated curves may be realized by rescaling the frequencies
(x27r = w(a'U)) with the third power of the initial principal quantum number (n~).
One finds that the three situations correspond to values which are extremely close to
each other, namely 0.45, 0.44, and 0.43 au respectively. This hints at the existence
of a relation between the various curves.
We will show that this type of scaling has a physical significance and is in fact
required under classical conditions. By writing the classical Hamiltonian for the
fully three dimensional classical hydrogen atom interacting with a linearly polarized
external field pulse, with electric amplitude F A( at), as the sum of the free kinetic
energy of the electron, the nuclear Coulomb potential energy and the potential
364 L.MOORMAN

100

Z
C: 50 9.923GJz
I- n = 67
«
N
Z

o
o 50 100
ELECTRIC FIELD AMPLITUDE rV
\ / ern '\I

100

".)
9.923
..'"
67 0.4538
26.426 .f8 0.#14
o 36.022 43 a.f3S5

o 0.05 0.10
SCALED AMPLITUDE (n4F)
Figure 4. Example of five ionization curve~ in region III for three different fre-
quencies that show classical scaling (see text). A:(top) absolute arnp/it·ude units,
B:(bottom) scaled amplitude units.
MICROWAVE IONIZATION OF H ATOMS 365

energy of the unbound electron in the external field:

H(r, p, t; F, w, a, ¢) = ~ 1P12 - I~ + F A(at)z cos(wt + ¢). (1)

In this A(at) is the normalized shape of the amplitude seen in the restframe of the
atom and a-I is the total time the atom is inside the cavity.
It is easy to check that this Hamiltonian has the following exact classical scaling
property with arbitrary constant 0: (¢ is invariant):

H '("" ..., I pi ,w,a

r,p,t; I ')
== H( a -2~ ~
r,o:p,a -3 t;a ip,aw,aa
3 3 ) ==a 2H(-+
r,p,t;
-+ )
P,w,a.
(2)
Due to the complete freedom of the value a one may rewrite any solution (r(t),p(t»
of H (say e.g. describing a Kepler orbit) as a solution (r'(tl),P'(t ' )) of H' [28,29].
We may for instance choose to define the primed system to be the system in which
F' == 1 (unit of amplitude), or we may choose Wi == 1 (unit of angular frequency).
Instead however we make another, more natural choice: We will choose to define the
primed system such that the initial classical principal action variable of the orbit
(I~)o [31] is Ii, i.e. one unit of action (1 au), when the field is not yet turned on.
We can use dimensional arguments as in [24]: The principal action variable has a
dimension rp, == M/ [31]' and therefore the principal action variable of the initial
orbit scales as:
(3)
The other two action variables for the initial conditions (1')0 and (Im)O scale with
the same factor because they have the same dimensions [31].
In classical considerations later (In)o will be abbreviated by 10 , Using Bohr-
Sommerfeld-Wilson quantization we may replace 10 by noli to translate expressions
from classical into quantallanguage (or no if au are chosen for all quantities), and
using Eq. (3) we may replace 0:0 by no (both dimensionless) in scaling relations.
The im port ant parameters which characterize the dynamics are now (F', Wi, a' ) ==
(n~P, n~w, n~a) in which F, wand a are in principle in arbitrary units but particular
convenient physical interpretations for the values of these scaled parameters can be
given if they are expressed in atomic units (see below). It is for the reason described
above that these parameters are called ciaHically scaled variables. For a field pulse
that rises slowly enough we assume that we may leave n~a unspecified. As for our
experiments one can in principle extract this information from Table 2.
The goal of the explanation after Eq. (2) above was to show that the scaled
parameters are a natural choice to label the solutions rescaled to a system where
I~ == Ii (or 1 au). There is another way of interpreting these parameters. In the
initial state with action 10 there are two time scales in the classical dynamics: The
internal Kepler frequency WK and the external driving frequency w. (There would
be more frequencies for instance if more electrons would be present). Therefore,
because one is free to scale the time parameter (see Eq. (2» these two frequencies
366 L.MOORMAN

can be reduced to one; namely their ratio. Therefore the classical dynamics for
hydrogen and one-electron ions depends only on one frequency ratio:

(4)

Although the ratio is dimensionless and thus does not depend on the system of
units, we clearly choose atomic units in the second ratio and therefore w in the
right hand side should be expressed in au also. Since time scales are important one
should expect the classical dynamics to behave differently for different values of ngw
« < 1 ; ;::::j 1 ; > > 1). Of course as can be seen from Eq. (1) we have introduced
a third timescale a-l. However since from Eq. (2) one is free to scale only one
parameter we have to require that the ratio a/w is preserved under scaling.
Quantum mechanically the ratio n~w has a very simple meaning: It represents
the ratio of the energy of the microwave photon to the average energy splitting of
the n states for the initial quantum number.

(5)

Similarly, we may interpret the classically scaled field amplitude as the dimensionless
ratio of the electric field amplitude that the electron experiences from the external
field and the mean electric field of the nucleus averaged over one classical Bohr orbit
about the nucleus:
(6)

In general, tested for the three frequencies as shown in our example in Fig. 4, we
can say that the 10% threshold amplitudes scale very well within a few percent;
these tests were done over a wide range of scaled frequencies (0.2 ::; n~w ::; 0.7).
This indicate that classical theory might be able to describe ionization curves in
this region and also seems to justify that the exact scaling of the field pulse (i. e.
including scaling of the switch- on and -off of the amplitude at the entrance and
exit of the cavity) was not neccesary for these scaled frequencies. In addition to the
scaling results just described there is much more evidence for classical behavior in
this region. Individual curves can be calculated by classical methods using Monte
Carlo simulations and directly compared to the experiments. This will be discussed
in section 6.

5. ON THE CLASSICAL-QU ANTAL CORRESPONDENCE

OF THE AVERAGE COULOMB FORCE ON THE ELECTRON

In the previous section we emphasized that the classically scaled amplitude can be
understood very easily by taking the ratio of amplitude and the averaged Coulomb
MICROWA VE IONIZATION OF H ATOMS 367

force Fe = Zr- 2 (in the initial, unperturbed state) on the electron in a classical
Bohr orbit around a nucleus of charge Z and used < r- 2 >c= 10 4 . The latter is
trivial by realizing that a Bohr orbit is a circular orbit in which the radius has a
fixed value during the motion say Irl = R = I~ao (ao = 1 in atomic units). This is
the reason we may rewrite the average:

< r -2 > c-<r

B!!l'r >-2_ [-4
c-o' (7)
The first step is in general not allowed, neither for elliptic Kepler orbits nor for
quantal states as we will see below. We will investigate the correspondence between
the average force on electrons in quantum states and classical orbits a bit further.

A. QUANTAL

Using the orthonormality of the spherical harmonics, the average strength of the
Coulomb force of the nucleus on the electron in a given quantum mechanical substate
Inolm > is found to be given by [34,35,36):

• Z 2 dr = (I
< F >Q=< nolmlZr -21 nolm >= 10"" Rnol2Rnoir 1 (Z3)
""2 ' (8)
o r + 2".1) no3 ao
in which Rnl are the radial wavefunctions and ao is the Bohr radius. It is clear from
this that for different states of fixed angular momentum I and arbitrary substate m,
for example an s state, < F >q decreases'" n0 3 and not as the classical situation
for a Bohr orbit described above. This would be relevant for instance for He (and
other many-electron atoms), where the initial state for experiments may be a S
state for different no [11,12].
The quantal states that correspond with Bohr's circular electron orbits are the
states with no radial nodes in the wavefunction when I = n - 1. The m = ±l states
are easiest to visualize as circular states, as they have their maximum electron
density in the azimuthal plane perpendicular to the quantization axis, and there
is a direction of charge current. The state with m = 0 is also easy to visualize
as its maximal electron density is aligned along the quantization axis, approaching
closest to a 1 dimensional state (Another way to view this is that it corresponds
to an ensemble of classical circular orbits lying in all orbital planes that contain
the quantization axis). In that case, with I = n - 1, from Eq. (8) we find indeed
< r- 2 >'" n04 for large no which corresponds to Eq. (7).
For the case of a statistical distribution of substates land m, we find the fol-
lowing dependence on no:

(9)

This has the same no dependence again as for the classical circular orbit except that
it is larger by a factor 2 due to all other states in the distribution that are bound
368 L.MOORMAN

stronger to the nucleus than the l = no - 1 (Bohr) state. Of course for a coherent
superposition of initial substates within one no value (,shell') which are observed
in modern atomic physics experiments the dependence on no may be very different
between the boundaries", no 3and '" no4.
The averaged energy splitting between neighboring states can be found from
the total energy Eno = - 2='11 (Z2) which is, due to the total degeneracy of the state
no ao
with principal quantum number no, the same for a statistical distribution of I and
m states. The average energy splitting around the initial state is therefore:

1 1
6.Eno = (1/2){(Eno - E no - I ) + (Eno +1 - Eno)} = n~ (1- 1/n8)· (10)

This is in fact closely related to Bohr's correspondence principle and from it we

see that the value no 3 in Eq. (5) is a good approximation. The first correction
exists of a term that is smaller by a factor no 2 which is typically in the order of
10- 3 in our experiments. The first relevant correction for a spinless particle from a
different origin is a relativistic correction that is smaller by a factor :J.2 ~
no
~3 . 10- 6
(fine structure constant as). Both corrections can be omitted for our purpose.

B. CLASSICAL

Now we will treat the classical derivation of the mean force < F >c for an arbitrary
(3D) Kepler orbit of a 'classical' electron of mass m = 1 au orbiting around the
nucleus in a potential V = -G/r. Although this can essentially be found in Born
(1924) [30] (for a later reference see [25]), we prefer to reintroduce a few concepts
and parameters that label Kepler orbits as it is not common to view an atom as a
classical object nowadays. Also, giving some details may help to understand what
we mean when we compare our experiments later in sections 6 and 7 with numerical
calculations that used a microcanonical distribution of classical initial states.
By using spherical coordinates we can define a set of angle-action coordinates
(r, 0, </>, I" 19 , I,) and from this 3 new action variables which are linear combinations
of the old ones can be defined that have more physical significance: In is the principal
action variable, meaning the action for which the frequency is "nondegenerate and
different from zero" (If no static fields are applied it is the only one that is nonzero ).
1/ is the total angular momentum and 1m is the component of the angular momentum
onto the polar axis [31]; consistent with modern notation we use 2rrI, [32,24] for Ji
in [30,25J.
The first two new actions, which are constants of the motion, can be related to
the energy of the motion, eccentricity and semimajor and -minor axis of the classical
elliptic Kepler orbi t:

E = (11 )
MICROWAVE IONIZATION OF H ATOMS 369

a
= FU1
= _.£
1-CJ

= G- 1 I 2
(12)

(13)
2E n

b = a~ = G-1llln . (14)
Since in our case with no 2: 24 (for a spinless particle as well as for the spin-orbit
interaction [36]) the first relativistic corrections are smaller by a factor o.~/no~2 .
10- 6 it is sufficient to use the non- relativistic approximation here. The orbit itself
in the orbital plane is described by: r = a(l - E cosu), in whichu is the eccentric
anomaly defined by WT = U - f sin u, where W is the fundamental frequency and T
a time or epoch measured from an instant when the particle was at the perihelion.
Born shows that for every elliptic orbit < r- 2 >= -di. This can directly be derived
without performing the integration from the second law of Kepler. We find that
the mean force on the electron in a Kepler orbit is inverse proportional to the area
(nab) enclosed by the ellipse:

G G3
< F >c= ab = III{ (15 )

This shows corresponding to the quantal treatment that for fixed angular momen-
tum I, and increasing In the mean force on a classical electron goes down as I;;J.
The first two action variables may be used for imposing Bohr-Sommerfeld-
Wilson rules to quantize the atomic motion by replacing In = nh with n = 1,2,· ..
and II = kh. In the old quantum theory k was 1, 2.... ,n. Modern semi-classical
quantization methods (Einstein-Brillouin-Keller) prescribe k = l + ~ with the or-
bital angular momentum quantum number l = 0, 1,2· .. , n -1 [31]. One should not
be mislead to identify the ~ as originating from the spin, as in both quantal and
classical treatments in this section the intrinsic spin is not taken into account. If the
spin was taken into account the quantization rule wodd instead be k = j + ~ with
the total angular momentum quantum number j = ~,~, ... , n - ~ that combines
the electron spin and the orbital angular momentum. In that case k = 1,2, ... , n
is integer valued as in the old quantum theory, which is the combined effect of the
two contributions just described: semi-classical quantization and spin. Each shifts
the energy of the quantized states, the first conceptually theoretically, the latter
through the spin- orbit interaction, which is a relativistic effect. The combined re-
sult however was that it did not shift the (free field) energy levels of the hydrogen
atom, although there were twice as many (doubly degenerate) states due to the spin
- now labeled by (n, j) instead of (n, l) - but allowed for different selection rules
consistent with the presence (and intensity ratios) of certain fine structure lines in
the observed Balmer spectrum of hydrogen ( For instance the absence of the b'- line,
and presence of the c-line by G. Hansen in 1925) . For an introduction, references
370 L.MOORMAN

and the continuation of this facinating story on the hydrogen spectrum and of the
development of the new quantum theory see [33].
Returning to the spinless electron we see that with I/ = (l + ~)n we find an exact
correspondence between the mean force in the quantized classical Eq. (15) and the
purely quantal situation of Eq. (8).
For a circular-Bohr orbit (t = 0; a = b) we obtain In = I/ leading to the familiar
result that was used in eq. (6), see also (7) (G = 1 au; In = Io):

F K'1?!tr _1_ B2}r ~ (16)

< >c - 113 -
/ 0
Ii'
n

The mean Coulomb force has therefore, classically, a pole for orbits that are a
line through the nucleus (I, = 0; to = 1), (where the "~" due to the semi-classical
quantization rule does not allow for a corresponding quantal state). Averaging over
all classical Kepler orbits for a fixed energy means averaging over a microcanonical
distribution as in [32] (here a uniform uniform distribution of 1'2) resulting in:

<< F >c > Mitroc.dillr= 10r 1 (

1 - to
2)-1/2. 4 2
(fo) d( E )
2
= If (17)

This pure classical result corresponds indeed to the quantal statistical average cal-
culated in Eq. (9). Finally, the classical Kepler frequency can be derived from the
energy in Eq. (11) WK = 1;;3 corresponding to Eq. (10) and used in Eq. (5)
From Eq. (8) and Eq. (15) we may conclude that the mean Coulomb force
on the electron, averaged over its orbit, is smallest for the circular- Bohr orbit,
classically among all Kepler orbits of the same energy, and quantally among all
substates of the same energy. We saw also that the mean force from the nucleus is
independent of m ( or Im) in the absence of a field. The reader is reminded here that
these statements about relative strengths do not hold for the total binding energy
for those states, as the latter is the sum of potential and kinetic (t.g. centrifugal)
energy. The goal in this paragraph, however, was to provide us with an internal
atomic measure for electric field amplitudes and frequencies (or internal clock).

6. LOW SCALED FREQUENCY EXPERIMENTS

Several H atom Dffi, DIP and SA experiments have now covered the wide range
of scaled frequencies 0.05 ~ n~w ~ 2.8. The curves were recorded at various fixed
microwave frequencies between 7.58-36.02 GHz for a large number of no-values
(Table 2). From these curves we extracted the classically scaled amplitude threshold
n~F{10%). Because of finite signal-to-noise ratios, the actual 0+% threshold field
amplitude could not precisely be located. However because the majority of curves
increased quite abruptly, except for occasional structures near threshold, the 10%
threshold provided a faithful representation of the onset of microwave ionization.
MICROWAVE IONIZATION OF H ATOMS 371

~ 0.16.--.---.--~--.---r-~--~---r--'---~--.--'
.-
Q)
r;:
..d 0.12. CL ",r\
rIl
Q) I \
I-. I \
..c:
E-< I \
J 0.08 I
I

o "EXPT
O:l

I
~ 0.04
o
~
Q)
"EXPT
~ O.OO~~--~~--~~--~~--L-~~L--L~

~ 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Scaled Frequency {for 9.92 GHz)
Figure 5. Clauically scaled ionization threshold a.mplitudes versus clauically scaled
frequency for 10% (0 ) and 90% (0) experimental ionization data taken at 9.92
GHz compared to the results of the SD clauical theory (broken line).

In Sects. 6.1 and 6.2 we will refer only to experiments and classical theory with
n~w ~ 1.1, which spans the low frequency (II) and the semi- classical (III) regions.
In Sect. 6.3, at higher scaled frequencies, n~w..2:. 1, we will see a very different
behavior.

6.1. CLASSICAL IONIZATION TlffiESHOLDS

Fig . .I) shows 9.9 GHz scaled amplitude threshold data (n~w) for 10% (circles) and
90% (squares) ionization probability as a function of scaled frequency. Each point
in the graph is an average of individual thresholds extracted from many curves
measured with both DIP and SA methods. Each set of curves yields two data
points corresponding to the two threshold levels. Connecting the data points for
the same threshold levels in Fig. 5 produces two curves which in the average decrease
with increasing scaled frequency while also displaying many local structures. The
structures appear as a series of local maxima. The scaled field values at which
they occur are found to be near 1,1/2,1/:3,1/4,1/5,1/6. There is also a maximum
which is most prominent in the 90% curve just above 1/2 (0.56), and another
which is most prominent in the 10% curve just below 2/5. The figure compares
these experimental data with 3D classical Monte Carlo calculations that closely
model the experiments [4,6,28,37,38,29,39,40,41]. The model employs a classical
microcanonical3D-distribution of orbits that correspond to the uniform distribution
of quantal substates [32]. It also uses the experimental envelope function, and the
372 L.MOORMAN

cutoff value nco The results of this model follow the general trend in the curve
very well. Below n~w = 0.8, apart from occasional exceptions, there is agreement
to within a few percent. We emphasize the quality of this agreement by noting
that the scaled values are a linear (not logarithmic) and ab$olute (not relative)
function of the threshold field amplitude Po. Moreover, the classical calculation
reproduces the experimentally observed local maxima as well. Looking in even
greater detail, it appears that the classically computed thresholds agree more closely
with the measurements for scaled frequencies between the local maxima than for
those precisely at the maxima.
It has been shown that these local maxima can be interpreted in a very direct
way by a much simpler classical model in one dimension. The Poincare sections of
the phase space [38] for such a ID model exhibit resonant islands and the overlap
of adjacent resonant islands is known to accompany the onset of ionization. Over-
lapping resonant islands (or "islets") can be linked to the experimentally observed
local stabilities as can be seen from Sanders et al. [42] who have done numerical cal-
culations for the 10% thresholds and compared their results to experiments done in
our laboratory. They find the distinct peaks at IIp (p = 1,2,3,4,5) as in the exper-
iments (Fig. 5), but also find 2/3 to be locally more stable. Similarly a calculation
based on the half width of the IIp primary islands by Blumel et al. reaching the
2/(2p -1) resonances reproduced the measured thresholds to better than 30% [43]
and an interpolation based on higher order perturbation theory, leading to solving
an implicit equation, was reasonably successful (their figure 2) in calculating the
relative threshold variation around these resonances. They showed that this result
can also be obtained from quantal perturbation theory. A very different theory, the
ID adiabatic low frequency quantal theory of Richards et al. [44] which takes only
two adiabatic states into account found resonances near the frequencies t~:~)\; (p a
small integer). This result agrees also well with that of the ID classical theory.

6.2. FULL IONIZATION CURVES

While the ID-classical dynamics is capable of reproducing quite well the experimen-
tal lO%-threshold fields for n~w ~0.05-1.2, it cannot reproduce the amplitude de-
pendence of the actual (3D) direct-ionization or survived-atom experimental curves.
Classical 3D Monte Carlo calculations, in contrast, are able to do this over large
parts of this scaled frequency range. But there are two exceptions: (i) Agreement
with experiment can break down for scaled frequencies on or near the classical
resonances of low order rational values. (ii) The classical 3D calculations can not
reproduce the distinct structures found in more than a dozen ionization or quench
curves for 0.05.:sn~w .:s 0.3 (the low frequency Regime (II)) [45,46,47]' including non-
monotonic structures (not shown here), steps, or changes in slope. Both (i) and (ii)
indicate the importance of quantal efFects.
Quantal (lD) calculations [45) done for the same range of scaled frequencies as
MICROWAVE IONIZATION OF H ATOMS 373

e:: 1.20 (b)

"'d
t"'.)
'-....
:2
E-<
O.BO
"'d
t"'.)
I 0.40
E-<
~
~O.OO~------~~--~~----~+-------------~

O.OB

__ 0.06
~
o
.-t

r.: 0.04
'-"

-to
I:l
0.02

O.OOL--L--L--L--L--L--L--L--L--L--~~--~~
O.B 1.2 1.6 2.0 2.4 2.B
n~ w (scaled freq. for W/21T = 36.02 GHz)

Figure 6. Curves have been drawn to guide the eye. A: (bottom) scaled 10 %-threshold
amplitudes for (0 ) experimental survived atom (SA) and (x, 0) S-D classical
calculations vs. scaled frequency. B: (top) ~, fractional differences between 0 and
x from (A). For 'vertical line segments in (A) and (B) see ref. /6}.
374 L.MOORMAN

above reproduce the experimental 10% threshold field strengths quite well. Two sets
of quantal ID calculations have explained the structures mentioned in (ii): Blumel
and Smilansky [45] linked them to the effect of occasional clustering of many avoided
crossings between Floquet (=quasi-energy) curves. Some avoided crossings served
to mediate transport to the continuum via so-called window states. Richards et al.
[44] associated the experimental structures in (ii) with resonances between a small
number of states in an adiabatic basis. That both sets of quantal ID results are
able to explain the existence of structures in an experiment with 3D atoms, c.q.
statistical mixture of substates, indicates that the 3D dynamics is linked to that of
the much more restricted ID problem in important ways. It may be expected that
such a link only exists for dynamics in relatively weak fields and is a consequence
of the hydrogen atom being degenerate, i.e. only one frequency being present, and
will probably not work for other systems ([52]).
The local breakdown of classical atomic dynamics near the resonances in (i)
is somewhat more subtle. It would be wrong to expect classical behavior based
orJy on the presence of large values of no in the experiments. Islands and other
structures in the Poincare sections of the classical phase space are of finite size;
one might conjecture that the structures in the classical Poincare section having
an area less than Ii would play an inferior role in the quantal atomic dynamics.
Climbing the hierarchy of higher order resonances in classical perturbation theory
leads to structures in the classical Poincare sections of ever smaller volumes. Thus.
one expects that only the largest, lowest order classical phase space structures could
correspond to experimental features. This is consistent with the observed behavior
in Fig. 5.

7. HIGH SCALED FREQUENCY EXPERIMENTS

For experiments with scaled frequencies rising beyond 1 the SA (Fig 6A bottom)
and the DIE (Fig 7A bottom) methods were employed. We found that the measured
10%-threshold amplitudes (open circles) were no longer well reproduced by classical
3D calculations (x and diamond). However, the experimental data continued to
show the effect of local stability, near e.g., f, i,~, f,~,~, and, perhaps, others. Some
of this behavior can be captured by the classical calculations, e.g., for n~w near f, f,
and ~, but not alL The experimental dip near %in Fig 6A is a clear counter-example
to the classical local maxima there. The reverse is also true: the experiment exhibits
strong local stability near 1.30, where there is little or none classically. The reason
is that the island associated with the relatively high order ~ classical resonance is
too small to have an important effect on the classical dynamics. Even so, recent
calculations do show an important relation to the quantal dynamics there (see
Section 8).
For n~w > 2, classical dynamics breaks down completely by systematically
underestimating the field required for ionization. Our experimental observation of
MICROWAVB IONIZAnON OF H ATOMS 375

.Q

,....... 1.20
E-'
't:l 1.2
t':l

~ O.BO O.B
't:l
t':l
I
E-' 0.40 0.4
.......
~ E-t
~
--
~O.OO
--
0.0 r::l
0

O.OB
(a)
W O.06 ,
--
II
o
..... r
II
I
r..:..~ 0.04
'Ito
s::
0.02

O.OO~~~~--L-~-L~--L-~~~L-~~
O.B 1.2 1.6 2.0 2.4 2.B
n! w (scaled freq. for c.J/21T = 36.02 GHz)

Figure 7. Curves have been drawn to guide the eye. A: (bottom) lame al Fig.
6A except 0 are experimental 'direct ionization, electron' (DIE) results. B: (top)
b., fractional differences (with Icale on the right vertical axil) between 0 and x
from (A); 0, fractional differences beteween DIE (Fig. 7A) and SA (Fig. 6A) 10%
threshold fields. For vertical line segments in (A) and (B) lee ref. 16}.
376 L.MOORMAN

the enhanced stability of the quantal atom in the "high frequency" region was the
first confirmation of theoretical predictions of this effect initiated by Casati et al.
[5,48,49,50,51] This will be discussed in detail in Section 8. The triangles Figs.
6B (top) and 7B (top) denote the relative difference between the experimental and
theoretical thresholds (both averaged over the cross section of the beam). The
fluctuating behavior seen in the SA data (Fig. 6A) indicates that the real (quantal)
experiment shows very strong stabilisation which is highly significant with respect
to experimental uncertainties.
In Fig. 7B (top) the open squares compare the two experimental methods in
a relative way: (DIE - SA)/SA. We remind the reader that the only dift'erence
between DIE and SA is due to the cutoff. The SA (Fig 6A) has a relatively low
cutoff, nc = 86 - 92, whereas the DIE (Fig. 7A) has a relatively high cutoff,
nc = 160-190. We see from the squares that the relative difference of the thresholds
obtained with the two methods fluctuates between 0 and 1.20. We interpret the
small relative threshold differences occuring for ngw':;'0.9, and ngw = 1.3; 1.6; 2.06
as the lack of final state population with n-values between the two cutoffs. On
the other hand the appreciable differences occuring at n~w = 1.5, and n~w = 1.9
indicate an appreciable population of the final states between the two cutoffs. We
will see later in Section 8 that this provides important additional information on
the dynamics.

8. THEORIES

We have already seen some of the results of a comparisons of theoretical calculations

with our experiments. Before continuing our discussions of specific theories that
have been applied to our experimental situation further, some general remarks on
possible theoretical approaches are becoming. There seem to be three relevant
questions here: The first is whether it is necessary for theoretical models to quantize
the microwave field (as in QED). The second question is whether in the semi-classical
approximation in which only the atom is quantized, a perturbation series expansion
converges for all orders in some perturbation parameter. The third question is if one
may use a completely classical model in which the atomic motion is also considered
classically.
With regard to the first question, about quantizing the microwave field. consider
the photon density per unit volume )? [53]. If this is much larger than 1 the
description of physical phenomena based on classical electrodynamics is reliable, and
we may say that we are in the semi-classical region. (Except for specific coherent
photonic state effects as for instance squeezing of light, bunching and anti-bunching
of photon statistics etc. which we will not consider here). For our cavities typical
values for this entity ~~3 range from 10 9 to 1016 per volume of wavelength. At
these photon densities the difference in the effect of a creation operator resulting
in a factor v'N + 1 and anihilation operator resulting in VN even for 100 photons
MICROWAVB IONIZATION OF H ATOMS 377

absorbed or emitted out of the field, would not be detectable compared to the
spontaneous quantum fluctuations of the field [53].
The second question has been answered in part by a recent proof [54]. It states
that for N-body atomic systems the Rayleigh-Schrodinger perturbation expansion
converges to the resonances of the AC-LoSurdo-Stark effect, defined as eigenvalues
of the complex scaled Floquet Hamiltonian. It is further mentioned that this is valid
for a "weak" external AC electric field and very much in contrast to the DC field
case, where such a perturbation expansion is known to be asymptotic divergent, i. e.
the sum diverges for fixed perturbation parameter if evaluated to increasing orders.
The relevant question seems to be if this finite convergence radius is large enough
to be applicable to our problem.
The third question, the relevance of classical models, is supported by having
high no Rydberg states in our experiments and using the correspondence argument.
In the previous section we already saw the success of completely classical calcu-
lations for the experiments discussed in the low scaled frequency regions (II, III).
However even where there is classical behavior for certain experimental conditions
one would expect that by improving the experimental situation to higher resolu-
tion and increasing its selectivity of for instance the prepared and detected state,
quantal dynamics is necessary to describe the experiments. It is in this sense that
the correspondence argument is not a mathematical theorem. For the other regions
accurate comparisons have to be made to find an answer whether classical dynam-
ics is sufficient to describe the experimental results at the level of accuracy (and
selectivity) at which the experiments are done at this moment.
Having concluded that quantization of the microwave field is not appropriete for
the description of our experiments and that semiclassical perturbation theory and
fully classical methods might work under specific circumstances, we begin our dis-
cussion of theoretical calculations which were directly compared to our experimental
data.
The experimental data was previously compared with the classical Monte Carlo
calculations of Richards following the regularization methods of Ref (29). As dis-
cussed in section 6.1 and 6.2., which we will not repeat here in detail, the agreement
was found to be very good at low scaled frequencies (0.05..s n~w ..s 1.1).
Blumel and Smilansky [23) have also investigated this scaled frequency region
with a 1D-quantal model. They used a basis of the unperturbed Hamiltonian for
the bound and continuum part of the spectrum, and neglected only continuum-
continuum transitions. The resulting set of integro-differential equations were solved
and resulted in two basic mechanisms responsible for ionization. The picture is a
two step process consisting of excitation to states inside a "window" followed by
transfer of probability to the continuum as a decay process. In the first mechanism
the population probability is transferred via Floquet (quasi-energy) states to so
called window states, which are highly excited states that decay with appreciable
rates to the continuum. The second mechanism explained the subthreshold non-
378 L.MOORMAN

monotonic structures as the clustering of avoided crossings of pairs of quasi energies.

Each coherent superposition of the Floquet states overlaps with the unperturbed
initial states and final (window) states allowing for an efficient transfer of population
to the window states.
Richards [55] proposed an expansion in ID- adiabatic states in which ionization
was represented phenomenologically by the addition of decay terms to the differ-
ential equations. Two, fOllr and eight state approximations were discussed in ref.
[44]. They were quite successful in calculating positions where the 3D experiment
showed non-monotonic resonances. The theory also suggests a link between these
non-monotonic (quantal) structures on individual ionization curves and the local
stabilities observed for the 10% thresholds that were previously always explained as
indications of resonance islands in Poincare sections of the phase space of classical
models.
For the high scaled frequency several theories have been proposed. Predictions
were made [5] that quantized atom effects would suppress ionization and therefore
raise the thresholds compared to those given by classical atomic dynamics calcula-
tions. One such theory is the localization theory for the sinusoidally driven Kepler
problem. The basis for this theory lies in the numerical observation that wavefunc-
tions did not spread indefinitely in the action parameter, but instead they localize
around the initial state after some time scale, i. e. their distribution function de-
cays exponentially away from no. This is in contrast to the continual diffusion that
occurs in classical dynamics for high scaled frequencies.
Localization theory represents a dynamical analogue of what is known as Ander-
son Localization in the field of condensed matter physics. The analogy stems from a
formal equivalence with the tightbinding model equations [56,5,49,50,51,57,2] (orig-
inally done for the so-called "kicked rotator" [58]) which attempt to describe the
effects of extrinsic disorder on electronic wavefunctions in spatially periodic poten-
tials. Localization theories, however, are theories for average behavior and do not
encompass mechanisms which would produce the rich structures observed in our
experiments.
A numerical comparison with our data shows that for the SA experiments with
nc = 89 - 92, the analytical curve resulting from Localization theory only crudely
reproduces the average of the experimental data [59,2] (not shown here). Indeed,
the experimental data display many robust deviations of up to 50%. For the DIE
experiments with nc = 160, localization strongly overestimates the average 10%
thresholds.
Further improvements designed to produce simplified theories which precisely
model the structures observed in the experiment were made using other approximate
treatments of the quantal atomic dynamics via maps. In particular, the quantum
Kepler map has been developed in both ID- and 2D-versions [51,60]. The 1D-
version has been used to model our 36.02 GHz survived atom experiment, including
the cutoff nr and the envelope function. It reproduces surprisingly well most of
MICROWAVB IONIZATION OF H ATOMS 379

the structures in the n~w -dependence of the 10% threshold fields n~F (10%) for
3D atoms. However, as n~w is varied by miniscule amounts, e.g., .!ln~w = 10- 7,
significant fluctuations of the thresholds are calculated. These were interpreted as
mesoscopic effects such as in solid state physics for so-called universal conductance
fluctuations. Averaging the Quantum Kepler Map over the relative accuracy with
which the atom can determine the frequency (10- 3 ) suppresses the effect of the
microscopic oscillations of the calculated threshold amplitudes.
In strong contrast with the above is the explanation offered by Leopold and
Richards [61]. They conclude from a comparison of their own calculations with
their quasi-resonant compensated-energy (QRCE) basis calculations that the QKM
is a reasonable approximation provided that the field is insufficiently strong to excite
states far from the initial state. On the other hand, they find that the QKM treats
the very highly excited states incorrectly which leads to an overestimation of the
ionization probability and to wild fluctuations with very small variation in the field
frequency whenever one of the quasi-resonant states lies very close to the continuum.
They claim that these fluctuations are therefore spurious in the sense that they are
a consequence of the approximation and not due to a connection between classical
chaos and quantum mechanics in the hydrogen microwave problem.
Jensen et al. [62] have made large quantal1D numerical calculations on a CRAY
that included both the cutoff, nC) and an approximate form of the envelope function
used in the 3D quench experiment. Their result of (5% ionization thresholds) agrees
quite well with the experimental 10% threshold fields obtained with 3D atoms.
In another paper, Jensen et al. [63] have argued that for high scaled frequencies
n~w > 2 the main reason for the disagreement between classical and quantal atomic
dynamics is that a dynamical selection rule picks out a few quantal states which
dominate the motion. In Richards et al [64] it is shown, numerically, that these
states provide a good representation of the quantal dynamics. Instead of an infinite
number of quasi-resonant states between the initial level and the continuum, they
confine their interest to a finite number of quasi-resonant states which play an
important role in reaching the continuum. A crude estimate of this number is given
by the binding energy of the initial state divided by the photon energy, (2n~tl /w.
This may be re-expressed as no/(2n~w ), that is the number of photons required to
reach the continuum.
Instead of working in the unperturbed basis, < zln >, Leopold and Richards use
a time dependent basis of states that diagonalizes the compensated energy [65]. The
domination of these QRCE states leads to strong selection rules. These are stronger
than those of the unperturbed basis, because the coupling between the QRCE states
is stronger than that of the virtual resonant states in the unperturbed basis. This
makes the description on the QRCE basis very accurate even for a small number of
states. For H(no = 80) atoms in a 36.02 GHz field, the number of quasi-resonant
states is only about 15. Leopold and Richards found their basis-state truncation
method to work reasonably well when n~w ,(,2. They [66,67,64] also considered the
380 L.MOORMAN

manner in which the classical limit of the quantal behavior might be reached in the
limit of large initial principal quantum numbers no for fixed scaled field strength
and frequency. It was concluded that our experimental 10% threshold fields for
n~w ~ 2.8 and no-values up to 80 were not within the classical atomic dynamics
region.
The selection of theoretical models briefly mentioned above is not at all an ex-
haustive list of the work that has been done in this field. While we have been forced
to limit our discussion to only a few of these, we would like to mention that alter-
native approaches exist. One of these employs the language of classical dynamics
using Cantori, which are KAM like boundaries with Cantor set properties. Clas-
sical stochasiticity will be suppressed when the phase space area escaping through
classical cantori each period of the electric field is small compared to Planck's con-
stant [68,69]. Others use quantum atomic dynamical approaches such as Floquet
(quasi-energy) calculations to show various aspects of the microwave excitation and
ionization problem [70,71,72,73]. In the next section we will focus on still another
very different and quite recent explanation that may well playa key role in bringing
us new insights on the connection between classical and quantal atomic dynamics.

9. QUANTAL SCARS OF CLASSICAL ORBITS

In recent years new links between quantal and classical dynamics have been sug-
gested by comparing certain transformations from QM eigenstates with unstable
periodic orbits in the phase space of the classical problem. Such a quantal state is
said to exhibit a scar of the classical unstable periodic orbit [74]. 'Scars' have now
been identified in several theoretical problems. First recognized in the Bunimovitch
stadium [74,75] they have been identified in other time independent problems as
the hydrogen atom in a uniform magnetic field [76], the problem of two coupled
anharmonic oscillators [77] and others [78,79]. Recently also the time dependent
problems such as the quantum kicked rotor [80] and the hydrogen atom in strong
fields [81,82,83] have exhibited scars.
We will concentrate here on a recently proposed explanation for some of the
local stabilities in our threshold curves on the microwave ionization of hydrogen
[81,82]. Calculating coarse-grained Wigner (or Husimi) transformations of quasi-
energy states (QES) in the time dependent H-microwave problem, Jensen et a/. was
able to compare certain characteristics of a wavefunction w(t) with the Poincare
section of a classical phase space representation for the same control parameters.
Similar comparisons had been done by Stevens et al. [84] who used coherent states of
the driven surface-state electron on a basis of zero field states and showed the results
in the (x, p) phase space representation. They found that for larger frequencies and
a fixed field strength, the quantum phase space evolution is restricted or localized
in contrast with diffusive motion present in the classical evolution. They found that
tunneling is a better characterization of the quantal evolution at larger frequencies
MICROWA VB IONIZATION OF H ATOMS 381

than diffusion.
Jensen et al. found that the inhibition of (quantal) transport is due to excitation
of individual quasi-energy states whose Husimi transformation is highly localized
near the classical unstable periodic orbits.
Using the action angle representation of the Hamiltonian, the coherent states
were chosen to be "Poisson-like" on a basis of unperturbed states with a complex
periodic dependence on the classical angle:
00

II, () >= L[An(a )JOne-aljI!2ei2n8In > (18)

,,=0
with squeezing parameter a which determines relative width of the wavepacket in
B versus I as is consistent with the uncertainty principle. These can be seen as
the coarse grained "probe" function in the procedure. A state Pn(I, B) =< nil, e >
would be peaked at action I = n as a 'squeezed' Poisson distribution with width
J
AI = (an + 1) I a 2 and uniformly distributed in B. For a general coherent state
the Husimi distribution is now simply defined by a projection of the wave function
on the probe functions:
p(I, B) = I < I, Blw > 12 (19)
with
(20)

Using this prescription they performed numerical experiments. They noticed that
for certain scaled frequencies for the (relatively) slow switch on of the field only
one QES state is excited inside the interaction region, e.g. at scaled frequency
n~w = 1.30, 97.8% in a single QES. They investigated how these generalized Husimi
transformations of QES compare to a phase space Poincare section. They overlayed
the contour lines of the generalized Husimi distribution with the Poincare section
using certain initial conditions. As can be seen from Fig. 8 there are no "contour"
lines above n = 72 for this coherent state which means a strong localization at low
actions. They found that scar like objects are more than exponentially "localized."
Moreover the generalized Husimi distribution seems to be linked to the geomet-
ric structures which are maximum near n~w = 1.3 (no = 62 at 36.02 GHz). This
represents a quantal local stability when compared to the 3D classical calculation.
Another feature that the scarring phenomena explains is the insensitivity of the
10% threshold field strength to the cutoff value nc that was observed in the exper-
iments [6]. Most intriguing is the seeming contradiction that quantum dynamics
is stabilized by scarring a subset of orbits that are classically unstable. Moreover
for the classical dynamics this subset of classical orbits does not seem to be very
important (it is of measure "zero") since there is no sharp local minimum near
n~w = 1.3 in the classical calculations of Fig 6A (bottom) and 7A (bottom). It is
presently not understood why the scarred wavefunctions are concentrated on
382 L.MOORMAN

..
:.
.' ., .' .. ..
r I
h'.
' II
: :." " __' ..
o •• , .1..
• • 0'

.' .
70

I
60

5 0 ,_ _ __

0.0 0.5 1.0

Figure 8. Contour lines of the Husimi transformed QES that is predominantly

excited for no = 62 with n~w = 1.3 and n~F = 0.05. It is shown here overlayed
on a clauical Poincare section. The heart shapes, a resonance island, surround a
classical stable /,uonance for n~w = 1.

the classical unstable periodic orbits, but the observation seems to promise to playa
key role in the understanding how quantal and classical dynamics are linked together
for a system for which the classical dynamics is known to be exponentially unstable.

ACKNOWLEDG EMENTS

The review on this research would not have been possible without the dedi-
cated cooperation of many colleagues involved with the experiments in the labora-
tory in Stony Brook. In particular I would like to thank M. Bellerman, Dr. E.J.
Galvez, A.F. Haffmans, Prof. P.M. Koch, B.E. Sauer, and S. Yoakum. I would
also like to thank many theoretical colleagues for stimulating discussions, especially
MICROWAVE IONIZATION OF H ATOMS 383

D. Richards, J. Leopold, B. Sundaram, R Jensen, R Bliimel, D.L. Shepelyansky,

B. Meerson, and A. Rabinovitch. In addition I thank D. Richards and S. Yoakum
for their readiness to carefully read the manuscript, their comments and many text
improvements, and RV. Jensen for providing us with Fig. 8. This research was
supported by grants from the Atomic, Molecular, and Plasma Physics Program of
the US National Science Foundation.

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[1] "Irregular" here means chaotic in the sense of exponentially unstable for finite
times, as opposed to infinite times (see e.g., T. Tel, 'Transient chaos', to be
published in 'Directions in Chaos, Vol. 3', Bai-Lin Hao (ed.), World Scientific,
Singapore). This is numerically proven for the system discussed in this chapter
by calculating the Liapunov exponents in [3] for the Kepler map that is believed
to be a good approximation of the 1D-ordinary differential equation model of
a hydrogen atom in a microwave field.

[2] Moorman L., Koch P.M. (1991) 'Microwave ionization of Rydberg atoms', Bai-
Lin Hao, Da Hsuan Feng, Jian-Min Yuan (eds.), Directions in Chaos, Vol. 4,
World Scientific, Singapore, Ch 2, to appear.

[3] Haffmans, A.F., Moorman, L., Rabinovitch, A., and Koch, P.M., 'Initial con-
dition phase space stability pictures of two dimensional area preserving maps',
in preperation

[4] Leeuwen K.A.H. van, Oppen G. v., Renwick S., Bowlin J.B., Koch P.M., Jensen
RV., Rath 0., Richards D., and Leopold J.G. (1985) 'Microwave ionization of
hydrogen atoms: Experiment versus classical dynamics', Phys. Rev. Lett. 55,
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[5] Casati G., Chirikov B.v., Shepelyansky D.L. (1984) 'Quantum limitation for
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[6) Galvez E.J., Sauer B.E., Moorman L., Koch P.M.; and Richards D. (1988)
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[7] Bayfield J .E., Casati G., Guarneri I., Sokol D. W. (1989) 'Localization of classi-
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[8] Koch P.M. (1990) 'Microwave ionization of excited hydrogen atoms: What we
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[9] Richards, D., Leopold, J.G. (1990) 'Classical ghosts in quantal microwave ion-
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[10] Koch P.M., (1990) 'Microwave excitation and ionization of excited hydrogen
atoms', in S. Krasner (ed.) "Chaos" perspecitves on nonlinear science', AAAS,
Washington, p 75-97

[11] Water W. van de, Leeuwen KA.H. van, Yoakum S., Galvez E.J., Moorman
L., Bergeman T., Sauer B.E., and Koch P.M. (1989) 'Microwave multiphoton
excitation of helium Rydberg atoms: The analogy with atomic collisions' , Phys.
Rev. Lett. 63, 762-65

[12] Water W. van de, Leeuwen KA.H. van, Yoakum S., Galvez E.J., Moorman
L. Sauer B.E., and Koch P.M. (1989) 'Microwave multiphoton ionization and
excitation of helium Rydberg atoms', Phys. Rev. A 42, 572-91

[13] Berry M.V. (1989) 'Quantum Chaology, not Quantum Chaos', Phys. Script.
40, 335-336

[14] Gutzwiller, M.C. (1990) 'Chaos in classical and quantum mechanics' Springer
Verlag, Interdiscipl. Appl. Math., Vol 1, New York

[15] Moorman L., Galvez E.J., Sauer RE., Mortazawi-M A., Leeuwen KA.H. van,
Oppen G. v., and Koch P.M. (1989) 'Two-frequency microwave quenching of
highly excited hydrogen atoms'in Phys. Rev. Lett. 61, 771-74

[16] Moorman L., Galvez E.J., Sauer RE., Mortazawi-M A., Leeuwen KA.H. van,
Oppen G. v., and Koch P.M. (1989) 'Two-freqeuncy microwave quenching of
highly excited hydrogen atoms' in 'Atomic Spectra and Collisions in External
Fields 2', eds. K T. Taylor, M.H. Nayfeh, and C. W. Clark, Plenum Press, 343-
57

[17] Koch P.M., Moorman L., Sauer B.E., Galvez E.J., and Leeuwen K.A.H. van
(1989) 'Experiments in quantum chaos: Microwave ionization of highly excited
hydrogen atoms' Phys. Script. T26., 51-57

[18] Koch P.M. (1988) 'Microwave ioniztion of highly excited hydrogen atoms: A
driven quantal system in the classical chaotic regime' in H.B. Gilbody et al.
(eds.), Electronic and Atomic Collisions, North-Holland, Amsterdam, p 501-16
MICROWAVE IONIZATION OFHATOMS 385

[19] Koch, P.M., Mariani, D.R. (1981) 'Precise measurement of the static electric
field ionization rate for resolved hydrogen Stark substates ' in Phys. Rev. Lett.
1275-78

[20] Banks, D., Leopold, J.G., (1978) in 'Ionization of highly-excited atoms by

electric fields: (I) Classical theory of the critical electric field for hydrogenic
ions' J. Phys. B: Atom. Molec. Phys. 11 37-46; and in (1978) '(II) Classical
theory of the Stark effect' J. Phys. B: At. Molec. Phys. 11,2833-43

[21] Koch, P.M., (1983) 'Rydberg studies using fast beams' in Rydberg States of
Atoms and Molecules, R.F. Stebbings and F.B. Dunning (eds.), Cambridge
University Press, New York, p 473-512

[22] Kleppner, D., Littman, M.G, Zimmerman M.L. (1983) Rydberg atoms in
strong fields', in R.F. Stebbings and F.B. Dunning (eds.) Cambridge University
Press, New York, p 73-116

[23] Blumel R. and Smilansky U. (1987) 'Microwave ionization of highly excited

hydrogen atoms', Z. Phys. D: Atoms, Molec. and Clusters 6, 83-105

[24] Landau, L.D., and Lifshitz, E.M. (1976) 'Course of theoretical physics, Vol. 1:
Mechanics' 3Td edition, Pergamon Press, Oxford, Ch. 3

[25] Goldstein, H.(1977) 'Classical Mechanics', Addison Wesley Inc., Reading, 12th
edition, Ch. 3.6

[26] Halbach K. and Holsinger R.F. (1976) 'SUPERFISH, a computer program for
the evaluation of RF cavities with cylindrical symmetries', in Part. Accel. 7,
213-22

[27] Sauer B.E., Leeuwen K.A.H. van, Mortazawi-M A., and Koch P.M. (1991)
'Precise calibration of a microwave cavity with a nonideal waveguide system',
Rev. Scient. Instr. 62, 189-97

[28] Leopold J.G., and Percival I.C. (1978) 'Microwave ionization and excitation of
Rydberg atoms', Phys. Rev. Lett. 41,944-7

[29] Leopold J.G. and Richards D. (1985) , 'The effect of a. resonant electric field
on a one-dimensional classical hydrogen atom' J. Phys. B: Atom. Mol. Phys.
18, 3369-94

[30] Born M., 'The mechanics of the atom' (1924), Republished by Frederick Ungar
publishing co., New York (1960).

[31] The principal action variable is In == IT + Ie + I.p, the total angular momen-
tum is I, = Ie + I"" and the component of the angular momentum onto the
386 L.MOORMAN

polar axis is 1m = 14>' with 21rIi = f Pidr, for i = r, 0, and ¢ (and no sum
convention) [30,32]. The quantization conditions become In = nh, I, = kti,
and in a magnetic field 1m = mho In the old quantum theory n = 1,2,··· is
the principal quantum number and k is the subsidiary (or azimuthal) quan-
tum number [30]. This would be k = 1,2,' .. ,n, however modern quantization
methods (Einstein-Brillouin-Keller) require k = l + 1/2 with I = 0,1,2, ... ,n,
in which the 1/2 does not refer to the spin but to the Maslov (or Morse) index
in semi-classical quantization. This index counts the number of classical turn-
ing points a encountered by a closed trajectory in the classical phase space (for
a more general description in terms of caustics see [79]). At each turning point
a phase-loss of 1r /2 (equivalent to 1/4 th of a wave) has to be taken into account.
Continuity of the phase of the wave function then leads to I = (n + ~a )h. For
example the 1 dimensional harmonic oscillator encountering two turning points
per period obtains for the same reason (but named after Wentzel- Kramers-
Brillouin) the well known 'zero-point' energy, as given in E. = (v + ~)h for
II = 0,1, . '. For further reading and how this can be understood from condi-
tions to be satisfied under a coordinate transformation of a pathintegral from
Cartesian into spherical coordinates, requiring a new term"" lh2 in the classi-
cal Hamiltonian, which adds to the angular momentum part ILI2 = 1(1 + 1)h 2
giving (I + ~)2h2, see p 203 and 212 etc. of [14].

[32] Percival I.C., and Richards D. (1975) 'The theory of collisions between charged
particles and highly excited atoms', in 'Advances in atomic and molecular
physics', Vol 11, pl-82

[33] Series, G.W. (1988), 'The spectrum of the hydrogen atom: Advances', World
Scientific Publishing Co, (or the 1957 original version of part I, by Oxford
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[34] Woodgate, G.K. (1980) 'Elementary Atomic Structure', Clarendon Press, Ox-
ford, 2n d ed., p 22

[35] Vol 3 of [24] (1977) 'Quantum Mechanics' 3,d edition, Pergamon Press, Oxford,
p. 120

[36] Bethe, H.A., Salpeter, E.E. (1977) 'Quantum mechanics of one and two electron
atoms', Plenum/Rosetta, Oxford, p.17; p.58

[37] Meerson B.I., Oks E.A., and Sasorov P.V. (1982) 'A highly excited atom in
a field of intense resonant electromagnetic radiation: I Classical Motion', J.
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[38] Jensen RV. (1984) 'Stochastic ionization of surface-state electrons: Classical

theory', Phys. Rev. A30, 386-97
MICROWAVE IONIZATION OF H ATOMS 387

[39] Leopold J.G. and Richards D. (1986) 'The effect of a resonant electric field on
a classical hydrogen atom' J. Phys. B: Atom. Mol. Phys. 19, 1125-42

[40] Jensen, RV. (1987) 'Effects of classical resonances on the chaotic microwave
ionization of highly excited hydrogen atoms', Physica Scripta 35, 668

[41] Richards, D. (1990) 'The Coulomb potential and microwave ionization', In-
ternational conference on the physcis of electronic and atomic collisions, New
York 1989, AlP, 54-64

[42] Sanders M.M., Jensen RV., Koch P.M. and Leeuwen K.A.H. van (1987)
'Chaotic ionization of highly excited hydrogen atoms', Nucl. Phys. B (Proc.
suppl.) 2, 578-579

[43] Bliimel R. and Smilansky U. (1989) 'Ionization of excited hydrogen atoms by

microwave fields: a test case for quantum chaos', Physica Scripta 40, 386-93.

[44] Richards D., Leopold J.G., Koch P.M., Galvez E.J., Leeuwen K.A.H. van,
Moorman L., Sauer B.E., and Jensen RV. (1989) 'Structure in low frequency
microwave ionization of excited hydrogen atoms', J. Phys. B 22, 1307

[45] Bliimel Rand Smilansky U. (1987) 'Localization of Floquet states in the rf

excitation of Rydberg atoms', Phys. Rev. Lett. 58, 2531-4

[46] Blumel R, Goldberg J., and Smilansky U. (1988) 'Features of hte quasienergy
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and Clusters 9, 95 and Bliimel, R, Hillermeier, RC., and Smilansky, U. (1990)
Z. Phys. D 15, 267

[47] Koch P.M. (1982) 'Interactions of intense fields with microwave atoms', Journal
de Physique Colloque 43, C2-187

[48] Casati G., Chirikov BY, Shepelyansky D.L., and Guarneri I. (1987) 'Lo-
calization of diffusive excitation in the two- dimensional hydrogen atom in
a monochromatic field', Phys. Rev. Lett. 59, 2927

[49] Casati G., Chirikov B.V., Shepelyansky D.L., and Guarneri I. (1987) 'Relevance
of classical chaos in quantum mechanics: The hydrogen atom in a monochro-
matic field', Phys. Rep. 154, 77

[50] Casati G., I. Guarneri L, Shepelyansky D.L. (1987) 'Exponential photonic lo-
calization for the hydrogen atom in a monochromatic field', Phys. Rev. A 36,
3501

[51] Casati G., I Guarneri I., Shepelyansky D.L. (1988) 'Hydrogen atom in
monochromatic field: Chaos and dynamical photonic localization' I.E.E.E.:
J. Quantum Electron. 24, 1420
388 L.MOORMAN

[52] D. Richards (private communication).

[53J Sakurai, J.J. (1978) in 'Advanced quantum mechanics' in series in advanced

physics, Addison-Wesley, 7fh printing, p.35 and p.38

[54J Graffi, S., Grecchi, V., Silverstone, H.J. (1985) in 'Annales de l'institute de
Henry Poincare - Physique theorique', Vol 42, p. 215-234

[5.I)J Richards D., (1987) J. of Phys. B At. Mol. Phys. 20, 2171-92

[56J Casati, G., Guarneri, I., Shepelyansky, D.L. (1990) 'Classical chaos, quantum
localization and fluctuations: A unified view' Physica A 163, 205

[57] Grempel D.R., Prange R.E., and Fishman S. (1984) 'Quantum dynamics of a
nonintegral system', Phys. Rev. A 29, 1639-47

[58] Fishman S., Grempel D., Prange R.E. (1982) 'Chaos, Quantum recurrences
and Anderson localization' Phys. Rev. Lett. 49, 509-12

[59J Koch P.M., Moorman L., Sauer RE. (1990) 'Microwave ionization of excited
hydrogen atoms: experiments versus theories for high scaled frequencies' in
a special issue on Quantum Chaos of 'Comments on Atomi- and Molecular
Physics', Vo1.25, pp. 165-183

[60] Brivio, Casati G., Guarneri I., and Perotti L. (1988) 'Quantum suppression of
chaotic diffusion: theory and experiment' Physica 33D, 51-57

[61J Richards, D., Leopold, J.G. (1990), 'On the Quantum Kepler Map' J. Phys. B:
At. Mol. Opt. Phys. 23, 2911-2927

[62J Jensen R.V., Susskind S.M., Sanders M.M. (1989) 'Microwave ionization of
highly excited hydrogen atoms: A test of the correspondence principle', Phys.
Rev. Lett. 62, 1476-79

[63] Jensen, R.V., Leopold, J., Richards, D. (1988) 'High- frequency microwave
ionization of hydrogen atoms' J. Phys. B: At. Mol. Phys. 21, L527-31

[64J Richards D., Leopold J.G., and Jensen R.V. (1988) 'Classical and quantum
dynamics in high frequency fields', J. Phys. B: At. Mol. Phys. 22, 417-33

[6.I)J Leopold, J.G., Richards, D. (1989) 'Quasi-Resonances for high frequency per-
turbations' J. Phys. B 22, 1931
[66] Leopold J.G. and Richards D. (1988) 'A study of quantum dynamics in the
classically chaotic regime' J. Phys. B: At. Mol. Phys. 21, 2179-2204

[67] Leopold,J.G., and Richards, D. (1988) 'Quanta! localization and the uncer-
tainty principle', Phys. Rev. A 38, 2660-3
MICROWAVB IONIZATION OF H ATOMS 389

[68] Mackay, RS., and Meiss, J.D. (1988) 'Relation between quantum and classical
thresholds for multiphoton ionization of excited atoms', Phys. Rev. A 37,4702-
7

[69] Meiss, J.D., (1989) 'Comment on "Microwave ionization of H- atoms: break-

down of classical dynamics for high frequencies" by E.J. Galvez et al.' in Phys.
Rev. Lett. 62, 1576

[70] Breuer, H.P., Dietz, K, Holthaus, M. (1988) 'The role of avoided crossings
in the dynamics of strong laser field-matter interactions' Z.Phys. D: Atoms,
Molec. and Clusters 8, 349

[71] Breuer, H.P., Dietz, K, Holthaus, H. (1988) in Z. Phys. D 10, 12; and (1989)
in J. Phys. B 22, 3187

[72] Breuer, H.P., Holthaus, M.,'Adiabatic processes In the ionization of highly

excited hydrogen atoms' (3rd paper)

[73] Wang, K, Chu, S-I. (1989) 'Dynamics of multiphoton excitation and quantum
diffusion in Rydberg atoms', Phys. Rev. A 39, 1800-1808

[74] Heller E.J. (1984) 'Bound state eigenfunctions of classically chaotic Hamilto-
nian systems: Scars of periodic orbits', Phys. Rev. Lett. 53, 1515-8

[75] O'Conor P.W., Gehlen J.N., Heller E.J. (1987) 'Properties of random superpo-
sitions of plane waves', Phys. Rev. Lett. 58, 1296-9

[76] Wintgen, D., and Honig, A. (1989) 'Irregular wave functions of a hydrogen
atom in a uniform magnetic field', Phys. Rev. Lett. 63, 1467-70

[77] Waterland, RL., Jian-Min Yuan, Martens, C.C , Gillilan, E., and Reinhardt,
W.P. (1989) Phys. Rev. Lett. 61, 2733-6

[78] Feingold M., Littlejohn RG., Solina S.B., Pehling J.S. (1990) 'Scars in billiards:
The phase space approach', Phys. Lett. A 146, 199- 203

[79] Berry M. V. (1989) 'Quantum scars of classical closed orbits in phase space',
Proc. Roy. Soc. London, A 423, 219-231; and in (1982) "Chaotic behavior
of deterministic systems', G. Iooss, RH.G. Heileman, and R Stora (eds.),
Proceedings of the Les Bouches Summer Institute, North-Holland, Amsterdam,
p 172

[80] Radons, G., Prange, R.E. (1988) 'Wave function at the critical Kolmogorov-
Arnol'd-Moser surface', Phys. Rev. Lett. 61, 1691-4
390 L. MOORMAN

[81] Jensen, R.V., Sanders, M.M., Saraceno, M., Sundaram, B., (1989) 'Inhibition
of quantum transport due to "Scars" of unstable periodic orbits', Phys. Rev.
Lett. 63, 272-5 and pre print NSF-ITP-89-1281 (Yale).

[82] Jensen, R.V., Susskind, S.M., Sanders, M.M. (1991) submitted to Phys. Re-
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[83] Jensen, R.V., Sundaram, B., (1990) 'On the role of "Scars" in the suppression
of ionization in intense high-frequency fields', Phys. Rev. Lett. 65, 1964-7

[84] Stevens, M.J., and Sundaram, B. (1989) 'Quantal phase space analysis of the
driven surface-state electron', Phys. Rev. A 39, 286-277
 The Electron 1990 Workshop
List of Participants
BANDRAUK, A. D. Universite de Sherbrooke Canada
Sherbrooke, Quebec

BARUT, A. O. University of Colorado USA

Boulder, Colorado

BOUDET, R. Universite de Province France

Marseille Cedex 3

BRUNEL, F. National Research Council Canada

Laser and Plasma Physics
Ottawa, Ontario

BURNETT, N. H. National Research Council Canada

Laser and Plasma Physics
Ottawa, Ontario

CAPRI, A. Z. University of Alberta Canada

Edmonton, Alberta

CHIN, S. L. Universite Laval Canada

Quebec, Quebec

COOPERSTOCK, F. I. University of Victoria Canada

Victoria, British Columbia

CORKUM, P. National Research Council Canada

Laser and Plasma Physics
Ottawa, Ontario

FREEMAN, R. R. AT&T Bell Laboratories USA

Electronics Research Department
Holmdel, New Jersey

GALLAGHER, T. F. University of Virginia USA

Charlottesville, Virginia

GRANDY, Jr., W. T. University of Wyoming USA

Laramie, Wyoming

GULL, S. Cavendish Laboratory

Cambridge

HAWTON, M. Lakehead University Canada

Thunder Bay, Ontario
391
392 LIST OF PARTICIPANTS

HESTENES, D. Arizona State University USA

Tempe, Arizona

HOLLEBONE, B. R. Carleton University Canada

Ottawa, Ontario

JAYNES, E. T. Washington University USA

Saint Louis, Missouri

JUNG, Ch. Universitat Bremen Germany

Bremen

KERWIN, L. Canadian Space Agency Canada

Ottawa, Ontario

KRUGER, H. Universitat Kaiserslautern Germany

Kaiserslautern

L'HUILLIER, A. Centre d'Etudes Nuclaires de Saclay France

Service de Physique des Atomes et
des Surfaces
Gif-Sur-Yvette Cedex 3

MARMET, P. Herzberg Institute of Astrophysics Canada

Ottawa, Ontario

McEWAN, J. The University of Kent

Canterbury, Kent

MOORMAN, L. State University of New York USA

Stony Brook, New York

MORGNER, H. Universitat Witten/Herdecke Germany

Witten-Annen

VAN DYCK, Jr., R. S. University of Washington USA

Seattle, Washington

WALLBANK, B. St. Francis Xavier University Canada

Antigonish, Nova Scotia

WEINGARTSHOFER, A. St. Francis Xavier University Canada

Antigonish, Nova Scotia

WILLMANN, K. Buchenbach-Unteribental Germany

 INDEX

This is both a subject and name index, it follows the alphabetical order except for
the entries under the topics: electron theory and QED and self -field QED, for
clarity we follow the page sequence as presented in these articles.

A Born interpretation, 27, 117

above-threshold ionization (ATl), Boltzmann distribution (thermal)
35, 218, 297f, 299, 302 262, 263, 276
with microwaves, 315, 316 bound state problems, 127, 133, 134
ATl electrons, 314, 315, 316 Bragg diffraction law, 33, 34
and vibronic structure, 192, 20lff bremsstrahlung, 120
299 inverse, 297
above-threshold photodissociation, stimulated, 35, 301
209, 213 see free-free transitions
action formalism, 110, 124, 137, 166 Broglie, Louis de, 83, 91
167 corpuscule. 89
algebra frequency. 33
Clifford, 22, 49 Brownian motion
Dirac, 25 line shape theory, 276, 277, 279
radial, 59ff 281
spacetime (STA) , 25ff, see STA Brown, L.S., 276, 283, 288
anomalous magnetic moment, 34, 161 and Gabrielse, G., 257, 261, 263
see: free electron g-factor
see: self-field QED C
antiparticles, 83, 91, 132 Casimir effect, 120
atom cavity
Electron exchange in, 303 effects, 265, 267
metastable, 301 modes, 267, 283
Rydberg, 296, 311, 354 central fields
states and "zitterbewegung", 33,34 new solutions of Dirac equation,
singlet/triplet states, 305, 335 49f, 52ff, 83f, 93ff
atomic see: H-like bound states
classical and quantum see: anomalous Zeeman effect
dynamics, 353 see: Darwin solutions
quantal interferences, 353 see: magnetic dipole moment
avoided crossing, 314 see: Yvon Takabayasi, ~ - 0
laser-induced, 192, 206, 213, 214 chaos
quantum, 220, 298, 352
B in quantum dynamics, 236, 237
Barut, A.O., 2, 16, 31, 33, 97, 155 scattering, 2l9ff
158, 159, 160 chaotic
beta "angle", see Yvon-Takabayasi dynamics, 352
parameter saddle, 228
Bell, J., 4 chaotic scattering
Berry phase, 31 classical. 220f, 236, 298
Bethe, H.A. and Salpeter, E.E., 70, 73 quantum, 220, 237
128, 131 field-modified, 2l9ff
Birrel, N.D. and Davies, P.C.W., 183 and nonlinear dynamics, 236
bivector, 22, 59, 84 classical scaling, 368
unit bivector, 23, 24 quantal correspondence, 366, 367ff
Bjorken, J.D. and Drell, S.D. 186 Clifford algebra, 22, 49
Bohm, D., 7, 32 collision
Bohr, N., 3, 4 complex, 341

393
394 INDEX

half, 299, 343 stabilization, 192

and AT!, 300, 302 dynamics
ionizing, 388 classical, 353
theory, 197, 207, 209 nonlinear, 220, 236ff
Coulomb field quanta1, 353
long-range, 302, 303, 350 quantum, 237
in Dirac's theory, 68
Compton E
cross section, 13 Einstein, A., 4, 15, 18, 107, 108, 171
effect, 120 172
scattering, 298 A-coefficient, 155, 159
laser-stimulated, 296, 300 -Rosen program "building elementary
current particles", 33, 171
Dirac, 15, 27, 38, 95, 122ff, 128 electron, 1, 6ff, 30, 31ff, 83, 105
number of electrons, 122 171, 183, 191, 239
probability, 124 detectors, 19, 296
Schr6dinger, l22ff diffraction, 8, 31, 32, 33
transitions, 78, 96 extended in space, 31
see: self-field QED fragments of (leptoquarks), 19
complex number z - UV, 23 free, 1, 18, 296, 311, 316
meaning of ~ 26 g-factor, 239, 240
creation/annihilation operators, 150 lie anomaly, 240, 282, 283, 285
history of, 105ff
D interference, 8f, 14, 31, 32
Darwin solutions,(H-atom) 27, 49, 71 magnetic moment, 32, 177
72 (Table), 77, 79, 83, 93ff, 96 mass, 33, 34, 88, 106, 115
97 model, 6, 31, 38, 46, 110, 174, 175
Dirac point particle, 30, 3lff, 109
algebra, 25 properties, 110ff
action, 151 radiative problem, 105
current, 15, 27, 38, 75, 96, l22ff self-energy, 105, 106, 107, 157
166 self-field, 124
equation, 6, 26, 37, 88-89, 91 self-interaction, 15, 32, 33, 150
132 size, 32, 171, 174
for central field, 49ff single, 14, 124, 239
geometrical interpretation, 24 spin, 2, 22, 32, 106, 109ff, 112
42, 84 177
and STA, 38ff, 42 structure, 107ff
quaternionic solutions, 91ff trajecteries, 32
spinor, 39 wave-packet, 18, 19
streamlines, 28, 30 wave-particle duality, 7, 115, 117
theory, 21, 29, 31, 33, 42, 68, 73 wave function, 1, 7, 22, 31
172, 239 electron theory and QED
geometry of, 22, 26ff, 89f history of, 105f
ZBW interpretation, 29 history of selfenergy, 106f
wave function, 26, 28, 30 structure and self-energy, 107f
Dirac-Maxwell theory, 171 the point electron: Lorentz-Dirac
divergencies equation, 108f
IR and UV, 127 classical relativistic spinning,
dressed 109f
electron, 192, 215 helix, natural motion, 109, 111
molecule, 191, 196ff, 20lff properties of the particle and
molecular ion, 192, 202, 211ff solutions, 110-114
vibronic structure, 191
INDEX 395
modelling mass and wave-particles G
duality, 115-117 Gabriels, G., 246, 266
electron in an external field, l83ff geometric product, 22
"quiver" and "drift" motion, 298 geonium, 239, 242, 253, 256
301, 302, 315, 318, 320 g-factor, See: free electron
electron at potential (EM) steps, 37 Gibbs, J. W., 24
40ff Goudsmit, S. and Uhlenbeck, G., 2
electromagnetic field (EM) gravitation theory, 171, 175, 176
interacting with leptons, 149 Gurtler, R., 73, 84
electronic exchange
in electron scattering, 305, 335 H
in Penning reaction, 344 Hamilton, W.R., 24
elementary particles harmonic generation
field theory of, 171, 172 ionization limited, 323
model of, 84, 175 multiple, 297, 321, 330
excitation single atom response, 326
simultaneous electron-photon, 298 many atoms response, 326
304, 333ff, 336ff power laws, 324
simultaneous ionization, 318, 319 polarizability of atom, 324
emission single electron, 301
spontaneous, 3, 33, 34, 97, 120 Hawking-Penrose, 176
126, 127, 134, 135, 140, 150, Heisenberg, W., 1, 6, 11
154, 157 helium, 2
stimulated, 299 Hestenes, D., 5, 8, 37,49, 80, 83, 92
EPR paradox, 4 hydrogen atom, 2, 49, 141-143
magnetic dipole moment, 49, 75ff
F -like bound states, 65ff
Fermi, E., 106 72 (Table)
Feynman, R., 37, 41 microwave ionizaton, 353
field hydrogen
auxiliary, 184, 186 molecule, 191, 299
driven process, 311 new bound states, 191
emission point (FEP) , 246, 248 molecular ion, 202, 211
free asymptotic, 188 hyperfine splitting, 141
ionization, 314, 315
self -, 150
theory of elementary particles inner product, 22
171-172 interference
history of, l72ff electron, 8, 31, 32
quantization, 2, 13 quantized-atom, 354
quasistatic, 313 ionization
fractal structure, 219, 221 diffusive, 354
free-free (FF) transitions, 219, 297 in microwave fields, 297, 3llff
299, 301, 302, 303, 306 353ff, 355ff
in Coulomb field, 303, 350 theoretical models, 376ff, 380
and scattering chaos, 2l9ff, 298 multiphoton (MPI) see MPI
frequency Penning, 297, 34lff
low - approximation, 233ff, 303 process, 355
see "soft photon" ion
positive, 184 molecular, 202, 213
Fulling, S.A., 183 negative, 296, 299
negative "temporary", 297, 299, 300
303
396 INDEX

J Maxwell, J.C., 24
Jaynes, E.T., 22, 30, 31, 38, 150, 155 Maxwell-Dirac theory, 16, 149, 160, 168
160 magnetic bottle, 243
Josephson microwave fields
ac effect, 288 ionization, 297, 298, 311, 353
linearly and circularly polarized,
K 297, 311ff
Kapitza-Dirac effect, 317 Minkowski, model of spacetime 22, 83
Kepler molecule
frequency, 355, 365 dressed, 201
orbit, 355, 367 stabilization, 192
Kibble, T.W.B., 300 motional sideband cooling (SBE), 260
Kinoshita, T., 287 monoelectron oscillator, 240
Klein-Gordon Equation, 9, 31, 41, 116 multiphoton
173 ionization (MPI), 21, 25, 295, 297
Klein paradox, 28, 38, 41ff, 42, 50 298
Kroll and Watson theory, 299, 303, 305 electron simultaneous excitation,
334, 337 304ff, 333ff
Kruger, H., 27, 34, 83, 91, 93, 94, 96 transitions, 192, 204
97 multivector, 25
muon, 32, 105, 106, 171, 172
L muonium, 141, 142, 143, 144
Lamb, W., 7, 31
Lamb shift, 34, 97, 120, 126, 127, 134 N
135, 140, 150, 154, 157, 168, 204f Negative energy states
laser interpretation, 132ff
atomic physics, 21, 295, 297 neoclassical theory, 19, 155, 159
assisted processes, 335 neutrino, 62
electron interaction, 21, 295ff Newtonian celestial mechanics, 3
induced avoided crossings, 206 nonlinear dynamics, 220
208, 214
induced free-free transitions, 297 P
299, 306 pair creation/annihilation, 119, 120
induced resonances, 206, 207, 212 160
214 parity, 50, 59
modified Penning ionization, 298 and multivaluedness, 62
246-248 particle-antiparticle, 2, 34
stimulated processes, 296, 300, 301 particle,
Labonte, G., 184, 186 classical models, 46
lepton, 178 and field, 171
modelling of, 174ff Pauli principle, 34
spin and magnetic moment, 177f Penning
lifetime electrons, 297, 306, 341ff
resonances, 297 angular distribution, 345
positronium, 141 energy spectrum, 349
decay rates, 127 ionization, 298, 306, 311, 341ff
Lorentz, H. A., 106 laser-modified, 341ff, 346ff
Lorentz-Dirac equation, 37, 43, 45, 47 trap, 240, 241, 242, 244, 250
113, 121 compensated, 245, 246, 280
orthogonal compensated, 280
M phase matching, 321
magnetic dipole moment, 49, 73ff, 78 factor, 328
79 function, 330
anomalous, Zeeman effect, 50, 73ff phenomenology, 1, 2, 21
INDEX 397

photoelectric effect, 21 quarks, 171, 17B

photoelectron spectroscopy lepto-, 19
nonlinear, 191, 296, 301, 319 quantal energy
photon, llff, 88 interpretation, 85
absorption, 296, 311 quantized radiation field, 141, 149
electron interaction, 304, 338 quaternions, 24
microwave, 311, 312, 319, 353, 355
momentum, 299, 302, 306 R
"soft", low frequency limit, 233 Rabi-Landau levels, 255, 256, 283
303 Rabi oscillations, 192
spin, 306 radiation
virtual, 204, 287 theory semiclassical, 149
and QED, 11 damping, 247, 263, 266
Planck's raction, 37, 43, 47
constant, 22, 85, 87, 116 radiative processes, 21, 34ff, l19ff
radius, 177 120 (Table)
scale, 176 in an external field, 124ff
ponderomotive effects 296, 300, 315 two-body system, 137ff
316, 317 two-fermion system, 140
potential step (EM), 37, 43f see: Self-field QED
Dirac electron at, 40 relativistic mass shift, 257, 263
Petermann, A., 287 relativity
Poincare, H., 20, 107, 172 principle, 3, 107
section, 224ff general, 171
polarization microwave rf trapping, 258, 27B
circularly and linearly, 311 Rich, A., 272
and rotating frame, 313 resonances
positron, 18, 28, 83, 89, 90, 168, 169 "temporary" or Feshbach, 297, 300
245, 247, 248, 249, 272 rotation
positronium, 120, l4lff frame and polarization, 313
para-, 143 duality, in Dirac theory, 83, 85ff
pseudoscalar, 24 100
and ZBW, 86ff
Q Rydberg
quantum electrodynamics (QED), 2, 11 states, 202, 296, 311, 354
13, IS, 141, 149, 240, 2B5ff electrons, 202, 296
bound state tests of, 142
explicit nonlinearity, 149 S
in cavities, 120 scattering
relativistic two-body, 127 chaos, 219ff, 230
history of, 105ff Compton, 296, 300
see: self-field QED electron, 297
quantum optics, 21 formalism, 197, 207
quantum field theory (QFT) , B7, 149 of light by free electrons, I, 8
and semiclassical theories, 149 11ff, 14
l51ff Thomson, 296, 300
quantum mechanics, 2, 3, II, 20, 31 trajectories, 219, 221, 230
Copenhagen interpretation, 3 scars, 353
quantum quanta 1 of classical orbits, 380ff
chaos, 353 shift detector (CSGE), 246, 256, 257
chaology, 354 sideband excitation (SBE), 248, 249
effects, 20 258, 259, 260
hall effect, 288
scattering chaos, 237
398 INDEX

·simp1eman's theory· Dirac, 39

free e1etron phenomena, 295, 302 central field, 50
306, 318 factorization, 51
self-field QED multiva1ued, 49, 50
radiative processes radial equation, 49, 50
overview of, 119 soliton, 33
Table I, 120 quantum, 171, 172, 178ff
development of, 120ff Sommerfeld, A.,
and classical electrodynamics, 120 fine-structure formula, 69, 102
166 Sommerfield, C., 287
Lorentz-Dirac equation, 121 spherical harmonics and spin, 53, 55
current, most importnat quantity, 64
122, 166 and monogenics, 58, 64
classical models of the electron, Stern-Gerlach
122 experiment, 3
Schrodenger and Dirac currents QED, effect, continuous (CSGE), 240
122ff Schrodinger, E., 1, 4, 7, 18, 21, 166
radiative processes in external Schwartz-Hora effect, 7, 35
(Coulomb) field, 124ff Schwinger, J., 287
one single equatin, 124
the action W, 124 T
energy shifts, 126 tau, 171
anama10us magnetic moment (g-2) or Thomson
lie and a., 127 cross section, 14
decay rates in Hand muonium, 127 scatterin, 296
QEM and relativistic two-body stimulated, 219, 230
system, 127ff trajectories
negative energy states, 132ff irregular, 354
calculation of (g-2), 134ff in phase space, 221
radiative processes for two-body scattering, 219, 230
systems, 137ff transition current, 78
two-fermion system, 140f£
bound state tests of QED, U
Table III, 142 Unruh effect, 120
positronium, hydrogen, muonium, uncertainty principle, 11
141-144
list of references, 147-148 V
mathematical procedures of, 164ff Vacuum, 183, 184ff
self f1utuations, 4, 150
-energy, 125, 156, 157 polarization, 13, 120, 126, 127
history of, 106ff 134
-field, 150 135, 140, 154, 156, 157, 168
-interaction, 15, 33, 49, 71, 150 van der Waals forces, 32
and Dirac theory, 33 variable bottle, 255, 265
semiclassical theory, 149, 151ff, 155ff
single electron, 7, 14, 239 W
problem, 34, 124, 239 wave packet, 6, 7, 8, 12, 13, 14, 15
singularities, 172 18f
spacetime algebra (STA), 5£, 22ff, 30 binding energy, 17, 19
37, 42, 47, 84 frequency, 19
structure of, 24 size and shape, 14
Solvay Congress, 4 Wigner-Weisskopf approximation, 154
spin angular momentum, 27 Wey1, H., 55
spinor
INDEX 399

Y
Yvon-Takabayasi parameter "~", 28, 39
45, 54, 60ff, 80, 83ff
~ - 0 solution, 49, 54, 60ff
"mysterious", 83, 97
physical interpretation, 28

Z
Zeeman effect
anomalous, 50, 73ff, 76
zitterbewegung (ZBW) , 1, 5, l4ff, 21
33, 35, 37, 46, 47, 51, 86, 113
114, 122
forces, 1, l6ff
frequency, 9, 15, 28, 33, 34
field, 33
interpretation of Dirac theory, 29
30, 31
radius, 34
resonances, consequences of, 33-35
resonant atomic state, 33
resonant momentum transfer, 33
electron diffraction, 33-34
Pauli principle, 34
radiative processes, 21, 34, 35
physical picture of, 19
 Fundamental Theories of Physics
Series Editor: Alwyn van der Merwe, University of Denver, USA

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