I'HVSI CAI REVIEW VOLUME 162, NUMBER 5 2& OCTOBER 1967

Quantum Theory of Gravity. II. The Manifestly Covariant Theory*

BRVCE S. DEWITT
Institute for Advanced Study, Princeton, New Jersey
Cwd

Department of Physics, University of North Carolina, Chape/ Hill, North Caroiinat

(Received 25 July 1966; revised manuscript received 9 January 1967)

Contrary to the situation which holds for the canonical theory described in the first paper of this series,
there exists at present no tractable pure operator language on which to base a manifestly covariant quantum
theory of gravity. One must construct the theory by analogy with conventional 5-matrix theory, using
the c-number language of Feynman amplitudes when nothing else is available. The present paper undertakes
this construction. It begins at an elementary level with a treatment of the propagation of small disturbances
on a classical background. The classical background plays a fundamental role throughout, both as a technical
instrument for probing the vacuum (i.e., analyzing virtual processes) and as an arbitrary fiducial point for
the quantum fluctuations. The problem of the quantized light cone is discussed in a preliminary way, and
the formal structure of the invariance group is displayed. A condensed notation is adopted which permits
the Yang-Mills field to be studied simultaneously with the gravitational field. Generally covariant Green s
functions are introduced through the imposition of covariant supplementary conditions on small dis-
turbances. The transition from the classical to the quantum theory is made via the Poisson bracket of
Peierls. Commutation relations for the asymptotic fields are obtained and used to define the incoming
and outgoing states. Because of the non-Abelian character of the coordinate transformation group, the
separation of propagated disturbances into physical and nonphysical components requires much greater
care than in electrodynamics. With the aid of a canonical form for the commutator function, two distinct
Feynman propagators relative to an arbitrary background are defined. One of these is manifestly co-
variant, but propagates nonphysical as well as physical quanta; the other propagates physical quanta only,
but lacks manifest covariance. The latter is used to define external-line wave functions and non-radiatively-
corrected amplitudes for scattering, pair production, and pair annihilation by the background field. The
group invariance of these amplitudes is proved. A fully covariant generalization of the complete S matrix
is next proposed, and Feynman's tree theorens on the group invariance of non-radiatively-corrected n-particle
amplitudes is derived. The big problem of radiative corrections is then confronted. The resolution of this
problem is carried out in steps. The single-loop contribution to the vacuum-to-vacuum amplitude is first
computed with the aid of the formal theory of continuous determinants. This contribution is then func-
tionally diRerentiated to obtain the lowest-order radiative corrections to the n-quantum amplitudes.
These amplitudes split automatically into Feynman baskets, i.e. , sums over tree amplitudes (bare scattering
amplitudes) in which all external lines are on the mass shell. This guarantees their group invariance. The
invariance can be made partially manifest by converting from the noncovariant Feynman propagator to
the covariant one, and this leads to the formal appearance of fictitious quanta which compensate the
nonphysical modes carried by the covariant propagator. Although avoidable in principle, these quanta
necessarily appear whenever manifestly covariant expressions are employed, e.g. , in renormalization theory.
The fictitious quanta, however, appear only in closed loops and are coupled to real quanta through vertices
which vanish when the invariance group is Abelian. The vertices are nonsymmetric and always occur with
a uniform orientation around any fictitious quantum loop. The problem of splitting radiative corrections
into Feynman baskets becomes more difhcult in higher orders, when overlapping loops occur. This problem
is approached with the aid of the Feynman functional integral. It is shown that the "measure" or "volume
element" for the functional integration plays a fundamental role in the decomposition into Feynman
baskets and in guaranteeing the invariance of radiative corrections under arbitrary changes in the choice
of basic field variables. The "measure" has two effects. Firstly, it removes from all closed loops the non-
causal chains of cyclically connected advanced (or retarded) Green's functions, thereby breaking them
open and ensuring that at least one segment of every loop is on the mass shell. Secondly it adds certain non-
local corrections to the operator field equations, which vanish in the classical limit 5-+ 0. The question
arises why these removals and corrections are always neglected in conventional field theory without apparent
harm. It is argued that the usual procedures of renormalization theory automatically take care of them.
In practice the criteria of locality and unitarity are replaced by analyticity statements and Cutkosky rules.
It is virtually certain that the "measure" may be similarly ignored (set equal to unity) in gravity theory,
and that attention may therefore be confined to primary diagrams, i.e., diagrams which contain Feynman
propagators only, with no noncausal chains removed. A general algorithm is given for obtaining the
primary diagrams of arbitrarily high order, including all fictitious quantum loops, and the group invariance
of the amplitudes thereby defined is proved. Essential to all these derivations is the use of a background
fie1d satisfying the classical "free" Geld equations. It is never necessary to employ external sources, and
hence the well-known difhculties arising with sources in a non-Abelian context are avoided.

1. INTRODUCTION Geld. Attention was focused on some of the bizarre

' 'N the Grst paper of this series' an attempt was made
features of the resulting formalism which arise in the
case of finite worlds, and which are of possible cos-
~ ~to show what happens when canonical Hamiltonian
mological and even metaphysicaL signidcance. Such
quantization methods are applied to the gravitational
*This research was supported in part by the Air Force Ofhce t Permanent address.
of Scientific Research under Grant AFOSR-153-64, and in part by s3. S. DeWitt, Phys. Rev. 160, 1113 (1967). This paper will
the National Science Foundation under Grant GP7437. be referred to as I.
162 1195
 8 RYCE S. D l.-. WI TT I62

prosaic questions as the scattering, production, absorp- This, however, is not the whole story, for the general
tion, and decay of individual quanta were left un- coordinate transformation group still has, even as a
touched. The main reason for this was that the canonical gauge group, profound physical implications. Some of
theory does not lend itself easily to the study of these these we have already encountered in I, and some we
questions when physical conditions are such that the shall encounter in the present paper. Others will appear
effects of vacuum processes must be taken into account. in the final paper of this series, which is to be devoted
A manifestly covariant formalism is needed instead. to applications of the covariant theory. If it were not
It is the task of the present paper to provide such a for these implications there would be little interest in
formalism. pushing our investigations further, for there is no
YVe must begin by making clear precisely what is likelihood that such "prosaic" processes as graviton-
meant by "manifest covariance. " In conventional graviton scattering or curvature induced vacuum
5-matrix theory (whether based on a conventional polarization will ever be experimentally observed. 4 The
Geld theory or not) "manifest covariance" means real reason for studying the quantum theory of gravity
"manifest Lorentz covariance. " In the context of a is that by uniting quantum theory and general relativity
theory of gravity the question arises whether it should one may discover, at no cost in the way of new axioms
mean more than this, since the classical theory from of physics, some previously unknown consequences of
which one starts has "manifest general covariance.
" general coordinate invariance, which suggest new in-
Here one must be careful. There is an important teresting things that can be done with quantum Geld
difference between general covariance and ordinary theory as a whole.
Lorentz covariance, and neither one implies the other. Our problem will be to develop a formalism which
Lorentz covariance is the expression of a geometrical makes manifest the extent to which general covariance
symmetry possessed by a system. In gravity theory permeates the theory. This will be accomplished by
it has relevance at most to the asymptotic state of introducing, instead of a Rat background, an adjust-
the field. As has been emphasized by Fock, ' the word able c-number background metric. Use of such a
"relativity" in the name "general relativity" has con- metric has the following fundamental technical advan-
notations of symmetry which are misleading. Far from tages: (1) It facilitates the introduction of particle
being more relativistic than special relativity, general propagators which are generally covariant rather than
relativity is in fact less relativistic. For as soon as space- merely Lorentz-covariant. (2) It reduces the study of
time acquires bumps (i.e. , curvature) it becomes radiative corrections to the study of the vacuum. (3) It
absolute in the sense that one may be able to specify makes possible the generally covariant isolation of
position or velocity with respect to these bumps, pro- divergences, which is essential to any renormalization
vided they are sufficiently pronounced and distin- program. (4) It renders theorems analogous to the
guishable from one another. Only when the bumps Ward identity almost trivial. (5) It makes possible,
coalesce into regions of uniform curvature does space- in principle, the extension of the theory of radiative
time regain its relativistic properties. It never becomes corrections to worlds for which space-time is not
more relativistic than Oat space-time, which is char- asymptotically Oat and which may even be closed
acterized by the 10-parameter Poincare group. and Gnite. These advantages are typical of what we
The technical method of distinguishing between the shall mean by the phrase "manifest covariance. Use "
Poincare group and the general coordinate transforma- of the phrase, however, is not to be understood as
tion group is to confine the operations of the latter implying that the simple trick of introducing a variable
group to a finite (but arbitrary) region of space-time. background metric makes everything obvious. The
The asymptotic coordinates are then left undisturbed generally covariant propagators will not be unique
by general coordinate transformations, and only the but will be choosable in various ways, analogous to
operations of the Poincare group (if that is indeed the the gauge choices in quantum electrodynamics, and
asymptotic symmetry group of the problem) are we shall have to undertake a separate investigation,
allowed to change them. The general coordinate just as in quantum electrodynamics, to verify that
transformation group thus becomes a gauge group the choice is irrelevant. This investigation turns out
which, although historically an offspring of the Poin- to be much more complicated than in the case of
care group and the equivalence principle, plays techni- quantum electrodynamics.
cally the rather obscure role of providing the analytic Of the Gve advantages listed above as stemming
means by which the Einstein equations can be ob- from the use of a variable background metric only
tained from a variational principle and their essential the Grst two will appear in the present paper. The third
locality displayed. '
argued /see S. Weinberg, Phys. Rev. 138, 8988 (1965)g that the
' V. Fock, The Theory of Space-Time and Gravitation (Pergam- general coordinate transformation group is simply a consequence
mon Press, New York, 1959). of the zero rest mass of the gravitational field and its long-range
' The content of the Einstein equations can be expressed in an character.
intrinsic coordinate-independent form only at the cost of introduc- 4 Although one might hope for some very indirect cosmological

ing nonlocal structures. (See, for example, Ref. 32). It can be evidence for such processes.
 QUANTUM THEORY OF GRA . V ITY. II

and fourth will be demonstrated in the following paper The language of graphs and the S matrix is much more
of this series, while the fifth remains a program for direct.
the future. It is not out of place here, however, to The latter language, embracing as it does many dif-
speculate brieQy on this ultimate program. As long ferent particle theories at once, is also much less
as the conventional S matrix is our chief concern it is dependent on the detailed Lagrangian structure of the
appropriate to choose a background metric which is field theory on which it is based. It assumes that virtual
asymptotically Oat. We shall see that Lorentz invari- processes may be described by an infinite set of basic
ance of the S matrix then follows almost trivially from diagrams, the combinatorial properties of which are the
the formalism, in the limit in which the background same for all field theories. In working out the details
metric becomes everywhere Minkowskian. Now it is of how this language is to be extended to the non-
obvious that scattering processes are also possible in Abelian case, we have attempted to develop it within
an infinite world which is not asymptotically Bat. In as broad a framework as possible. Every theorem in
such a world it should be possible to construct a this paper will therefore apply not only to the gravita-
generalized S matrix in which the conventional plane- tional field but also to the Yang-Mills field'which,
wave momentum eigenfunctions are replaced by wave like the gravitational field, possesses a non-Abelian
functions appropriate to the altered asymptotic invariance group. '
Section 2 begins with the introduction of a notation
geometry. The asymptotic geometry itself would be
which is sufficiently general to embrace all boson field
fixed by choosing the background metric appropriately.
theories and at the same time condensed enough to
In a closed wor1d no rigorous S matrix exists. The reduce the highly complex analysis of subsequent sec-
continuum of scattering states is replaced by a regime
tions to Inanageable proportions. A table is included
of discrete quantization, and, as we have seen in I,
to facilitate comparison of the condensed notation with
the wave function of the universe may even be unique.
the detailed forms which the various symbols take in
It may be conjectured that the forma1ism most ap- the case of the Yang-Mills and gravitational fields.
propriate to this case is obtained by choosing the back- The notation is particularly useful in dealing with the
ground metric to be rot a c number but rather an
second functional derivative of the action, which plays
operator depending on a small number (e.g. , owe) of
the role of the differential operator governing the prop-
quantum variables similar to the operator E represent-
agation of infinitesimal disturbances on an arbitrary
ing the radius of the Friedmann universe studied in I.
background field. It is also useful in dealing with the
These variables would be quantized by the canonical
higher functional derivatives, which are the bare vertex
method, while the full q-number metric would continue
functions of the theory. The problem of the quantized
tb be treated by manifestly covariant methods. (Con-
light cone is discussed in a preliminary way in Sec. 3,
ditions of constraint would, of course, have to be im-
and its relationship to the "nonrenormalizability" of
posed on the latter metric to take into account the fact the theory is noted. Attention is called to the various
that some of its degrees of freedom have been trans-
roles of the background metric, one of which is to define
ferred to the background metric. ) The resulting "
the concepts of "past" and "future. Green's theorem
simultaneous use of both the canonical and covariant
for an arbitrary differential operator is then derived.
theories might help to reveal the relationship between
Section 4 introduces a notation for the basic struc-
them.
tures governing the action of the invariance group on
As has been remarked in I, no rigorous mathematical
the field variables. The relationship between manifest
link has thus far been established between the canonical
covariance and linearity of the group transformation
and covariant theories. In the case of infinite worlds
laws is emphasized. In Sec. 5 it is pointed out that the
it is believed that the two theories are merely two infinitesimal disturbances themselves are determined
versions of the same theory, expressed in di6'erent
only modulo an Abelian transformation group. This
languages, but no one knows for sure. The analysis of
group, which is the tangent group of the full group,
radiative corrections has turned out to be of such afI'ects only the field variables but not physical ob-
intricacy that the covariant theory has had to be servables. The latter are necessarily group-invariant.
developed completely within its own framework and Infinitesimal disturbances satisfying retarded or ad-
independently of the canonical theory. Although the
structure of the covariant theory is suggested by the 5 C. N. Yang and R. I . Mills, Phys. Rev. 96, 191 (1954).
formalism of field operators, and hence maintains a few
"
The term "invariance group, as used in this paper, will
always refer to the infinite dimensional "gauge" group of the
points of contact with conventional field theory, the theory, and not to the finite dimensional (&10) asymptotic
language of operators is dropped at a certain key stage isometry group, which is undetermined a priori. It is not hard to
and c-number criteria are thenceforth exclusively em- show that the Yang-Mills field and its "gauge" group can be
given a metrical interpretation which suggests a physical kinship
ployed to maintain internal consistency. It turns out between the Yang-Mills and gravitational fields which is closer
that the language of operators is a peculiarly unwieldy than the formal mathematical similarities between them alone
one in which to discuss questions of consistency when indicate. LSee B. S. DeWitt, Dyssamscal Theory of Groups aruE
Fields {Gordon and Breach Science Publishers, Inc. , New York,
the invariance group of the theory is non-Abelian. 1965), problem 'I/, p. 139.g
 B RYCE S. DEWITT 162

vanced boundary conditions can be computed with the section. The lemma is used again in Sec. 11 to prove
aid of corresponding Green's functions provided sup- that the non-radiatively-corrected amplitudes for scat-
plementary conditions are imposed. For convenience tering, pair production and pair annihilation by the
these supplementary conditions are chosen in a rnani- background field are group-invariant. "Group in-
festly covariant way, but their essential arbitrariness variance" here implies invariance under group trans-
is emphasized. formations of the background field, under gauge changes
Use of the covariant Green's functions in connection of the propagators, and under radiation gauge changes
with Cauchy data for infinitesimal disturbances is in the asymptotic wave functions. The amplitudes are
discussed in Sec. 6, and the fundamental reciprocity also shown to satisfy a set of relations which are the
relations of propagator theory are established. Transi- relativistic generalizations of the well known optical
tion from the classical to the quantum theory is made theorem for nonrelativistic scattering.
via the Poisson bracket of Peierls (see Ref. 20), which Construction of the full S matrix of the theory is
is determined solely by the behavior of infinitesimal begun in Sec. 12. The field operators are separated into
disturbances. The reciprocity relations are used to show two parts, a classical background satisfying the classical
that Peierls' Poisson bracket satis6es all the usual field equations, and a quantum remainder. Vacuum
identities. Section 7 introduces the important concept states associated with the remote past and future are
of the asymptotic fields, which obey the 6eld equations defined relative to the background field. Vacuum matrix
of the linearized theory. From the asymptotic fields elements of chronological products are constructed by
one can construct asymptotic invariants, which may varying the vacuum-to-vacuum amplitude with re-
be used to characterize completely the physical state spect to the background field. It turns out that all
of the field. The asymptotic invariants are conditional physical amplitudes can be obtained in this way
invariants, i.e. , invariants modllo the field equations. despite the fact that the variations in the background
It is emphasized that their commutators (i.e., Poisson field are subject to the constraint that the classical
brackets) are nonetheless well defined. A direct proof is field equations never be violated. The well-known
given that the asymptotic invariants satisfy the com- difhculties arising with the use of external sources in
mutation relations of the linearized theory, a result a non-Abelian context are thus avoided. When no in-
which is nontrivial when a group is present. This result variance group is present the vacuum matrix elements
is used in Sec. 8 to construct the creation and annihila- of chronological products are expressible in terms of
tion operators for real (i.e. , physical) quanta in the functions having the combinatorial structure of tree
remote past and future. The detailed structures of the diagrams. Use of these functions constitutes an essential
asymptotic Yang-Mills and gravitational fields must part of the program for constructing the S matrix as
be investigated separately, but a condensed notation given in this paper. Since these functions are initially
(for the asymptotic wave functions) is again introduced, defined only in the absence of an invariance group,
which embraces both 6elds at once and emphasizes however, we are at this point forced to abandon the
their similarities. A table is included to facilitate the strict operator formalism. Section 13 displays the struc-
comparison. The quanta of both fields are transverse ture of the S matrix and its unitarity conditions when
and diGer only in spin. States are labeled by helicity, no invariance group is present. Section 14 then begins
which is readily shown to be Lorentz-invariant.
the long and intricate task of generalizing this struc-
ture to the case in which a group is present. Aside
Continuing the uniform treatment of the two 6eMs,
from an invariance lemma which is used to suggest the
Sec. 9 shows that the asymptotic commutator functions
desired generalization, the important proof of this sec-
of both can be expressed in a standard canonical form.
tion is the tree theorem. The tree theorem says that the
A special notation is introduced for the projection of
lowest-order (i.e. , non-radiatively corrected) contribu-
the canonical form into the physical subspace. With
tions to any scattering process can always be calculated
the aid of this projection two distinct Feynman prop-
agators are defined relative to an arbitrary back- by elementary methods, using any choice of gauge for
ground field. Both serve to describe the propagation of the propagators of the internal lines and any choice of
field quanta in nonasymptotic regions as well as at gauge for the external-line wave functions. The result
in6nity. One is manifestly covariant but propagates will be independent of the gauge choices provided all
nonphysical as well as physical quanta; the other prop- the tree diagrams contributing to the given process
agates physical quanta only but lacks manifest are summed together.
covariance. The latter is used in Sec. 10 to define the There remains only the question of the vacuum-to-
external line wave functions which enter into the ulti- vacuum amplitude itself. Since all radiative correc-
mate definition of the S matrix. These functions serve tions can be obtained by functionally differentiating
to generalize the asymptotic wave functions to the this amplitude with respect to the background 6eld,
case in which an arbitrary background field is present. a proof of its group invariance would complete the
They satisfy a number of important relations following proof of the invariance of the entire S matrix. The real
from a fundamental lerrima which is proved in this problem, however, is to constrict the amplitude, and the
 QUANTUM THEORY OP GRAVITY. II

invariance criterion must therefore be used as a guide in the classical limit A — + 0. The question arises why
rather than as an a posteriori consistency check. these removals and corrections are always neglected in
Section 15 pauses briefly to review the question of conventional field theory without apparent harm. It is
Lorentz invariance, to point out that the theory should argued that the usual procedures of renorma. lization
also be invariant under changes in the speci6c variables theory automatically take care of them and that in
with which one works, and to comment upon the utility practice the criteria of locality and unitarity are re-
of using c-number language exclusively. Section 16 placed by analyticity statements and Cutkosky rules
then plunges into the main problem. The single-loop (see Ref. 52). A detailed investigation of these cor-
contribution to the vacuum-to-vacuum amplitude is rections when a group is present is undertaken in Sec.
computed with the aid of the formal theory of con- 20. The two-loop Feynman-basket decomposition of
tinuous determinants, and various alternative forms for the preceding section is appropriately generalized and the
it are given. There is no ambiguity about this contribu- result is reexpressed in terms of covariant propagators,
tion, and its group invariance is readily demonstrated. including the fictitious quanta. It turns out that the total
This contribution is functionally di6erentiated in two-loop amplitude is obtainable from a set of covariant
Sec. 17 to yield the lowest-order contribution to primary diagrams (containing Feynman propagators
single quantum production by the background field. only, and hence o8-mass-shell contributions in all
The latter splits into two parts, one involving the lines) by a process of removing noncausal chains and
covariant propagator for normal quanta and the other adding nonlocal corrections, which is completely
involving the covariant propagator for a set of fI, ctitiols analogous to that of the no-group case. Moreover, the
qlmta which compensate the nonphysical quanta that primary diagrams, taken together, are group-invariant
the first propagator also carries. The fictitious quanta as they stand, independently of the tree theorem. This
are coupled to real quanta through asymmetric vertices suggests that even when a group is present the non-
which vanish when the invariance group is Abelian. causal chains and nonlocal corrections may be neglected
With the aid of the fundamental lemma of Sec. 10 and as in conventional field theory. The problem therefore
a collection of new identities it is shown that the becomes one of 6nding a general algorithm for obtain-
fictitious quanta can be formally avoided by replacing ing the primary diagrams of arbitrarily high order, in-
the covariant propagator by the noncovariant one cluding all fictitious quantum loops. The remainder of
which carries physical quanta only. The covariant Sec. 20 is devoted to the construction of such an algo-
propagators, however, are needed for the practical rithm. The generator for the algorithm is a Feynman
implementation of any renormalization program. functional integral for the vacuum-to-vacuum ampli-
The lowest-order radiative corrections to the tude, which includes fields representing the fictitious
e-quantum amplitudes are analyzed in Sec. 18. These quanta. The group invariance of this integral is explicitly
amplitudes split automatically into Feyemm baskets, demonstrated, and the 6ctitious quanta are shown
i.e. , sums over tree amplitudes (lowest-order scattering formally to obey Fermi statistics despite their integral
amplitudes) in which all external lines are on the mass spin. No physical criteria are violated, however, since
shell. The tree theorem then guarantees their group the fictitious quanta never occur outside of closed loops.
invariance. This invariance can be made partially Finally, the rules for inserting external lines into the
manifest by converting from the noncovariant prop- primary vacuum diagrams are given, and the asym-
metric vertices contained in the fictitious quantum
agator to the covariant one, and the fictitious quanta
loops are shown to have a uniform orientation around
again make their appearance.
each loop.
The problem of splitting the radiative corrections
into Feynman baskets becomes more difficult in higher
2. NOTATION. INFINITESIMAL DISTURBANCES.
orders, when overlapping loops occur. This problem
BARE VERTEX FU5'CTIOJKS
is approa, ched in Sec. 19 with the aid of the Feynman
functional integral. When no invariance group is present A quantum field theory begins with the selection of
it is shown that the "measure" or "volume element" for an action functional 5. If the theory is local this func-
the functional integration plays a fundamental role in tional is expressible in the form
the decomposition into Feynman baskets and in
guaranteeing the invariance of the vacuum-to-vacuum
amplitude under arbitrary changes in the choice of
S=— Zdx, = dx'dx'dx'dx',
dx— (2.1)
basic field variables. The "measure" has two effects.
Firstly, it removes from all closed loops the eoncalsal —
where Z the Lagrangian (density) — is a function of
chains of cyclically connected advanced (or retarded) the dynamical variables and a finite number of their
Green's functions, thereby breaking them open and in- space-time derivatives at a single point. Various criteria
suring that at least one segment of every loop is on such as covariance, self-consistency of the 6eld equa-
the mass shell. Secondly, it adds certain nonlocal cor- tions, the existence of the vacuum as a state of lowest
rections to the operator Geld equations, which vanish energy, and positive definiteness of the quantum-
1200 8 RYCE S. DzNITT
mechanical Hilbert space in practice drastically limit Suppose the form of the action functional suGers the
the possible choices for Z. However, many different following change:
choices exist for the Lagrangian of a. given field. Thus
it is always possible to add a trivia, l divergence to the 5-+ 5+ed, (2.3)
Lagrangian without changing the field equations at all.
where e is an infinitesimal constant. Such a change may
Moreover, the field variables may be replaced by
the
be thought of as being brought about by weak coupling
arbitra, ry functions of themselves; this replaces field
to some external agent. The coupling produces an in-
equations by linear combinations of themselves. Finally,
finitesimal disturbance 8y' in the field, which satisfies
even the number of field variables is not unique; for
the linear inhomogeneous equation
example, alternative La, grangians may be found leading
to field equations which express some of the variables in S "8q»'= —~A (2.4)
terms of derivatives of others. What is important is that
the choice of I.agrangian is basically irrelevant to the That is, y'+by' satisfies the field equa, tions of the
development of the theory of a given field and shouM system 5+ed if y' satisfies those of the system 5. The
be determined only by convenience. The quantum undisturbed field y' may be regarded as a background
theory of a given Geld must be constructed in such a Jkld upon which the disturbance 5&p' propagates. The
way that it is invariant under changes in the mode of concept of the background field proves to be a useful
description of the field. one in the cova, riant theory, and will occur repeatedly
It will prove convenient in what follows to adopt a in what follows.
highly condensed notation. The field variables (assumed For local theories the quantity S;; has the form of a
here to be real) will be denoted by y', " and commas linear combination of 8 functions and derivatives of 8
followed by indices from the middle of the Greek functions, with functions of the Geld variables and their
alphabet will be used to denote differentia, tion with re- derivatives as coefTicients. In Eq. (2.4)
spect to the space-time coordinates. The first part of plays the role of a linear differential operator with
5„, therefore

the Greek alphabet will be reserved for group indices, variable coefficients. The reader mill find it useful to
to be introduced presently. Primes will be used to consult Table I, which lists the explicit forms which this
distinguish different points of space-time; they will also and various other abstract symbols of the general
appear on associated indices, or on field symbols them- formalism take in the cases of the Yang-Mills field and
selves, when it is desired to avoid cumbersome explicit the gravitational Geld, respectively.
appearances of the x's. In most cases, however, the In the case of linear theories S,,; corresponds to a
primes will be simply omitted. This corresponds to linear differential operator with constant coeKcients,
making the indices i, j, etc. do double duty as discrete and the higher functional derivatives S,;,k, etc. , vanish.
labels for Geld components and as continuous labels over In nonlinea. r theories the higher functional deriva, tives
the points of space-time. That is, an index such as i will are known as bare vertex functions They descr. ibe the
really stand for the quintuple (i, x', x', x', x') and the basic intera, ctions between finit disturbances, the prop-
summation convention for repeated indices will be agation of which, as will be seen later, provides a direct
extended to include integrations over the x's. The classical model for the quantum S matrix.
significance of the indices thus becomes almost purely It is frequently convenient to introduce a further con-
combinatorial. When this notation is employed it is densation of notation, namely to make the replacement
necessary to remember that expressions such as 3f,, are
really elements of continuous ma, trices and that the (2.5)
symbol 8', involves a 4-dimensional 8 function.
and to drop the indices altogether. Equa, tions (2.2)
For most purposes the form of the field equations is and
(2.4) a, re then replaced by
more important than the value of the action functional.
Therefore, the domain of integration in (2.1) is un- Sg=0 (2.6)
important; when otherwise unspecified it is to be under-
stood as being large enough to embrace all points a, t
which it may be desired to perform functional dif-
ferentiations. Functional differentiation with respect to 525' — &Ay, (2.7)
the Geld variables will be denoted by a comma followed
respectively. If the basic Geld variables are properly
by one or more Latin indices. Thus the Geld equations
chosen the number of nonvanishing bare vertex func-
will be expressed in the symbolic form
tions is finite in the case of both the Yang-Mills and
S„=o. (2.2) gra, vitational fields. Thus, for the Yang-Mills field we
have S = 0 for e) 4 when the field variables are chosen
7 In this
paper no restriction is imposed on the range of Latin as in Table I, while for the gravitational Geld we have
indices. Other conventions, to the extent they overlap, are the
same as in I. S„=Qfor ~&9 jf the quantities &p~=~5i~8&I~ g"" are
162 QUANTUM THEORY OF GRAVITY. II 120i
TABLE I. Expressions for the Yang-Mills and gravitational helds corresponding to quantities appearing in the abstract formalism.

Abstract
symbol or
equation Corresponding expression for the Yang-Mills field Corresponding expression for the gravitational field

p=0, 1,2, 3 n=1 ~ ~

n Ppv=gfsv 'QIbv j P&v =Op1p2p3b Ppy= Pv

F P @vs7 = g'" &'&m~,

S—
„— =R",' Rp. =R. ;—
r„. „,
—A;
pa„v= Aa„„+Cap,p„A~,. = det(—
g.,),
R„,:— ;„—
A &2&R

The indices p, v are raised and lowered by means of the

g
= r„.
o'= 1
r „+r„.
„':, ~r„,— ~r„,,
v+g"r p gpv r)'
'

Minkowski metric g„, —diag( —1,1,1,1) and its inverse

= The indices p, v, p, 0.,
2g (gpr
are raised and lowered by means
7
g&v. The indices 0. , p, p are raised and lowered by means of the metric tensor g„,and its inverse g&v.
of the Cartan metric, In the remaining entries of this table the symbol (4)R is
'yap= C C&agC p& — replaced simply by R.
and its inverse y p. The c's are the structure constants
of a compact n-dimensional semi-simple Lie group, and
the constant c' is chosen so that det(y p) = 1.
O=S, ;
b =b
p"'—
0=5S/bA
pb "b(xx')
„=
——P b', „. b
0=bS/brp„, = — l/2(Rp" —
g— 'gp"R).
—,

"'=-'(b b '+b, b„')b(x,x').

S, ;; b2S/bA ppbAÃv, = b ppp, v'. aa bpap ";po+—cap„t&pab,api" be/by b~, , —1gl/2(gppgv'k+gphgvp gpvgpl)gca

1gl/2(Rpv 1gpvR)b pa'r'

+gl/2(Rp bpva'r'+Rv bppa'r' lRbpvr'r'
—-,'g p"R 2"b, &"")
The infinitesimal group parameters are functions bP(x} The infinitesimal group parameters are the functions
which assign to each point x a corresponding bp(x) appearing in the infinitesimal coordinate
infinitesimal transformation of the generating I.ie transformation x&= x&+bP. Under inner automorphisms
group. Under inner automorphisms they transform they transform as contravariant vectors. Note that
according to the adjoint representation of the full group. group and coordinate indices coincide in the case of the
general coordinate transformation group.
R R"„p—=—bap „,
,. b"p =
—bapb(x, x') RIbvop = oIba';v uvo';Ibb gpfb&T'= gfs&r&y&p~ )
By'=R' 5$
Semicolons denote invariant differentiation. A field
~ v p. = —&4;.—~b:~= —g~.. ~P —g-~E . ~ —g~.~k .
Semicolons denote covariant differentiation. A field
.
quantity q which has the group transformation law quantity p which has the group transformation law
g +— G +g~a
where the G are the generators of a matrix representa-
~v = — v, ~~$"+G"~ v»P, = — .
v: ~~$"+G"~v»P:.
where the G"„arethe generators of a matrix representa-
tion of the generating Lie group, is defined to have the tion of the linear group, is defined to have the covariant
invariant derivative derivative
&;bt= ttg'. bs+Ga~ p;Ib= &tg, fb+G o~btv
Invariant difFerentiation leaves transformation properties Covariant difFerentiation adds one covariant index. It
intact. It has the commutation law has the commutation law
p; bbv p;vs GaF bbv p p; pv p;ytb — G TRIbyo'
S Rs =
—0 —2[g'/2(Rp" ——,'g p"R)]. —
= 0. .

„.
g',

This identity is a consequence of the antisymmetry of This identity results from contracting the Bzanchi
F „vand of the structure constants c& p. F transforms identity
R„,'; p+R„p,T, „+Rp„,
„„

according to the adjoint representation of the group and ', „=—

0
also satishes the cyclic identity
Fa~'o+F vo ~+F .—
= o.
which can be verified by straightforward computation
using the fact that the Riemaee tensor
forms as a mixed tensor of the fourth rank.
trans- R„„'
bRa„p /bA ""„- x') b(x, x") = c l, b'p. b'„„-""—
= c p„b„"b(x, /b vip-, "
bR„„, bp. v'"";)b— "; bi.'—"
—, b"—„bp"2"'"''b,"—
;;v
C p c pr. = c p, b(x, x')b(x, x")
— cp. .
. -=b,b, .(x,x'}b(x,x") -bp, b, „(x,
x")b(x,x')

v "p

p ap' =
pap' = ~ap'
sp ap' 'o
a1gl/2(bpva'r'
rp'v=

P„;=gl/'(b„; :+R„b.
g1/2g
, ;)-.
lgpvb pa'r')

'y -lap ~-lap' = gap' ~-lbtv' = gbsv'g P-1/2

Fa"ppv =—&a"pp";o'+2c'apF~"o&e'pp" I/pva'r' —rgl/2(gppgv'k~gplgvp gpvgpX)
X (& ),'"''"—2Rf, s) «~'" "' —Rvt, &«"'"'
+2R '
«g &o'T'
g &R $
b«o'T'

+pg )RB„' ')

The last five terms inside the parentheses may be omitted
when the held equations are satisfied.
1202 BRYCE S. 0 H W I TT
chosen as the basic field variables. ' With the conven- the characteristic surfaces are in fact unaBected by the
tional choice of Table I the number of nonvanishing fields themselves; only in the case of general relativity
gravitational vertex functions is inGnite. does the background Geld exert an inQuence.
For a local theory a typical term in
product of e —
5„involves the
1 8 functions or derivatives of 8 functions.
In the quantum theory 5& is not only a differential
operator but also a quantum-mechanical operator. In
In momentum space with a constant (e.g. , flat) back- gravity theory the position of the light cone thus be-
ground field these reduce to a single 8 function, which comes a q number. Critics of the program to quantize
expresses the conservation of momentum of the n field gravity frequency ask "What can this mean?" A good
quanta taking part in the elementary process described answer to this question does not yet exist. However,
by the vertex in question. The calculation of speciGc there are some indications where the answer may lie.
processes is usually most conveniently performed in mo- We have seen in I that the canonical formalism can be
mentum space; the development of the general theory, developed to a considerable degree without the question
however — in particular, the demonstration of the arising. This is particularly true when the discussion is
covariance of renormalization procedures —is best done carried out in the "metric representation, "
in which
in coordinate space. the metric appears as a c number. It is also true in
Because of the commutativity of functional differen- Leutwyler's analysis" of transition amplitudes as
tiation the bare vertex functions 5,;;&... are completely Feynman sums over classica/ histories. Where do these
symmetric in their indices, and S„.
; corresponds to a ana, lyses break down, or rather, where must they be
self-adjoint linear operator. When employing the nota- supplemented by more sophisticated reasoning? They
tion (2.5) we may regard the symbol Ss as actually must be modified at precisely the point at which it be-
representing this operator. Note: The abstract notation comes necessary to account for radiative corrections
must be used with a measure of caution because the and field renormalization.
associative law of matrix multiplication does not always In the covariant theory we shall not make use of a q
hold. If C' and 4" are two functions which do not vanish number 5~. Our approach will be that of perturbation
rapidly at infinity, the value of the expression O'S, ;,0'&' theory, with all its limitations. We thereby gain, how-
may depend on which implicit integration is performed ever, the possibility of working exclusively with c num-
first. This ambiguity may be removed by using arrows bers. The background field will play two roles simul-
to distinguish the two possibilities: C 52% and C $2%'.' taneously. Firstly it wiH. serve as a classical reference
The present discussion will be limited to boson Gelds. point about which the quantum Auctuations may be
For the extension of the formalism to the- case of assumed to take place. Secondly it will serve as a useful
fermion Gelds, which involves anticommutative dif- technical instrument. By varying the back. ground field
ferentiation and antisymmetric vertex functions, the we can reproduce the effect which individual Geld
reader may consult the reference given in Ref. 6. quanta have on a variety of fundamental processes,
This reference contains detailed proofs of some of the including the laws of propagation (i.e., on the light cone)
important theorems to be stated in what follows. We By allowing these effects to superpose nonlinearly we
achieve the full 5-matrix expansion, including all radia-
shall therefore restrict ourselves here to sketch-proofs
or simple statements of these theorems but will take the
"
tive corrections. The only limitation is that we never
occasion to improve their presentation. consider more than a Gnite number of quanta at once.
The total perturba, tion series is never summed, and
thus we never determine the answer to Pauli's specu-
3. THE QUANTIZED LIGHT CONE AND THE
lation' that quantization of gravity may yield an
DEFI5ITION OF TIME. GREEN'S THEOREM
intrinsic cuto6 by "smearing out" the light cone, which
In a standard hyperbolic wave theory the opera, tor would at the same time be the deGnitive answer to the
52 deGnes a class of characteristic hypersurfaces which question of the meaning of a q number 52.
separate spacelike from timelike directions. To this A clue to the eventual answer may perhaps be found
operator therefore falls the task of providing the in the fact tha, t quantum gravidynamics is not, by
de6nition of time. Since 52 generally depends on the standard criteria, , a renormalizable theory. It is not
background Geld it is evident that the background field difficult to see that the strongest divergences (which,
may play a role in this deGnition. For most Geld theories, from a perturbation point of view, are responsible for
the nonrenormalizability) are precisely those which
""g„„—
An alternative choice of basic variables is g q„„, which arise from the fluctuations of the light cone. It may be
yields S„=Ofor l&11. Neither choice, however, can in general hoped that these divergences will some day prove to be
avoid an in6nity of nonvanishing bare vertex functions if other
fields, necessarily coupled to the gravitational held, are present. summable to a Gnite correction embodying Pauli's
Peres has proposed to treat the latter case by a method which
makes use of an additional constraint. LSee A. Peres, Nuovo ~0
H. Leutwyler, Phys. Rev. 134, 31155 '1964).
C,

]
Cimento 28, 865 (1963). It should be remarked, however, that
the presence of an infinity of bare vertices does not pose an
"Hence the present formalism takes into account all nonlinear
e8ects, classical as well as quantum, and is not merely the theory
essential difficulty for the theory, as we shall see. of a linearized held on an arbitrary background.
&
The symbol denotes transposltjon. i'%. Pauli, Helv. Phys. Acta Suppl. 4, 69 (19SQ.
162 gUANTU THEO RY pF GRA V& TY ' 1203

. q . ratic com-
cutoff P resent ev' ence pn th ;s ~atter, as ~e.
signlfican ceof
~
te pl anck 1eng
we]1 as the
Wl'll be discusse d
w
. gp it 1S a
p

bination o f delta
certain hpmogeneou
'
functlpnns and their eriva, tlVe ' It is
~

the lna 1 pap

R er of this se npt h d to show th t the self-ad3oinntness pf $2 lm lies
e assume, nes two classes
asse of tnne-
S gj"= gP rip ~ (3 2)
ke dlrec tions, ' pne of wwhich will be
g ~~+fQf 8 eassume urt rthermore
e Mpr g ener»ly we e
eometrical
(O'F '4&'' — ~'F
"F— I
';@')dx'
npntrivia
hav»g these prope rties exist . npt pnly1 «m
grpunds Rr e gbsoN ) t ey)y classlca. n
~
ro a ma, the- dS gl(@i'fteetpt% ) (3.3)
matical bu ].so from R physical p Oint Pf Vle w A spa, ce
pf inhnlte voplume h as the capa cjt for n ingnlte
amou " o ~4
d hence can serv
Rn.
rve as th classical
for Rny d 'ff erentla 1 operator I"i'
. .
~, regard esss o
pf whet
lt is self-R d jpint t «n
and fo n Rlr 0f functlo»
. ~ity The
t boundary con the same @'. @'; regarrdless of their behav ior Rt 'n
time « "te .thst the rea
'
f nite, is so '
different» 1 p p eratpr .„hpwever hast h esy mmetry
huge that i t is e ective t y classica 1 For e„ample, y
nouncing o y slig h ty
' . ~

l the infinitete p recisionn usuallv '

lf ad&oint case
.
If Z 's he bou da, te doma & then (3.3)
ascribed to the energy R"d momentu mlabelso fS matrix implies
elements, an d by using the terms rem«~ o d remote
~ '
futN« i a relative sense,' we can have an eff ective
S- Rtrix theory w hich lc is extremeel y p reclse, based on a dx dx ' C'J g. 'O'
' —C»'IlI' ',
backgroun
bound
.
asymptotica ll y pat at the
Q ite but large eg o
' 0

The rcmp t pas

n
ast a, nd rempt 'll be denoted
r spec tlvely. If t ~ p ace-time po»t
dx dx i~@i
rt
f ., „@
p j
'jrl
3 4)
. associate wlith an jndex zzlies to th e future of
whic»s as
Z

' ~
'rected surface
e directe ent of &. If
e epece lke
' hypersur face & wee sshall write iz g g. If wh re "- . ' finlty we
~
the as« iy we write z ~ If, rea i. 1 tive tp two an d @' vanish su anciently»p' 'dl t spatial ln unity
. . . 0bttainn pnlettlng, 0 expan d w ithout limit,
poln ts associated Rnd j,
resP ectivett',
space-lde h yP ersurfa, ce cann be found suc
' t'
and & + &. then we wrritei ~ ~ It is ppssible to have —C F+=~ f @dZ„.
both z j '
Rnd 2 g z slmultaneou us] y . In t h is case the
associate P PlntS areeseparated
S b y R space ]'ke 1 e interval
@ FQ—
fi ~o 3.5

~
Evident 1y 00 ~ and ~ —00m for all z ~
Here the condense d nota, tlon h ass been emP lp y ed, and
and Z. d puublearrow ~
has eenn laced above @to em
'
Consider now tthee following exp x resslon: phaslze that as a dl erentla 1 opeperatprlthas comppnen tss
h'
whlc h ac act to the rigri ht as we 1] s comp ents w h lch
'
~ ~ ~', ' 4')dx',
~
actt too the left. Inn a similar mann
nner we wriritee
ann

-
e 8-functionn ccharacter
a of S„, C S2%-C S24=~ iC s&%'dZ„. (3.6)

4. P PHYSICAL
ANCE GROUP)
ES AN
OBSERVABLE gD MANIFES
f t'o C'
COVARIANC
h t the
integra e
his implies h

. ..,....
d' The invariance gr ooth th e Yang-Mills and
itself must be expressl 'bl e o
g""""'"'
Abelian.
e . In thhea b sra th h
(c's„,'4&' —c»'s.
&'S ;. ; '
';+')dx' u trans formation

ds dxx (4 ' s~erjtr%~

; &" , , (3.1) bq'=E'~8@ or
or, more simply,
nn. . . Y.) 7, 466 (1959;
f I wa Report o. SUI 61-4, 1961 un
'4 It js for t his
i reason
9 State
u p ublished).
it
a University
identi6cation of porn s
(i.e.t infinite).
'
.-"d.4,
ure for w ich no
h';, th f
 BRYCE S. 0 E%'I TT
Here the hP are the infinitesimal group parameters and representation generated by the matrices R';, while
the R' are certain linear combinations of the 8 function a 6eld index in the lower position will correspond to the
and its derivatives, with coeKcients depending on the contragredient representation. Similarly a group index
q's (see Table I). As functions of the x's the 8P are in the upper position will correspond to the adjoint
assumed to be differentiable and to vanish outside a representation of the group, while one in the lower posi-
finite domain of space-time but to be otherwise arbitrary. tion will correspond to the contragredient representa-
The group property is expressed in the form of a func- tion. The adjoint representation of the Yang-Mills
tional differential condition on the R': group is the infinite direct sum of the adjoint representa-
R' R&p —R'p R& —
= R' c~~p, (4.2)
tion of the generating Lie group taken repeatedly over
the points of space-time; the adjoint representation of
where the c& p are the structure constants of the group, the coordinate transformation group of general relativity
which in turn satisfy is that of a contravariant vector Geld.

ap=0 (4.3)
Both the R', ; and the structure constants c& p are
c age py+c pic ya+c gee ~
homogeneous quadratic combinations of the 5 function
A functional A of the p's is regarded as a physical and its derivatives, independent of the q's. In the
observable if it is a group invariant. The condition for theory of radiative corrections we encounter the ex-
this is pressions R', ; and ct' p which are mathematically
A, ;R' —
= 0. (4.4) meaningless, involving the 8 function and its deriva-
tives at x'=x. We shall find it necessary to assign
The action functional in particular is a group invariant: vanishing values to these expressions in order to main-
S,;R' —
= 0. (4.5) tain internal consistency of the theory;

By functionally differentiating the latter identity we

R' „-=0, cl' p —0. (4.10)
learn that under a group transformation the Geld equa-
The reasonableness of these assignments may be
tions are replaced by linear combinations of themselves:
made apparent by noting the transformation laws of
85;=5;;,8 — ' bP, = —S,RJ;8P, (4.6)
p'= 5;,R—, the quantities in question. Both transform contragredi-
ently to the adjoint representation of the group. In the
and hence that solutions go into solutions. We also case of general relativity they are therefore covariant
learn that 52 is a singular operator, at least when the vector densities of unit weight" and may be presumed
6eld equations are satisGed, for it then possesses the to vanish by virtue of the fact that space-time has no
R' as zero-eigenvalue eigenvectors of compact support; metric-independent preferred directions. In the case of
ST=0. the Yang-Mills group they may be presumed to vanish
by virtue of the fact that the corresponding quantities
With the conventional choice of field variables given vanish for the generating Lie group which, for physical
in Table I the dependence of the R' on the &p's is linear reasons is necessarily compact.
inhomogeneous, so that R',
;I, vanishes. This has im-
portant consequences for the manifest covariance of S. BOUNDARY CONDITIONS) SUPPLEMENTARY
the formalism. For example, by repeatedly differentiat- CONDITIONS, AND GREEN'S FUNCTIONS
ing (4.5) we find for the transforms, tion law of the uth
vertex function Suppose the infinitesimal change (2.3) in the action
—(5;;,...; R';, +. functional corresponds to alterations in the structure
55;,...;„= of the physical system which are limited to a finite
+5,;,... ;„„R',
;„)5P. (4 8) region of space-time. The functional A will then be
Similarly, from (4.2) we find constructed out of field variables evaluated at points
within this region, and its functional derivative A, ; will
gR'„=(R'p, ;R& R*,c'p ) 8&p.
'
— (4 9) vanish at points outside. Under these circumstances we
These simple linear laws permit the transformation may distinguish two particular solutions of Eq. (2.4)
which are of special importance, the returded and
character of many quantities appearing in the formalism
cd~aeced, denoted by 8~ q' and 8~+q', respectively,
to be inferred at once from the positions of the Geld and
characterized by the boundary conditions
group indices attached to them. In general, when in-
troducing new quantities, we shall be careful to insure (5.1)
that they obey the following transformation laws, of
which (4.8) and (4.9) are special cases: A quantity
Because of the singularity of the operator S2 the
bearing several indices will transform according to the
direct product of a corresponding number of (con-
tinuous) matrix representations of the group. A field
"The 8 functions in Table I are to be regarded as densities of
zero weight at their Grst argument and unit weight at their
index in the upper position will correspond to the second,
 QUANTUM THEORY OF GRAVITY. II 1205

conditions (5.1) do not determine the 8g+y' completely, (i.e., differential) operator. Condition (1) insures that P
but only modllo transformations of the form will be self-adjoint. Condition (2) maintains the manifest
&~+a"=&~+a'+R' &8 (5.2)
covariance of the formalism by insuring that Fp will
transform according to the law suggested by the position
Since such transformations can be additively super- of its indices. Condition (3) enables (5.8) to be replaced
imposed, they constitute an AbeliarI, "gauge" group for by the stricter relations
infinitesimal disturbances. Unlike the situation which
holds for the familiar gauge group of electrodynamics,
G+~&=0 for a & P, G ~~=0 for P +n .(5.10)
the scale of these transformations varies from point to In addition to the matrix y;; we shall also introduce a
point owing to the dependence of R' on the back. ground matrix y p for the purpose of lowering group indices.
Geld. This fact is responsible for all of the formal com- Like p;; it may be chosen in a completely arbitrary way
plications which arise in the quantum theory of non- except for a single essential requirement. The require-
Abelian gauge fields. ment in this case is that y p shall be nonsingular and
Although Eqs. (5.1) do not suffice to determine the possess a unique inverse p ' p, which may be used to
8z+p' completely they provide unique physical bound- "
raise group indices. It is then not dBTicult to show that
ary conditions. Because of the invariance condition the m. atrix
B„E' =0 the disturbance produced in any physical
observable 8 is unaffected by the transformation (5.2): F;; = S,g+ R; y—' &R; p (5.11)
is nonsingular, provided (as is true in the cases of
4+& =& 4+v"=&,'4.+v'=~~+& (5 3) interest) the R' constitute a complete set of zero-
Nevertheless, in practice it is a convenience to restrict eigenvalue eigenvectors of 5„;
having compact support.
the h&~p' by adding further conditions known as Although the arbitrariness of y p, lik. e that of y;;,
supplementary colditiorls. must be stressed in the general formalism, it is again a
As the standard form for supplementary conditions practical convenience (and for the same reasons) to im-
we shall choose pose three additional conditions, similar to those im-
posed on y;;: (1) that y s shall be symmetric in its
E. 8g+ y'= 0 (5.4) indices; (2) that it shall have the group-transformation
law suggested by the position of its indices; (3) that it
(5 5)
shall be such as to make F;, correspond to a local
where';; is a matrix which may be used to lower 6eld (differential) operator.
indices" and which is arbitrary except for a single In the case of the Yang-Mills and gravitational 6elds
essential requirement, namely that it be such that the it turns out that if all of the above conditions are
operator corresponding to the matrix satisfied then only one additional requirement, namely
—E; E'p
F p= (5 6)
that the Green's functions of Ii and P shall have the
weakest possible singularities on the light cone, leads
shall be nonsingular and have unique advanced, and to choices for y;; and y p which are unique up to a con-
retarded Green's functions G~ p satisfying stant factor. These are the choices shown in Table I.
— They are the generalizations, to the case of arbitrary
p G+&p g p (5.7) background fields, of the well-known Lorentz and
lim G+~&= lim 6+~~=0. (5.8) DeDonder conditions of the corresponding linearized
0!~+o0 p~~oo theories. Any other choices lead to more singular Green's
functions.
If the supplementary conditions (5.4) are not initially
satisfied they may be made to hold by carrying out a
%e note that the supplementary conditions are here
imposed on the in6nitesimal disturbances rather than
transformation of the form (5.2), with
on the fields themselves. The di6erences between this
8@=G+~&R; pay+ p', (5.9) approach and that of more familiar formulations of
gauge theory will become apparent as the discussion
and the 8~+y' thus restricted will generally be unique.
progresses.
Although the arbitrariness of y;; in the general When the supplementary conditions (5.4) are satis-
formalism must be stressed, it is nevertheless a great
fied, Eq. (2.4) may be replaced by
convenience in practice to impose the following three
additional conditions: (1) that y, ; shall be symmetric in F;;by&= —eA, ;, (5.12)
its indices; (2) that it shall have the group transforma-
which has the unique advanced and retarded solutions
tion law suggested by the position of its indices; and (3)
that it shall be such as to mak. e F p correspond to a local 8g+q'= eG+'&A . (5.13)
"If p;; has a unique inverse this inverse may be used to raise "The Cartan metric y p=——c& qc'p~ cannot be defined for an
field indices, but this is not -essential. in6nite dimensional group, and hence cannot be employed here.
 BRYCE S. 0 E%' I T I
the G+'& being the Green's functions of F, satisfying retarded egect of A ol B eqgals the advanced effect of B
oe A, aed mice versa. Although the use of (6.8), which
F;I,G+~&'= —b;& (5.14) holds when y and y are symmetric, is the easiest way to
and Lin virtue of condition (3) above) prove these relations, it is to be emphasized that since
they involve physical observables (invariants) only,
G+'& ='0 for i & j, G '&'=0 for j 6i .(5.15) these relations are independent of such conditions. In
particular it can be shown explicitly that by+8 and
6. CAUCHY DATA AND RECIPROCITY RELA- b~+2 remain invariant under arbitrary changes in the
TION'S. THE POISSON BRACKET &'s, including changes which destroy the synunetry and
group-transformation properties of the y's.
Instead of studying disturbances which are produced
Another important relation which can be obtained
by physical alterations in the system it is frequently of
is the following:
interest to consider disturbances which originate at
inanity and which satisfy the homogeneous equation RG+y= G+R, (6.11)
S2by=0. (6.1) which is proved by making use of (4.7), (5.6), (5.11), and
notation. ) If the the kinematic structure of the Green's functions. Since
(We here employ the supercondensed
supplementary condition
(4.7) generally holds only when the background field
satisfies the field equations, it is important to remember
(6 2) that Eq. (6.11) holds only in this case. The transpose
of Eq. (6.11) may be used in a straightforward way in
is imposed $cf. (5.4)g then these disturbances also satisfy
combination with (4.4) to show that the solutions (5.13)
F8rp=0, F= Ss+yRy—'R y (6.3) of the equation for in6nitesimal disturbances are con-
sistent with the supplementary conditions which were
and the value of by is determined throughout space- used to get them in the first place. Equation (6.11) also
time if it and its derivatives are known over any space- 6nds repeated use in the theory of radiative corrections.
like hypersurface Z. With the aid of Eq. (3.5) it is not
The above results provide the starting point for a
dificult to derive the following integral realization of covariant theory of the Poisso~ bracket. In the canonical
these facts:
theory equal-time Poisson brackets are defined for
arbitrary functions of the g„„andtheir conjugate mo-
(6.4) menta, and the physical Hilbert space of the quantum
theory is determined by constraints imposed on the
state vectors. In the manifestly covariant theory
where Poisson brackets are de6ned only for observables, and
(6.5) hence it is possible in principle to work within the
One has only to make use of the kinematics of the G+ physical Hilbert space from the very beginning. "More-
and to assume that they are left inverses of —F": over, the covariant theory makes no distinction between

G+F= —1, CF=O. (6.6)

equal-time Poisson brackets and others.
The definition, which is due to Peierls, is"
From (6.3) and the arbitrariness of the Cagchy data (A, B)= D~B D~A, — —(6.12)
fl'8&p it then follows that they are right inverses as well: where

FG~= —1, FC= 0. (6.7)

—hm
DgJ3= e 'bg 8 (6.13)
g-+0

Equations (6.4) to (6.7) hold regardless of the sym-

With the aid of (6.5) and (6.10) this may be converted to
metry of F. If its self-adjointness is taken into account,
the following additional laws are obtained: (A, B)=Ar CBr. (6.14)
G+ =G (6.S) Peierls' dehnition makes immediately manifest the
fundamental role played by the Poisson bracket in the
(6.9)
theory of mutual disturbances in measurement pro-
Combining these laws with Eqs. (5.3) and (5.13), one cesses, and provides the most natural bridge to the
obtains the important reciprocity relations quantum comlnutator and the uncertainty principle.
In its quantum form,
b~+B=eBr G+Ar —eAr Ger=4 A, (6.10)
which be loosely expressed in the words, the PA, B)=i(D„B Deb) =iAr GBr—
, (6.15)
may
'9 Since a complete operator formalism does not yet exist for
"Kronecker 5's and 5 functions are replaced by the unit symbol the covariant theory this idea will not be fully developed here.
soR. E. Peierls, Proc. Roy. Soc. (London) A2&4, t43 (1952).
1 in the supercondensed notation.
 QUANTUM THEORY OI" GRAVITY. II 1207

it allows one to derive in a straightforward manner the The Grst three terms of the expanded form vanish on
variational formula account of (6.9) and the commutativity of functional
diQerentiation. In order to show that the fourth term
likewise vanishes an expression for the functional
which, in Schwinger's hands, has been used to derive all derivative of G must be obtained.
of quantum electrodynamics. Here, and in the future, The desired expression is a special case of a general
we use boldface to distinguish quantum operators from relation obtained by varying Eq. (5.14). Under an
the corresponding functionals of the classical back- arbitrary in6nitesimal variation 8Ii in the operator Ii
ground field. In Eq. (6.16), A') and IB') are eigen-
I
the G+ suffer variations satisfying
vectors of A and B, respectively; the field variables out (6.1g)
of which A is constructed are assumed to be taken at
points all of which lie to the future of the points at which, taking into account kinematics, has the solution"
which the variables making up B are taken; and 5S,
which represents a change in the functional form of the (6.19)
action, is assumed to be constructed from field variables Therefore,
taken at intermediate points.
G+ij G+iep G+ bj
We shall make no use of Eq. (6.16) in this paper,
Grstly because in the absence of a complete operator Gyia(S b+g gba+. g gba )G+bj
theory we cannot be sure how to order the factors —G+iaS Gkb j+G+iag GkalgjS
occurring in A, B, etc. , and secondly because it is
necessary in a generally covariant theory, to handle the
+R'PG+~ Rb", G+bj, . (6.20)
problem of the relative temporal location of the opera- in which (5.11) and (6.11) have been used.
tors A, B, and bS in a completely intrinsic way. Instead Breaking 0 up into its advanced and retarded parts
of attempting to alter the form of the action functional and inserting (6.20) into (6.17), we see that in virtue
we shall develop alternative techniques based on varia- of the group invariance of A, 8, and C, only the terms
tions of the back. ground Geld. involving the third functional derivative of the action
It is worthy of note that the Poisson bracket is deter- survive. These terms, however, cancel among them-
mined solely by the behavior of ijifbrbitesijjbal disturb- selves, as may be seen by writing them out in the form
ances. Since the commutators of the quantum theory
completely determine the physical Hilbert space, this
.C .L(g+ia g—ia)(g+ jbg-kc G jbg+kc)-
suggests that the quantum theory is obtained merely
+(G+jb G jb)(g+kc—g—ia G kcg+ia)-
by appending a theory of in6nitesimal disturbances to +. (g+kc g—kc)(g+iag jb G —
iag+jb)— ]S (6.21)
the classical theory. Such a view is defective in that it in which use has been made of (6.8).
ignores (a) the factor ordering problems arising in the
We 6nally remark that Peierls' Poisson bracket, being
definition of the quantum operators (which like their
defined for pairs of invariants, is itself a group invariant.
classical counterparts are involved in nonlinear Geld
More precisely, it remains unchanged not only when a
equations) and (b) the existence, in the quantum
group transformation is performed on the background
theory, of nonclassical phase effects which manifest 6eld but also when a transformation of the form (5.2) is
themselves in viitual processes and radiative correc- cor-
performed on the in6nitesimal disturbances,
tions. Nevertheless, if the word "inGnitesimal" is
responding to an arbitrary change in the y's and
modified to "6nite but small" we shaH see that this view
hence in the supplementary conditions. The demon-
accords quite well with the perturbation theoretic ap-
stration is straightforward and will be left to the reader.
proach to quantum Geld theory. Moreover, because of
the uniqueness of the formalism which emerges, it will
'7. CONDITIONAL INVARIANTS AND
appear that the exact theory is already completely
determined by the behavior of in6nitesimal disturbances. ASYMPTOTIC FIELDS
Peierls' Poisson bracket satis6es all of the usual The functional A appearing in (2.3) must be a group
identities. The only one which is not immediately invariant. Otherwise the equation (2.4) for infinitesimal
evident is the Poisson-Jacobi identity. For any three disturbances will not be consistent with the singularity
observables A, 8, C, we have condition (4.7). The invariance condition (4.4), how-
ever, need not hold as an identity but may hold in
(A, (B,C))+(B,(C,A))+(C, (A, B))
=A „6" (B 0'kC, k),i+B,jG". (C, gk'A, ~), i
consequence of the field equations. That is, (4.4) may
be replaced by an identity of the form
+C Qkl(A .QijB,)
= A „iB, A, Q' =
—S„ib' .
;C, k(C"G"+G"Ck')+A „B,
; iC.k
)((Qjkgil+Gkl/ij)+A B,C (GkiPj. i+Qilfjk)
+A, ;B,jC, k(G"C'k +Gi"G , +bidiG kii) . ij(6.17) 2' Kinematics
in (6.~9).
assure the associatiativity of the matrix product
 8RYCE S. DHWITT
%hen the a's are nonvanishing A will be called a of (4 7)., (5.11), (6.11), and (7.3), we have
conditional invariant.
Poisson brackets are as unique and well-de6ned for
S2 &p =Ss P (Fp YORO/0 Rp 70)GO (Sl S2 P)
conditional invariants as they are for exact invariants.
—(1+ypROCO+Rp )St= 0. (7.5)
Therefore any invariant, whether conditional or exact,
It
is to be noted that this equation holds regardless
is an observable. The chief tool for proving these state-
of the choice of the y's. In fact it can be shown that the
ments is the lemma
only eRect of a change in the y's is to produce a gauge
SOC = —yR(G+ —G-)R, (7.2) transformation of the asymptotic Gelds, having the
form
which is a corollary of (6.6), (6.7), and (6.11).With its
by+=Rpht + (7.6)
aid it is straightforward to show that the Poisson
bracket of two conditional invariants is itself a condi- A group transformation (4.1) of the 6eld O0 has a similar
tional invariant and that transformations of the form eRect. Thus
A ~ A+S„a', 8~ 8+S,;b' leave the Poisson bracket by+= 8y G— St (Fp
0+$8— p 'Rp
ORE—
y— 70)8(pj
unaRected. Evidently observables are deGned only
modnlo the field equations. = —RpGp+Rp ypby (7.7)
An important class of conditional invariants are those which takes the form (7.6) with"
which can be constructed out of the asymptotic Jields.
The asymptotic Gelds are deGned by g'+= Gp+R—
p ypRbh (7.8)
~+i= yi G +ij(s . s . oyp) For this reason the asymptotic 6elds can be used to
construct group invariants by the dozen. One has only
O' G—
( 's, ,prep -"q'+ . )
p"'— (7.3) to introduce a set of Geld-independent quantities
the notation here being based on the formal expansion I&;, I&;... satisfying
of the action IgRp —0, Imp= 0. (7.9)

2!
'
S . o~i~j+ S . Op~.i~j~k+. . . .
31' (7.4)
and then de6ne
A+ Igpk 8+ Igp (7.10)

The index 0, in either the upper or lower position, in- Since (7.7) holds only when the 6eld equations are
dicates that the quantity to which it is a%xed is to be satis6ed the latter quantities, as well as all functionals
evaluated at the zero point y'=0, which, with the con- of these quantities, are conditional invariants.
ventions of Table I, corresponds to Qat empty space- In practice it is very easy to Gnd differential co-
time. In Eq. (7.4) terms linear in the 00's are absent efGcients I~, I~ with the desired properties, and sets
since q'=0 is a solution of the field equations, and of quantum invariants A+ 8+ forming complete
~

constant terms are irrelevant. commuting sets in the physical Hilbert space are readily
If the amount of "energy" contained in the 6eld is constructed. In this way the quantum states may be
6nite, e.g. , if the Geld has the form of one or more uniquely deGned by the asymptotic behavior of the
Geld.
essentially 6nite wave packets" (which inevitably
Poisson brackets for the invariants A+, 8+. . are
spread in both past and future), then the fields rp+ and
— determined in a straightforward manner with the aid
y will coincide with q in the remote future and past,
of the easily veriGed identity
respectively. The quadratic dependence of the leading
term in the expansion of S~ — S2py ensures that the 60= (1 —Gp"U)G(1 —UGO ),
diRerence between p and y+ will behave like the
potential due to a distribution of charge which becomes where
more and more diRuse in the remote past and future. = F—Fp.
U— (7.12)
Because of the spreading of the 6eld the eRect of non-
linearities diminishes with time, and we anticipate that Thus, substituting (7.3) into (7.10) and using (6.8)
the asymptotic 6elds will satisfy the linear equation and (6.14), we 6nd
S2py+=0. The formal proof is innnediate. Making use
(A+, 8+) = IgL1 —Gp+(Ss —SOP) j
~In the quantum theory one speaks of matrix elements be- XGL1 — (S2 Ss )Gp jIB . (7.13)
tween analogous "wave-packet" states, and then the same argu-
ments apply. In this case, however, a wave function renormaliza- ~ The set of transformations (7.6) forms an Abelian group for
tion constant Z'" should be attached to y+ in Kq. (7.3). For the asymptotic fields. It is to be emphasized that the relation
simplicity we shall omit such constants both here and iri our later (7.8) between the parameters Oi+ and Og raises no issue of attempt-
discussion of the 5 matrix. The. reader should supply the missing ing to map an Abelian group on a non-Abelian group, for the
Z's when needed. 5f+, unlike 5(, depend on q through the presence of the factor R.
162 QUANTUM THEORY OF GRA VI TY. II 1209

If Ss —Sss were the same as F —

Il s then (7.11) would be Here the a's and e's are functions of the 3-vector p,
immediately applicable. The diRerence between the two and the 4-vector p„satisfies
quantities involves R's and Rs's. With the aid of (6.11)
the Eo's can be brought to bear on the I' s, yielding (p")=(Ep) p'=o, E= Ipl (87)
terms which vanish on account of (7.9). The R's on the The e's themselves are the usual complex helicity
other hand must be moved in the opposite direction. polarization vectors satisfying
Using (4.7), (6.11), and —
SssGs+Ia =In it is not
dificult to see that this leads to a set of terms which e+ e+=0, e~ ep=i,
mutually cancel. Hence, 6nally p ep —0, N ep=0, (8.8)
(A+, 8+) =I~CsIa . (7.14) where e~ is an arbitrary timelike unit vector;

8. THE LINEARIZED THEORY. ASYMPTOTIC (8.9)

WAVE FUNCTIONS. HELICITY AND The 3-vectors Re e+, Ime+, and p, in that order, are
LORENTZ I5'VARIANCE required to form a right-handed system. In the case of
the Yang-Mills Geld the X„areeigenvectors of an ap-
The asymptotic 6elds satisfy the field equations and
propriate complete set of commuting matrices within
Poisson bracket relations of the so-cal1.ed linearized
the adjoint representation of the generating Lie group,
theory derived from an action functional of the form
and the index r labels the corresponding internal states. '4
~5, ;,'y+'q+&. Since the linearized theory is well under- The X's may be chosen real and independent of p,
stood we may at this point con6dently pass to the
satisfying
quantum theory in order to get our bearings on the
ultimate goal; a covariant 5-matrix theory. Our first X~aXea ~re Xrag~P +aP (8.10)
task is to construct, from appropriate asymptotic in-
where y is the Cartan metric of the generating group.
j
variants, creation and annihilation operators for in-
coming and outgoing 6eld quanta.
If the quantum version of Eq. (7.14) is now used to
In the case of the Yang-Mills and gravitational Gelds compute corrimutators of the asymptotic invariants
the simplest and most important invariants are, re- F+~„„and R+„„„at
different space-time points, and if
the function Gs, which is determined by the zero-Geld
spectively, the asymptotic curl and the double curl
(Riemann tensor): forms of the operators F defined in Table I, is subjected

F+ „„=—
A+ „,
„,„—„, A+ (8 1)
to a Fourier decomposition, then it is straightforward
to show that the a's and their Hermitian conjugates'5
satisfy the following unique commutation laws:
/kVfr 2 ('P /l6, P1+ 0 Vr /if 9 sr, Pl/ 'P Vd, l4 )1. (8 2)
La+a, a+a]=0, [a+~,a+a ]=~ca (8 11)
Both of these quantities have the linear structure (7.10)
with differential coefficients satisfying (7.9). Using which identify them as annihilation and creation opera-
the well-known cyclic differential identities satisfied tors, respectively. Here the capital Latin indices are
by these invariants (see Table I), as well as the prop- used as schematic labels for the states of the cor-
agation equations responding quanta. The symbol 8~~ is to be understood
aF+ „„=0, (8.3) as the product of a 8 function of the 3-momenta and a
Kronecker delta in the helicity and internal states.
CI R+sv sr = 0 p (8.4) If the quanta of the Yang-Mills or gravitational Geld
it is straightforward to derive the following Fourier are able, through Geld nonlinearities and exchange of
decompositions: additional quanta, to bind each other into stable com-
posite structures, then additional creation and annihila-
tion operators for these structures will have to be intro-
F+~ =i(2') '—
~s
p — (a+,+e+„+a+„e
$p ) . duced. Although nothing is presently known about such
possibilities, we do know that the complete set of all
—e „))X„~e'r
P„(a+„+e+„+a~, *(2E) '~sdp' such operators will determine the physical Hilbert
+Hermitian conjugate, (8.5) space. No other operators are needed for constructing
observables. In fact, if group arbitrariness is made
,== —'(2e-) '"
R+„„. —, Lp, p, (a++e+„e~ +a+ e „e,) explicit the creation and annihilation operators suSce
for the Geld variables q' themselves. Comparing (8.1)
+p„p,(a++e+„e+,+a+ e,e,) ~ The internal states are e in number, where e is the dimen-
p, p, (a++—
e e+++a+ e „e,) sioI)ality of the generating group.
"Hermitian conjugation is here denoted by *. The symbol 1'
p„p,(a+~e+, e+ +—a+ e „e,)5e'r *E '~sdp' will be reserved as an abbreviation for -*
where denotes an
additional matrix transposition in a vector space other than thy
+Hermitian conjugate. (8,6) quantum-mechanical Hilbert space,
1210 BRYCE S. DHWITT I62
TABLE II. Expressions for the linearized Yang-Mills and gravitational Gelds corresponding to quantities
appearing in the abstract formalism.

Abstract Corresponding expression for the

symbol Yang-Mills Geld Corresponding expression for the gravitational Geld

e „„(x,
p)-=(2x) '"x„"e
v'(2~)
„

Rp —i "pP„
6 e(n&, P +n-P&)

Bp —&~"oP~ e(n&.p—+n f&r n& f& )—

= (2n.)
p"„(x,p) — "'x„" (p~. (x, p)) —
= (2x)-'"(e+e e e pe p)
v'(2~) v'(2~)
p'p 0 0 0
s *p ps(p, p'} 0 pp 0 0 h(p, p')
0 0 (P p)'
.0 0 2(P p)'
0 0 0.
0 1 0 0
0 0 0 8(p, p')
s-s(p, p')
.
0 0 pp
p p
0,
Qp(+) G&&&+&~„e'„— =
peen„„Gp&+&(x, x') =
Gp&+&&cv~'r'— (np~nvr+nprnvo npvnar)Gp&+ (x, x')
gp(+) Gp&+& e'=y~eG&&&+&(x, x') —n~"Gp&+& (x,x')
G&&&+»"'=

f ~i+ ' (a—a')

Gp&+& (x, x') =- 8'p, =dppd p'd p'd p'
d p—
(2 )'~. & &

The hypercontour C(+) runs along the real axes in the p', p', p' planes and forms a closed loop in the p'
plane surrounding the pole at +E.

and (8.2) with (8.5) and (8.6) we see, in particular, that and
the most general form for the asymptotic 6elds q+' is S2oRo= 0. (8.16)
p&+'=u'~a+~+u'~~a+~*+Rp' (+~ The latter relation, combined with the locality of Eo,
(8.12) in fact permits one to infer, without computation, the
&&p+ = ua++u*a+*+Rp(~, vanishing of the integrals (8.14) as well as of

where the u's are the functions indicated in Table II —i

and the (+'s are completely arbitrary Hermitian func- Ro so"Romp=0 (8.1/)
tionals of the creation and annihilation operators.
The u's appearing in Eq. (8.12) may be regarded as Equations (8.14) and (8.1/) imply that, as representa-
wave functions for the asymptotic states. Using the tives of the asymptotic states, the u's need be defined
explicit forms given in Table II, one may verify by only up to a gauge transformation u u+Rpt p, which ~
direct computation that they satisfy the following leaves Eqs. (8.13) unaffected. In actual practice the u's
important orthonormality relations: are restricted by a supplementary condition, namely
the zero-field analog of (6.2):
i u so u~p=0) i ut s—
peudZ„= 1, (8.13) 0.
Ro™you= (8.18)
When this condition holds, the u's satisfy
u sp"RpdZ„=O, i dZ„=O,
Rp speu— (8.14) Pou= 0 (8.19)
in addition to (8.15), and Eqs. (8.13) may be replaced
where the hypersurface Z is completely arbitrary except by
that it must be asymptotically spacelike, and where 1
is the super-abbreviation for 8~~. The Z independence
(8.20)
of these relations follows inunediately from (3.4) to-
gether with
(8.15) The validity of the latter orthonormality relations
 QUANTUM THEORY OF GRAVITY. II
follows from the easily verified identity 9. THE CANONICAL FORM OF THE COM-
MUTATOR FUNCTION. THE FEYNMAN
f~ =qX~~ 'R -~ ~Rq 'X~-~,
s~— (s.21) PROPAGATOR
where the matrix (X&)' has the form Although the e's, in virtue of Eqs. (8.13), (8.14),
and (8.17), may be regarded as forming a complete
(X"')-.o =--b-s b."' (8.22) orthonormal wave basis for the operator S2P, they do
not form such a basis for the operator Fp. Fp possesses
for the Yang-Mills Geld and
additional, nonphysical wave functions having ortho-
=——
(X"')„„, b„;o„"'8„,b„
— "' (8.23) norrnality properties more general than (8.20). The s I'
define only the physical subspace of such functions.
for the gravitational Geld. In the case of the Yang-Mills and gravitational Gelds
The supplementary condition (8.18) does not yet
completely determine the I' s. Equations (8.18) to
(8.20) remain unaffected by gauge transformations
u~ I+Rpt p for which Fop 0. To— obtain the u's of
and Bp' e „"
it turns out that a complete basis for Fp is obtained
simply by adjoining to the I's the functions Ep' e
where the v's constitute any complete
basis for the auxiliary operator Fp.
Table II a further condition must be imposed, of the
form
Foe=0, (9.1)
whence also
Ro Qol —0 a (s.24) Fo~oe= 0 Fo~oe= 0 (9 2)
Many diferent choices for Bp can be made which lead By straightforward computation one may verify that
to the same I' s. It will turn out to be a convenience to in addition to (8.20) we now have relations of the form
choose Bo in the particular way indicated in Table II
where its momentum-space forms, as well as those of
Ep, are given. The 4-vector p„appearing in the ex- i) Rp f "Rp5dZ
p )tlV,
pressions for Bp is defined by

p. = p.+2&.&"p. , (8.25) i i)tR—

p f,~R,i)dZ„=sY, (9 3)
where e„ is the tiro. elike unit vector of Eqs. (8.8)
and (8.9). It is easy to see that p„, p„,
like is null. In with all other similar "inner products" of the functions
analogy with the terminology employed for null hyper- u, Epv, Bpv and their complex conjugates vanishing.
surfaces p„may be called a characteristic vector and Since the matrix
p„ the bicharacteristic of p„relative to n„.
The presence of e„ introduces a nonrelativistic ele-
ment into the formalism, the effect of which must be
1 0 0
0 0 E (9 4)
determined by asking for the changes in the n's and a' s .
0 Ã 0
produced by changing ~„. It sufFices to consider in- is symmetric, an orthogonal basis for Fp can be found if
finitesimal changes be„ leaving (8.9) invariant. If desired. However, since (9.4) turns out to be a non-
Eqs. (8.8) are to remain invariant in form, one readily positive-definite matrix, the positive normalization
Gnds that the e's must suffer the corresponding changes (8.20) cannot be extended to the entire basis. It proves
convenient not to insist on complete orthogonality but
he+&= Wibble+" &(e+. be),
(n p) 'p— (8.26) to leave the basis as given. In this form it will be called
a cGROszccl &@sos.
where by is an arbitrary infinitesimal angle. If, in A particular choice of v's for the Yang-Mills and
addition, the form of the decomposition (8.12) is to gravitational Gelds is given in Table II, along with the
remain invariant the a's must be multiplied by phase
corresponding matrices E. Using the table it is straight-
factors e+'"& where s, the spin, is 1 for the Yang-Mills forward to verify that the function Go, which appears
field and 2 for the gravitational field, the + sign or- in the commutator of asymptotic invariants, may be
sign being chosen according as the helicity is positive given the following canoyucal decompositiol:
or negative. The f'irst term on the right of (8.26) pro-
duces inverse phase changes in the I's while the second gp —gp
(+) +.gp (—) (9.5)
term, produces a gauge transformation which may be —Gp — — 6 (—)
Gp + Gp(+) (9.6)
absorbed into the last term on the right of (8.12). The
phase changes produce corresponding changes in the igp'+)=NNt+Rpi)X 'vtRp +RpviV ' i)tRp . (9.7)
elements of the S matrix but leave transition probabili-
ties unaffected. Observationally, therefore, the classiGca- .~6In the -case of massless fields having spins greater than 2
tion of states according to helicity is Lorentz-invariant. these functions do not-suKice to complete the basis.
 8 RYCE S. DEW ITT 162

The function Go(+) is called the positive energy flection. and (9.5), which lead to the boundary conditions
In a theory with no gauge group iGp&+), regarded as an —Gp&+)'J
Herrr~itian matrix, must be positive semidefinite if a
Gp'J= i + Jj )
— —
state of lowest energy the vacuum is to exist. In the —G, (—)6 j%j- (9.18)
present case iGp&+) need be positive semide6nite only
in the physical subspace. Since the physical subspace is
These conditions may be generalized so as to be appli-
represented by the functions I
and N~ we see that this cable to nonzero background fields. In the general case
the Feynman propagator is de6ned as that Green's
requirement holds. It will be convenient to introduce a
function which, as a function of its 6rst argument, has
special symbol for the projection of Co into the physical
subspace: only positive energy components in the remote future
and only negative energy components in the remote
Qo
—Qo(+)+go(—) (9.8) past. These boundary conditions suQice to yield the
variational law
(9.9)
bG=G SIC G, (9.19)
(9.10)
and the expansions
The importance of the canonical form for Co lies in
G=Go(1 —UGo) '=Go+GoXGo (9.20)
the presence of the Rp s. It is easy to see, for example,
that in virtue of (7.9) the quantum version of (7.14) = (1—UGo) 'U= U+ UGoU+
X— . (9.21)
bamediately reduces to
The variational law (9.19) has exactly the same form
[A+,8+ j=iIgS os, (9.11) as Eq. (6.19) for the advanced and retarded Green's
functions. The Feynman propagator has, in addition, a
which is obviously consistent with the decomposition
symmetry not possessed by G+, namely
(8.12). Other more important uses of the canonical form
will be encountered later. Go =Go, (9.22)
For completeness we record the following additional
relations satisfied by the quantities thus far introduced: which follows from (6.8), (9.6), (9.16), and the (assumed)
self-adjointness of F. The Feynman propagator and
—Ro yoRo=Ro yoRo,
Fo= (9.12) its complex conjugate may therefore be characterized
as the only Green's functions which, when regarded as
yo 'Zo yoRov=vM 'E, (9.13) continuous matrices, obey all the rules of finite matrix
iGp&+&=vM 'vt, (9.14) —
theory a characterization which may serve to define
them uniquely even when the condition of asymptotic
fatness does not. hold and S-matrix theory ceases to
'L v jo"vdZo =0 ) i vt fo—
~vdZp=M ) (9.15) exist. In a Bat Euclidean 4-space Ii has only one unique
Z Z
inverse (Green s function) which vanishes asymptotic-
e+„e „+e+„e
))„„=— „
ally, and the Feynman propagator is obtainable from
this inverse by analytic continuation to Minkowski
space-time, the "direction" of the continuation being
correlated with the direction in which time is chosen to
Here is the positive energy part of the function
Gp~+&
"Qow". In this sense the Feynman propagator may be
Go+ —Go and Eq. (9.13) assures consistency of (6.11) —
Ii, its complex conjugate
regarded as the inverse of
with the decompositions (9.7) and (9.14). The explicit
being obtained by analytic continuation in the alterna-
form of the matrix M appearing in (9.13), (9.14) and
tive direction.
(9.15) is given in Table II for the particular v's which are We now record for later use a number of identities
adopted there. The operator go& is related to Fo in the involving the various Green's functions, which are
same way that fo" is related to Fo. The identity (9.16), derivable by straightforward algebraic manipulation of
which follows from Eqs. (8.8) and (8.25), is used re- previous equations:
peatedly in the verification of the decompositions (9.7)
G+= Go+(1 —UGo+) '= Go++Go+X+Go+, (9.23)
and (9.14).
In the classical theory a dominant role is played by X+=—(1—UGo+) 'U= U+UG+U, (9.24)
the Green's functions G+. In the quantum theory this
role is usurped by the Feynman propagator. For zero X=- (1&X+Go(+)) 'X+, (9.25)
GeMs the latter is defined by —
1 aXGo(+) = (1aX+Go(+)) ' (9.26)
(9.17)
1+XGo= (1&X+Go(+)) '(1+X+Go+)
(9.27)
the equivalence of the two forms following from (6.5) =(1—UGo) ',
 QUANTUM THEORY OF GRAVITY. II
1 —UGp+ = (1—UGp) (1WXGo&+&) S(k) —(1+So~X )Sp(+)(1 ~X~Sp(k))-i
= (1—UGo)(1&X+Go&+)) ' (9.28)
&(I+ X+@o+) (9 42)
G —G+ ~G(+) (9.29) So So p — (Sp(+) Sp(-)) (9.43)
G(+) = (I+Go+Xk)Go&+)(I~X+G (6))—i Xg*= —Xp(So&+' —
Xg — Sp' ') Xg*, (9.44)
X (1+X+Gp+) (9.30a)
Sg —S~*=—(1+SogX~) (Sp'+' —So' ')
= (1+GoX)Go&")(1%XGo&+)) ' X(1+X,*So,*). (9.45)
X (1+XGo), (9 30b) The only difference is that So~, S~, X~, unlike Gp, G,
Gp —Gp~ ——(Gp&+& —G& &), (9.31) X, are nonsymmetric, which accounts for the & signs
attached to them. From (6.8) and (9.9) it follows that
X—X*=—X(G, —Go — X*,
&+& &
&) (9.32)
G— G*= —(1+GoX) (G — — p&+& G&)
&
&)
S~-=S, S,-=S, X,-=X . (9.46)
X (1+X*GoP) ~ (9.33) Ke must evidently ask what difference it makes if we
use S instead of S+ as a replacement for G. In order
Equations identical in form with these are satisfied by to show that it in fact makes no difference we must
the corresponding functions Gp+, G+, Gp, G, Gp(+), Grst develop the formalism somewhat further.
G&+), X'+, X', and U associated with the operator F.
In the theory of the S matrix the function G plays 10. EXTERNAL-LINE WAVE FUNCTION'S.
the role of the propagator of Geld quanta. When an FUNDAMENTAL LEMMA
invariance group is present this function suGers from
a fundamental defect, namely, it propagates non- Consider the following functions:
physical as well as physical quanta. For purposes of
defining "external-line wave functions" (see Sec. 10)
(10.1)
and checking the unitarity of the S matrix (which is
defined only between real physical states) it is con-
venient to introduce alternative functions which prop- In virtue of Eq. (6.4) these functions satisfy
agate real quanta only: Ff+=0 (1o.2)
—G+wS &+),
Sg= (9.34) and reduce to the asymptotic wave functions I
in the
S (6) = (I+Go+Xk)So(k) (I ~X+So(+))—i remote future and past, respectively. If (as is always
&&(1+X+G,+). (9.35) assumed) 8 is based on a choice of y's which cor-
responds to the same supplementary conditions (6.2)
The use of these functions, however, destroys manifest as those which are imposed on the I's LEq. (8.18)j
covariance and, when divergences are present, is then the f+'s will also satisfy the equations
limited to formal arguments. In actual calculations the
functions G, G+, G, G+ must be employed to assure R yf+=0, Spf+=0 (1o.3)
consistency of renormalization procedures. One of our By making use of the combination law
tasks will be to show how to pass formally from one set
of functions to the other.
The functions S+, S&+), etc. satisfy a list of identities Gf»GdZ„, (10.4)
similar to those satis6ed by G, G(+), et ut. : I
= Go+~So&+),
Sop— (9.36) which is a special case of (6.4), and taking note of the
symmetries (6.9) and f» = —f»(cf. Eq. (3.2) as well j
S =S, (1—US, )-i
(9.37) as the fact that f» reduces to fp" in the remote past
= So~+SopXgSop, and future (because the background field is then dis-
Xg=—(1 —USpp) 'U= U+USgU, (9.38a) persed to a state of infinite weakness), one may show
that the f+'s constitute two distinct complete ortho-
= (1+X+So&+&) 'X+, (9.38b) normal bases for infinitesimal disturbances on a non-
iWXgSo&+) = (iaX+So&+)) ', (9.39) vanishing background:

1+X~Sop= (1&X+So'+)) '(1+X+Go+) f+ f"f dZ»= ' f ~ "f dZ»=

(9.40)
=(1—USpp) ',
(10.5)
1—UGp+ —(1 —US op)(1WXgSo&+))
= (1—USp~)(1&X+So&+)) ', f~dZ„= i f+t s —
i f+tf»— "f+dZ„=
(9.41) Z Z
 "
The f~'s are basis functions for "classica'1. waves. satisfy
In the quantum theory a different basis, satisfying
boundary conditions which tak. e pair production into
account, must be employed. The method of construct- From this it follows that
ing the latter basis will be most clear if we 6rst obtain IiRg+= 0. (10.13)
an alternative form for the f+'s. Taking note of the
kinematic structure of G we may rewrite Eq. (10.1) Since the functions Rg+ coincide with Rpv in the remote
future and past, respectively, we may write
Rg+ = (1+Go+X+)Rpo. (10.14)
~G+fp udge„
i From (10.11) we may also write

= G+(Fp Fp)u= —
—G+Fpu Rog+ = Ro(1+Go+X+)o. (10.15)
= (1+Go+X~)u, (10.6) Subtracting (10.15) from (10.14) and making use of
the zero-field form of (6.11), we obtain
and, in view of the supplementary condition (8.18), also
(R —Rp) g+ = Gp+(X+Rp —ypR pyp '2'+) o. (10.16)
f+ = —G "Spou. (10.7)
The desired lemma then follows on applying the opera-
These forms suggest that the modified functions which tor Fp.
we seek are
X+Roo = y pR py p 'X+o F-p(R p) g+.
R— —
(10.17)
f= —GS&o—
u= GF pu= (—
1+GpX)u, (10.8) —
The quaritity (R Rp)g+ appearing in the last term
in which the Green's functions G+ are replaced by the of (10.17) vanishes at infinity rapidly enough so that
Feynman propagator. However, such functions are integrations by parts may be performed when it appears
inappropriate for the following reason: In the remote as part of a larger expression. This means that the
past they possess not only components from the physical operator Ii p attached to it may act in either direction.
basis I but also nonphysical components which have Therefore, mak. ing use of the supplementary condition
been "scattered backwards in time" and which appear (8.18), as well as o Rp yoRp —v Fp ——0, we immediately
because the quantity X has nonvanishing matrix ele- obtain the corollaries
ments between physical and nonphysical states.
The desired functions are obtained from (10.8) by I X+Rp'v = 0 ) v Rp X+RE=0, (10.18)
substituting S~ for G: which hold also when the u's and/or o's are replaced by
= —S~Soou= —S~Fpu= (1+SppX~)u
fg — (10.9a) their complex conjugates. Referring to Eqs. (9.10)
and (9.38b) we see that these corollaries in turn imply
= (1+Go+X+) (1+ So'+~ Xg) u, (10.9b)
u gyRpo =0 o Rp XyRpo = 0, etc. (10.19)
the final form being obtained through use of (9.39) and
(9.40). In virtue of the decomposition (9;10) it is ap- Next, by algebraic manipulation of Eqs. (9.36), (9.37),
parent that these functions can be expressed linear and (9.38) we find
furcation-sas
combinations of the functions f+ and their complex —X =X+(So+—So )X,
X+ (10.20)
conjugates. They therefore satisfy
Ff~=0, R —
pl~ 0, Spfg=0. (10.10) S+—S =(1+S~X+)(S~—S~)
X(1+% S. ), (10.21)
The f~'s are called exteruat 1iue waoe It can. —
be shown that they differ from the f's of Eq. (10.8) S~ S~= &o—So (10.22a)
by an amount which cannot be expressed as a -gauge = —iRpH, " 'v~R —sgp~lV ' ~~R
transformation. The difference between f+ and f, +iRoo*iV 'o Rp +iRpo*N 'o Rp, (10.—
22b)
however, caw be so expressed, and the & signs are
therefore. physically irrelevant. For the proof of this in which use has been. made of the canonical decomposi-
we now derive a fundamental lemma. tion (9.7). These results may finally be combined with
We first introduce the functions (9.40), (10.14), and (10.19) to yield
g+= (1+Go+X'+)u, (10.11) f+
—f-= (1+S~Xj.)(S~—S~)X u
which are related to the e's of Table II in the same way
that the f+'s are related. to the u's, That is, they coin-
cide with the v's in the remote future or past, and
= (1+Go+X+)(1+So(+)X+)—i

=R(
XRo( ioN 'ot+ io*N 'o —
—
)Rp g
ig+N 'ot+ig+*N 'o )R—p of u,
—
(10.23)
u—
162 QUANTUM THEORY OF GRA V I TY. I I

showing that the two functions indeed diHer froid one to vanish at inf&nity, the a'symptotic wave functions
another only by a gauge transformation. and the zero-point field remain unaffected by group
transformations of the background field. Only the
11. AMPLITUDES FOR SCATTERING, PAIR PRO- 'Green's functions G+, G, etc. change. Owing to the care
DUCTION, AND PAIR ANgfIHILATION BY THE which has been taken to construct these functions in a
BACKGROUND FIELD. THE OPTICAL manifestly covariant manner we may write at once
THEOREMS WHICH THEY SATISFY.
PROOF OF THEIR GROUP bg+ii —(Ri Gkkj +R'j G+io)bga (11.9)
INVARIA5'CE
which may be inserted into
uter
Another important relation may be obtained by in-
X+ = ~O~G+~0 (11.10)
serting (10.22b) into (10.20) and using (10.19):
u (g+ —g )u=0, u'(X+ —X )u=0. (11.1) This in turn may be inserted into

From this it follows that the quantities bg &6) —Gp+bX+N p&+) (1+X+go &+i)—i(1+X +Gok)
I= u—)' (11.2)
~ (1+;gokXE)go &+i (1~X+@o&6) )—ibXEN &+i
X (1&X+So&+&) '(1+X+Go+)+ (1+Go+X~)
V—
= utX~u*, (11 3) X No'"'(1+X+So'+') 'bX+Go~, (11.11)
A=—u X~u, (11.4) which follows from (9.35). Owing to the boundary
are independent of the ~ signs, showing once again the conditions on the b$ the arrow on one or the other of
irrelevance of the signs. the Fp's in (11.10) may always be reversed. As a result
I, V, and A are, respectively, the amplitudes for the second term of (11.11) vanishes, while the first and
scattering, pair production, and pair annihilation of field third terms together yield
quanta by the background field. More precisely, they
are the amplitudes for these processes when it is assumed bN &+) ii (Ri iN—
&+i&i+Re &@&6) i&)b(a (11 12)
that the quanta themselves do not interact with one which shows that 0&+' and S~ have the same trans-
another but behave as the quanta of a model 6eld formation law as G+ and G. Inserting this transforma-
theory with action functional — ', 5;,P')&'.
tion law into
By making use of (9.46), as well as (9.44) a, nd its
transpose, one easily verifies that these amplitudes bled = FobNPo,
satisfy the following relations:
and noting that one or the other of the arrows is again
V=V, A. =A. , reversible, we immediately get the desired result:
I—It =i (IIt+ V Vt) =i (I&I+Ate), .
M =0, 8V=O, Q. =O. (11.14)
A —Vt=i(AIt+I Vt) =i(VtI+I*A), (11.7)
With the aid of (10.9a) we also get, in a similar manner,
V = i(.It V+htI ) .-(11.8)
ht=i (IAt+ VI*)—
b4'=R'-, ;f. «. , (ii.is)
Equations (11.5) express the Bose statistics satisfied by
the 6eM quanta; Eqs. (11.6), (11.7), and. (11.8) are which will prove useful later.
relativistic generalizations of the well known optical The demonstration of invariance under changes
theorem for nonrelativistic scattering. The latter equa- in the y's is more complicated. We first note that
tions play an important role in the verification of the in- order to preserve the supplementary condition
unitarity of the 5 matrix, as will be demonstrated later. (8.18) under a change in the y's, the u's must suffer
The amplitudes I, V, and A are not only independent the gauge transformation"
of the ~ signs but are group-invariant as well. In the
bu=RpGpRp bypu=gp ypRpyp 'Ro byou
present formalism group invariance has three distinct
aspects: (1) invariance under group transformations of WRogo&+~Ro byou ~ (11 16)
the background field; (2) invariance under changes in
Froni this, together with (9.9) and (9.10), it follows that
the Green's functions, as well as in the asymptotic
wave functions I, resulting from changes in the y's; and b@o'+'=Go+yoRoyo 'Ro bromo'*'
(3) invariance under gauge transformations of the u's
for which the gauge parameters to satisfy Fpt p=0. 'r +No'+'bvoRovo 'Ro vogo+~Rogo'+'Ro byoSo'+'
—&Ro . (11.17)
Since the parameters b& of Kq. (4.1) are required WSp&+&bypRoGp&

"Changes of types {2) and (3) together yield the most general '.The e's may also su8er an additional change yl=gpt0
gauge transformation of the e's. where Ppb/0=0. See Eq. (11.30) ff.
 BRYCE S. 0EWITT
With the aid of (6.19), (8.18), and (936) this yields and (10.19). By making use also of the identities

8$oy=So~bFoSoy —RoGo+ Ro byoSo +& S~yR = RG+y, (11.27)

—So&+&bypRpGo& &Ro, (11.18) (1+So+&+)Zoo= R(1+Go+X'+)v, (11.28)
which are proved, respectively, with the aid of (6.11),
(9.35), (10.3), (10.6), and (9.40), (10.14), (10.18), we
8Fo=bvoRpvo 'Ro po+roRob7o 'Ro po may recast expression (11.26) in the form
+7oRovo 'Ro byo. (11.19) bf, =R)G+R-b~ f,~(1+&.",+X+)
Equation (11.18) may be used with (9.37) to obtain XGo&+&Rp byoSo&+&$~uj. (11.29)
8$~= $~8FoSp+Sgb USp
There remains only to show the invariance of I, V,
and A under gauge transforDiations
W(1 —SopU) 'RoGo'+'Ro 87oSo'+'(1+ US+)
bu= Robe p, (11.30)
~ (1 So~U)-&So&+&bypRoGo&-&Ro-(1+ US~), (11.20) —
where bi p satisfies Ppbf'p 0. This— is an almost
, however,
where immediate consequence of Eqs. (9.38b) and (10.18)
and will be left to the reader. An explicit form for the
bU= 8F —8FO, (11.21) change in the f+'s can be obtained by first decomposing
bi o into the &&'s of Eq. (9.1);
Rby 'R y+yRf 'R by.
8F=byRy 'R y+y— (11.22)
(11.31)
Equation (11.20) may in turn be used with (9.38a) to
where the 9's are certain coefficients. Use of (10.9b)
obtain
and (10.14) then yields
8X~= (1+XgSop) 8F(1+SogXp) —8Fp —8FoSo~Xg 8 f~= Rg+Q. . (11.32)
p 8Fo+ (RpG&+&Rp byoSo&+&
We note that Eqs. (11.29) and (11.32) both leave the
+So&+'8VoRoGo& 'Ro )X+. (11 23) validity of Eqs. (10.10) undisturbed.

Wenownotethatinvirtueof 8.18 Eq. 11.16 may

be reexpressed in the form 12. VACUUM STATES RELATIVE TO THE BACK-
GROUND FIELD. ABANDONMENT OF THE
bu=SopbFpu&RoGp&+&Ro bypu. (11.24) STRICT OPERATOR FORMALISM.
CHRONOLOGICAL PRODUCTS
We also note that u 8Fpu=0 and, in virtue of (10.10), AND TREES
f+ 8Ff+=0. Therefore, In order to build up the S matrix we begin with the
8+= bu g+u+u bg~u+u Xybu vacuum. The vacuum state is customarily defined by
the condition
=Wu (%~Romp'+'Rp byo@o'+'X~
a+(0) =0, (12.1)
+ X/$0'+'byoRoGo' 'Ro Xy+byoRpGO'+'Ro X/
where the a+'s are the annihilation operators of the
+X~RoGo&+&Ro byo)u, (11.25) decomposition (8.12). In this state no field quanta are
which vanishes by (9.14) and (10.19). Similarly, 8V=O
present, and the background field itself vanishes (flat,
empty space-time). It will be noted that no & signs
and 81=0.
have been affixed to the symbol 0), thus implying that
As a byproduct of this demonstration we again get a ~

transformation law for the f~'s. Thus, using (10.9a)

and the fact that bS2=0 as well as S2'8N=0, we dnd Gpk ~
FIG. 1. Graphical repre-
sentation of the bare I-
8fg= —8$gSoou= —8$~Fpu ~I)al ~ +p point functions for I =3,
4, 5, 6. The symbol P in-
= SybF fy~(1+Soy~y) (Ro~o&+&Ro bpoSo'+& sok&~s *+ )+ +PI~
pg dicates that the indices
associated ~vith the ex-
+So&+&byoRpGo&-&Rp ) (1+XgSo~)FQu e +p ~+ pi{)
I I ternal
perm uted
lines are to be
just suKciently
=S~yR7 'R by f~~(1+S-pgXp) + peo +p45
to yield complete sym-
metry. The numerical sub-
XRpGo&+&Ro bypSo&+&X~u, (11.26) script indicates the number
of permutations required in
the final expression resulting from use of (9.2), (9.14), each case.
162 QUANTUM THEORY OF GRAVITY.
The states 10,Woo) are functionals of the classical
in the course of time. "
if no quanta are initially present none will be produced
background. Because the background is capable of pro-
Instead of working with 0).we shall find it convenient
I
ducing and absorbing pairs, triplets, quadruplets, etc.
to work with relative eucuts 0, & ~ ), defined by
I
in individual elementary processes, any number of
quanta may eventually be produced, and hence the two
n+10, ~ )=o, (12.2) states are not identical. Our chief concern will be to
study the response of the vacuum-to-vacuum amplitude
where the annihilation operators e+ are based on a
separation of the total Geld q into a classical back- (o, oo IO, —ao) to variations in the background Geld.
Schwinger" has used external sources for this purpose
ground y, satisfying the classical 6eld equations (2.2),
and has shown that all physical processes can be com-
and a quantum part P satisfying the same commuta-
puted once the vacuum response itself is known. There
tion relations as rp. The classical background is always
is a well-known difficulty, however, in using sources
assumed to contain a Gnite amount of "energy" and
when a non-Abelian invariance group is present, namely,
hence it not only superposes linearly with P in the
remote past and future, where both satisfy the asympto- group invariance requires the source to depend on the
Geld. By working with a "free" background Geld we
tic field equations (7.5), but it also disperses ultimately
avoid this difhculty.
to a state of infinite weakness. Ke may therefore write"
Suppose now the background GeM seers an in-
P= %+4~ (12.3) 6nitesimal change 8y which satis6es (6.1) so that the
Geld equations are maintained. Since the total Geld
sr+ —q)k+ Pk (12.4) operator q does not depend on which classical Geld is
chosen as the background the operator P must suffer
y+ = ua++u*u+*+ Ref+, (12.5) (modulo an irrelevant group transformation) an opposite
$+= un++u*n+a+Re((+ f+),— (12.6) change:

a+ = a++ a+. lip = —8p, lin+ = —8a+. (12.8)

(12.7)
This produces changes in the vacuum states satisfying
w Vl7hen dealing with massless bare (i.e. unrenormalized) quanta
,
one must be cautious in asserting that the vacuum is stable. For (n+ —g|s+)(IP, ~ ~)+ g IP, ~ oo)) =0 (1,2.9)
example, the I.agrangian 2 = — xsqr „rp,& — -', py' — (1/24)Xs4 (X)0)
appears to describe a self-coupled massless scalar field satisfy- or
ing the usual condition (y)=0 in the vacuum. However, one
hnds in fact (y)= — 3p/k The Lagrangian should be rewritten
ss(g„g— &+m, 'y.')+'gg' (1-/24)— hy4, where p= y — (y) and (12.10)
m2=3p2/2X, to display the fact that as long as p, /0 the actual
quanta carry mass. This result shows up in another way if one By making use of the orthonormality relations (8.13),
attempts to compute the self-decay rate of the quanta on the as- (8.14) and the decompositions (12.5), (12.6), it is easy
sumption that they are massless. Because of the possibility of
having the momenta of massless quanta all parallel, conservation to see that the unitary transformation which yields
arguments cannot be invoked to exclude the decay, and, contrary (12.10) is
to a widespread impression, phase-space arguments do not suKce
but must be investigated in detail. It turns out that the decay rate
into softer quanta is infinite. The infinity arises from diagrams
with internal lines. Such lines, when not in closed loops, are neces- +
$+ ™so"~y+d~„l0,oo). (12.11)
sarily on the mass shell. Nevertheless, it is the y' term of the
Lagrangian which gives the trouble and not the q4 term. When
p =0 phase-space limitations prevent the dangerous diagrams from Hence, remembering that f+=P and @+= y in the
contributing, and the decay rate then vanishes.
In the case of the gravitational held the work of Brill (see
remote future, that P = $ and y = q in the remote
Ref. 13) and others sho~s that Hat space-time really is the state past, and that so&= s& in bath regions, we have
of lowest energy, i.e., that bounded asymptotically vanishing
deviations from Qatness correspond to an iecreuse in energy.
Similarly the condition (A„~)=0 (modmlo a gauge transformation)
holds in the ground state of the Yang-Mills field, as follows from
the positiveness of the term r'F;;F~'& (i, j=1, 2, 3) in the H'amil-
tonian when the generating group is compact. As for the stability
of the quanta themselves, it turns out that in the graviton case
the relevant matrix elements all vanish when the quanta have
parallel momenta. The reason is that the coupling is always of
the derivative type, which introduces momentum vectors having (12.12)
vanishing contractions with the polarization tensors. In the Yang-
Mills case the cubic term in the Lagrangian yields a matrix We have now reached a critical point. From here on
element which likewise vanishes on account of derivative coup- the description of the quantized Yang-Mills and gravita-
ing. The matrix element of the quartic term does not vanish,
but in this case phase space limitations prevent the decay. tional fields in terms of operators must be dropped. No
Gravitons and Yang-Mills quanta are therefore both stable. one knows (or at any rate no one has yet shown) how
~ Because of the linearity of the group transformation law (4.i)
to develop a consistent operator language for these
P transforms according to the homogeneous law 5qV'=R', ,$28& .
From (7.7) it follows that the asymptotic helds transform ac-
— A
cording to bye+ = Roao+Ro pPf. "J. Schwinger, Proc. Nat. Acad. Sci. U. S. 37, 452 I'195j.).
 BRYCE S. DEWITT j.62

at the same time manifestly covariant further variational formulas. We first compute
Gelds which is
and useful for calculations. " a(0, ly'lo, — )=(o, icy'lo, —
%hat we shall do is to retain the operator language
only for Gelds which possess no invariance groups. +Z(~(o, - I
s'&) ~"&~'Io, --)
After developing the theory of such Gelds to the point +&(0, I
y')y"sA'I o, — ). (12.15)
at which all statements can be made in c-number. Here the $'& are eigenvectors of the complete set of
language we shall then modify these statements in
j
I

commuting operators P', including =i, taken over a

such a way as to become applicable to the Yang-Mills
hypersurface Z;, and the summation is to be extended
and gravitational fields. over all the eigenvalues. If the variation (12.15) is due
To achieve maximum simplicity we shall assume not to a change in the background field we have
only that the Geld q possesses no invariance group but
also that its components all commlte with ore another at (0' —&v')(l 0')+&I 0'&) = 0"(I 0'&+~I y'&), (12.1&)
01
the same space tinze p-oilt In p. ractice this limits us to
scalar Gelds possessing vertex functions S,;;I,... which (~'—s")& Is'&= &~'I s'&, (12.17)
involve no derivative couplings. However, it in no way where P' is restricted to Z; .In view of (12.14), the unitary
limits the member of scalar fields embraced by the transformation which yields this is
symbol q' nor the algebraic complexity of their mutual
couplings. Hence the abstract notation is still appro- a y'&=
I
—s p s "spd&„lp'). (12.18)
priate, and the combinatorial (i.e., diagrammatic)
aspects of the theory are identical with what they will
be for the fields of actual interest. It is by studying the Making use also of (12.8) and (12.11) we therefore get
combinatorics that we shall be led to a self-consistent ~(o, Iy'lo, — )= —~i *(0, lo, — ) —scabs,
x(o, -lz(~~')Io, —-), (12»)
„

general theory.
The chief advantages of the restriction to scalar
Gelds and nonderivative couplings are that the ordering where T denotes the chronological product.
of factors in the Geld equations becomes immaterial,
33 Since the field q' now has no invariance group the
operator Ss is nonsingular, and Eqs. (12.12) and (12.19)
chronological products can be defined unambiguously,
may be rewri. tten in the forms
and the operator 6'&' which appears in the conunutator

(12.13)
«, - l~ lo, --)=G (~S'~. )
G'&'
X(0, ~ lo, —~), (12.20)
reduces to the c-number function of the background
field when the space-time point associated with the
index i is in the immediate vicinity of that associated
(0 " I
2'(&'&') Io — &= I
—sG"+G"
Qto"
GjV
i hy'J
with j.
The latter simplification has the consequence
that X(0, Io, — ), (12.21)
the Feynman propagator being used because of the
8'v 'd&„= ',
g ', &')™s+; &v (12.14) boundary conditions specified by the relative vacua.
Continuing in this way we obtain an infinite set of
equations, all of which are comprehended in the
where Z; is any spacelike hypersurface containing the
generating-functional formula
space-time point associated with i.
~— ) " )t
With these simplifications we are ready to obtain QQ

(0 "lz'(0""e'")Io-
n=0 gg!
'~The most beautiful attempt at such a language is that of
S. Mandelstarn /Ann. Phys. {N.Y.) 19, 25 (1962)g. lay propagat-
ing local frames from infinity along; intrinsically defined paths,
Mandelstam is able to deal exclusively with operators which are
=exp i P—
n=s
); .
pgt
. ), Gr
j
coordinate-invariant and hence possessed of unique commutation
relations. Mandelstam's formalism is on the borderline of being
practical, but unfortunately becomes excessively complicated
beyond all but the simplest calculations. A choice of paths is
Xexp X;G'& 0, ~ 0, — ~,
ultimately equivalent to construction of an explicit gauge, and
the freedom to work with local (differential) rather than rionlocal
(integral) gauge conditions, is to be preferred if at all attainable.
where the X's are arbitrary variables and the G" " are
~ Under these restrictions the usual practice of "normal defined by
ordering" is unnecessary as far as the formal theory is con-
cerned. The residue obtained on converting from ordinary to Gil in= Giu'I .Gin-Nim-+ Gin-lie
normal ordering can always be lumped with vertices of -lower
~ ~
(1 2 23)
order. /pal $yjn-2
162 QUANTUM THEORY OF GRAVITY. II i219
It is easy to verify that the operators G'&8/bq&& tudes are given by
commute with each other, and from this it follows that
the G" ' '"
are completely symmetric in their indices. (Ag' A„',oo lA) ~
A„„—
oo),
These functions, which are known as the bare n;poi yt
functions, have a well-known graphical representation = Q +(m, n; l)bing'Ag' ' ' bAg'Ag
which is illustrated in Fig. 1 for the cases x=3, 4, 5, 6. L 0

Feynman propagators are represented by lines and bare

vertex functions 5 by vertices or forks with nz prongs.
X(A~(' A„',~ [A)+) .A, —~), (13.5)
The lines are joined together at vertices in the same where'4
ways that the propagators in the explicit expressions '
for the G'&'' '&'s are coupled to vertex functions by (A, A '~oo IAg'''A oo)
dummy indices. It is easy to see that the diagrams
—( i)m+nujq, 1 e. . . uj, sg .
~

g o. . . . o

I2'(y"" y"y' " y'-) l0 — )

n, ~gill Pnle
making up G" ' "~ are obta, ined from those for G"
' "& '
X(0
by inserting an additional external line in all possible
ways. G" ' "
is therefore expressible as the sum of all
XS,y~;~ ' ' 'S, o; u' gz' ' 'u' (13.6) g,
distinct trees having n branches, the indices attached and where the symbol I' in (13.5) indicates that the
to the latter being permuted just sufficiently to yield expression following it is to be summed over all distinct
complete symmetry. permutations of the A's and A "s, the subscript
A tree is any diagram which has no disconnected parts
but which is divided into two disconnected parts by m tn!
cutting any line. A tree therefore possesses no closed
(m, u; l)—=
—l)!(u—I)! (13.7)
(m
loops. 1A'e shall see that the first factor on the right-
hand side of (12.22) describes all the lowest-order or denoting the number of permutations required in each
bare scattering processes. The radiative corrections, case. It is important to realize that the LSZ method is
which involve closed loops, are all contained in the formally applicable even when an invariance group is
remaining factors. present, and hence Eqs. (13.5) and (13.6) hold in the
general case. This is because the creation and annihila-
13. DEFINITION OF THE 8 MATRIX. tion operators, in virtue of (8.13), (8.14), and (12.6),
ITS STRUCTURE IN THE ABSENCE are unambiguously defined by
OF A5' INVARIANCE GROUP
The 5 matrix,like the vacuum states, may be defined u so"$~p~ e+= ut Fo&gdZ„. (13.8)
relative to the background 6eld. It then has as elements
the amplitudes The only diKculty is that we do not yet know, inthe
(Ag' A„',~ lAg A, —oo), general case, how to calculate the chronological pro-
ducts appearing in (13. 6).
pent.
where In the restricted case of scalar Gelds with nonderiva-
tive couplings Eqs. (12.22), (13.5), and (13.6) permit
(131)
us to write the following compact expressions for the
If the possibility of stable composite structures is scattering operator:
ignored, the above states form two complete orthogonal
oo
bases in the physical Hilbert space, and the scattering y+ii. . . y+~~g . . o. . .g . . o
amplitudes may be regarded as the matrix elements,
S ~
p
%=2 nt
with respect to either basis, of the unitary operator
X(0, ~ lT(f~'i. . . f~'") l0 —oo) (13 9a)
=g
S—
00

IA, " A„,—~)—(A, " A„,~l. (13.2) I

-(—1)" . . pk g . . o. . . g, j
m=0 nI expl iQ . oGix.
nt
Here an implicit summation-integration is to be under-
stood over the repeated A' s. It is easily verified that —$+'S;JoG ' b
S satisfies Xexpl l(0&~ l0, —~):. (13.9b)
B+= S-'B-S, (13.3)
where the B's are any asymptotic invariants l j;
cf. (7.10) 3'In Eq. (13.6) the reader should remember to insert a re-
normalization factor Z '~~ with each wave function u. {See
B+=Isg+. —(13.4) Ref. 22). An alternative procedure is to choose for the action
functional S the "pre-renormalized" action, in which all "counter
terms" have been inserted in advance. The operators S,@0 and
By the standard Lehmann-Symanzik-Zirrunermann wave functions I
will then automatically contain the necessary
(LSZ) method one can show that the scattering ampli- Z factors in (13.6) without the need for making them explicit.
1220 8RYCE S. DEVVIT Ii

The colons indicate that the creation and annihilation alone in order to check that the formalism yields an S
operators making up the $+'s are to be normal-ordered. matrix which is unitary. This is one of the important
An alternative and very useful version of (13.9b) is advantages of working with an arbitrary background
Geld.
(Ai' A„',~ l&i &,—~)
14. THE 8 MATRIX IN THE PRESENCE OF A5'
~ ~ ~ ~ ~
IHVARIANCE GROUP. THE TREE THEOREM
~GADDI+ ~QfA+~+ ~@Ay ~O'A~
Consider the operator exp(po'8/bq') appearing in
(,, . Eq. (13.10). By taking into account the fact that go'
Xexpl 'iX—;;)too'goo~+i Q o" .Po'"
t;, ...;„@— I depends on the background Geld through its dependence
on the f's of Eq. (13.13), it is not difficult to show that
the effect of this operator, when acting on any functional
xexpl $o —
I(0, oo IO, —co ) (13.10) of the background field rp, is to replace oo by q+Q
Ss 'I —a~a+~0 where P is obtained by iteration of

where the t's, which will be called tree functions, are the
bare m-point functions with their external lines removed: =go''+G iP 'S1

-=2gt
'" ..
'". y~ i . y'ia. . (14.1)
f;,... ;„=(—1)"S—;„, S„„;„G"
"", (13.11)
,
'

Equation (14.1) has three remarkable properties.

and where
First, its iterated solution yields, as coefficients, all of
goo
=
—un+ u*u*, (13.12) the tree functions:

rt o= i',
&)rt os= frr+fr—
(1+Go— (13.13) 1
4' —4o*+G" Z — f" - do*'. 4o™ (14.2)
the f's
being the functions defined by (10.8) and the =2 e!
f's the corresponding functions with u replaced by u*.
The above expressions can also be used to obtain the Second, if p satisGes the classical Geld equations then
hierarchy of conditions on the scattering amplitudes so does y+p. Third, and most important, the second
which follow from the unitarity of the S matrix. Thus, property holds evenin the presence of an invariance
,

inserting (13.9a) into StS=1 and reordering operators group, provided the definition (13.13) is generalized to
into normal products, one Gnds
=
~o— f.~+t.&*'~*+&f, (14.3)
00
(y'&. y'")*c . c . (y" . . y'") where the f~ are the functions (10.9), the f~&*' are
m-0 gf obtained from these by replacing u by u*, and f is
—~-i (W'—W'+) 7 (13.14) arbitrary. "
The Grst property may be veriGed by straightforward
Z —Z(-1)'~-.
00
iteration and term-by-term comparison. The second
(~" ~'-~" property is obvious; the third, however, requires special
n~0 Qt
l=O
drscusscon.
)('c. , c. . ()ir. . . rI)in)st+1. . . $&m) —0 We first rewrite Eq. (14.1) in the form
= 1,2,
ns ~, (13.15) 4'= 4'o+G[9oj(Sr[so+4 j—Ss[o )4'), (14.4)

—distinct —arrangements in which the functional dependence of the various

ns!/(ns i)!f! of the 0's, and . l. .t Th

y'")=,
f t d t h t th 1
where

(0-12(~""~'-)IO- 0=So[v jo)o, (14.5)

)
(y' ~ ~
(13.16) 0 S [ $/[ (14.6)

—S2'NN~S2',
c= (13 17)
O=R [yjSi[yj, (14.7)

W==i ln(0, ~ IO, —~). —

(13 18) O=R [oo+QjSi[io+pg
(& [o3+~i 4)Si[—
v+&j, (14.8)
The equation SS = 1 leads to identical conditions.
We may note that Eqs. (13.15) are not independent of s' Since f+ and f dier from one another by a gauge transforma-
(13.14) but can be obtained from it by functional dif- tion PEq. !t0.23)g we do not bother to put & signs on po. The
ferentiation. For this reason it suffices to verify (13.14) difference can always be absorbed into the term Rg.
162 QUANTUM THEORY OF GRAVITY. II 122i

permit us to write where bt' is any change in f which one may wish to in-
— G—[p](S&[o&+P] S—o[o&3') ) clude along with the change in y. Equating the right-
0= So[&]f O' A hand sides of (14.15) and (14.16) we therefore get
=S,[o 14 (F[o]—v[o]R[o l~ '[~]R [o]v[o])
—
GR—yRgh) '(GR byg+G&+&R bygo
X G[oo](S&[o +4]-So[( ]4) bp=(1
= S,[q+4]+&[o]R[&]G[q]RTo ]S,[q+y] 0—
+R byRf+bf'). (14.17)
= (I —~[& ]R[~]G[o]R& 4)S&[&+4], (14.9) It is straightforward to showin a similar manner that
the gauge transformation (11.30) in the u's also pro-
in which the analog of (6.11), with G+ replaced by G, duces a change in g of the form (14.11), with bg given
has been used. The factor in parentheses in the final in this case by
expression is generally nonsingular. Hence it may be
removed, yielding the desired result bg= (1 —GR yRig) '(g+Qa+g+*Q, *a*) . (14.18)
Si[p+P]=0. (14 10) In both cases we can rewrite (14.11) in the form
It is to be emphasized that this result depends in no b(~+~) =RI ~+~]br, (14.»)
way on the choice of p's used in the definition of the since the background field remains unaffected.
Green's function G. In fact we can show that a change We may ask what happens if the background field
in the p's produces only a group transformation of the itself suffers a group transformation. Here it is con-
p's, of the form venient to assume that the i of (14.3) transforms ac-
b4 =R[o+4]bk. (14.11) cording to the adjoint representation of the group; any
portion of it which does not transform in this way can
We first take the variation of Eq. (14.4) and rearrange be lumped with the bf' of (14.17). It then follows from
the result in the form (4.9) and (11.15) that Pp suffers the transformation
bPp —bP —bG[p](S&[o +y] —So[o ]y) b4o'=R', ,A'b& . (14.20)
G[&]—(S,[„+q]S, ]—)b4, [, (14.». )
The tree functions, on the other hand, transform in a
We then insert (11.21) into (9.19) and make use of the contragredient fashion, i.e., in precisely the manner
analog of (6.11) to obtain indicated by the downward position of their indices.
—RGb~GR This is because they are built from Feynman prop-
bG=Gb~RGR +RGR byG. (14.13) agators and bare vertex functions by simple contrac-
Next we remember that tions of indices, and because we have taken care to
construct the propagators in a manifestly covariant
So[v+4]R[o+4]=0, (14 14) way. From this and Eq. (14.2) it follows that P trans-
which results from functional differentiation of (14.8) forms like pp.
and use of (14.10). Finally we note that the operator in bp'=R' y~bP (14.21)
the Anal parentheses in (14.12) can act in either direc-
tion. This is because of the fundamental assumption Hence
which is always implicit in the use of decompositions of
the form (14.3), namely that the a's are such as to give
b(~'+e') =R'-bt +R'. ,e'b&. =R'.[~+y]bP
, (14.22)
pp the character of a wave packet. The difference be- which has again the form (14.19).
tween So[oo+P] and Sp[p] therefore vanishes suf- We now have the following lemma: If A[&p] is any
ficiently rapidly at infinity to make reversal possible. invariant fotnctional of the background field then A[y+y]
Writing R[y+P]= R+R&P, and making use of (14.4), remains cotnpletely unchanged Under all the invariance
(14.6), (14.11), (14.13), and (14.14), we now have transforrnations of the theory.
—RGR —So@) The above results suggest that when an invarjance
bop= bg bpG(Siftp+tg

= Rb)+Ri&bg —RGR by(g

bp-
G(So[o&+4] —
Pp)— So)
group is present Eq. (13.10) for the S-matrix ampli-
tudes may be generalized to

+G(F yRy='R y)Rigbp- (Ai' ~

A„',oo (Ai ~
A, —oo) =
=R(1—GR qR, 4)b& RGR bv(y yo). — (14.15)—
But from (11.29) and (14.3) we have wI
—
X ~ ~ ~
exp~ oi&p'&Coo'4oo'+i Q—
-o
byo= R(GaG&+&)R bq(@o Ri.)+Rbf', (14.16a) 80!g& 80!g nl
—bf +(1+Go+X'+)[Go&+&Ro byogo&+&gala
bf'= l
X t' - '.4 o" 4 o'"+&~[o&+o»] gjI', (14.23)
+Go' 'Ro byp@o' 'Xy*n a*], (14.16b) ) - a~a
~

~O
 DENSITY
f222 B RYCE S.
where W is de6ned by (13.18) and where and 8go Ss —0, the latter of which holds under (11.25)
and (11.29), we see that this variation vanishes. Since
goo= —
Nrr+uece*+Rol o, (14.24) the n's in the decomposition (14.3) are completely
l o being an arbitrary gauge parameter satisfying arbitrary it follows the every term irt the sstm ore the left
Popo=0. of (14.25) is fully grostp irtvtt-ritJnt:
The demonstration that this is indeed the case is a
(14.26)
task which falls into two parts. First, we must obtain
an explicit form for W[tto) which, because the vacuum- This result is known as the tree theorem. "
to-vacuum amplitude is a physical observable, must be The tree theorem provides a very useful check on the
invariant under changes in the y's as well as under accuracy of lowest-order scattering calculations. One
group transformations of p. Second, we must verify the simply replaces any one of the external-line wave func-
group invariance of (14.23) itself. The first of these tions by 8 and looks to see if the resulting amplitude
tasks is the most difficult and will be carried out in sub- vanishes. Since scattering calculations involve lengthy
sequent sections. Here we accomplish the second. algebraic expressions, mistakes are often discovered in
In view of the lemma stated above, group invariance this way. In applying the test it is important to remem-
of the term W[oo+g] in (14.23) follows from the in- ber that ct/ the diagrams which go to make up a given
tree amplitude must be added together. They are riot
variance of W[oo] itself. Invariance of the term in g+
follows from the invariance of the amplitudes I, V, and individually invariant. "
A, which has been proved earlier. Only the terms in-
volving the tree functions require further investigation. 15. LORENTZ INVARIANCE. INVARIANCE UNDER
These terms are manifestly invariant under group CHANGE OF VARIABLES. QUANTUM VERSUS
transformations of the background field. We may re- CLASSICAL SCATTERING
mark that because the tree functions are obtained by
Space-time in 5-matrix theory is assumed to be
iteration of Eq. (14.1), which involves the ordinary
asymptotically fiat. A fiat space-time has group-
Feynman propagator, it is the ordinary Feynman prop-
theoretical properties not possessed by a general mani-
agator which is used for the internal lines of the tree
fold, namely Lorentz invariance. In S-matrix theory the
diagrams. However, because of the transformation law
Poincare group must appear as an asymptotic invariance
(11.12) the invariance of the tree terms would not be
spoiled if the functions S~ were substituted for G. As
group. "
If the zero point of the gravitational Qeld were chosen
a matter of fact, it can be shown that this substitution —
differently in this paper corresponding to a manifold
leaves the tree terms unaffected, and that although the
propagator G is the most convenient one to use in
with some other group of isometrics —then the formalism.
would have a different appearance, since the pertinent
practical calculations, the propagator N~ could be used
for the internal lines instead. " physical questions to be asked would not involve the
scattering of plane waves but something else instead.
In order to show that the tree terms are also invariant
It would still be necessary to make an independent
under changes in g and Po of the form (14.11), (14.16),
check of the theory for invariance with respect to the
etc. , we observe that in virtue of i(14.2) and, (14.4) we
underlying isometry group, because the origin of such a
may write — —
group in particular, of the Poincare group is distinct
from general coordinate invariance.

—1)!t;, ...; @o"

4o'"= 4o ~(4 —4o) It is quite easy to verify the Lorentz invariance of
s=s (rs the present theory, much easier than it would be to
check invariance under any other asymptotic or under-
=So (S [p+4] —S4) lying symmetry group. This is because, with the usual
(14.25) choices of field variables (Table I), Lorentz invariance is

The variation of the right-hand side of this equation R. P. Feynman, Acta Phys. Polonica 24, 697
37

has the form 3' The test is usually carried out in momentum (1963).
space. Since the
wave-packet assumption is implicit in the Fourier transformation
hPo (S [o+y]—Ss4)+do (Ss[o+&]—Ss)~o, process it is then no longer necessary to worry about conditions
of reversibility of the order of various operations. In fact, the
whole test reduces to an algebraic exercise.
in which use has once again been made of the rever- "It has been pointed out by Sachs that the asymptotic
sibility of the operator Ss[&p+Q] — Ss[oo]. Using Eqs. invariance group of gravity is actually much bigger than the
Poincare group. LSee R. K. Sachs, in Relativity, GrottPs ortd
(14.10) and (14.14), as well as the equations Po So=0 Topology, edited by C. DeWitt and B. De%itt (Gordon and
Breach Science Publishers, Inc. , New York, 1964l.g We make no
attempt here to investigate this larger group, the existence of
3' Since the functions gq are not symmetric there is a problem which seems to be related to certain conformal invariance prop-
of relative orientation of the internal lines. The orientation must erties of the theory. Ke remark, however, that such an in-
be that which results from the iteration of (14.1) with G re- vestigation might yield important new insights into the properties
placed by 5+. (See Ref. 46}. of S-matrix amplitudes.
 QUANTUM THEORY OF GRAVITY. II 1223

manifest -in both the Yang-MiBs and gravitational That these changes must leave invariant the term in
cases. The only point which really needs checking is X~ of the amplitude (14.23) follows from the fact that
the invariance of the theory under changes in the time- g+ refers to disturbances which propagate without
like unit vector e„which is used to define the bichar- mutua1 interaction. The theory of such disturbances is
acteristic p„of
Eq. (8.25) and the asymptotic wave identical with that of infinitesimal disturbances on the
functions N. But we have already seen from Eq. (8.26) back. ground field, and it does not matter what back-
that changes in e„
lead to changes in the I's which are
compounded of (1) gauge transformations of the form
ground variables are chosen to represent them. This
reasoning also leads to the simple transformation laws
(11.27), which have previously been shown to leave the /~
theory inva, riant, and (2) phase transformations. The
fg"~= fg'~, (15.6)
phase transformations alter the scattering amplitudes
on1y by phase factors and leave the probabilities them-
selves unchanged. Therefore, as long as we use helicity
assignments for the initial and final states the theory. 4o"= 4o', (15.7)
is indeed. Lorentz-invariant.
/pi
The following question, however, arises: Suppose we provided we require p, ; to transform like S, ;; LEq.
were to replace the basic field variables of the theory by (15.2)) so that the Feynman propagator suffers the
arbitrary nonlinear functions (or local functionals) of change4'
themselves. Would we then still arrive at the same
quantum theory by the methods outlined here, even 8p 8p ~
6/ jj Gkt (15.8)
though such a change of variables would generally
$~k
destroy the manifest covarianceP In particular, would
the scattering amplitudes remain unchanged' Less obvious is the invariance of the tree terms. This
We must remark that not all nonlinear transforma- is because the bare vertex functions, and hence the
tions destroy manifest covariance. For example, in the tree functions, unlike Pe', do not transform in a simple
case of gravity the change of variables + y„„—
y'„„or fashion. (See Eqs. (19.29), (19.30), and (19.31).g It is
nevertheless true that when the tree functions are
s& ~, a6ects neither the manifest Lorentz invariance nor multiplied by pe's, as in (14.23) or (14.25), the result is
the linearity of the general coordinate transformation invariant. To see this we note that the right-hand side
laws. However, we need not consider these cases of Eq. (14.25), in terms of the new variables, becomes
separately, as it is just as easy to consider the general @&' (St'5&'+rtr'j Sr'$')= —— 'S2'Q'
qP
case directly.
It is not difficult to see that a change y' —+ q" from
one set of basic 6eld variables to another produces the
following changes in the various quantities appearing
in. the theory4': Since the wave-packet assumption is always implicit, the
nonlinear teens in the P's inside the parentheses vanish
at ininity rapidly enough so that for them the arrow on
S„.
—S,,
'= - = 0, (15.1)
S;; may be reversed. Expression (15.9) therefore re-
duces immediately to (14.25), and we have, for all rt) 3,
= S,~i,
-.

S„—
= S,
8p'5p'
+S,
ar
$~k g~l

8p"8p' B(p'5y'
~
$2~k g~k g~l
, (15 2) A"r" 4,'"=t;,... 4
' 4 *. (15.10)
a result which, in each individual case, can also be
verihed by a straightforward but nontrivial computation.
e E ap (15.3) There remains to be discussed only the term in 5.
/pi Since 8' is a physical observable its value must remain
(8p" unaBected by changes in the mode of description of the
(15.4) 6eld. Its functional form must therefore adjust in such
k5pr e a way that
~'Lv '3= ~L~j (15.11a)~

4'+ — 4'4 "+' ' ' (15.5) which, together with (15.5), implies
'b(p& 2 I Bp'Bp~

"For convenience
II'Lv'+4'] = ~Le +4 j. (15.11b)
it will be assumed that q" =0 when q'=0 and
that the transformation is one-to-one analytic at the zero point "Any other transformation law for p;; would simply add a
so that-. series such as (15.5) have a nonvanishing domain of gauge term to (15.6) and (15.7}.The y invariance of the theory
convergence. has already been demonstrated.
 8 RYCE S. D EW I TT
We cannot, however, give a proof of this since we do not e's and e*'s, respectively, we may then write (assum-
yet possess a formal prescription for constructing 5' out ing (olo)=1)
of the basic building blocks of the theory, viz. , the bare
(a'l S(i)"S(i) la) = e'(~&'& ~&'&*&{at la){ol Fl 0), (16.2)
vertex functions and Green's functions. What we shall
in fact do is use (15.11) as one of several interlocking where o, and n~ are the eigenvalues and where
requirements which will ultimately serve to define 5'
in a unique manner. It turns out that (15.11) leads to a
F= ex—p[ 'P(e+a)
ia— (n +n ) V'(n+n)
xpi—

rather interesting and previously unknown result which 'It

opia— ta*] exp[i(e'+a')Ia+xpi(et+a')
can be translated into the q-number language as follows: X V(a*+a*)+', ia-Aa] . .(16.3)
When operator field equations exist (e.g. , when no
Unitarity requires
invariance group is present) they must necessarily
contain eoelocal terms, which vanish in the classical 2 ImW(i) —ln(0 F 0). (16.4)
~
l l

limit A 0, in order that the theory be invariant under Since 5'{~~ is independent of the eigenvalues o. and n~
changes of variables. We shall discuss later the reasons it should be possible to simplify the right-hand side of
why such terms are not normally considered. this equation by setting these eigenvalues equal to zero.
The reader will have noted the ease with which To show that this is indeed the case we first compute the
fundamental theorems may be proved now that the commutators
theory has been expressed completely in c-number
language. The c-number language has also the effect of [e,F]= F[iIa+ i V(n*+a*)], (16.5)
emphasizing similarities between the classical and [F,n'] = intP —
[ i (e +a—) V']F. (16.6)
quantum theories of wave scattering. From a classical
point of view the function &t&represents a finite disturb- Each of these commutators may be used to reexpress the
ance on a background q, and the tree functions describe other in the form
the self-scattering which it seers. The differences [e, F]= [iIa+i V (a*+a*)+VI*a*
between the classical and quantum theories arise from
the existence, in the latter, of the radiative correction + VV'(e+a)]F, (16.7)
term W[&)&+&&)], which has no counterpart in the —
[F,et]= F[ iatP i(e +a )— V"
classical theory, and from the fact that it is not the +a I Vt+(et+at) VVt], (16.8)
retarded or advanced Green's function which is used
but the Feynman propagator, with the result that P is from which, with the aid of the optical theorems (11.6),
complex instead of real. (11.7), and (11.8), we obtain
Fn+iVe*F= (1+iI)[(1—iP)e
16. FIRST APPROXIMATION TO THE VACUUM-
TO-VACUUM AMPLITUDE. PROOF OF ITS
iPa iAta*— ]F, —(16.9)
GROUP INVARIANCE ntF —iFe Vt= F[et(1+iI)+iatI+ia Jt]
We come now to the most diKcult part of the theory; X (1 —iP) . (16.10)
the determination of the functional 8' which describes From these equations it follows, after factoring out the
all radiative corrections or so-called vacuum processes. (1+iI) and (1 —iP), that
We do this first for a fictitious system defined by the
action functional ipS, , ;P'P' and then later extend the o= (ol (1—iIt)e —iP —icosa*]F o) l

results to the real system. It is clear, from the point of = {0l ([—iP(e+a) —iA'a*]F+[e, F]}l0)
view of perturbation theory, that the fictitious system
=~(0IFlo)/~-', (16.11)
provides a first approximation to the real system. This
approximation will be denoted by the subscript (1). and
Since the quanta of the fictitious system do not 0= (0l F[et(1+iI)+iatI+ia A) 0) l

interact with one another the tree functions all vanish, =(Ol (F[i(et+ t)I+in-~]+[F, et]}lO)
and the scattering operator reduces to = s{ol Flo)/sa, (16.12)
S(i) = '. exp(iW«&+ pip+ X+f+): (16.1a) which is the desired result.
=:exp(iW(i)+ie+tIe+ With the eigenvalues n and nt set equal to zero
+ ,'ie+t Ve+*+-',in+ A-n+ ):, (16.1b) Eqs. (16.7) and (16.8) become
which is obtained by reexpressing (14.23) in the format (1—VVt)eFo= Fpe+iVe*Fo, (16.13)
of (13.9b). The functional W(» will be determined by Fpet(1 —VVt) =etFp —iFpe Vt, (16.14)
the requirement that S«& be unitary.
The + signs in (16.1b) are irrelevant and may be Fp—= exp( —'in Vte)
—,

dropped. Introducing right and left eigenvectors of the X exp(-', iet Vee) (16 15
 gUANTUM THEoRY oF G~Avi~Y. ii i225

whence the identity

I(OI FoI 0&ysP= ;-i(-OI« F-oI 0) det(1 —Vvt) = det(1+iI) det(1 —iP)
= —)i(OI&t(Fpo( =det(1 —ZtX), (16.21)
+i&ttvFp) 0) I

&((1— ' which follows from (11.6). We note that these results
PV) insure that the vacuum-to-vacuum probability lies
= ~p
V(1 —PV) '(0I FpI0), (16.16) between 0 and 1:

b(OI F, 0&/av=-', i(OIF~*d o&

0& I(o oo lo —oo)(r&I p=e '™~&»
I

=-,'i(1 —Vt V)
I

—iFovtrr)nt = det(1 —Vt V)"'&1. (16.22)

'(OI e*Fo I 0&
=-', (1 —Vt V) 'Vt(OI FpI 0). (16.17) We note that they also permit, with a suitable choice of
phase, the complete identi6cation
Under variation of V and Vt (caused, for example, by
W(r&
—— rpi —
ln det(1+iI) .
a variation in the background field) we therefore have
In order to compute lowest-order radiative correc-
2 ImI&W(r&= 8 ln(OI FpI0) tions it is necessary to perform functional differentia-
=-', trLV(1 — Vtv) '8vt+(1 —Vtv) —'VtSvj tions on 8'(~). For this purpose it is convenient to re-
express 8'(~~ in a different form. We Grst recall that a
= ——', tr ln(1 —Vt V)
&&
formal determinant like (16.23) may be expanded by
= —Ii ln
-', det(1 —Vt V), (16.18) the Fredholm method in terms of traces. Remembering
the cyclic invariance of the trace and making use of
which, with the boundary condition: 5"~&~=0 when (9.10) and (9.39) we may therefore write
V=0, may be integrated to yield det(1+iI) = det(1+iv'&+I) = det(1 —I+So(+&)
2 ImW(r&= — ln det(1 —VtV).
rp (16.19) = det(1+X+No(+&) —'. (16.24)
If we had used the condition S(r&S(r&t=1 we would We next compare this determinant with
have arrived at the result det(1+X+Go&+&)—'
2 ImW(r& ——o ln det(1 —AAt) (16.20) which contains the sects of both physical and non-
physical quanta. Using the canonical decomposition
instead. The two results are, however, identical in view (9.7), the fundamental lemma (10.17), and Eqs. (9.2),
of the transposition invariance of the determinant and (9.13), (9.14), and (10.18), we have

det(1+X+Gp&+&) = detI 1 iX+(eoot+RovN — 'vtRo +RovN )j

' vtRo
1 igtX+I — —iltX+Rps —iltX+Rp» '
=det —iN 'vtRp X+I, —
1 iN 'vtRp X+Rpv iN vtR—X+R vN o
'
. —i vip X+I —ietRp X+Ep~ 1—ivtRp X+RovN

'1 —igtX+I — '

=det —iN 'vtRp X+I
0
2+v'—iEiN~X+Bp»
1 iM 'vt— 'strap X+Rp~/ '
0 0 1—ie~X+e3f '.
= det(1+X+8 p(+&) det(1+2+Go(+&)'. (16.25)

det(1 — det(1 —UGp)

From this we obtain at once
W(r& =-', i ln det(1+X+Go'+&)
=kiln
det(1 —
UGp)

UGp+)
i ln
det(1 —OGp+)
. .27b)
(16—

(1+X+Go(+'&) . (16.26)
i ln det— detG detGpo detGo+ detG+'
= —gi1' ln (16.27c)
Other forms for lV (&) may be obtained by making use detop detG detG+ detGp+'
of Eqs. (9.27), (9.28) and their analogs for Gp+, Go,
etc., namely The last expression must be used only formally, as the
determinants of the Green's functions themselves do
det(1+XGp) not really exist. 4'
8'(g) = —~i ln
det(1+ X+Go+) 4'The determinants det(1 —UG0), det(1 —UG0+), etc. do not
exist either. However, the divergences which they contain are
det(1+1'Cp) removable by renormalization procedures. These . divergences
+i ln (16.27a) may be shown to contribute only to the real part of W'(1) and
det(1+2+Go+) hence do not acct the vacuum-to-vacuum probability (16.22).
 1226

Since the matrix I

is group-invariant 8"~~~, as given
by (16.23), is invariant. This invariance must also
hold for the forms (16.26) and (16.27), and this pro- —
2
I
i

vides us with a useful consistency check on our results.

Expressions (16.26) and (16.27) are manifestly in-
variant under gauge transformations of the I' s, since 0 o

they do not even depend on the I' s. Their invariance

—
I —— I
under changes in the p's may be verified with the aid of i 1 i
2 2 tw I 2

(14.13) and (b)

bJi=E ByR. (16.28) FIG. 2. Lowest-order diagrams for the single-quantum-produc-
tion amplitude. Diagrams (a) and (b) refer to the cases in which
Thus we have infinite-dimensional invariance groups are, respectively, absent
and present. Lines terminating in dots represent external-line
detG wave functions. Lines bearing arrows represent virtual quanta
8 ln — = —tr(F()G) —2 tr(G1)F) on the mass shell. Dashed lines represent fictitious quanta. The
asymmetry of the vertices from which fictitious quanta emanate
detG' is indicated by the obliquity of the angle at which the solid
lines are attached.
tr(FGByR—
GR FRG87GR—

+FRGR ()yG+2GR oyR) first produces two or more virtual quanta which, after
various interactions with each other and with the back-
tr(pe '— — —
"oyGR ) = tr(y '()7), (16.29) ground 6eld (involving scatterings both forwards and
backwards in time) proceed to coalesce into a single
and similarly
quantum via elementary vertex interactions. From
detG+' Eq. (14.23) it is easy to see that the amplitude for this
S ln = —tr(7-'bq), (16.30) process is
detG+
(A, ~!0, —~ ) = ie'wW;f~(*)'~. (17.1)
with corresponding expressions for the zero-point
For simplicity we ignore the vacuum processes
quantities, whence 88"(~) = 0.
described by the exponential and replace t~(*) by the
To verify invariance under group transformations full wave packet ps, we may regain individual ampli-
of the background field we use (11.9), obtaining
tudes by functional differentiation with respect to the
ii ln detG+= —tr(F8G+) n's when desired. In lowest-order perturbation theory
= —F "(R' sG+"'+R', sG+") 5& the amplitude then becomes
= 2R' 8f (16.31) 4 s'~'(i) (= s&(G'" G+")F s —
@ o'—
Similarly,
+i(G & G+ &)F p—
„(t)s'
~1g ig, „G(+)js ,

() ln detG+ = —tr(FSG+) +sit'o*'(I'( )t)+ I'(t)') )G'+' ~ (17 2)

Fp(c, sG—
+@'+c~,sG+~') f)P where
=2c 7 ()p. (16.32) = R'-A~s.
~( '))s— (17.3)
We may now either use (4.10) or else note that these When no invariance group is present the second term
variations will be exactly cancelled by identical ex- on the right of Eq. (17.2) is absent, and the ampli-
pressions coming from the G's and G's of (16.27c). In tude may be given the graphical representation depicted
either case we have HV(~)=0, which completes the in Fig. 2(a). The line terminating in a dot represents the
consistency check. external-line wave function, and the solid line bearing
We remark that no special significance is to be an arrow represents the function G(+»~. The arrow may
attached to the use of the advanced Green's functions be assumed oriented in the direction "k to j"43 and
in Eqs. (16.27). Because of the transposition invariance serves as a reminder that the virtual particles associated
of the determinant the retarded Green's functions could with it are om the @sass shell, as follows from the fact
be used just as well. that G'+ satisfies the homogeneous equation IiG +) =0.
When an invariance group is present the function G&+)
17. SINGLE QUANTUM PRODUCTION. propagates nonphysical as well as physical quanta, and
FICTITIOUS VIRTUAL QUANTA the second term on the right of (17.2) appears in order
to compensate for the unwanted quanta. Feynman,
The simplest example of a physical process which who was the first to call attention to the need for this
can be classed as a radiative correction or a closed-loop
effect is the production of a single quantum by the ~ Unlike the Feynman propagator the function G(+»'~ is not
background field. In this process the background field symmetric in its indices.
162 QUANTUlut 'I IIEORY OF GRAVITY. II 1227

extra term, has referred to the auxiliary propagator G Q —~G(+) ~Q(+)

G(+) which occurs in it as the propagator for fictt'tiols = (I+GoX) (Go —Sop) (1+XgSop)
qguntu. 44 In the case of the Yang-Mills Geld the Gctitious = (I+So~&+)(Go—Sp~)(1+XGo) (17.6a)
quanta constitute a set of massless scalar particles
which transform among themselves according to the = (1+So*Kg) (Go —Sop) $1 —Xg(Go —Qp )1-'
adjoint representation of the group. In the case of X (1+%~Spy) . (17.6b)
gravity the Gctitious quanta are massless vector
Further reduction of these expressions is most easily
particles.
carried out by direct formal expansion of the bracketed
It is to be noted that the Gctitious quanta are needed factors. Using
only when the invariance group is non-Abelian. In the
Abelian case the vertices V(;)p to which they are Gp Q~ — Go(+)+So(+)
coupled vanish. This is one of the reasons why quantum =iRpvN vtRp +iBpvN ' vtRp, (17.7a)
electrodynamics, with its Abelian gauge group, fails Go —Qp =Go( ' —So' '
to provide a satisfactory training ground for studies in =iRpv*N 'v B() +iBpv*N ' v Ro (17.7b)
quantum gravidynamics. Another peculiarity of the
which follow from (9.7) and (9.10), and
vertices V(;~p is their lack of symmetry with respect
to the group indices. Although they appear in a sym- XgRpv = X~Rov, (17.8)
rnetric combination in (17.2) they do not always appear
which follows from (9.38b) and (10.18), we find, with
thus in more complicated processes. Their asymmetry ls
the aid of (9.13), (9.14), and the lemma (10.1/),
indicated in Fig. 2(b) by making the solid lines attached
to them join the dotted lines at an oblique angle. The (Go —Sp+) Ll —Xp(Go —So+)P'
dotted lines represent fictitious quanta and the presence =iTRo(1+Go(+)2+) 'vN 'vtBo +RpvN ' vt
of the arrows indicates that the propagator G&+) rather X (I+X'+Gp(+&)Ro — Ro(1+G()(+)X+)—'vN —'
]
than G is to be employed. The sum of the three diagrams '
XvtBo X~BovN vt(1+2+Go(+))Ro (17.9)
appearing in Fig. 2(b) gives the full production
amplitude. (The corresponding formula with +
signs replaced
—signs can be obtained from this by transposition.by)
Explicit calculation of the amplitude leads to diver-
gences which must be handled by the methods of re- Inserting this into (17.6b) and using the analog for
normalization theory. For this reason use of the the functions Gp, G, etc. of Eq. (9.27), together with
manifestly covariant functions G&+) and G(+) is essential. (I+SoyXy)Rpv=R(1+Go+X'+)v, (17.10)
From a purely formal standpoint, however, the func-
which follows from (9.40), (10.14), and (10.18), we
tions Q(+', which propagate only physical quanta,
obtain
suggest themselves as natural replacements for 6&+);
they should in principle permit one to avoid dealing G Q+ — G(+)+Q(+)
with the Gctitious quanta. That is, we expect that it =iRQ++iQ R RP+R, —
(17.11a)
should be possible to rewrite (17.2) formally in the G Q — G(—) Q(—)
simpler form =t'RQ +iQ+ R RPM, —
(17.11b)
4o'W(t) „= &pi(/)o'S„; S(+»,.
g, (17.4) where

in which the propagators G&+~ no longer appear.

= (I+G(g)vN 'vtBp (1+X+So+),
Q+— (17.12a)
Equation (17.4) can in fact be shown to follow from Q
—
= (I+G(g)v*N 'v Bp (I+X Sp ), (17.12b)
(16.23). We shall here show its equivalence to (1/. 2)
directly. For this purpose we must Grst assemble a
P+= F= (I+GpX)vN — —
'vtBp X+BpvN ' vt
number of fundamental identities. X (1+/Go) . (17.13)
We begin with Eqs. (9.20), (9.21), (9.37), and (9.38), Before Eqs. (17.11) can be used to compute the effect
which, after a certain amount of algebraic manipulation, of replacing G(+) by Q(+' some special properties of
yield the functions Q+, F~ must be derived. First we note
that Eq. (9.2) permits us to write
X—X~ = X(Go —Soy) Xy = X~(Gp —Qpy) X (17.5a)
= X~(Go —So~) L1 —X+(Go —Qoy) j-'X~, (17.5b) F(1+Go+X+)Bpv= —FG+FpRpv=FpBv=0. (17.14)
From this, together with (4.7) and (5.11), it follows that
44 R. P. Feynmsn, Proceedhngs of the 1962 Warsato Colferertee
Fy 'R y(1+Go+X+)Bpv
ort the Theory of Grave'tatter) (PWN-Editions Scientifiques de
Pologne, Warszawa, 1964). =R F(1+Go+X+)B,v=0. (17.15)
 8RYcE s. DE%ITS
By the laws of pr opagation, taking into account namely
boundary conditions in the remote past and future,
this in turn implies
(17.31)
iR y(1+Go+X+)Rpi)= (1/Go+I'+)yp Rp gogo&)
= (1+Go+X+)()M-'N, which relate bare vertex functions diQering in order by
(17.16)
unity. We give the steps of the present derivation
in which (9.13) has been used in obtaining the second without comment:
form. Analogous reasoning, combined with (8.18), leads
;;~,'S, .,g(+& o
to
y 'R y(1+Go+X+)I= (1+Go+X")yo''Ro (17.17)
,iPo'-S„;o(G&+»'+iRi
ohio'(S,
.
Q+"'+iQ 'R"

which also follows from (10.3) and

yoN,

(10.6). Equations ,'iyo'S, i-ioG&+)&'+ '@oi(S,;-oR& „+S,

+ oiR". ;+S, oR",, )Q "'
.
R) F~~— eRoe)
';;Ri. p)Qp ' ,

(17.16) and (17.17), together with (9.40), then yield ,

+ pido'(S, )oR'a, i+S,';R'a, o)Fp R'p

f iR y(j+NoyXy)Boo=(1+Go+++)i)M iE . (17.18) =-,'i&t oiS; oG&+»' ', i(t)o'R eRi„,G&+).e
From this, with the aid of (9.14) and the analogs for +,'i(t p'RoeR' „G& )~e (1.7.32)
Gp, G, etc. of (9.27) and (9.30a), we obtain
In view of the symmetry relation G&+' = — G' &

Q~yRy '=y —'R yQ =iG&+&, (17.19a) Lcf. (9.6)] the last line reduces iri)mediately to the
right-hand side of (17.2).
Q yRy '=y 'R yQ+ = iG&— &. (17.19b) Aside from the 6ctitious quanta, Eq.
eliminating
Ke also have the equations (17.4) has the important advantage of yielding an
immediate formal proof of the group invariance of the
FQ+ =0, (17.20) amplitude pp'W&i), ;. To see this we note that S,,)o is
FQ~=0, (17.21) identical with the tree function t;, I, . Therefore, in view
of (17.30) the right-hand side of (17.4) appears as
FP'~= 0, (17.22) a su)N ower tree ampHtldes in which all of the external-
line wave functions refer to physical quanta on the
which are immediate consequences of
mass shell. Group invariance of the total amplitude
follows immediately from the tree theorem. This
F(1+IIpyXy)Rpi)= FgyFoRpi)=FpRpi)=0, (17.23)
possibility, namely of reducing all amplitudes to sums
F(1+Gpss)i)= =0.
FGFo'v= Fo()— (17.24) over tree amplitudes so that group invariance is assumed
"
by the tree theorem, was 6rst suggested by Feynman.
These equations, combined with (5.11) and (17.19), We shall now see how it works in more complicated
give us processes.
Q~So ——WiC&+)R y (17.25)
For completeness we record here also the following 1L MULTIQUANTUM PROCESSES.
FEYNMAN BASKETS
useful and readily veri6ed identities:
S2G= —1—
Next in order of complexity are the lowest-order
yRGR, (17.26)
radiative corrections to the amplitudes for scattering,
S,G&+) = —~RG(+)R, (17.27) pair production, and pair annihilation by the back. —
ground 6eld. 4' These are obtained by functionally
I&&+)yR=O, (17.28)
di6erentiating the amplitudes of Fig. 2 and using the
S Q(k) —0 (17.29) variational law
Q&+) Q&-)-= jf (1+jr)-if &*)-. (17.30) 8G&+) = G&+) SFG+G8FG&+) aG&+) SFG&+), (18.1)
The last identity, which is obtained with the aid of which follows from (6.19), (9.19), and (9.29). When no
(9.10), (9.42), (10.9a), and (11.2), shows explicitly invariance group is present the result is
that the functions 8(+) propagate real quanta on the 4'o 4 p ( II( )iij+SjkG", 'II (i), 'i) = oi4o'4o', t io)G
mass shell only.
We are now ready to employ (17.11) in the verifica-
+ i(y it G(+)kmG(+)alt .4& j (] 8 2)
tion of (17.4). In this, as well as in many similar but which has the graphical representation shown in Fig.
more complicated derivations later to be stated without 3(a). We see immediately that Feynman's idea works;
proof, repeated use is made not only of (4.7), (14.5),
(17.25), and the other identities collected above, but 45When the background 6eld vanishes these reduce to the
also of a hierarchy of identities following from (4.8), self-energy corrections to the 1-quantum propagator.
162 QUANTUM THEORY OF GRAVITY. II 1229

From Eq. (17.28) and the explicit form

+ +p +~i
(a) t'&F= ttSs+t'&(pe 'R y), (18.6)
it then follows that
t"I ( +~i Pi ~2 Pgk
P
8$&+&=$~(t&Ss+yRy '8R y+yRy 'R i&y)$&+&
+Q&+&(bS,+&&yRy —'R y+ybRy —'R y)$~
$&+&sS,Q
(b)

FIG. 3. Lowest-order radiative corrections to the 2-quantum Q 5S ${y&+$&k&5S Q ~$&k&t&S $&+&

amplitude. (a) Invariance group absent. (b) Invariance group +RG+(f&R y+R 5y)$&+&
present.
+ Q&+& (t&yR+yt&R) C+R . (18.7)
the total amplitude appears as a sum of products of When this law is used for the purpose of generating
tree amplitudes. In the 6gure the diagrams have been Feynman baskets it turns out that the last two terms
grouped into sets corresponding to the tree structure, never contribute anything owing to the presence of
i.e., to the two terms of Eq. (18.2). These sets are the R's. One finds also that the Q~'s in the first two
known as Feyeman baskets. The key method in develop- terms may always be replaced by G's, a result which is
ing the general theory of radiative corrections of directly related to the previously mentioned possibility
arbitrarily high order will be to take diagrams having a of using G or Q+ interchangeably for the internal lines
given topological structure and reassemble them into of tree diagrams. 46
Feynman baskets.
Another fact which is useful in computations is that
The corresponding amplitude when an invariance
in performing functional differentiations one may skip
group is present may be obtained by three distinct
over any p's which occur. Terms involving functional
methods: (1) functional differentiation of (17.2); (2)
functional differentiation of (17.4); and (3) replacement derivatives of y's conspire mutually to cancel in any
of G&+& by Q &+' in (18.2) and use of the identities (17.11), observable amplitude. This is a consequence of the p
(17.19), (17.27), (17.31), etc. All yield the same result, invariance of the theory.
namely There is, however, one possible source of worry which
needs to be disposed of. In passing from an expression
ys'yet(W «& „;+S&sG"W , &»&), like (17.2), say, to the expression (17.4), one makes use
jt. . Q&+&kl
Lsy &y

'y 't,.„, Q&+& Q&+& t „y '

(18 3a)
of (6.11) and many other identities which depend on
the background field equations being satis6ed. The
right-hand sides of (17.2) and (17.4) are therefore not
t" G&+&~'r+t G&+& "sG+ &"&'t
identical but are equal only modulo the 6eld equations.
—2V(;)pG &G(+»'V(g;)~ —2V(p;) G tG(+)&'V(,g)g They differ by an expression of the form a' S,;. One may
—V(~') pG'+'"G'+'"V (») v —V (~') G'+'"G'+'"V(v~) ask what happens to this difference when it gets dif-
S.rtsG—(I " &«&&&+ I'&t»&~)G'+'sj
~

(18 3b) ferentiated. The answer is that, in the passage from the
one-quantum amplitude to the n-quantum amplitude,
which has the graphical representation shown in the combinations in which the functional derivatives
Fig. 3(b). In each case the derivation is straightforward
but tedious. Obviously, the amount of computational 4' KVith the identities which we now have at our disposal it is
labor involved in converting from Q&+' to the functions straightforward to show that the theory of tree functions may
G(+) and G(+) mounts rapidly as the complexity of the be based on S~ rather than G. One replaces Eq. (14.4) by
underlying tree diagrams increases.
In functionally differentiating either the external-
4+ =4o+8 g (Sicko +4 +'j Sot +)—
and obtains S&Ly+p~J=O, in complete analogy with (14.10).
line wave functions or the physical propagators it is The corresponding tree amplitudes are then obtained from
(14.25) by replacing @ with os~. To show that this replacement
necessary to have a variational law for $&+& analogous leaves the tree amplitudes unaffected we write
to (18.1).This is obtained by inserting (6.19) and (11.10) &t
—&&,= —GSo4+ +o4, = —GSo (P —P,) —(G —&S&,)So&t,.
$
into (11.11) and then using (9.23) and (9.35), which
yields
tlQ &6& —G+5PQ &6&+$&+&5PG+~$&k&&&PQ&k&
—Q~JFQ&+&+$&6&f&'PQ~~Q&+&t&PQ&+& (18.4)
is solved by

The quantity ft+ —

iR&&+(R ~R~ 'R v) (—
@0 vanishes at indnity
y~ yo)—
In view of (4.7), (5.11), (10.10), (14.3), and (17.11) this equation

rapidly enough so
that the operator in parenthesis can be reversed. From (17.19)
and (17.20) we therefore get
and, incidentally, gp= WRG'+ R y(&&&~ —
P— go) = WRG +&R y(y — yo),
(18.5) from which the invariance of (14.25) immediately follows. ..
1230 8RYCE S. DEWITT 162

of this difference occur always add up to zero. These etc. By making use of (2.2), (10.10), (17.26), and (17.31)
combinations, in order of increasing complexity, are one readily veriles in each case that these combinations
vanish. It is not hard to show, in fact, that this is to be
+ S„;)j,
(a', ,S,, a' lt'0
expected as a, corollary of (14.10).
[a',; sS, ,+2a',;S„&+a' S,;,0 %e close this section by recording the contributions
+(a', )S,~+a'S, 'i)G'"S. ,pj's'p(0', (18.8) of 5"(&) to the three- and four-quantum amplitudes:

its'40 yp'(W(1) „;0+3t;,lG'"W(1), 0+top(G'"W(t), „)=, i40i4p (tp'(t;;plmg+)'m+3t;;lmg(+)'"Q(+) 0

+2t,.l Q(+) nt jg(+)0 t
t„p
N(+)ql) (18 9)
$0 4 0 4 0 4 0 (W (1),ij lcl+ 6tijmG W (1),nki+4tij pmG W (1),nl+3tij mtklnG G W (1),yq+ tij qlmG W(1),n)
'y jy Pypl(t, jul g(+) +4t,,p ((()(+) ng(+) q t l+3t, g(+) 17(y(+) q t y j
12t, (r(I(+) „g
yt t,
(+) q ig(+)
+6t,. „Q(+)net Q(+) qrt„., pg(+) «t lg(+) nm) (18 10)

The corresponding diagrams when no invariance group Connected diagrams having two or more closed loops
is present are shown in Fig. 4. The grouping of the correspond to higher-order radiative corrections. A
amplitudes into Feynman baskets is again evident. diagram having e-independent closed loops is said to be
The task of reexpressing (18.9) and (18.10) in the of the mth order.
general case in terms of 6, G, G&+', G'+' will be left to Consider the set of all connected nth-order diagrams
the reader as a (rather lengthy) exercise. The reader which contribute to a given scattering amplitude. By
may also enjoy discovering the simple rules of dif- repeated functional integration one may remove the
ferentiation which lead in a step by step fashion from external lines. The resulting vacuum diagrams represent
Eq. (17.4), through Eqs. (18.3a), (18.9), and (18.10), the eth-order contribution to kV, which will be denoted
to the lowest-order radiative correction to the general by W(n)
e-quantum amplitude. The basic topology of the vacuum diagrams and the
numerical coeKcients to be attached to them are the
19. HIGHER-ORDER RADIATIVE CORRECTIONS. same for all field theories. For purposes of orientation we
USE OF THE FEYNMAN FUNCTIONAL begin with the case in which no invariance group is
INTEGRAL TO CONSTRUCT A present. The Feynman functional integral may then be
CONSISTENT THEORY used as a convenient formal expression for the vacuum-
The functional derivatives of S'(~~ are represented by to-vacuum amplitude:
diagrams each of which has only a single closed loop.
(0, ~10,—~)=,' l.l, (19.1)
'; '$+$+P,'()'+P, g+P, )+P$+P, g~p, giPP
—, W[p ]=u [(qj —w[0j, (19.2)

+2i Pq +P~ +P6 + 2iP2 e'" ' = expi(S[(p+y] —S[(p]—S,~[00] ')

(a)
Xt) [pq+pjdlt (19.3)
21 + +P4 +P4 +P6 +P4 +P +P +P d(t =—II d4'. (19.4)
+ P4 +P4 + PI2 +PI2 + P6 + P6 + P6
Here 6 is a density which serves to define the functional
+PI2 +PI2 + PI2 + P6 +PI2 +PI2 +PI2 volume element. It will be chosen in such a way as to
maintain invariance of the theory under the variable
+P +P2 +P24 +P +P~ +P2 transformation (15.5).
It is not difficult to show that Eq. (19.3), when
+ 21 P4. +P4 + PI2 +PI2 +PI2 +PI2 +PI2 supplemented by the statement

+p, +p„+1p +p +p„+p

(o "12'(~[kj)10 —")= ' "' ~[47
+PI2 + I2 2' I2 I2 24 + 21P6
tb) x e~i(S[0+yj —S[p]—S„[0j')
Pro. 4. Lowest-order radiative corrections to (a) the three-
quantum amplitude and (h) the four-quantum amplitude in &&~[~+~jd~, (19.5)
the absence of an invariance group. When an invariance group
is present all the-mass-shell propagators G&+&are replaced by Q&+). yields the hierarchy of Kqs. (12.20), (12.21), (12.22).
 QUANTUM THEORY OF GRAVITY. II

The formal identity

—
lo, — ) {exp'(S[q +47 S„[q7y')
g* S—
[q 7
(o,
/pi
iraq' = 0,
x~[q+y7}&y— (».&)
[
combined with the condition S,, q7)= 0 on the back-
ground field, suggests that the operator fmld equations
of the theory may be written in the form
[q ]if)' ) 6[q)+y7} s,,—
s, l— k[q ]y expi(s[q)+II)7
T(s, ,[q)+ f7 —i{lnh[q)+ ill]},; ) = 0. (19.9)
On the other hand we expect that they may also be ex-
pressed in the simpler "classical" form
1
o, I
y'lo, — (19.6) o=s [ +y]=s yj+ —s ",yq
2f
—ao)
g'k Gjl (0 oo 10 +—S, " k4l0j"0'+ (19.1o)
3l
the manifest Hermiticity of which follows from the
symmetry of the coeKcients (bare vertex functions).
Equation (19.10) will, in fact, turn out to be not quite
right; it cannot be reexpressed in the form (19.9) and,
'""lG" {yj p'(S[q+it] —S[q 7 moreover, it is not form-invariant under transforma-
g k
tion of variables. However, we shall adopt it tentatively
and then correct it later.
S, i[q'7g')h[q'+@]}+{ib'k S.kl[q)7&'lf)j} The term in 6 in Eq. (19.9) may be regarded as
arising from the process of converting from ordinary to
chronological products, and may be computed on this
basis. In rearranging factor sequences we need to know
the commutator [P', )j].
'r For this purpose we take the
G'(o, lo, — &+(o, ITS'yj) lo —" commutator of (19.10) with $k and And that the result
is solved by
etc. , in which functional integrals of total functional
derivatives are set formally equal to zero. In fact, (19.11)
Eqs. (12.20) ef al. , can be used to derive Eqs. (19.3) 6"= 6"+8",krak+ (I/2!) 8",ki&kP'+ . . (19.12)
and (19.5), showing once again that the technique of
varying the background 6eld is completely equivalent The algebra is straightforward. Here we work only up
to (but of wider applicability than) more familiar to the order needed in discussing S'~~~, more eKcient
methods employing external sources. methods of procedure will be given in the next section.

Further algebra yields

0'0 j T(4*0')— "=zG+"—
= zi'j(j, z) &—
zg+ij+zg+ij yk+rzg+ij T(alkyl) rgij klg+kl+. . . (19.13)
Pj)k T(gi)'jf k) —zg(j Q) G+ijPk+zg(fz ~')PkG+ij+zg(1z j) 6+ikiIlj+zfl(j Iz)gjG+ik+zi}iG+jk
—zg+ijffk+zg+ikPj+zg+jkPi+zg+ij T(glgk)+zg+ik lT(jill j)+zg+jk T(glori)
G+ij G+lk G+jk gli+. . . - 14)
(19.

PiP jgkgl T (Qi$ j$k$l) zg+ijT(iIjkgl)+zg+klT(pip j) G+ijg+kl+zg+ikT($)gl)+ zg+llT(ffjpk) g+ikg+jl

+iG+"T($'$k)+ig+jkT($'$') G "G+jk+ ~ —(19.15)

"Here we proceed purely formally and ignore the fact that in a local theory all the P's of Eq. (19.10} are evaluated at the
same space-time point.
 8RYcE s. DE% TT jt

where e(i, j) is the temporal step function"

for i + j
for j +i. (19.16)
These results permit us to reexpress Eq. (19.10) in the form
o= 2'($, L9+8)+%$ jk{ig+"+bg "S. b.g '"0'+big+"S,
+lig+jaS g+be$
. . b.dg "L2(4'0')+ ig+"]
g+ j»P2'(Peed)+ig+cd+igcd]-+. . . }+»$ .. (3ig+jkgl+3ig+ja$ G+bkg((I ceI l)
g+jaS b G+bk(g+cl+Qcl)-+. . . }+1bS . , G+j»2'(jtjl jIjm) 1$ . Q+jkg+lm+. . . (19 17) ,

The terms following T($,;((ej+ g) in this equation exp(ir('j(»L 0 ])

are almost, but not quite, expressible in the form
—iT((lnh[p+itl]}, ;) of Eq. (19.9). What is missing exp iqS, ;& ' ' detG+ d
is a term having the following structure:

,', S—lkg+, j Sb,G,+"S,jdg, +jkC'd (19.18) (detG) ' j'

=z (19.23)
If this term is added to Eqs. (19.10) and (19.17) we find
B,Lij]=(detG+) 'j'exp( @kg+"6+k"G '"s
6is—
siS—;klG+ "G+"+ ) . (19.19)
~
l„„ (detG+) ' j'

where Z is a numerical constant determined by the

lattice spacing used in the definition of the functional
j).P(j&+(I&]= (detG+) ' ' exp( 'G+"— S-;,ky» integral, but independent of the background 6eld.
1G+iaS G+bjS . alkyl 1g. +ijS . . alkyl The Green's function G makes its appearance in (19.23)
L&S . . G+itG+~G-I;nS owing to the Feynman boundary conditions assumed
1iS . »lg+ijg+. kl+. . .) (19 20) in the Gaussian integral.
Expression (19.18) is the Grst of an infinite sequence Writing
of correction terms, which must be discovered by
laborious computation. These terms maintain the formal 8'= Q 8'(„), llj= Q r(t(„), (19.24)
Hermiticity of the Geld equations $e.g. , (19.18) is real) n 1

but are not mathematically well defined. Like the terms

of 6 they involve Green's functions with coincident (19.25)
arguments and hence cannot be properly discussed
and making use of (19.20), (19.21) and the hierarchy of
apart from renormalization theory. However, they may
be regarded as possessed of certain formal properties. equations generated by

(0, ~ T'(expi), y') 0, —~ &(,)

Owing to the kinematics of the Green's functions they
I I
depend only locally on the 6elds, and in the case of
scalar fields with nonderivative couplings they may be = exp(iW(»+-', iX;X,G"), (19.26)
regarded as vanishing by virtue of the commutativity
of 6eld components at the same space-time point.
Ke are now ready to compute 5'~2~. We 6rst reexpress
Eq. (19.3) in the form detG detGO+
5"(y) = —2i ln (19.27)
e' «' = (exp'(l(i)LO])(0, ao T (expi[(1/3!)S;,kf'f j$k,
(
detGO detG+

+ (1/4 )I$, ekA lAj'+ ] gT k

$ . . (Gilgjmgka+2G+ilg+jmgka-
X~'L~, g)10,—~&(». (19.21)
3gilQ+jmg ka)$ i(gij —
g+ij)$ . .„GklS
Here d'Ly, i!e] is ELp+g with the factor (detG+) 'jk
removed, and X (Gma G+ma) »$ „(Gij g. +. ij)(gkl G+kl)
(0, ~ I2"(~Lb]) I0, —"&(i)
minus the same terms evaluated with y = 0. (19.28)
= Lexp( —im(i&LO])]
— 'S,
A Lqb] exp(i — tV'Pije')
Equation (19.27) is observed to agree with (16.27c)
X (detG+) 'jkdP, (19.22) for the case in which no invariance group is present.
"The step function need not be defined for spacelike separa- Expression (19.28) for W(k), on the other hand, is
tions of i and j but must be handled with care when the two
space-time points coincide. Fortunately it disappears in the anal
still not quite right; it fails to be invariant under the
forms of Eqs. (19.13), (19.14), and (19.15). variable transformation (15.5). To Gnd out how it
162 QUANTUM THEORY OF GRAVITY. II 1233

changes we make use of the transformation laws

S„-j=S, b (19.29)
by" by"

S„;k'=S,ab.
bya byb byc
by't by'& by"
+S, abl,
abye b2yb
-+, byn
+
(by'* by"by", by" by', by", by'
bkyb bya b2yb

&y'*&y"/
(19.30)

byd bya byb b2yc bya byb b2yc bya byb b2yc
S,ijkl S,eben +S,abc + +

+,
by" by" by" by" by" by" by"by" by" by" by"by" by" by" by"by"
bya byb b2yc bya byb b2yc bya byb b2yc b2 ya b2y b b2ya b2 y b
+ + +Sb +
by by by by by by by by by by by by by by by by by by b by

b2yn

by by
b2yb

by by
bya

by
bbyb

by by by
, +,bye

by
bbyb

jby , by by '+
bye

by
b8yb

by by 'by '
byn

by
bbyb

by 'by jby ~)
19.31

,
from which terms in S, have been omitted owing to the W(2) and 6 (to second order) are
fact that the background Geld obeys the classical field gr — 2
S . . (Giigjmgkn+. 2G+iig+jtng-kn
equations. These laws permit us to infer $cf. (15.8)) 3G(lg+jmg-kn i gilgjmg+kn)S
by by by" by" L(gij G+ij)S . . GklS (Gtnn g Hnn)
G"'= Gkl G+j'ij G+kl (19 32) —32S itki(G" G+")(G— k' G+k') m—
inus the
byk byl byk bye
same terms evaluated with y=0, (19.35)
whence it follows that 8'~~) is invariant. For 8'~2~, on 'j' eXp( + "G+j G k"S,lm„
the other hand, we Gnd, by a straightforward but DLy7= (detG+) 82iS,;jkg—

tedious calculation, +
(1/48)8S . .„Gilgjmg+knS
t kl—G+"g "'+
,'iS;— ) . (19.36)
~ ~
,
i
II'(2)'Ly'j —II (2) Pyl= —S,ab. C''(" '
b y by by
The introduction of the term (19.34) brings a qualita-
24 by"by" by~ by' tively new element into the theory. It adds to the
by~i byjj by~k by~i
operator field equations (19.10) a term of the form
b2ya b2yb
T(I'(», ;(y+p]) which, unlike (19.18), depends jboj8-
48 by"by'~by' by" by' by" by' by locally on the Gelds and is nonvanishing even for scalar
fields with nonderivative coupling. This implies that
minus the same terms evaluated with y=0, (19.33) within the framework of local field theory there exists
no covariant ordering of the factors of the operator Geld
showing that Eq. (15.11a) is violated. equations which maintains form-invariance of the theory
The violation, however, is not very great. Relative under arbitrary (local) transformations of variables.
to the large number of terms involved in the calcula- Such a conclusion, however, presupposes a definition of
tion and the large amount of cancellation which takes "locality" which, because of its formality, is perhaps not
place between them, expression (19.33) represents a very useful. Of greater importance are the conditions of
very small residue. One suspects that it can be easily analyticity on scattering amplitudes which ought to
eliminated by the addition of a suitable term to (19.28). hold if certain conditions of causality (conventionally
The desired term should be real, so as not to disturb assumed to follow from "locality" ) are to be valid.
the vacuum-to-vacuum probability, and should be built The "derivations" of this section are purely heuristic
out of quantities, such as Green's functions and bare (since one is dealing with the unrenormalized fields)
vertex functions, which already exist in the classical and there is evidence that the surgery e6ected by
theory. It is not diKcult to verify that there is only standard renormalization techniques (which, when
one second-order expression with the necessary prop- applicable, is implicit also in dispersion theory) removes
erties, namely from the theory precisely the formal nonlocality repre-
(1/48)S . . Qilgjmg+knS sented by V(». We shall return briefly to this question
in the next section, where alternative, more systematic
(1/48)S .. og ilg jmQ+knS 0 (19 34)
methods for treating the higher-order radiative correc-
We therefore conclude that the Gnal correct forms for tions are discussed.
1234 BRYCE S. 0 E%'I TT 162

20. NONCAUSAL CHAINS. FEYNMAN BASKETS are those which con. tain overlapping loops (just as in
FOR OVERLAPPING LOOPS. GENERAL renormalization theory). Let us therefore consider erst
ALGORITHM FOR OBTAINING THE the simpler diagrams in which no loop touches any
PRIMARY DIAGRAMS TO other loop in more than a single point. By referring to
ALL ORDERS Eq. (19.35) and to Figs. 2, 3, and 4 it is not dificult
to see that, as far as these diagrams are concerned, the
If, in Eq. (19.3), the density functional 6 is set equal correct expression for W is obtained from that for W
to unity then all the terms drop out of Eqs. (19.27) and
simply by removing the eoncalsul chains from all loops.
(19.35) save those which involve the Feynman propaga-
tor G only. The resulting functional will be denoted by By "noncausal chain" we mean any cyclic product of
advanced (or retarded) Green's functions connecting a
sequence of points of which the last is equal to the 6rst.
W= —-,'i ln detG
1Gijg . , GH$
,', 5—,;, G*tG~G~"S,
Gmn
—I, )„..
15' . „GijGkl+..
Such cyclic products necessarily vanish except when all
the points coincide, and hence they depend only locally
minus the same terms evaluated with q = 0. (20.1) on the background field. In the case of scalar 6elds with
nonderivative coupling they may be formally set equal
The basic topology of vacuum diagrams is already con- to zero. In the general case they must be explicitly
tained in the terms of this series. Each term corresponds removed. 4~
to what will be called a primary diagram, composed of The diagrams with overlapping loops cannot be
bare vertices and Feynman propagators only. The treated so simply. Here the diKculty is twofold. First,
primary diagrams of orders 1 through 3 are shown in the noncausal chains enter in a more complicated way
Fig. 5. In these diagrams the terms with q =0 are to and, except in the case of W(2), there is no unique way
be understood as already having been subtracted out. of removing them. Second, the removal of noncausal
In most applications one is not interested in the chains by itself does not sufBce to lead to invariant
vacuum-to-vacuum amplitude itself but only in its amplitudes.
functional derivatives, which yield the radiative cor- The situation may be described more fully thus: At a
rections to scattering amplitudes. The terms with certain point in the process of removing noncausal
q =0 make no contribution to these amplitudes, being chains from a given primary diagram one must stop;
essentially constants of integration. Therefore, no no further noncausal chains remain. At this point the
attempt has been made to represent them pictorially. diagram no longer contains closed loops composed of
The terms of Eqs. (19.2/) and (19.35) which are Feynman propagators only. At least one segment of
missing from (20.1) are topologically similar to the
every loop consists of a "free" propagator 6'+& or 6& &.
primary diagrams. They differ only in the replacement That is to say, the removal of the noncausal chains
of various Feynman propagators by G+, G, and G. The "breaks open" all the closed loops, and the result is
question which presents itself is how these replacements representable as a sum over tree diagrams with all
are to be made in the general case and with what external lines on the mass shell. However, the particular
coefricients. trees which are obtained, and the coeKcients attached
It is evident from the analysis of the preceding sec- to them, generally depend on which noncausal chains
tion that the diagrams which cause the most trouble are removed first and on what oriemtatioe one chooses
to assign to them. In the more complicated diagrams
there is not even a unique way of averaging over
orientations.
One may nevertheless ask whether there is a "correct"

".-" -"CCO -'*

C
way of removing noncausal chains. The answer is yes,
but it must be determined separately in each individual
case by a computation which is as complicated as those
+—I
l6
I
8 8 of the preceding section; no simple general algorithm
has so far been found. Moreover, even when the non-
—
~C -' CO-C causal chains have been properly removed the resulting
tree diagrams cannot yet be assembled into Feynman
+—I
1
I
+~~1 +~6l [ )
baskets. "Nonlocal terms" beginning in lowest order
l6
with F(2), have also to be discovered and added.
To gain an appreciation of the complexities which
arise the reader may try his hand at decomposing W(3),
remembering to take into account the contribution
which I'~2~ makes in this order, through its presence in

FIG. 5. Topology of higher-order radiative corrections. Primary 49 For lV(1) this means subtracting ln detG+ from ln detG; the
diagrams of orders 1 through 3. No jnvariance group present. llutter is represented by the simple circle in Fig. 5,
162 QUANTUM THEORY OF GRAVITY. II 1235

the density functional A. %e shall content ourselves

here with the decomposition of W&si.
i+-
In this case it turns out that although the removal
of noncausal chains can be carried out in various ways s
I
~+ s M, + s ~, + g cK,~ + e ~j I P I

the end result is always the same. Thus the three prop-
agators of the 6rst diagram for Wt&i in Fig. 5 may,
with the aid of Eqs. (9.17), be decomposed as follows:

GSGSG=GSG+SG +G SG+SG+G+SGSG
—G-SG+S G- —G+S G+SG-+G&-& 8 GS G&-& ~
r-.
+Gi+)Sg(+)Sg gSG(+lSG( —)+G(—)Sg(+lSG(—) 8w~ r i g 8

G(+)Sg(+)Sg(—) (20 2)
+me~ s i.
The erst five terms on the right of this equation yield
noncausal chains. If they are subtracted one obtains
the first term on the right of Eq. (19.28). We have
FrG. 6. Second-order vacuum diagrams when
already seen that this expression is not quite right; we an invariance group is present.
must add the quantity Fts&, obtaining (19.35) as the
correct full expression for 8"(~). It is then straight- the sum of the 23 terms which are depicted in Fig. 6.
forward to verify that t/t/'(~) has the following decom- It tlrns out that these terms can alternative/y be obtained
position into Feynman baskets: from the pnmary diagrams of Fig. 7 by removing non-
Lt &&G(+) rjg. . (+) st+ (I/48) t, &G(+). jtg(+&m causal chains and adding the nonlocal "correction"
)&G'+) ~"t~~„— —
'3 "~G&+)"G&+»'~G'+& "~t~~„minus the I'tsl= (I/48)~, js&"~j G+'"5', i -—(I/24) V(-) p
same terms evaluated with q =0, (20.3) )((g(+)~&+g(—)~&) (g(+)Pr+G( )Pv)g+— jjV
a result which admits of immediate extension to the —(1/24) V&, (g(+)~&+G( )as)g+Pv—grjV
case in which an invariance group is present: —(1/24) V(;)PG+ '(G&+i»+6&—&»)G'&V(„)s, (20.5)
W — Lt. s[g(+). jN(+)&'t+(1/48)t. ggj(+) j. &N(+)m which is a generalization of (19.34). In this case the
XN+l~~t, s t. , „S+ 'iS+ imS+';~st „„minus noncausal chains must be removed in a maximally
the same terms evaluated with q =0. (20;4) symmetric manner which gives equal weight to both
dotted and solid lines and to the various distinct orienta-
Several observations may now be made. First, the tions of the diagrams.
possibility of decomposing the vacuum-to-vacuum Now it is a remarkable fact that W&», as given by
functionals into Feynman baskets is closely related the primary diagrams of Fig. 7, is already group-
to unitarity of the 5 matrix; unitarity statements such invariant as it stands. It is not only invariant under
as Eq. (13.14) involve sums over tree amplitudes of
"
precisely the form (20.4). Second, although the require-
group transformations of the background 6eld, which is
obvious from its manifestly covariant construction,
ment that the theory be invariant under transformations but it is also y-invariant as well. The latter assertion
of variables has led us to functionals which decompose
may be veri6ed by a straightforward but tedious com-
into Feynman baskets, it is clear from the tree theorem
(14.26) and the invariance statement (15.10) that we
putation which makes use of Eqs. (4.2), (4.3), (4.10), "
(6.11), (14.13), (16.28), (17.26), and (17.31).
could instead have started from decomposability itself This result shows that combinations of tree ampli-
as a criterion for the discovery of "correction" terms tudes are not the only group-invariant quantities in
such as Y(2), and thereby obtained vacuum-to-vacuum the theory and suggests that the method of decomposing
amplitudes which are not only invariant under trans- diagrams into Feynman baskets, and the forrnal com-
formations of variables but group-invariant as well.
Evidently the various consistency requirements of the
theory fit together in an interlocking fashion, and it
appears that the imposition of one will yield the others
also. This makes it possible to consider alternative
approaches to the theory of radiative corrections.
One such approach is arrived at by reexpressing ~ P+

(20.4) in terms of the manifestly covariant propagators

G, G&+&, G, G&+&. Using Eqs. (4.2), (5.6), (17.11), (17.19)& FIG. 7. Primary diagrams of order 2 when an invariance group
(17.26), (17.27), and (17.31) one finds, by rather is present. (Dashed lines without arrows represent Feynman
A
intricate and tedious algebra, that (20.4) appears s.s propagators 6 for Gctitious quanta. }

. 'OAn explicit veri6cation that (20.4) satis6es unitarity has "Equations (4.10) are never needed except when dealing with
been carried out (unpublished). primary diagrams.
1236 8 RYCE S. 0EWITT 162

plications which go v ith it, can be avoided. Indeed in they never begin or end on solid lines. (2) In addition~
conventional field theory one works with the primary to the bare vertices S the only vertex which is needed
diagrams from the beginning and never bothers to is V&;&e. Vertices such as R",;y~iR'e, ; at which more
remove the noncausal chains. In the case of non- than one solid line meets a dotted line never occur. (We
overlapping loops it is easy to see why one nevertheless shall see later that they do not even occur when external
gets correct results. It is a standard procedure in mo- lines are inserted into the vacuum diagrams. ) (3) The
mentum-space calculations, after all terms of an solid lines which enter a given fictitious quantum loop
integrand have been brought to a common denominator all do so with the same orientation around the loop.
of the form (h'+2p 0+6 i0)— ", to perform a rotation (Remember they enter obliquely. ) This means, for
through 90' in the k' plane and thereby to convert example, that the combination V(;~pG &Gi"G'&V~»)g
from Minkowski space to Euclidean space for the sub- does not appear in Fig. 7.
sequent evaluation. When the integral is convergent the It is remarkable that the condition of y invariance
procedure is legitimate, but when the integral diverges alone sufFices to determine all the higher-order radiative
— —
a part the arc at infinity is lost which can be shown corrections. By going through the computation for
to correspond exactly to a noncausal chain. Moreover, W&» one is easily convinced that the same procedure
since the arc is at infinity in momentum space its coa.- gives unique results to all orders, with no ambiguity
tribution is necessarily "local" in space-time and would about coeKcients. However, it is extremely tedious to
in any case be removed by renormalization, e.g. , with carry out the computations required, order by order,
the use of regulators. and one naturally asks whether or not a short cut can be
In the case of overlapping loops eonloca/ renormaliza- found, Fortunately it can.
tions, i.e., renormalizations within momentum sub- One introduces a fictitious system described by the
integrations, must be performed in order to get rid of action functional ~tF@p'p&'+F eg* ge, where the field
the well-known overlapping divergences. Although a P' is of the commuting type and the fields g~ and g*~,
complete analysis of the overlapping case remains to which create and annihilate the fictitious quanta, are
be carried out there is considerable evidence that here of amticormrmltimg type One the. n computes W from the
too renormalization absorbs the "correction" terms, formulas
which now include not only noncausal chains but also
the "nonlocal" quantities I"(~), etc. One expects that W[9'] = w[9&j- w[0j, (20.6)
the decomposition of radiative corrections into Feyn- expiw[qg= (expiw&t&[05)(dety) ''
man baskets is in effect replaced by analyticity state-
ments, and that the unitarity of the S matrix is secured &&(0, -
T(-pi[I«-»e*-~'e'+(1/3')&, ;;.~'~'~'
by the famous Cutkosky rules. " I

+(1/4 )~, o~~4*4'0'0'+" 3)10,—~)&», (20.7)

As a working procedure we shall therefore assume,
just as in, conventional field theory, that it sufFices to where the subscript (1) indicates that the evaluation is
deal with the primary diagrams alone. Although much to be carried out with reference to the fictitious system.
work remains to be done to establish this assumption The vacuum-to-vacuum amplitude of the fictitious
with complete rigor, it is then quite easy to construct system itself is to be understood as defined without the
a manifestly covariant quantum theory of gravity removal of acausal chains. Thus
(and/or the Yang-Mills 6eld) which is unique to all 1/2
orders of perturbation theory. One has only to discover
(0, ~ —~)&t& ——(det~) (20.8)
what diagrams have to be added to those of Fig. 5,
~0,
(detyo) 'I' expiWu&[&pj,
etc. , in order to obtain y-invariant vacuum-to-vacuum
amplitudes, and this problem has been completely W&»[yg= w&t&[&j —wo&[oj, (20.9)
solved.
The solution of the problem for the case of W~i& is (detG)'"(dety) "'
given in Fig. 7. The diagrams of this figure can be dis-
expiw(t&[q j= (20.10)
"
covered in the following way. One adds to the Wtt&
detG

diagrams of Fig. 5 other topologically similar diagrams, with no factor (detG+)/(detG+)'~' appearing in (20.10).
involving the fictitious quanta in all possible ways, The factor (dety) '~' or its inverse is inserted into
each with an arbitrary coeKcient, and then adjusts the
coefFicients so that the total expression becomes in-
variant under changes in the y's. In the process one
discovers the following facts, which hold to all orders:
(1) The 6ctitious quanta always occur in closed loops;
"R. E; Cutkosky, J. Math. Phys. I, 429 (1960). When the
Eqs. (20.7), (20.8), and (20.10) so as to make w&t&
y-invariant [see Eq. (16.29) j.
The anticornmuting character of the fields Q~ and
implies that the fictiIioies qgarita are ferr&iioris
It is this property which enables them to play a com-
pensatory role in the theory. For example, it is what
causes detC to appear in the denominator rather than
'4.
Cutkosky rules are applied to divergent diagrams it is always
assumed that the divergence is6rst removed by regulators. The
nonacausal chains are therefore
automatically excluded. ~The usual relation between spin and. statistics obviously
~ B. S. DeWitt, Phys. Rev. Letters 12, 742 (1964). need not apply to these quanta.
162 gUANTUM THEORY OF GRAVlTV. 1237

the numerator of Eq. (20.10). It is also what restricts " where the ),; are ordinary c-number-variables and the
the fictitious quanta to appearing only in closed loops.
The explicit evaluation of expression (20.7) is carried
), "X
* are variables from an anticommuting number
system. The determination of the higher-order primary
out with the aid of the hierarchy of equations generated diagrams then becomes a straightforward exercise. In
Fig. 8 we show, for example, the diagrams which must
(detpo)'&' be added to those of Fig. 5 in the case of WIs&.
(O, oo (
T (exp(ili. ;$'+iX* Q +i/* X )) ~0,—~) I I The proof that formula (20.7) yields 7-invariant
(detv)'" vacuum amplitudes to all orders may be carried out by
= exp(iW&q&+ ~~iX;X G"+i7Ie XsG'&) (20.11) 6rst rewriting it in the form

expiw[q ]= Z'
ex& i(lV.~X.X'+V'';~'~'+(1/3 )S;;.~'~'~"+" ) . dXdg
J' exp(iF pP* P~)dPdP

expi(S[p+p] S[p— S,;[—

q]y ] + ', q.ex-.xs+ ', R;.R-, It'qv')
dXdg, (20.12)
J' exp(i&-~4' 4')4d4*
where the factor g' is a normalizing constant for the variables;
functional integrals, the Geld X is introduced in order XIa Xa+ )Xa yIi pi+ $yi Pa+ /Pa
/&a
to generate the necessary factor (dety) 'is, and 5 is a
non-self-adjoint operator deGned by yea yea+ g ea (20 15)
5 II= OV P+ V(;)II&',
R;II =
—J' (20.13) %e then choose these changes in the following way:
—
(R' =R' +R', ,Q'=—R' [q+Q]. (20. 14)
1
~ 1nPb~ (20.16)
We next place primes on all the 6eld symbols X, P, 8P'= IR' 5$ (20.17)
f, P* in (20.12) and (20.13).This is purely a formal step 8'@—
'II(~V VR'v V(I'—
)&~V)4'" (2o 1g)
which changes nothing. However, we may regard it as
corresponding to actual changes in the integration /pea —gu pey$$s (20.19)
"oP= P'(-', R—
,bf '»~ s+-b~;;R~ &)gII~ (20'. 20)

--I
8
I
/
) --I
I
4
J
'L
]
where g is the Feynman propagator" for the operator
5, and the by;;, by™p are to be regarded as arbitrary
I
inGnitesimals.
4
I
) --I + l OI
I
We now compute the effect which these changes
produce on the numerator and denominator of the big
-—
s
i i--I'I'('
I 4
I
c. ——
I
2 ~&
& I I
I
~
&+ —It
I

4
integrand in (20.12). By making use of the identity

V(ej)II(R V{yg)p(R ~=c ggy8'sp (20.21)

~
r y
I ("~- ~ --I I

4 4 and the fact that S[y+P] is invariant under (20.17),

+I,
——
I
4I
' 1 . 1--I',

)
I
8
+2I g
w
I 4 r
we Gnd
t'(l~. sx.x~+l&;,~'e~+(1/3I)S;;.~'~'e'+
=y sx bxi'+R; R; P'bye'= ,'x"8y px&-
)".
+qVR;"8PPII .
%1~ /
g+-2I I
=-'X by pX~+ 'P'R 5y ' &R sQ'-
+P'R, R" 8ys, g', (20.22)

=& sP'V'+&-8* ~8+4* V( ')As~4'

I
l
rI (R' 8y "R'II+. (20.23)
--I
6 But these are just the changes which these quantities
would suffer under the changes by;;, by p in the y's.
FIG. 8. Diagrams which must be added to 8'(3) of Fig. 5
when an invariance group is present. ~6 See J. Schwinger, Proc. Nat. Acad. Sci. U. S. 48, 603 (1962).
"The presence of the Feynman propagators in Eqs. {20.18)
~~That ln detG appears in with opposite signs for bosons g and (20.20) indicates that we are dealing here with a special class
and fermions was Grst pointed out long ago by Feynman. It is of nowlocal transformations of variables. Transformations of this
a consequence of the familiar minus sign which goes with closed class are permissible because of the Feynman boundary condi-
fermion loops [R. P. Feynman, Phys. Rev. 76, 749 (1949)g. tions implicit in the functional integral.
1238 BRYCE S. DEWITT
Therefore, if we can show that the functional Jacobians This establishes, in particular, the legitimacy of the
of the transformations (20.16), (20.17), (20.18), and scale transformation
(20.19) cancel in (20.12), it follows that (20.12) is
y-invariant. Psg ~ ~Pay') PaP ~ ~&nP ~ (20.31)
These Jacobians are obtained by first computing Under this transformation we have

R;~R, ~ —+ kR;~Rp ) S~p —+),S~p) (20.32)

=B p+2v "Bv»p, (20.24)
bx~ and in the limit )( ~
0 Eq. (20.12) reduces to the con-
ventional but ambiguous formula
—= B';+R',;BP+6V'. B(BP) (20.25) tl—
expi(() [yl = Z expi~ S;,(t 'P'
2f
=B p 8"—
6V'»(»'~R'p V(—
i') pBV) (20.26)
g P
+—
3f
S.';~4'4'0"+ id&, (2o 33)

=B p+c where
'V pBP (20 27)
gp
Z=Z dx (20.34)
B(B~-)
'Ribs —
'"-y p(o By)R p
pg—
Expression (20.12) evidently removes the ambiguity
and may be regarded as the depnitioii of the integral
+B'AV (m)»8" (20 2g) (20.33) when an invariance group is present.
There remains only the question how to insert ex-
Invoking Eqs. (4.10) we then get

B(x', ~'),
=1+-', tr(y 'By)+R' BP+6V
B(B~-)
ternal lines into the primary vacuum diagrams. When
we dealt with W instead of W' the insertion was ac-
complished by simple functional differentiation. Now
B(x,y) that the noncausal chains are left unremoved and no
=1—6V' By; R'pbP +6V BPV(p;)„g», (20.29) correction terms are added we must proceed some-
what more carefully.
BS",S") We have remarked earlier that expression (13.6)
=1—() p6V'p(By;;R' —V(„;)Bp)+c» bp for the 5 matrix in terms of chronological products
BQ,~') holds even when an invariance group is present. Ke
B(x',y') were nevertheless forced to use it in a very circuitous
(20.30)
B(x,y) manner, by restricting it -to the case in which no group
is present and then generalizing its c-number con-
Since the latter Jacobian is independent of and P*N f sequences, because. we previously had no direct way
it can be removed from the integral in the denominator of calculating the chronological products. Now we have.
of (20.12), whereupon it cancels with the Jacobian for Following the example of Eq. (19.5) (but ignoring
the X and P integrations. The y invariance of formulas the density functional 6 since we are here dealing with
(20.7) and (20. 12) is thus proved. primary diagrams) we may set

(0, [T(A[fj)[0,— )=exp( —A()[0])Z' A[/]

xp 'X xp+'-~'8*0'+(1/3))S. ' 0'0'4'+
( —
.)
dxdqb. (2035)
J' em (i~-8* WP)44*
Then Eq. (19.6) is replaced by

(0, ~y'~0, — )=exp( —ie[0j)Z'8 ' ~

I'; y"+— B

X— dxdg (20. 36a)

J' exp(PS pP* PP)digdij~

=exp( —iw[0])Z'O)g'& )
1
—S' gag'+ — S' )) O'Q'(tv+
f
V(-) p4*VP em(i~ 4*»&')@'d&*
J' p( e8x'.Ai'VP)44'*
2t 31
exp'(-', y, px~xP+-', F„,
P"g'+(1/3!)S,„gg'P'qV+ )
X dxdp, (20.36b)
J' exp(i&. r4*%")d4dP
 QUANTUM THEORY OF GRAVITY. II

where we now use the propagator 8+ in place of G in front of the integral so as to obtain correct external-line wave
functions for the S matrix.
The only vertices which get inserted by the factor in square brackets in (20.36b) are the bare vertices S~ and
Vt;&e. Therefore (O, co $'~0, —co) may be expressed in the compact form (12.20), but with G replaced by S~,
~

provided the symbol 8/5 p is no longer taken literally but is understood to yield GSsG when acting on G and S~+t
when acting on S,
to have no effect on V &„;&s,and to insert (in all possible ways) into any 6ctitious quantum loop
"
merely one more vertex V&„&p Itaving lhe same orientatt'ott as att the other vertices already irt the loop. With this
understanding it is easy to see that (20.35) then yields also Eqs. (12.21), (12.22), and (12.23), with the modi6cation
G ~ N~ applied to all external lines. Chronological product amplitudes de6ned in this way may be used directly
in (13.6) to calculate the S matrix.
The consistency of these simple rules with previously obtained results is readily checked. For example, if non-
causal chains are reinserted into Figs. 2(b) and 3(b) the resulting primary diagrams for the lowest-order radiative
corrections to the one- and two-quantum amplitudes are precisely those obtained by the present prescription. We
note in particular the suKciency of the vertices S„and V~;~ p and the uniform orientation of the latter around any
6ctitious quantum loop.
"It will be noted that the operators S/by', when redefined in this way, are still commutative.

PHYSICAL REVIEW VOLUME j. 62, NUM BER 5 25 OCTOBER &967

Quantum Theory of Gravity. III. Applications of the Covariant Theory*

BRvCE S. DEWITT
Institute for Advanced Study, Princeton, itIeto Jersey

Departnient of Physics, University of North Carolina, Chapel Hill, Iiorth Carol&tat

(Received 25 July 1966; revised manuscript received 9 January 196'I)

The basic momentum-space propagators and vertices (including those for the fictitious quanta) are
given for both the Yang-Mills and gravitational fields. These propagators are used to obtain the cross
sections for gravitational scattering of two scalar particles, scattering of gravitons by scalar particles,
graviton-graviton scattering, two-. graviton annihilation of scalar-particle pairs, and graviton bremsstrah-
lung. Special features of these cross sections are noted. Problems arising in renormalization theory and the
role of the Planck length are discussed. The gravitational Ward identity is derived, and the structure of
the radiatively corrected 1-graviton vertex for a scalar particle is displayed. The Ward identity is only one
of an infinity of identities relating the many-graviton vertex functions of the theory. The need for such
identities may be eliminated in principle by computing radiative corrections directly in coordinate space,
using the theory of manifestly covariant Green's functions. As an example of such a calculation, the con-
tribution of conformal metric fluctuations to the vacuum-to-vacuum amplitude is summed to all orders.
The physical significance of the renormalization terms is discussed. Finally, Weinberg s treatment of the
infrared problem is examined. It is not difBcult to show that the fictitious quanta contribute negligibly to
infrared amplitudes, and hence that Weinberg s use of the DeDonder gauge is justified. His proof that the
infrared problem in gravidynamics can be handled just as in electrodynamics is thereby made rigorous.

1. INTRODUCTION' canonical or Hamiltonian theory and the other on the

'N the Grst two papers of this series' two distinct manifestly covariant theory of propagators and dia-
- mathematical
grams. So far no rigorous mathematical link between
approaches to the quantum theory
the two has been established. In part this is due to the
of gravity were developed, one based on the so-called
kinds of questions each asks. The canonical theory
leads almost unavoidably to speculations about the
*This research was supported in part by the Air Force OfBce meaning of "amplitudes for diferent 3-geometries" or
of Scientific Research under Grant AFOSR-153-64 and in part by
the National Science Foundation under Grant GP7437.
"
"the wave function of the universe. The covariant
ts Permanent address. theory, on the other hand, concerns itself with "micro-
B. S. DeWitt, Phys. Rev. 160, 1115 (1967); preceding paper, processes" such as scattering, vacuum polarization, etc.
ibid. 162, 1i95 (1967). These papers will be referred to as I and
II, respectively. The notation of the present paper is the same as Some of the questions raised by the canonical theory
that of II, which should be consulted for the definition of un- were explored in I. In this third and 6nal paper of the
familiar symbols, e.g. , S„ for the n-pronged bare vertex and
V(;)p for the asymmetric vertex coupling real and fictitious series we examine some of the consequences of the
quanta. covariant theory.