QFTlectures
QFTlectures
QFTlectures
L. Vanzo∗
Dipartimento di Fisica dell’Università di Trento
Italia
Contents
1 Relativity and Quantum Theory 2
10 Poincaré Symmetries 32
13 Massless Particles 46
1
18 Interactions and Collision States 59
21 Perturbation theory 69
25 Scattering 80
A Polarization vectors 90
B Dirac theory: non relativistic limit and the fine structure of hydrogen 91
B.1 Pure Coulomb field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Part of its physical meaning is that for time-like separation −ds2 ≡ dτ 2 is the square of
the proper time interval, namely the reading of a standard clock. The metric defines a
non degenerate scalar product
(u, v) = ηab ua v b = η ab ua vb = ua v a
some equivalent expressions were emphasized. We write it also in the form u · v; the
squared norm of a u is denoted u2 . It is negative for time-like vectors, zero for light-like
(or null) vectors and positive for space-like vectors.
E1: Show that the scalar product is non degenerate.
The Lorentz transformations are isometries of the space-time metric, i.e. they send
x → Λ · x (compact notation, matrix product is understood) with Λ satisfying the ten
conditions
Λan Λbm ηab = ηnm (2)
2
This group will be denoted by O(1, 3). It follows from this that
The set of elements with unit determinant and Λ00 ≥ 1 is the proper orthochronous Lorentz
group, at the same time the connected component of the identity. We denote it by L↑+ .
E2: Show that L↑+ is a subgroup of the general Lorentz group O(1, 3).
The spatial inversion
1 0 0 0
0 −1 0 0
P= 0 0 −1 0
(3)
0 0 0 −1
and the time inversion
−1 0 0 0
0 1 0 0
T = 0 0 1
(4)
0
0 0 0 1
do not belong to L↑+ . The subgroup of elements with unit determinant is called the proper
group and contains the matrix PT , besides the ordinary proper orthochronous Lorentz
transformations. The full isometry group of Minkowski space also contains translations,
mapping x → x + a. The combined action is
0
x =Λ·x+a
0
and we denote it by x = (Λ, a) · x. Then one obtains the composition law (verify)
0 0 0 0
(Λ, a) · (Λ , a ) = (ΛΛ , Λ · a + a) (5)
In particular, the identity element is (e, 0) (e is the identity matrix) and
The group of all such elements (Λ, a), with the given product, is the so called Poincaré
group, the composition law describing it as the semi-direct product of L↑+ with the trans-
lation group T4 .
Special relativistic dynamics forbids action-at-a-distance interactions and, basically for
this reason, it really requires infinitely many degrees of freedom. For example, if to a
body we apply N contact forces simultaneously at N different points, to describe the mo-
tion we need N degrees of freedom to begin with. The only consistent relativistic system
with a finite number of degrees of freedom probably is a system of N non interacting
particles, having at most localized pointlike collisions from time to time.
Special relativity is also a classical theory; thus one cannot use it, as it stands, to compute
relativistic corrections to atomic spectra, for example, or to scattering cross sections at
high energies between elementary quantum particles.
UNITS in relativity will always be chosen so that c = 1.
3
QUANTUM THEORY’s stage is the space of unit rays in a separable, infinite di-
mensional Hilbert space H, where by a unit ray we mean a set of vectors of the form
Ψ̂ = {αΨ; Ψ ∈ H, |α| = 1}. The space of unit rays in H will be denoted by H. c
The state space of quantum mechanics is thus an infinite dimensional complex projective
space, or the union of two or more spaces of this kind called superselection sectors. These
can appear when the states are partitioned into classes such that no linear superposition
can be prepared among states in different classes. For example, all physical states are
believed to be eigenstates of the electric and barionic charge operators, Q and B, i.e. it
is believed (and experimentally verified) that it is impossible to prepare a linear super-
position of states with different values of electric charge or barionic number. However
mixtures are not forbidden, in other terms only the relative phases of superselected states
cannot be measured. It seems also impossible to prepare a superposition αΦ0 + βΦ1/2 of
a spin zero particle with a spin-1/2 particle: indeed a 2π rotation sends the state into
αΦ0 − βΦ1/2 , but this belongs to the original ray if and only if either α = 0 or β = 0.
Thus it is believed that all physical states are eigenstates of the operator (−1)F , where
F counts the number of fermions in the state. If there are several superselection sectors,
Ĥi , it will be convenient to introduce the direct sum
H = ⊕i Hi
but keeping in mind that linear superposition of states in distinct superselection sectors
have no physical meaning. AsP already mentioned one can formP instead density matrices,
or mixtures, of the form U = i xi Pi , with each xi > 0 and i xi = 1, where the Pi are
projection operators over sub-spaces in Hi .
Observables are self-adjoint operators on H; if superselection rules exist then not all
self-adjoint operators are observables. For example, the projection operator on a physi-
cally non realizable state is not an observable. Moreover, since every physical state is an
eigenstate of the operators Q, B and (−1)F , then these operators will commute with all
observables.
It is characteristic of quantum theory that measuring an observable A in a system pre-
pared in a given quantum state Ψ does not give the possible values with certainty, but
only a statistical distribution. The expectation value on a large number of identical mea-
surements is the real number (Ψ, AΨ).
The main defect of quantum theory is its non relativistic character; thus one cannot use
it, as it stands, to compute quantum corrections to relativistic phenomena, if not to a
limited extent. A first step into a quantum relativistic theory would be to find the quan-
tum version of the Lorentz transformations, i.e. to find all unitary representations of
the Poincarè group on Hibert spaces. Thus one would like to know the correspondence
(Λ, a) → U(Λ, a) into unitary operators acting on the space of states. This problem was
first solved by E. Wigner in the Thirties, and we will summarize his work later in the
course. For the time being we content ourselves with few highlights into the difficulties
one encounters if a naive construction of relativistic quantum mechanics is attempted.
UNITS in quantum theory will always be chosen so that ~ = 1.
A RELATIVISTIC WAVE EQUATION is expected to propagate fields with finite
propagation speed less than or equal to c. However, quantum theory and stability re-
quirements also need positivity of energy and a certain notion of localizability, specifically
4
the possibility of admitting wave functions compactly supported1 , but these requirements
seem severely conflicting.
Let us begin with a free particle; the Schrödinger equation
∂Ψ 1 2
i =− ∇ Ψ ≡ HΨ
∂t 2m
is the transcription of the classical energy-momentum relation E = p2 /2m into a differ-
ential form. It does not propagate with finite speed. For instance, the amplitude to go
0
from x to x in time t is easily computed to be (verify)
r 2 0 0
0 −iHt m x − 2xx + x 2
< x |e |x >= exp i
2πit 2mt
0 0
and this is non vanishing for any t, x, x , even for those such that |x − x | > t (space-like
separation).
E3: find the amplitude for propagation in three dimensions. p
Let us try the relativistic energy-momentum relation E = p2 + m2 : we obtain the
pseudo-differential equation
∂Ψ √
i = −∇2 + m2 Ψ (6)
∂t
E4: find the probabilty amplitude in this case at spacelike separation. Show that it is non
vanishing but exponentially decreasing.
For eq. (6) there is a very simple result: it does not admit solutions with spatially compact
support.
PROOF Let Ψ(x, t) be vanishing outside a compact spatial set. Then the Fourier trans-
e
form, Ψ(p, t), and its time derivatives are analytic functions in the complex p-space. But
from (6) we get
e
∂ Ψ(p, t) p 2
i e
= p + m2 Ψ(p, t)
∂t
showing that the time derivative cannot be analytic everywhere; also, the branch cut
cannot be eliminated by multiplication with an analytic function. I
Thus, for example, even if Ψ(x, 0) has the support within a ball with finite radius R,
after t = 10−10 sec the wave will be possibly non zero at some point 10 light years away
from the center of the ball. The probability density Ψ(x, t)Ψ(x, t) will also be non zero,
showing a manifest violation of causality with finite probability. Here the conflict with
locality (reflected into analyticity) and the positive energy condition (requiring the branch
cut) is at the root of the result that finite propagation speed cannot be achieved in this
model.
OUR NEXT step is to consider (as Schrödinger himself did) the Lorentz covariant
Klein-Gordon (KG) equation
η ab ∂a ∂b φ − m2 φ ≡ (∂ 2 − m2 )φ = 0 (7)
This is an hyperbolic partial differential equation (you recognize the wave equation for
m = 0) which has a well posed Cauchy problem and also admits initial data with compact
1
i.e. vanishing outside of a compact set.
5
support.
E5: Find a formula for the solution of the KG equation with initial data φ(0, x) = u(x),
∂t φ(0, x) = v(x).
Plane wave solutions are functions of the form a exp(±ik · x); applying the KG operator
one obtains the mass-shell condition (verify)
k 2 + m2 = 0
√
giving k 0 = ± k2 + m2 ≡ ±ωk . Interpreting k as the four-momentum of a quantum par-
ticle gives then the unpleasant result that particles with negative energy exist. Requiring
only positive energy in the momentum space support implies the general structure
Z
e
φ(x) = φ(k)θ(k 0 a
)δ(k 2 + m2 )eika x d4 k
where φ(x, 0) is the initial data. The same must be true not just at t = 0 but also for
any sufficiently small t, say t ∈ [0, ), since the propagation speed is supposed to be finite.
Thus
Z
J(t) = fR (x)φ(x, t)dx = 0 t ∈ [0, ) (9)
2
this is the set of points p in space-time such that all past directed timelike or null curves through p
intersect the support of the initial data set.
6
but for larger t > , J(t) will be possibly non zero and real. By (ii) the function J(t)
is analytic in the lower half complex t-plane, namely J(t − is) is analytic for s > 0 and
real on the real time axis. Let us now recall the Schwarz’s reflection principle3 : Given
a function f (z) analytic in a domain D of the upper (or lower) complex plane, having a
boundary segment on the real axis, and let the function be real on this segment. Then
there is a unique analytic continuation to a reflected domain D ∪ D such that in this
domain f (z̄) = f (z). Thus J(t − is) is actually analytic in a larger vertical strip at least
as wide as the domain of definition of the putative solution φ(x, t). But J(t) = 0 in
[0, ) so by another well known result in the theory of analytic functions, J(t) vanishes
everywhere. Hence φ(x, t) vanishes outside K ∀t and φ(x, t) will not exist. I
It follows from this result that any tentative relativistic Schrödinger equation must run
into conflicts with one or more of the stated assumptions.
REMARKS: There are some indipendent arguments which help to understand the fail-
ure of relativistic wave functions to obey locality, positivity of energy and finite propaga-
tion speed all at a time. According to the standard Copenhaghen interpretation the main
use of the wave function is to furnish a probability density for the position variables. But
this makes sense only if the spatial position variables are in the joint spectrum of quantum
operators representing the position measurements. Now Lorentz symmetry mixes time
and spatial coordinates thus electing time to the role of a quantum observable, namely the
reading of a quantum clock. It would be most natural to suppose that the four position
operators X a be accompanied by four momentum operators P b fulfilling the canonical
commutation relations
[X a , Pb ] = iδ ab
so also the joint spectrum of momenum operators is translationally invariant. But the
physical P a must lie in the forward light cone P 2 ≤ 0, P 0 > 0, hence X a pa cannot be
self-adjoint. Thus it is difficult to give a physical meaning to a quantum wave functions
on space-time in a relativistic theory. On the other hands there is no problem to define
wave functions in momentum space.
7
given how these coordinates are to be measured. Space-time coordinates are then external
parameters much like the status of time in quantum mechanics: they simply denote the
event at which an observable is measured which, consequently, becomes a field operator.
This idea has far reaching consequences: for example one can construct local observables,
i.e given a space-time region O one has naturally the set A(O) of all observables having
support within O. Also in ordinary quantum mechanics, although not in its usual for-
mulation, one can localize observables in the same sense, but in a relativistic theory a
0
new very restrictive requirement appears: if a space-time region O is space-like separated
0 0 0
from another one O. i.e. if (x − x )2 > 0 for any x ∈ O and x ∈ O , then
h 0
i
A(O), A(O ) = 0 (10)
because causally disjoint experiments must give uncorrelated results. Another bonus will
be that the negative energy field components now become negative frequency components,
causing no more troubles than expanding a signal using exp(iωt) instead of exp(−iωt).
Finally, the idea is closer in spirit to general relativity theory, according to which space-
time coordinates can have no meaning independent of observation.
THE FIELDS BEING OPERATORS, how should we set up the commutation rela-
tions? To illustrate the ideas let us consider a field on a two dimensional cylindrical world
with periodic spatial boundary conditions. The field equation will be that of a “massive
closed string”, topologically a circle S 1 ,
∂2Φ ∂2Φ
− + − m2 Φ = 0
∂t2 ∂x2
with Φ(t, x) = Φ(t, x + 2π). The general solution will be
0
X
Φ(t, x) = An (t)ei2πnx + A0 e−imt + A∗0 eimt (11)
n∈Z
where for any operator A, we denote by A∗ the adjoint operator. The field equation
gives now the amplitudes as solutions of infinitely many harmonic oscillator differential
equations
Än + ωn2 An = 0, ωn2 = 4π 2 n2 + m2 (12)
We assume the field operator is Hermitian, Φ∗ (t, x) = Φ(t, x); then A−n (t) = A∗n so we
can write the solutions in the form
√
An (t) = An e−iωn t , n > 0, ωn = 4π 2 n2 + m2
and put A−n = A∗n . The quantum prescription for the harmonic oscillators is of course
the set of commutation relations
[An , A∗m ] = f (n)δnm , [An , Am ] = 0, n, m ≥ 0 (13)
The factors f (n) can be fixed by using canonical quantization, i.e. the fact that the scalar
field can be put in the form of a hamiltonian dynamical system, but we shall not dwell on
8
these points at the moment. More important is the understanding of the Hilbert space
structure. It is here that a substantial difference appears with systems of only finitely
many degrees of freedom: the oscillator algebra (13) admits many unitarily inequivalent
representations whereas in the finite dimensional case there is only one, up to equivalence.
But there is one particular representation which is distinguished by its clean and direct
physical interpretation.
THE FOCK REPRESENTATION starts by defining the vacuum state as the state
annihilated by all lowering operators An , n ≥ 0
An Φ0 = 0, n≥0
I 0 and declare that kΦ0 k2 = (Φ0 , Φ0 ) = 1.
We call H0 the one-dimensional Hilbert space CΦ
The ∞-many states (we agree that A−0 = A∗0 )
Φn = A−n Φ0
span the Hilbert space H1 . The energy operator of the nth oscillator being Hn =
ωn (A−n An + 1/2), we define the total Hamiltonian4 as
X
H= ωn A−n An
n≥0
9
3 The Quantum Scalar Field, Preliminary Facts
THE OBJECT of our study will be the Hermitian scalar field obeying the KG equation
(∂ 2 − m2 )φ = 0. Positive/negative frequency solutions are easily found (factors of π are
for our convenience)
Z
d4 kθ(±k 0 )δ(k 2 + m2 )φe(±) (k)e±ika x
(±) −3 a
φ (x) = (2π) (15)
The operators φe(±) (k) multiply the functions e±iωk t so are to be interpreted as raising
(superscript minus) an lowering (superscript plus) operators for a continuum of oscillators
labelled by the continuous four-vector k. Then
0
[φe(±) (k), φe(±) (k )] = 0 (17)
The remaining commutators
0 0
[φe(±) (k), φe(∓) (k )] = (2π)3 2ωk δ(k − k ) (18)
will be inferred later.
We recall that k 2 = ka k a = −(k 0 )2 + |k|2 , θ(x) is the step function and the Dirac delta
function is
1
δ(k 2 + m2 ) = [δ(k 0 − ωk ) + δ(k 0 + ωk )]
2ωk
p
where ωk = |k|2 + m2 . The measure
is invariant under all transformations in L↑+ , i.e. dν(Λ · k) = dν(k) for all Λ ∈ L↑+ . Clearly
Z Z
d3 k
f (k)dν(k) = f (ωk , k)
2ωk
so that d3 k/2ωk is also Lorentz invariant. Let V̄+ the forward light cone in momentum
space, i.e. the set of pa such that p2 ≤ 0, p0 > 0, and M+ the positive mass-shell of
timelike momenta pa with positive p0 . The Hilbert space, H1 , of all square-integrables
functions on M+ w.r.t. the Lorentz invariant measure is very important: the invariant
scalar product is
Z Z
¯ d3 k
0 2 2 4
(f, g) = f (k)g(k)θ(k )δ(k + m )d k = f¯(k)g(k) (19)
M+ 2ωk
To each element (Λ, a) of the Poincaré group there correspond a unitary operator U(Λ, a)
acting on H1 so defined
[U(Λ, a)f ](k) = e−ik·a f (Λ−1 k) (20)
10
E6: Show that U(Λ, a) is a unitary representation of the Poincaré group, i.e. it also
satisfies
0 0 0 0
U(Λ, a)U(Λ , a ) = U(ΛΛ , Λ · a + a)
i.e. Λ · k = k. For these one has, evidently, [U(Λ, 0)f ](q) = f (q). Saying that rotations
have no effect on a particle state at rest, where there is no orbital angular momentum, is
exactly what is meant by saying that a particle has zero spin.
Z
d4 kθ(k 0 )δ(k 2 + m2 )f (k)φe(−) (k)e−ika x
∗ −3 a
a (f ) = (2π) (22)
Correspondingly
Z
φ(f ) = d4 xφ(x)f (x)
must obey the KG equation in each variable, e.g. in x. To fix it uniquely we impose the
following initial data
∂∆(x − y; m)
∆(x − y; m)|x0=y0 = 0, = −iδ(x − y)
∂x0 |x0 =y 0
11
which are equivalent to the so called canonical commutation relations
[φ(0, x), φ(0, y)] = 0 (25)
∂Π(x0 , x) δH
0
=−
∂x δφ(x0 , x)
where
Z
1
H= d3 x(Π2 + |∇φ|2 + m2 φ2 )
2
The functional derivatives are defined as the coefficients of δΠ(x), δφ(x) in the first
variation
Z
δH = d3 x [A(x)δφ(x) + B(x)δΠ(x)]
∆(x; m) = −∆(−x; m)
but spatially even, ∆(x0 , x; m) = ∆(x0 , −x; m), hence temporally odd, ∆(x0 , x; m) =
−∆(−x0 , x; m). It is also Lorentz invariant in the following sense (translational invariance
is obvious): for Λ ∈ L↑+
∆(Λ · x; m) = ∆(x; m)
12
thus from the oddness of ∆(x − y; m) it readily follows the important locality property of
the commutator
[φ(x), φ(y)] = 0, for (x − y)2 > 0 (29)
Had we used anti-commutators
0 0
[φe(±) (k), φe(∓) (k )]+ = (2π)3 2ωk δ(k − k ) (30)
we would have obtained
[φ(x), φ(y)]+ = ∆+ (x − y; m) + ∆+ (y − x; m) (31)
instead of the difference as in (27), and locality would have been lost. Note that the
commutator alone
[φ(+) (x), φ(−) (y)] = ∆+ (x − y; m) (32)
is non local.
WE SAY that a field theory is local, or satisfies the principle of locality, when the fields
either commute or anti-commute at spacelike separation
[φA (x), φB (y)]± = 0 for (x − y)2 > 0 (33)
The so called quantization of the scalar field by means of commutation relations has given
us a local field theory. From the fact that scalar fields describe spin zero particles, we see
that quantum relativistic locality implies a particular case of the celebrated spin-statistic
theorem of Lüders-Pauli, namely that spin zero particles must be bosons.
THE FOCK REPRESENTATION of the oscillator algebra (17), (18) is readily in-
troduced. First we redefine the operators using a different normalization, i.e.
A(k) = (2π)−3/2 (2ωk )−1/2 φe(+) (k) (34)
and its adjoint. Then
h 0
i 0
A(k), A∗ (k ) = δ(k − k )
where
3
f = √d k
dk
2ωk
The vacuum Φ0 is the state annihilated by all A(k), A(k)Φ0 = 0 or A(f )Φ0 = 0, and
H0 = CΦI 0 is a one-dimensional Hilbert space. Many particle states are obtained acting
with creation operators on the vacuum: the improper states
Φk1 ,k2 ,...,kn = A∗ (k1 ) · · · A∗ (kn )Φ0 (36)
13
represents systems of free identical particles of mass m and four-momenta k1 , . . . , kn such
that ki2 + m2 = 0 (which is the reason why we call them “particles”). The normalizable
state
has particles with wave functions fi . Since all operators commute the particles are bosons.
Since the states are completely characterized by giving four-momenta the particles have
zero spin.
Thus the theory predicts that spin zero particles must obey the Bose-Einstein statistic
(they are bosons), a very important conclusion.
A simple computation gives the scalar products
X
(Φp1 ,...,pn , Φk1 ,...,kn ) = δ(p1 − kσ(1) ) · · · δ(pn − kσ(n) ) (38)
σ
where the sum is over all permutations of n elements. In this way the right hand side is
symmetric in all its arguments, as it was the left hand side.
The one-particle Hilbert space generated by all Φk , k ∈ M+ is the space H1 introduced
above; the Hilbert space HN generated by all Φk1 ,...,kN is the tensor product
N times
z }| {
HN = H1 ⊗S · · · ⊗S H1 (39)
The operators A(k) and A∗ (k) acts on the improper states as follows: by definition
i.e. they add a particle to the list in the state. Taking the adjoint in the scalar product
(38) one obtains
n
X
A(k)Φk1 ,...,kn = δ(k − kj )Φk1 ,...,k̂j ,...,kn (41)
j=1
F = ⊕∞
N =1 HN (42)
and are called, respectively, annihilation (or destruction) and creation operators. They
are defined at least on the dense domain of F consisting of all sequences with only finitely
many non zero entries.
14
so that
Z
(Ψ, Φ) = dν(k1 ) · · · dν(kN )Ψ(k1 , . . . , kN )Φ(k1 , . . . , kN ) (44)
E9: Determine the action of the operators A(f ), A∗ (f ) on the wave functions.N
The unitary representation U(Λ, a) in Eq. (20) can be defined to act on the multi-particles
wave functions in the following obvious way
P
[U(Λ, a)Ψ](k1 , . . . , kn ) = e−i j kj ·a
Ψ(Λ−1 k1 , . . . .Λ−1 kn ) (45)
and these operators are unitary in the above scalar product (recall that the integration
measure is Lorentz invariant). As to H0 it defines the trivial representation5
(∂ 2 − m2 )φ(x) = J(x)
Let
Z
d4 k 1
∆ret (x − y) = − 4 2 2 0
eik·(x−y) (47)
2π k + m + ik ε
the retarded Green function satisfying
(∂ 2 − m2 )∆ret (x − y) = δ(x − y)
15
Compute the retarded function explicitly for the zero mass case.N
We also introduce the (less used) advanced Green function
Z
0 −3 sin(ωk x0 ) ik·x
∆adv (x) = −θ(−x )(2π) d3 k e
ωk
which vanishes for x0 > 0. The general solution approaching a free incomig field at remote
past is then
Z
φ(x) = φin (x) + ∆ret (x − y)J(y)d4y (48)
Writing
Z
φ(x) = (2π) −3/2 f
dkA(t, k)eik·x
one finds
Z t
i 1 0
iωk t ˜ 0 0
A(t, k) = Ain (k) + √ e J(t , k)dt e−iωk t (50)
(2π)3/2 2ωk −∞
Z t
∗ i 1 0
−iωk t ˜ 0 0
+ Ain (k) − √ e J(t , −k)dt eiωk t (51)
(2π)3/2 2ωk −∞
where
Z
˜ k) =
J(t, J(t, x)e−ik·x d3 k
In the limit as t → ∞ the field approaches φout so we identify the out annihilation
operators as
Z ∞
i 1 0
iωk t ˜ 0 0
Aout (k) = Ain (k) + 3/2
√ e J(t , k)dt (52)
(2π) 2ωk −∞
or
i 1 e
Aout (k) = Ain (k) + √ J(ωk , k) (53)
(2π)3/2 2ωk
where the full Fourier transform appears
Z
e
J(ωk , k) = e−ik·x J(x) d4 x
The out creation operators are just the adjoint of the destruction operators. Since they
differ from the in operators by a c-number shift they share the same commutation rela-
tions. It follows that an operator S must exist such that
16
E 12: Show that
Z
S = exp i(2π) −3/2 f e e ∗
dk J(ωk , k)Aout (k) + J(−ωk , −k)Aout (k) (55)
Show that
Z
S = exp i(2π) −3/2 f e ∗
dk J(−ωk , −k)Aout (k)
Z
−3/2 f e kJk2
× exp i(2π) dk J (ωk , k)Aout (k) exp − (56)
2
where kJk2 is the invariant squared norm of J.I
The S operator is unitary: SS ∗ = S ∗ S = 1. Hence the in and out operators generate
respectively in and out isomorphic Fock spaces, but this is not a general rule since there are
inequivalent field representations. The operator S is an example of a scattering operator,
the so called S-matrix.
since
Z
N= A∗out (k)Aout (k)d3 k (58)
is the number operator counting the number of particles in a given state. Its eigenstates
are just the multi-particle states defined above. Using (53) in < N > one obtains
Z
−3 dk e
< N >= (2π) |J(ωk , k)|2 (59)
2ωk
Note that this integral is the invariant L2 -norm of J in the space H1 . Also, by applying
Aout (k) to the in-vacuum we obtain
i 1 e
Aout (k)Φin
0 = 3/2
√ J(ωk , k)Φin
0 (60)
(2π) 2ωk
i.e. it is an eigenvector of the out annihilation operator; states like this are called coherent
states.
E 13: Show that if [A, A∗ ] = 1 and Afλ = λfλ , then
∗
fλ ∝ eλA f0 , Af0 = 0
17
Compute the the norm kfλ k. Hint: use
which is valid whenever [A, [A, B]] = [[A, B], B]; show that if kf0 k = 1 then kfλ k =
exp(−kλk2 ) N
Using the result of the exercise we see that
Z
in
Φ0 = Z exp i(2π) −3/2 f J(ω
dk e k , k)Aout (k) Φout
∗
(61)
0
THE STATES of the quantum field carry a unitary representation of the Poincaré group.
We now pause and discuss in general terms the Poincaré algebra, namely the algebraic
properties (commutators) of those transformations that are close to the unit operator.
First we have infinitesimal Lorentz transformations
Λab = δ ab + ω ab , |ω| 1 (64)
where ωab + ωba = 0; infinitesimal translations will be described by infinitesimal four-
vectors εa . Correspondingly we put
i
U(1 + ω, ε) = 1 + ω ab Jab − iεa Pa + · · · (65)
2
18
The operators J and P include important observables, like the relativistic expressions for
angular momentum, momentum and energy. They are called the generators of the unitary
transformations. We now consider the operator
U(Λ, a)U(1 + ω, ε)U(Λ, a)−1 = U (Λ, a)(1 + ω, ε)(Λ−1, −Λ−1 a)
and equate from left to right the coefficients of ω ab and εa . In this way one finds
Next we choose (Λ, a) itself infinitesimally close to the the identity; equating again from
left to right the coefficients of ω ab and εa we get the Poincaré algebra
i[Jab , Jmn ] = Jan ηbm − Jam ηbn + Jbm ηan − Jbn ηam (68)
E 15: Fill in the details of our discussion and compute the Poincaré algebra.N
It follows the familiar operator for pure translations
c
U(1, a) = e−iac P
Thus H is the Hamiltonian, P the linear momentum, J the angular momentum and K
the boost generators. Letting c → ∞ one obtains a central extension of the Galileian
algebra, since from (72) [Ki , Pj ] = imδij , where m is the mass.
E 16: Consider a non relativistic free particle with wave function ψ(x, t). Under transla-
tion [T (a)ψ](x, t) = ψ(x − a, t) defines a unitary operator and ψ(x − a, t) solves the free
Schrödinger equation. Under Galileain boosts we may put, in general,
Show that there exist a choice of phase such that [T (v)ψ](x) solves the free Schrödinger
equation if ψ(x, t) does. Using this result, show that the generators, a · P for translations,
v · K for boosts, satisfy the above commutation relation. N
19
A SET of operators closing the abstract Poincaré can now be obtained with the quantum
field as follows. We define the formal operators
Z
P = d3 k k a A∗ (k)A(k)
a
(74)
Z
3 ∗∂ ∂
Jij = i d k A (k) ki j − kj i A(k) (75)
∂k ∂k
Z
3 ∗ ∂ ki
J0i = d k A (k) iωk i + A(k) (76)
∂k 2ωk
E 17: Let {fn } an orthonormal basis in H1 and define P a f (k) = k a f (k); show that
X
Pa = [P a ]nm A∗ (fn )A(fm )
nm
and similarly
Z
J0j Ψ = f L0j f (k)A∗ (k)Φ0
dk
∂f (k) ∂f (k)
[Jij f ](k) = iki − ikj
∂kj ∂ki
20
Operators like P a and Jbc , with all creation operators to the left of all annihilation op-
erators, are said to be normal ordered. It is not difficult to show that the corresponding
expressions in terms of fields is
Z
Pa = Tea0 (x)d3 x (77)
Z
Jab = xa Teb0 (x) − xb Tea (x) d3 x (78)
where
1
Teab = ∂a φ(−) ∂b φ(+) + ∂b φ(−) ∂a φ(+) + ηab ∂c φ(−) ∂ c φ(+) + m2 φ(−) φ(+) (79)
2
The density Teab (x) is non local, hence it seems that energy and momentum and angular
momentum do not admit local operator densities. However the integral of the bilinears
(±±) 1
Tab = ∂a φ(±) ∂b φ(±) + ηab ∂c φ(±) ∂ c φ(±) + m2 φ(±) φ(±)
2
over d3 x vanish identically, so we can add them to Teab and then integrate over space
(−−)
without changing the operators. The operator Tab creates pair of particles out of the
(++) e
vacuum, while Tab destroy them; adding to Tab we obtain the local tensor density
1
Tab (x) =: ∂a φ(x)∂b φ(x) + ηab ∂c φ(x)∂ c φ(x) + m2 φ(x)φ(x) : (80)
2
where the colon means that the expression is normal ordered with all creation operators
to the left of all annihilation operators; that is, first the field is splitted as φ = φ(+) + φ(−) ,
then substituted into Tab , or any other polynomial in the fields and their derivatives, and
finally the resulting polynomials is rearranged into normal ordered form. The normal
ordered tensor satisfies
∂ a Tab = 0
(4) when smeared with a test function f , Tab (f ) is a well defined self-adjoint operator
on the Fock space.
21
More generally, Wick polynomials at x are polynomials in the fields and their derivatives
at x which are normal ordered according to the procedure just described. They can be
obtained from ordinary products at non coincident points by means of subtractions and
limiting procedures. For example it is easy to verify that
: φ(x)2 := lim φ(x)φ(y) − [φ(+) (x, φ(−) (y)] (81)
x→y
E 19: Show that (Φ0 , φ2 (x)Φ0 ) is infinite. Show that smearing with a test function does
not help. N
The last term is equal to the vacuum expectation value W (x, y) = (Φ0 , φ(x)φ(y)Φ0), also
known as the two-point Wightman function. One can also normal order product of fields
at non coincident points, for example one may consider : φ(x1 ) · · · φ(xn ) :. For n = 2 one
has
so that (Φ0 , : φ(x)φ(y) : Φ0 ) = 0. By means of Wick polynomials one can give a precise
meaning to products of fields at the same point. Finally one can consider expressions of
the form
Z
F (x1 , . . . , xn ) : φ(x1 ) · · · φ(xn ) : d4 x1 · · · d4 xn (83)
which can even be defined for F (x1 , . . . , xn ) in the space of tempered distributions, and
study their algebraic properties as a sort of “Wick calculus”, but we shall not dwell on
such fine points except for a remark. This is that for F (x1 , . . . , xn ) with support in some
bounded set O, the above expressions are canditates for local observables in the theory
which may be interpreted as physical measurement procedures performed within O.
THE LOCAL OPERATOR (80) is the quantum version of the classical energy mo-
mentum tensor of the scalar field. Its existence and conservation law comes about because
the Lagrangian density of the field
1 1
L(x) = − ∂a φ∂ a φ − m2 φ2 (84)
2 2
has the following Poincaré invariance property: for any (Λ, a) in the Poincaré group
0 0
L(φ (x), ∂a φ (x)) = L(φ(Λ · x + a), (∂a φ)(Λ · x + a)) (85)
0
with φ (x) = φ(Λ · x + a). Note that (85) is not tautological since
0 b ∂φ(y)
∂a φ (x) = Λ a
∂y b |y=Λx+a
22
E 20: Prove the invariance property of the action. N
0
Taking Λ = 1 and a = ε infinitesimal, we have to first order φ (x) = φ(x) + εa ∂a φ, so the
left hand side of (85) is, to first order in ε,
δI a b ∂L
L(φ(x), ∂a φ(x)) + ε ∂a φ + ε ∂a ∂b φ
δφ(x) ∂∂a φ
where
δI ∂L ∂L
= − ∂a (87)
δφ(x) ∂φ ∂∂a φ
is the variational derivative of the action, whose vanishing gives the Euler-Lagrange equa-
tions for the extremals of the variational problem δI = 0.
The right hand side of (85) is just L evaluated at the translated point x + ε, so we can
Taylor expand and write it as
L(φ(x), ∂a φ(x)) + εa ∂a L
Equating the two expressions we get the following Noether identity for translations
δI a
ε ∂a φ + εb (∂a T ab ) = 0 (88)
δφ(x)
where
∂L
T ab = Lδ ab − ∂b φ (89)
∂∂a φ
It follows that if the field equations are satisfied we have the four conservation laws
∂a T ab = 0 (90)
J abc = xb T ac − xc T ab (92)
∂c J cab = 0 (93)
then tell us that the stress tensor must be symmetric: Tab = Tba .
23
6 Some Rigorous Results
[See. R. Streater and A. Wightman, PCT, spin, statistics and all that, Princeton Se-
ries.]
A LOCAL POLYNOMIAL in the field operators in a set O is an expression of the
form
XZ
P =c+ F (x1 , . . . , xn )φ(x1 ) · · · φ(xn )d4 x1 · · · d4 xn (94)
n≤N
where F is a rapidly decreasing smooth test function with supp F ⊂ O . Sum and products
of local polynomials in O are local polynomials in O, as well as the adjoint operators P ∗ .
Thus the set of such operators is a ∗-algebra, the algebra of all local polynomials in O,
and will be denoted by A(O). There is now a very surprising result: let us denote by
A(O)Φ0 the set of all vectors in Fock space of the form P(φ)Φ0 , where P(φ) is in A(φ).
Then one has the Reeh-Schlieder theorem, that
A(O)Φ0 is dense in F .
We shall not give a detailed proof of this theorem, only an outline. Let Ψ be a state in
the orthogonal complement of A(O)Φ0 and consider the matrix element
W(x1 , . . . , xn ) = (Ψ, φ(x1 ), . . . , φ(xn )Φ0 )
which is a tempered distribution6 . Recall that the Fourier transform of the field has
support in the momentum forward cone p2 < 0, p0 > 0, so the function W is actually
analytic in the wedge, Tn , of points zk = xk − iyk , such that yk ∈ V + , the future light
cone7 . In fact the factors exp(ipa z a ) appearing in the Fourier transform are exponentially
damped at large y a since each pa is time-like. But for xk ∈ O, W(x1 , . . . , xn ) = 0, so
taking appeal to the “edge of the wegde” theorem we conclude that it vanishes for all
real x1 , x2 , . . . , xn . It follows the Ψ is orthogonal not only to A(O)Φ0 but also to any
other state obtained from the vacuum by repeated applications of the smeared field and
its monomials for any test functions, not just those with the support within O. Hence
Ψ = 0. I
A second interesting result comes directly from the Reeh-Schlieder theorem. It is that in
a local theory A(O) does not contain annihilation operators if the causal complement,
0
say O , of O, in non empty. That is, if for a T ∈ A(O) one has T Φ0 = 0, then T = 0. In
fact by the chain of relations
0 = P (φ)T Φ0 = T P (φ)Φ0 (95)
0
which are valid whenever P (φ) ∈ A(O ) by the local commutativity of the fields, and the
Reeh-Schlieder theorem, it follows readily that T = 0.
Remarks: one can say that in a local theory there exists no operators counting the particles
in any fixed bounded region in space-time, contrary to naive expectations. At the same
time it shows that the localization properties of particles are to be taken with great care.
The Reeh-Schlieder theorem, here discussed in a free field theory, is actually valid in any
interacting theory whose Wightman functions share the same analyticity properties of the
free theory. The properties of a field theory which guarantee this are called Wightman’s
axioms.
6
This follows from Schwarz’s nuclear theorem.
7
Actually the analyticity domain is much more larger.
24
7 Complex Fields and Anti-Particles
THE CONSTRUCTION of the real scalar field is not the only way to build a local
field. Suppose we have a particle p with annihilation field φ(+) (x), and also another
(+)
particle p̄ with the same mass and spin but annihilation field φc (x). We assume the
normal commutation relations for both p and p̄, that is
(+)
φ (x), φ(−) (y) = φ(+) (−) +
c (x), φc (y) = ∆ (x − y; m) (96)
(±)
φ (x), φ(±) (y) = φc(±) (x), φc(±) (y) = 0 (97)
and moreover8
[φ(±) (x), φ(±)
c (y)] = [φ
(±)
(x), φc(∓) (y)] = 0 (98)
Then we may form the local field (see below)
φ(x) = φ(+) (x) + φ(−)
c (x) (99)
for which
φ∗ (x) = φ(+)
c (x) + φ
(−)
(x) (100)
and is not equal to φ(x). The classical model is of course a complex scalar field with
Lagrangian
L = −∂a φ∗ ∂ a φ − m2 φ∗ φ (101)
and Euler-Lagrange’s equations of motion
(∂ 2 − m2 )φ = 0, (∂ 2 − m2 )φ∗ = 0 (102)
When two particles enter the definition of a local quantum field in the way just explained,
one says that each is the anti-particle of the other. Taking for definiteness p as the particle,
we will call p̄ the anti-particle of p. In the special case of the Hermitian scalar field, the
particle is its own anti-particle.
This definition of anti-particle is a meaningful one. For example, given a pair of particles
A, B with equal mass and spin we may form the local field
(+) (−)
φAB (x) = φA (x) + φB (x)
(+) (−)
which however is not local with φ∗A (x) = φA + φĀ , Ā being the anti-particle of A. In
the same way φ∗AB (x) is not local with φA (x), hence φAB and φA and their Hermitian con-
jugates cannot appear at the same time in the interaction Hamiltonian, or in any other
observable in fact, if we insist on the locality principle.
Theories with the same representations of the relativity group but with different associa-
tions of particles with anti-particles are different, in the sense that the observables which
can be associated to space-time regions have different locality properties.
Finally we have
[φ(x), φ(y)] = 0, [φ(x), φ∗ (y)] = ∆(x − y; m) (103)
where the ∆(x − y; m) is the usual Pauli-Jordan causal function. Clearly if the operator
algebra (96) involved anti-commutators in place of commutators we would have lost the
locality encoded in ∆(x − y; m). Thus we have again a spin-statistic theorem: the spinless
charged particles created by φ∗ (x) must be bosons.
8
This is valid only in the absence of interactions.
25
THE FOCK space of the complex field is easily constructed; first the field is expanded
into quantum oscillators
Z
1 d3 k
φ(x) = 3/2
√ [a(k)eikx + ac (k)e−ikx ] (104)
(2π) 2ωk
obeying the operator algebra
0 0 0
[a(k), a∗ (k )] = [ac (k), a∗c (k )] = δ(k − k ) (105)
These span the tensor product of Hilbert spaces H n ⊗ Hcm ; the direct sum
M
H n ⊗ Hcm = F ⊗ Fc
n,m
is the Fock space where the quantum field acts. The charge operator
Z
Q = d3 k(a∗ (k)a(k) − a∗c (k)ac (k)) (107)
The current
↔
J a = i : φ∗ ∂ a φ : (109)
26
is also invariant, i.e. I[φ] = I[eiλ φ]; let us takes the infinitesimal version of the symmetry,
which is
δφ(x) = iλφ(x), δφ∗ (x) = −iλφ∗ (x) (113)
L = −∂a φ∗ ∂ a φ − m2 φ∗ φ − V(|φ|)
one obtains immediately Eq. (109), apart from the normal ordering.
where Aa is the electromagnetic field and e a coupling constant (the electric charge). Show
that L is invariant under the extended gauge transformations
1
δφ = iε(x)φ, δφ∗ = −iε(x)φ∗ , δAa = ∂a ε(x) (117)
e
If ε(x) is constant we have the usual U(1) symmetry. Find the expression of the electric
current.
27
0
1. the fields φA (x) e φA (x) obey the same equations of motions
0
2. the transformation is local, meaning that φA (x) is uniquely determined from φA and
a finite number of its derivatives evaluated at the point θ−1 (x).
and Qa = ελ Qaλ , where ελ are small parameters specifying “the direction” of the trasfor-
mation. We also assume the Lie algebra condition
σ
[δλ , δρ ] = Cλρ δσ
although this is not strictly necessary, and we speak in this case of a continuous group of
symmetry transformations. Let us recall the formula of the variational derivative of the
action
δI ∂L ∂L
= − ∂a (119)
δφA (x) ∂φA (x) ∂∂a φA (x)
Solutions of the equation δI/δφA (x) = 0 are called extremals. We have then
NOETHER THEOREM
There exists a set of currents Jλa (x), one for each generator δλ , such that
δI
δλ φA + ∂a Jλa (x) = 0 (120)
δφA (x)
Furthermore the currents is
∂L
Jλa = δλ φ(x) + θλa (x)L − Qaλ (121)
∂∂a φ(x)
and it follows from eq. (120) that for an extremal it satisfies the continuity equation
∂a Jλa (x) = 0
TO PROVE this let us momentarily assume that ελ is a function of xa . Then since the
action is invariant when ελ are constants, its variation must have the following form
Z
δI = Jλa ∂a ελ d4 x
with no higher derivatives of ελ since the action depends only on the fields and their first
order derivatives. This establishes the existence of the currents Jλa .
For an extremal δI = 0 for any δφa (x) (including the symmetries with space-time depen-
dent parameters) but from our formula this is only possible if ∂a Jλa = 0.
28
To obtain eq. (121) we expand the left hand side of (118) to first order in ελ . One obtains
(note that δλ ∂φ = ∂δλ φ)
δI ∂L
δλ L = δλ φ(x) + ∂a δλ φ(x)
δφA (x) ∂∂a φ(x)
δλ L = −θλa (x)∂a L − L∂a θλa + Qaλ (x) = −∂a (θλa (x)L + Qaλ (x))
The second term comes from the Jacobian: if θ(x) = x + ελ θλ (x) then to first order in ελ
−1
∂θ
λ a
∂x = 1 − ε ∂a θλ
X a −→ X̂ = X a σa
where
1 0 0 1 0 −i 1 0
σ0 = σ1 = σ2 = σ3 =
0 1 1 0 i 0 0 −1
det X̂ = −ηab X a X b = −X 2
29
so there exists a Λ(α) ∈ L↑+ such that
αX̂α∗ = [Λ(α)]ab X b σa
Clearly Λ(α)Λ(β) = Λ(αβ), Λ(α) = Λ(−α) and Λ(±1) = 1. To find Λ(α), we use the
formula
1
Xa = Tr X̂σa
2
Then
1
Λ(α)ab = Tr[ασb α∗ σa ] (122)
2
Note that
1 1
Λ(α∗ )ab = Tr[α∗ σb ασa ] = Tr[ασa α∗ σb ] = Λ(α)b a = [Λ(α)T ]ab (123)
2 2
Clearly Λ maps SL(2, C) in L↑+ ; but in fact tha map is onto. First we consider some
example. Let
−iφ/2
e 0
Uφ =
0 eiφ/2
0 0 0 1
30
for which Mu∗ = Mu ; computing Mu X̂Mu∗ , one finds that Λ(Mu ) is a special Lorentz boost
along x3 with velocity v = tanh u, that is
cosh u 0 0 sinh u
0 1 0 0
Λ(Mu ) = L3 (u) =
0
0 1 0
sinh u 0 0 cosh u
Now each proper orthochronous Lorentz transformation is a product of two rotations and
a boost like this, and each rotation R can be written as a product (a classical Euler
theorem)
R = R3 (φ)R2 (θ)R3 (ψ)
of rotations around x3 , then x2 then again around x3 , each one being of the form Λ(α)
with α ∈ SL(2, C).
I Also the boost L3 (u) is in the range of the homomorphism Λ so that,
I onto L↑+ .
putting all together, it maps SL(2, C)
The connectivity of SL(2, C) I can be seen from the polar decomposition: α = u exp h,
where u is unitary and h is Hermitian with zero trace. Indeed since α∗ α is positive definite,
we can set
1
h = log α∗ α, u = α(α∗ α)−1/2
2
Hermitian matrices with zero trace fill in the space IR3 while SU(2) is isomorphic to S 3
(the 3-sphere). So SL(2, C)I = IR3 × S 3 , which is simply connected.
ADDENDUM: suppose we look at the subgroup of SL(2, C) I which fixes a given four-
vector. For k = (m, 0, 0, 0) this is the set of matrices such that
∗ m 0
αk̂α = k̂, for k̂ = (124)
0 m
or αα∗ = 1, i.e. the SU(2) subgroup. Hence this is also called the stability group, or the
little group, of a particle at rest.
For the null vector k = (κ, 0, 0, κ) a similar computation gives
−iθ/2
e z
α= z = α + iβ (125)
0 eiθ/2
Finally let β(p) the element of SL(2, C)
I such that Λ(β(p)) ≡ L(p) sends k = (m, 0, 0, 0)
to a general four vector p: L(p)k = p. The solution is
β(p) = (p̂/m)1/2 = (m−1 pa σa )1/2 (126)
Thus Λ(β(p)) is the pure Lorentz boost mapping q = (q 0 , q) into
0 0 v·q
q 0 = γ(q 0 + v · q), q = q + (γ − 1) 2 v + γ q 0 v (127)
v
where v = p/p0 and γ = (1 − v 2 )−1/2 . Observe that p̂ is a positive matrix, its determinant
and trace being respectively −p2 and 2p0 , both positive for a physical p.
If k = (κ, 0, 0, κ) then a solution is
q q
p0 +p3 −iφ/2 p0 −p3 −iφ/2
e e
β(p) = q 2κ q 2κ (128)
p0 −p3 iφ/2 p0 +p3 iφ/2
2κ
e 2κ
e
31
where θ and φ are the polar angles of p: p = |p|(sin θ cos φ, sin θ sin φ, cos θ).
10 Poincaré Symmetries
In that case Qa = 0 and L does not depend explicitely on xa . The fields form a linear
representation of SL(2 C)
I (that is to say we have a transformation law from one inertial
system to any other) and (repeated indices summed over)
0
φA (x) = D BA [Λ(α)]φB (Λ(α)−1 (x − a))
∂L
T ab = δba L − ∂b φA
∂∂a φA
Thus we recover the stress energy tensor and its conservation law, ∂a T ab = 0. The integral
Z
P = − T ab nb dx
a
over a space-like hyperplane, with normal na , is independent on the choice of the hyper-
plane and can be interpreted as the total four-momentum of the field. Also
Z Z
Eξ = − P na dx = T ab na ξb dx ≡ −P a ξa
a
a ∂L
Jbc =i [Σbc ]BA φB (x) + T ab xc − T ac xb
∂∂a φA (x)
32
We recognize a contribution to the orbital angular momentum density, so the first term
can be interpreted as the intrinsic angular momentum, or a sort of spin density. The
quantities
Z
c
Jab = − Jba (x)nc dx
are constant in time and are connected with the generators of the Lorentz group. Note that
J0k depends explicitly on the time so that they will not commute with the Hamiltonian.
This is a consequence of the more general definition of symmetries in field theory, which
are not required to be represented by operators (or functions of the canonical variables in
the classical version of the theory) that commute with the Hamiltonian.
a
The four-divergence condition ∂a Jbc = 0 implies readily
∂L B
Tbc − Tcb = i∂a [Σbc ] A φB (x)
∂∂a φA (x)
Hence if the representation corresponds to fields with intrinsic angular momentum then
the stress tensor will not be symmetric. Nevertheless it is possible to write the last
equation as a symmetry condition
33
The last one implies that [D, P 2] = P 2 , so that dilatation symmetry is only possible if all
masses vanish or if the mass spectrum is continuous.
INTERNAL SYMMETRIES: here tipically one has a multiplet of fields such that
δφ(x) = ελ Tλ φ(x) and δx = 0, so that there are no derivative (transport) terms in the
variations and the matrices Tλ generate a Lie algebra of some compact group
ρ
[Tλ , Tσ ] = Cλσ Tρ
In the canonical formalism the Poisson brackets of the conserved “charges”, or the oper-
ators in quantum theory, will generate the Lie algebra of the symmetry group
Z
ρ
Qλ = Jλ0 (x) dx {Qλ , Qσ }P B = Cλσ Qρ
But do they exist? An explicit realization of gamma matrices, called the chiral represen-
tation is as follows
0 0 1
γ = −i = −iβ (134)
1 0
k 0 −σ k
γ = −i (135)
σk 0
34
where each entry is a 2×2 matrix, either the null matrix, the identity or the Pauli matrices
σ k as indicated. So, for example
0 0 0 i
0 0 −i 0
γ 2 = −i
0 −i 0 0
i 0 0 0
Inspection of the these matrices show that the further relations hold
(γ a )† = −βγ a β, {γ 5 , γ a } = 0, Tr γ a = 0 (136)
αv = exp (v · σ/2)
α = exp (v · σ + iθ · σ)
35
so it has two doubly degenerate eigenvalues. The spinor space is a 4D vector space, say
S, which splits into two invariant subspaces
1 ± γ5
W± = S
2
whose elements are called Weyl spinors. Hence we write a spinor in the chiral represen-
tation as
φ
Ψ= (140)
χ
We use the convention to write the components of φ as φr , r = 1, 2 and call it a covariant
spinor: from (138) we see that under Lorentz transformation φr → αsr φs (sum under-
stood). A covariant dotted spinor is an objects transforming according to the complex
conjugate representation
and εrs its inverse. Then εαε−1 = [αT ]−1 so if we define contravariant spinors by rising
the indices with the “spinor metric” ε we get
Then the first will transform according to (αT )−1 and the second according to (α∗ )−1 .
Thus from (138) we must write the component of χ as χṙ and call it a contravariant
dotted spinor: its transformation law is
with the sum again understood. The fundamental representation φ → αφ and the complex
conjugate representation of SL(2, C) I are inequivalent and any other two-dimensional
irreducible representation is equivalent to one these. The symmetric tensor product of
any number of fundamental and/or complex conjugate representations exhaust all finite
dimensional, irreducible representations of SL(2, C).
I Its elements are multi-components
spinors
separately symmetric under permutations of dotted and undotted indices: this represen-
tation is indicated by the symbol (k, j) and has dimension (k + 1)(j + 1). The covariant
Weyl spinor is in (1, 0), the complex conjugate is in (0, 1), (1, 1) is the vector representa-
tion, and so on. But note that in this description the Dirac spinor is an element of the
reducible representation (1, 0) ⊕ (0, 1).
36
A DIRAC FIELD is a space-time function with values in spinor space such that under
inhomogeneous Lorentz transformation
0
Ψ (x) = S(α)Ψ(Λ(α)−1(x − a)) (141)
A quantized Dirac field is such a function with operator values, as will be described soon.
The adjoint spinor is Ψ̄ = Ψ† β; from (137) we see that the transformation law is
0 0 0
Ψ̄ (x ) = Ψ̄(x)S(α)−1 , x = Λ(α)x + a
Note that by thinking to Ψ as a column vector, then Ψ̄ is a row vector and the product
above is thus consistent with the general rules of matrix multiplication.
This transformation law together with eq. (139) has the immediate corollary that several
spinor bilinears transform as tensor representations of the restricted Lorentz group. For
instance
4
X
Ψ̄Ψ ≡ Ψ̄α Ψα
α=1
37
so only with the anti-commutator of the γ a taking the value 2η ab will the equation reduce
to KG.
REMARK: The first order character of Dirac’s equation obviously permits the equivalent
writing
i∂0 Ψ = HΨ
for some operator H. It was precisely to have such a Schrödinger like equation in a Lorentz
invariant theory that prompted Dirac to choose for H a first order differential operator,
and thus to write his famous equation.
E 24: Found H from the covariant form of Dirac equation.
As a simple consequence of the Dirac equation it follows readily that the vector current
J a = Ψ̄γ a Ψ (143)
where Ψ = (φT , χT )T (a T means transposition, i.e. Ψ, φ and χ are all column complex
vectors, the latters with two components, the former with four).
The zero mass limit decouples them into a pair of independent Weyl equations. Taking
the Dirac adjoint of the Dirac’s equation we further get
←
(−Ψ̄ ∂/ +m)Ψ = 0 (147)
E 25: Deduce (147) from the Dirac equation. Hint: use eqs. (136).
The Dirac equation is derivable from a variational principle that will now be discussed.
First a warning: in the full quantum theory the Dirac field is an anti-commuting field
obeying fermion statistics, thus certain operations on the Lagrangian like taking deriva-
tives with respect to a field component should be taken with some care. But for the
purpose of discussing the action principle we can dispense with such fine points and ac-
cordingly we shall treat the Lagrangian as a classical objects. The Dirac equation being
38
first order in the field derivatives, and the Lagrangian being a Lorentz scalar, a suitable
form will be
L = −Ψ̄(∂/ +m)Ψ (148)
where the sign has been chosen to ensure the positivity of the Hamiltonian.
E 26: By treating Ψ and its adjoint as independent variables, deduce the Dirac equation
and its adjoint from the above Lagrangian.
From the Lagrangian one finds the Noether currents describing stress energy tensor and
the total angular momentum density; the stress tensor is
T ab = Lδ ab − Ψ̄γ a ∂b Ψ (149)
The energy density and momentum density follows
H = −Ψ̄(γ · ∂ + m)Ψ, P = iΨ† γΨ (150)
The angular momentum density is
c
Jab = iΨ̄γ c Σab Ψ + xa T cb − xb T ca (151)
We will make use of these formulas in the quantum theory.
TO STUDY the general solution the Dirac equations we first determine the plane wave
solutions, which have the general form
a a
F+ = u(p)eipa x , F− = v(p)e−ipa x (152)
for positive and negative energy, respectively. Then the spinor amplitudes will satisfy the
algebraic equations
(i/p + m)u(p) = 0, (−i/p + m)v(p) = 0 (153)
Non trivial solution will exist only if p2 +m2 = 0 because off the mass shell the determinant
of the algebraic systems are non vanishing. In the rest frame where p = (m, 0, 0, 0) the
equations becomes
(β − 1)u(0) = 0, (β + 1)v(0) = 0
so there exist two linearly independent solutions for u and v of the form
ζ ξ
u(0, σ) = , v(0σ) = (154)
ζ −ξ
where σ is a discrete label with two values. Since (i/p + m)(−i/p + m) = p2 + m2 = 0 we
can write the general solutions in the form
−i/p + m
u(p, σ) = p u(0, σ) (155)
2ε(Ek + m)
(i/p + m
v(p, σ) = p v(0, σ) (156)
2ε(Ek + m)
39
where the denominators are normalization factors; these normalization conditions are
Lorentz invariant
ū(p, σ)u(p, λ) = δσλ , v̄(p, σ)v(p, λ) = −δσλ
ū(p, σ)v(p, λ) = v̄(p, σ)u(p, λ) = 0
The completeness relations read
2
X
+ [−i/p + m]rs
Prs ≡ ur (p, σ)ūs (p, σ) = (157)
σ=1
2m
2
X
− [i/p + m]rs
Prs ≡ vr (p, σ)v̄s (p, σ) = (158)
σ=1
2m
In fact both members of these equations project onto the positive (res. negative) energy
subspaces, so they coincide. The useful relations are easily demonstrable
0 ka 0
iū(k, σ)γ a u(k, σ ) = iv̄(k, σ)γ a v(k, σ ) = δσσ0 (159)
m
Using these results we may write the general solution as
X Z d3 k r m
Ψ(x) = 3/2
[A(k, σ)u(k, σ)eikx + B ∗ (k, σ)v(k, σ)e−ikx ]
σ
(2π) E k
40
c+
[Ψc− (x), Ψ (y)]± = ∓[−γ a ∂a + m]∆(+)
m (y − x) (165)
so that
[Ψr (x), Ψs (y)]± = [−γ a ∂a + m]rs ∆(+)
m (x − y) ∓ ∆(+)
m (y − x) (166)
(Ψr are the components of Ψ). Hence anti-commutators are locals but commutators aren’t,
and the spinor field is not osservable. Operators that are even functions of the field will
commute at space-like separation. Summarizing, the only non trivial anti-commutator
will be
This result is the content of the spin-statistic theorem for the free Dirac field: it implies
that multiparticles states, obtained from the vacuum by repeated applications of creation
operators A∗ (k, σ) and B ∗ (k, λ), will be completely anti-symmetric under exchanges of
the particle labels qk = (pk , σk ). For instance
Φq1 ,q2 = A∗ (k1 , σ1 )A∗ (k2 , σ2 )Φ0 = −A∗ (k2 , σ2 )A∗ (k1 , σ1 )Φ0 = −Φq2 ,q1
whose sole role is to restrict the accessible region of phase space to the surface χ1n =
χ2n = 0, and if the dinamical variables do not have vanishing Poisson brackets with the
constraints there is an inconsistency.
The procedure to deal with this situation is due to Dirac himself. In short: suppose to
have a number of constraints 10 χN (x) = 0, functions of the canonical variables. The
matrix of Poisson brackets {χN (x), χM (y)} is invertible11 , so that there exists a matrix
CN M (x, y) such that
XZ
{χN (x), χM (z)}CM L (z, y)dz = δN L δ(x − y)
M
Now Dirac defines a new bracket, {F (x), G(y)}D , the Dirac bracket, by means
XZ 0 0 0
{F (x), G(y)}D = {F (x), G(y)} − dz dz {F (x), χN (z)}CN M (z, z ){χM (z ), G(y)}
M,N
Here F (x) e G(y) are functionals of the canonical variables in the indicated points and
all brackets are computed at fixed equal times. The Dirac brackets satisfies
10
N denotes a pair {a, n}, where a counts the independent constraints and n the components of each
constraint.
11
Poisson bracket shoild be computed ignoring the constraints.
41
{F (x), G(y)}D = −{G(x), F (y)}D
L = −Ψ(∂/ +m)Ψ
The theory being undoubtedly relativistic invariant, we would like to clarify the trans-
formation laws of the states and the definition of the generators of the Poincaré algebra,
abstractly defined by eqs. (71)-(73). We start from the required transformation law of the
field under Poincaré transformations
U(α, a)Ψ(x)U(α, a)−1 = S(α)−1 Ψ(Λ(α)x + a) (168)
As it happens in field theory, this law determine the operators U(α, a) up to phase. The
result is as follows: first we restrict to Lorentz transformations, the translations being
relatively trivial to analyze. Then for any p ∈ V̄+ and any α in SL(2, C) I we introduce
the matrix
W (α, p) = β(Λp)−1αβ(p) (169)
where Λ is a shorthand for Λ(α) and β(p) = (p̂/m)1/2 . We recall that this is the SL(2, C)
I
matrix such that Λ(β(p)), shortened as L(p) from now on, is the Lorentz transformation
mapping the standard four-momentum q = (m, 0, 0, 0) into p, i.e. L(p)q = p. We may
note two important things about W (α, p), also known as the Wigner matrix: (i) since
c
W (α, p)W (α, p)∗ = β(Λp)−1 αβ(p)β(p)α∗β(Λp)−1 = β(Λp)−1m−1 Λpβ(Λp) −1
=1
42
(σ 2 is the second Pauli matrix), that is to say
1 0 0 1
0 −1 0
u(0, 1) = , u(0, 2) = 1 , v(0, 1) = , v(0, 2) =
1 0 0 −1
0 1 1 0
one has
2
X
S(α)u(p, σ) = [W (α, p)]λ,σ u(p, λ) (170)
λ=1
E 27: Prove the above equation; find its analogue for the spinor v(p, σ).
Using eq. (170) into eq. (168) one obtains the action on the creation/annihilation operators
s
2
∗ −1 (Λp)0 X
U(α)A (p, σ)U(α) = [W (α, p)]λ,σ A∗ (Λp, λ) (171)
p0 λ=1
s
2
−1 (Λp)0 X
U(α)A(p, σ)U(α) = 0
[W̄ (α, p)]λ,σ A∗ (Λp, λ) (172)
p λ=1
where W̄ is the complex conjugate Wigner matrix. Identical equations hold of course for
the anti-particles operators B ∗ (p, σ) and its adjoint. These equations immediately tell us
that if the vacuum is Lorentz invariant (it is) then the one-particle improper states, either
A∗ (p, σ)Φ0 or B ∗ (p, σ)Φ0 , will transform according the rules of the spin-1/2, mass m > 0
irreducible unitary representation of the Poincaré group
s
2
(Λp)0 −iΛp·a X
U(α, a)Φp,σ = e [W (α, p)]λ,σ ΦΛp,λ (173)
p0 λ=1
Eq. (173) shows that if α leaves p invariant, that is if Λ(α)p = p and thus it is a rotation,
then the linear span of the states Φp,σ is the carrier of the fundamental representation of
SU(2), which has spin 1/2. In particular, in the rest system where p = q = (m, 0, 0, 0)
one has
2
X
im·a0
U(α, a)Φ0,σ = e αλ,σ Φ0,λ
λ=1
This equation says that Φ0,σ has energy m, Φ0,1 has z-component of the spin equal to 1/2
and Φ0,2 has z-component of the spin equal to −1/2, i.e. that by putting sz = −σ + 3/2
one has
1
J3 Φ0,σ = sz Φ0,σ , sz = ±
2
43
and of course J 2 = 3/4 on both states and their linear combinations. In a relativistic
theory what we call spin of an elementary system is precisely the non relativistic notion
of spin as applied in the rest system, or in a less prosaic way, the order of the irreducible
representation of SU(2) to which the states of the system at rest belong.
REMARK: Eq. (173) remains valid for a massive particle of spin j under the replacement
of W (α, p) with D (j) [W (α, p)], the (2j + 1) × (2j + 1) dimensional matrices of the spin-j
representation of SU(2). Massless particles require a slightly more elaborate treatment
due to the fact that the stability group of a null four-vector is not SU(2) but the non
compact group of Euclidean motion in dimension two (see S. Weinberg’s book, Vol.I,
Ch.[2]).
The above construction showed how the unitary operators in the Dirac theory acted on
the physical states. Now we describe how the generators can be expressed as operator
functions of the Dirac field.
The four-momentum, or translations generator, is guessed easily: it is given by
XZ
a
P = k a [A∗ (k, σ)A(k, σ) + B ∗ (k, σ)B(k, σ)] d3k (174)
σ
just like as for the scalar fields. This is because it is the only operator which has as
eigenvectors the multi-particle states with eigenvalues that are sums of the single particle
four-momentum, without terms involving more than one particle at a time. It is reassuring
that the Hamiltonian derived via Noether theorem from the Lagrangian (148)
Z
H = − d3 x : Ψ̄(γ · ∂ + m)Ψ :
together give eq. (174) as desired. The densities of these operators (the quantities to be
integrated) are the normal ordered form (see remark below) of the components T 0a of the
energy-momentum tensor
T ab = Lδ ab − Ψ̄γ a ∂b Ψ (175)
Since L = 0 for solutions of the Dirac equation, one can also define T ab using only the
second term of this equation. The other generators are a little bit more complicated in
Dirac’s theory than in scalar field theory.
E 29: Show that the Noether currents following from the Dirac Lagrangian are
c
Jab = iΨ̄γ c Σab Ψ + xa T cb − xb T ca (176)
44
With some effort one finds
Z 2
md3 k X † ∗ † ∗
Jij = u (k, r)Σij u(k, s)A (k, r)A(k, s) − v (k, r)Σij v(k, s)B (k, s)B(k, r)
k 0 r,s=1
Z X
3 ∗ ∂ ∂
+ i dk A (k, r) kj i − ki j A(k, r)
r=1,2
∂k ∂k
Z X
3 ∗ ∂ ∂
+ i dk B (k, r) kj i − ki j B(k, r) (177)
r=1,2
∂k ∂k
Z 2
md3 k X †
J0i = 0
u (k, r)Σ0i u(k, s)A∗ (k, r)A(k, s) − v † (k, r)Σ0i v(k, s)b∗ (k, s)B(k, r)
k r,s=1
Z X
3 ∗ 0 ∂ ki
+ i dk A (k, r) k + A(k, r)
r=1,2
∂k i 2k 0
Z X
3 ∗ 0 ∂ ki
+ i dk B (k, r) k + B(k, r) (178)
r=1,2
∂k i 2k 0
To obtain these non trivial results one has to use the identities
0 ε h i
ū(p, σ)iγ 0 Σab u(p, σ ) = (Σab )σσ0 + (2m(m + ε))−1 (i ∂/† +m)(pa γb − pb γa ) 0 (179)
m σσ
where the matrix elements are to be computed using the zero momentum spinors, that is
for some operator A,
0
Aσσ0 = u† (0, σ)Au(0, σ )
: A(kσ)A∗ (k, σ) := −A∗ (k, σ)A(k, σ), : A(kσ)B ∗ (k, σ) := −B ∗ (k, σ)A(k, σ) (181)
and so on. More generally, one can consider polynomials : P (Ψ(x), Ψ∗(x)) : with any
even number of fields: one starts by substituting the field decomposition Ψ(x) = Ψ+ (x) +
Ψc− (x), with Ψ̄(x) = Ψ̄c+ (x) + Ψ̄− (x), into P (Ψ(x), Ψ∗ (x)) and then rearranging the Ψ(±)
fields as was described, with all creation operators (or negative frequency components) to
the left of all annihilation operators (or positive frequency components) and providing a
45
sign factor ±1 each time there occur an exchange of field components. As an example,
we may consider (notice the sign of the second term)
: Ψr (x)Ψ∗s (y) := Ψ+ c+ − + c− c+ c−
r (x)Ψ (y) − Ψs (y)Ψr (x) + Ψr (x)Ψs (y) + Ψr (x)Ψs (y)
−
= Ψr (x)Ψ∗s (y) − [Ψ+ − ∗ ∗
r (x), Ψs (y)]+ = [Ψr (x)Ψs (y) − (Φ0 , Ψr (x)Ψs (y)Φ0 )]
Note that the limit as x → y will exists due to the subtraction of the vacuum expectation
value. The gauge invariant polynomials (under the action of the obvious U(1) symmetry)
13 Massless Particles
We saw that for massive particles of spin j 12 the transformation law
s
j
(Λp)0 −iΛp·a X [j]
U(α, a)Φp,σ = e D [W (α, p)]λ,σ ΦΛp,λ (183)
p0 λ=−j
with Λ ≡ Λ(α) and where D [j] [u] are the unitary matrices representing a u ∈ SU(2) in
the spin-j representation of this group, defines a unitary irreducible representation of the
Poincaré group relative to the scalar product
0
Φp,σ , Φp0 λ = δσλ δ(p − p )
Here the group SU(2) appears as the stability group of the special time-like momentum
k = (m, 0, 0, 0) which represents a particle at rest.
With certain very important differences the same formula holds true for massless particles
except that D [j][W (α, p)]λ,σ is to be replaced with the appropriate representation of the
stability group of a standard four-momentum k = (κ, 0, 0, κ), which represents a massless
particle moving in the z direction (there is no rest system for such a particle). For k
we already determined the stability group (also called “the little group”): it is the set of
elements of the form
−iφ/2
e ze−iφ/2
w(z, φ) = , z = α + iβ (185)
0 eiφ/2
12
We stress that with this expression we only mean that the state label is a four-vector on the positive
mass-shell together with a discrete spin quantum number, without committing ourselves to thinking in
terms of pointlike objects.
46
With the appropriate choice of the matrix β(p) for the null case, see eq. (128), the Wigner
matrix still preserves the null vector k, so it must be of the form (185) with the angle φ
equal to some function Φ(W ). One can verify that Λ(w(z, φ)) = L(α, β)R3 (φ), where
1 + η α β −η
α 1 0 −α
L(α, β) = β
, η = (α2 + β 2 )/2 (186)
0 1 −β
η α β 1−η
and
1 0 0 0
0 cos φ − sin φ 0
R3 (φ) =
0 sin φ cos φ
(187)
0
0 0 0 1
The stability group of massless particles is also isomorphic to the group of matrices
cos θ − sin θ α x x cos θ − y sin θ + α
w = sin θ cos θ β ; y → x sin θ + y cos θ + β
0 0 1 1 1
where the action on column vectors is indicated: we see that it is the group of motions of
Euclidean 2D space. We set w(0, φ) = r3 (φ) e w(z, 0) = L(z). Then
we need now the matrices D[W (α, p)]λ,σ that furnish a representation of the little group,
so we need the algebra of this group. We then let z = α + iβ e φ to be infinitesimal; then
to first order in these parameters we have
47
We can diagonalize simultaneously A and B
and the unitary operators which represent the abelian part of the little group will be
U(L(z)) = eiαA+iβB
such that
U(L(z))Φφab = exp i Re e−iφ z(a − ib) Φφab
and
J3 Φk,σ = σΦk,σ
where Φ(w) is the rotation angle around the three axis appearing in the factorization
w(α, p) = r3 (Θ(w))L(z) of the Wigner matrix.
Looking further at r3 (Θ(w)), we see that r3 (4π) = 1, so in a faithful representation
U(r3 (4π)) = 1; this gives immediately exp(i4πσ) = 1, or
n
σ=± , n = 0, 1, 2, · · · ∈ IN
2
A state with positive (negative) elicity is said to be right-handed (resp. left-handed) and
viceversa. For zero mass particles then, chirality is the same thing as elicity, though this
is not true for massive particles.
48
14 Electromagnetic Fields, Preparation
THE ACTION of classical electrodynamics in rationalized units is
Z Z
1
I=− Fab F d x + Aa J a d4 x + IM [ϕ, ϕ∗ ]
ab 4
(192)
4
where IM is the action of all existing charged fields
Fab = 2∂[a Ab] (193)
and J a is the conserved electric current. For a scalar field13
↔
Ja = ieφ∗ ∂a φ − 2e2 Aa |φ|2
Ja = ieΨ̄γa Ψ
or the sum over all spinors and scalar fields the theory contains. The action is invariant
under the combined local gauge transformations
Ψ → eieθ(x) Ψ, Aa → Aa + ∂a θ (194)
This is usually achieved by the “minimal coupling prescription”, whereby in the matter
action the partial derivatives of the charged fields are replaced with the so called covariant
derivatives
∂a ϕ → Da ϕ = (∂a − ieAa )ϕ (195)
defined so as to transform covariantly under gauge transformations, unlike ∂a ϕ which
transform inhomogeneously
E 31: Will you see an analogy with general relativity? What is the analogue of the gauge
transformations and what is the field needed to restore covariance?
E 32: Prove the validity of the transformation law
Da ϕ → eieθ Da ϕ
49
The field equations are the Maxwell equations
or in terms of potentials
From this and current conservation we deduce that no conditions are imposed by the field
equation on the scalar field ∂a Aa , which can thus be freely specified. In fact Eq. (198)
cannot be solved uniquely for Aa due to its gauge invariance. It is then customary to
“choose a gauge”, namely to select one rappresentative potential from each equivalence
class of gauge equivalent potentials by means of certain conditions imposed to Aa , such
that one can solve the field equations. For example the Lorenz gauge condition
∂ a Aa = 0 (199)
gives 2Aa = −Ja which is clearly solvable. Note however that a further gauge transfor-
mation Aa → Aa + ∂a χ, with 2χ = 0, has no effects on either the Lorenz condition or the
field equation. Clearly there is a lot of freedom in the gauge choices: here is a partial list
E 33: Write the field equations in the Coulomb gauge; solve them explicitely for A0 (x)
with or without charges.
A. Fab (x) and Aa (x) are defined as operator valued distribution in a linear space H
50
and in H there is a sesquilinear form < ·, · > with respect to which U(Λ, a) are
unitary. As to Aa (x), we only assume the law for translations
THEOREM14 In any local QFT of fields Fab (x), ϕα (x) in which (A), (B) and (C) hold,
the assumptions
∂ a Fab = 0
and
imply
A relativistic theory with no vacuum correlations is very likely to be trivial. In fact, under
the further assumptions
E on the set of vectors, say D0 , obtained from the vacuum by applying polynomials
in the smeared field Fab (f ), the form < ·, · > is positive semi-definite
∂ a Fab = 0
imply
J These difficulties persist in the presence of charges. Recall that a charged field is one
such that there is an operator effecting U(1) phase transformations, i.e.
51
in which q is the electric charge.
THEOREM16 In any QFT in which there is a local charged field ϕ in a H equipped
with a non degenerate sesquilinear form < ·, · >, Maxwell equations
∂a F ab + J b = 0, eabcd ∂a Fbc = 0
0
cannot hold as operator equations in H. If H ⊂ H is stable under M b = ∂a F ab + J b , the
form is non-negative and
0
< Φ, M a (x)Φ >= 0
0 0 0
for all Φ, Φ ∈ H , then < ·, · > cannot be strictly positive on H nor non-negative on H,
0 0 0
unless < Φ, ϕ(x)Φ >= 0 for all Φ, Φ ∈ H . J
Finally we recall the following
THEOREM17 The vector potential Aa (x) can be defined as a local and Lorentz-covariant
operator only in a Hilbert space with indefinite metric.
∂ a Fab = −∂a ∂c Ac
or
2Aa = 0 (205)
There are actually a whole class of Gupta-Bleuler gauges, all connected by the standard
0
formula Aa = Aa + ∂a f .
There are also “improper gauge choices” whose potentials are not connected to Gupta-
0
Bleuler or Coulomb gauge by the usual formula Aa = Aa +∂a f . They rest on the following
modification of Maxwell equations: one modifies the action of the free EM field to read
Z Z
1 ab 4 1
I=− Fab F d x − (∂a Aa )2 d4 x (206)
4 2α
16
R. Ferrari, L. Picasso and F. Strocchi, Comm. Math. Physics, 35, 25 (1974).
17
A. Wightmann and L. Garding, Arkiv Fisik 28, 129 (1964).
18
As noted before we recall that it is necessary to choose a gauge if we want to quantize the EM field
via the potentials Aa .
52
The field equations are
1
∂a F ab + ∂a ∂b Ab = 0 (207)
α
or
1
2Aa − 1 − ∂a ∂b Ab = 0 (208)
α
From these it follows that ∂b Ab is a free scalar field: 2∂b Ab = 0. For any finite α the
equations are solvable but of course they do not coincide with Maxwell theory unless
∂b Ab = 0.
Now the choice α = 1 is called the Feynman gauge, the limit case α = 0 is the Landau-
Khalatnikov gauge. Except that for α = 1, we may note that
22 Aa = 0 −→ k 4 Ãa (k) = 0 (209)
instead of the usual wave equation, hence the Fourier transform of Aa is a linear com-
bination of δ(k 2 ) and δ 0 (k 2 ). There is no standard gauge transformation connecting the
Feynman and Landau gauges, so their equivalence in the quantum theory is non trivial.
E 34: Put a delta function source to eq. (208) and solve the equation in momentum space.
That is to say, invert the matrix
a 2 a 1
Mb = k δb − 1− k a kb (210)
α
Display its form in Feynman (this is trivial) and Landau gauges, respectively. Show that
there is no gauge transformation mapping each one into the other.
0
The fact that not all gauge transformations are of the special form Aa = Aa + ∂a f , means
that in the quantum theory the notion of gauge invariance must be very different from
that in the classical Maxwell theory.
RETURNING to Gupta-Bleuler scheme, Eq. (205) can now be solved, the most general
solution having the form
Z 3
X
d3 p
Aa (x) = p ea(λ) (p)A(p, λ)eip·x + h.c. = A+ −
a (x) + Aa (x) (211)
(2π) 3/2 0
2p λ=0
(λ)
where p2 = 0 and ea (p), λ ∈ {0, 1, 2, 3}, are a set of four polarization vectors defined
as functions on the forward null cone C+ = {k a |k 2 = 0, k 0 > 0}, and satisfying the
orthogonality and completeness relations19
(σ)
e(λ) · e(σ) = η λσ , ηλσ ea(λ) (k)eb (k) = ηab (212)
REMARK: Different choices of polarization vectors correspond to “internal” (i.e. not
acting on the space-time components of the vectors) Lorentz transformations
53
and can always be compensated by an internal rotation of the field operators
σ
Ā(k, λ) = Λ−1 λ A(k, σ) J
(1) (2)
We will always choose two of them, say ea and ea , to be orthogonal to k:
k a e(1) a (2)
a = k ea = 0 (213)
E 35: Show that there can be at most two four-vectors which are orthogonal to a given
null vector and that both must be space-like.
(σ)
We also introduce the dual basis of contravariant vectors ea(λ) (k) = η ab eb (k)ησλ (more on
this later), such that
η λσ ea(λ) (k)eb(σ) (k) = η ab , ηab ea(λ) (k)ea(σ) = ησλ (214)
We define the quantum potential by simply imposing the ordinary oscillator algebra to
all four creation/annihilation operators
[A(k, λ), A∗(p, σ)] = η λσ δ (3) (k − p), [A(k, λ), A(p, σ)] = 0 (215)
The vacuum will be the state annihilated by all destruction operators
A(k, λ)Φ0 = 0 (216)
The Fock space of states will be constructed in the usual way by applying strings of
creation operators to the vacuum for all four polarizations; a tipical improper state is
thus of the form
and for it the commutation relations will imply Bose statistics. We denote by H the
space generated by all these improper states: we will see in a moment that it carries an
indefinite scalar product.
The particles created by these operators are to be called photons for obvious reasons.
Photons created by A∗ (k, λ), λ = 1, 2, are called transverse, or physical photons, while
those created by A∗ (k, λ), λ = 0, 3, are called respectively scalar and longitudinal photons.
They will be considered as unphysical, although without them a covariant formulation of
electrodynamics would not be possible.
Finally, a computation also gives the local commutator
[Aa (x), Ab (y)] = ηab ∆(x − y) (217)
where
Z
1
∆(x) = d4 k ε(k 0 ) δ(k 2 ) eik·x
(2π)3
is the causal Pauli-Jordan function for zero mass. We have restored covariance and locality,
but note that the 0-component of the vector potential has a commutator with the wrong
sign. Evidently we will have to eliminate from the physical spectrum the fictitious scalar,
or temporal, photons created by the operators A∗ (k, 0), since the vectors
Z
Ψf = dkf f (k)A∗ (k, 0)Φ0 (218)
54
have actually negative norm
Z 3
2 dk
kΨf k = − |f (k)|2 (219)
2|k|
as was anticipated by the general theorems stated above. That is to say, H is equipped
with an Hermitian sesquilinear form20 < ·, · > not positive definite. By the same token,
since one cannot impose the condition ∂a Aa = 0 as a strong operator equation, Gupta-
Bleuler (GB) proposed to use it in the weaker form21
(+)
∂ b Ab (x)Ψ = 0 (220)
0
The subspace selected by the GB condition is called the physical state space, say H , and
within it < f, g > is positive semi-definite; throughout the physical subspace we have
then
0
Ψ, ∂ b Ab (x)Φ = 0, ∀Ψ, Φ ∈ H (221)
showing that the Maxwell equations are recovered as weak operator equations for matrix
elements between physical states
0
Ψ, ∂b F ba (x)Φ = 0, ∀Ψ, Φ ∈ H (222)
Notice that this is not the analogue of the Bohr-Sommerfeld “large quantum number”
condition, since it holds even for states populated by zero or very few photons. Of course
for states populated by billions of photons it always makes sense to define an average EM
field obeying Maxwell eqs.
E 36:Show that an improper state is physical if and only if
Show that this condition depends on the choice of polarization vectors, but is otherwise
Lorentz invariant.
00
TO GO further we need to discuss a bit the structure of the physical subspace. Let H
0
be the subspace of H consisting of vectors with zero norm. One example of a zero norm
vector is
0
∂ a A(−)
a (f )Φ, Φ∈H
20
Sesquilinear means < f, g > is linear in g and anti-linear in F ; Hermitian means < f, g >= < g, f >.
21
The stronger condition ∂ b Ab (x)Ψ = 0 was proposed earlier by E. Fermi but it is inconsistent.
55
In a space with semi-definite metric the null vectors are always orthogonal to all, null and
non null, vectors. To see this we note that the usual Schwarz inequality
| < f, g > | ≤ kf k · kgk
is still valid in the presence on null vectors (though it is not if there are vectors with
negative norm). Hence if either f or g are null while the other has positive norm the
scalar product obviously vanishes.
E 38: Prove the Schwarz inequality for a space equipped with a semi-definite metric.
It follows that all matrix elements between physical states are unaffected by the change
00
Ψ → Ψ + χ for any χ ∈ H ; in other words the physical Hilbert space of transverse
photons can be defined as the quotient space
0 00
Hph = H /H (224)
We can realize this construction directly in terms of wave functions in momentum space.
That is we may describe the one-particle Hilbert space H 1 as the space of vector functions
Φa (k) on C+ , equipped with the indefinite sesquilinear form
Z
< Φ, Ψ >= d4 kδ(k 2 )θ(k 0 )Φ̄a Ψb η ab (225)
01
Note that this is negative on time-like Φa ; the subspace H is defined as the set of
transverse vectors, such that
k a Φa (k) = 0 (226)
This condition eliminates the timelike but not the null vectors, such that Φa Φa = 0,
correponding to states with zero norm. But a null physical states must be of the form
Φa (k) = f (k)ka , so the equivalent relation is Φa (k) ' Φa (k) + f (k)ka and the physical
1 0 00
Hilbert space Hph = H 1 /H 1 is the set of transverse vector functions modulo those
1
which are proportional to ka . It is a simple matter to see that throughout Hph the form
< ·, · > is positive definite and defines a true scalar product. The Poincaré group acts on
H as expected for a vector function
[U(Λ, a)Φ]b (p) = e−ip·a Λbc Φc (Λ−1 p) (227)
See also the next section. Now the n-particle photon states are elements of the symmetric
tensor product
n times
z }| {
H n = H 1 ⊗S H 1 ⊗S · · · ⊗S H 1
An element is a tensor function Φa1 a2 ...a3 (k1 , k2 , . . . , k3) symmetric in all arguments and all
tensor indices, with natural norm (we use dν(k) as a shorthand for the Lorentz-invariant
measure on the forward light cone)
Z
< Φ, Ψ >= dν(k1 ) · · · dν(kn )Φ̄a1 ...an Ψa1 ...an (228)
all tensor indices being raised with the relativity rules. Physical elements satisfy transver-
0
sality conditions on all tensor indices; null vectors are of the form Φa1 ...an = kaj Φa1 ...an for
at least one index aj . The quotient is the space of physical many-photons states. The
Fock space is defined by taking the direct sum of all the spaces H n , and so on. We leave
to the reader to complete the details.
56
17 Relativistic Transformation Laws
The states A∗ (k, λ)Φ0 carry a linear representation of the Poincaré group which can be
defined by
1/2
∗ (Λp)0
U(Λ, a)A (p, λ)Φ0 = 0
e−iΛp·a [W (Λ, p)]λσ A∗ (Λp, σ)Φ0 (229)
p
where W (Λ, p) = L(Λp)−1 ΛL(p) is the usual Wigner matrix which leaves invariant the
standard four-momentum k = (ω, 0, 0, ω) and L(p)k = p. This operator is unitary on H
with respect to its sesquilinear Hermitian form (which appeared from the representation
of the field algebra). The effect on creation/annihilation operators is then
1/2
−1 (Λp)0
U(Λ, a)A(p, λ)U(Λ, a) = e−iΛp·a [W (Λ, p)]λσ A(Λp, σ) (230)
p0
1/2
∗ −1 (Λp)0
U(Λ, a)A (p, λ)U(Λ, a) = [W (Λ, p)]λσ A∗ (Λp, σ) (231)
p0
With a suitable choice of polarization vectors (see Appendix A) it acts on the field co-
variantly, i.e.
U(Λ, a)Ac (x)U(Λ, a)−1 = Λca Aa (Λx + a) (232)
Using W (Λ, p) = L(α, β)R(Θ) (v. (186) e (187)) we have
U(Λ, a)A(p, 0)U(Λ, a)−1 = η [A(Λp, 0) − A(Λp, 3)] (233)
+ α A(Λp, 1) + β A(Λp, 2) + A(Λp, 0)
22
The measure is
f= d3 p
dp p
(2π)3/2 2p0
.
57
0
map the operators A(p, λ) → A (p, λ) as
0 0
A (p, 3) + A (p, 0) = A(p, 3) + A(p, 0) + 2iκθ̃(p) (237)
0 0
A (p, 3) − A (p, 0) = A(p, 3) − A(p, 0) (238)
0 0
A (p, 1) = A(p, 1), A (p, 2) = A(p, 2) (239)
so that the physical subspace is invariant. All these gauges are equivalent to Gupta-
Bleuler gauge; however remember that there are more general gauges which do not have
this property.
From Eqs. (233), (234), (235), (236) we see that the restriction of the operators U(La, a)
to Hph is a pure rotation on the transverse states in accord with the general analysis of
the behaviour of massless particles under Poincaré transformations.
E 39: Show that the restriction of the field to Hph is
A(x)|Fph = Â(x) + ∂α+ (x) (240)
where Â(x) is the transverse part
Z 2
X
d3 k
Â(x) = √ e(λ) (k)A(k, λ) eikx + h.c. ∂ · Â = 0 (241)
(2π) 3/2 0
2k λ=1
and the transformation law take the form
A(p, 1) → A(Λp, 1) cos Θ+A(Λp, 2) sin Θ A(p, 2) → −A(Λp, 2) sin Θ+A(Λp, 1) cos Θ
E 40: Show that
1. the four-momentum operator is
Z
P = d3 k k a ηλσ a∗ (k, λ)a(k, σ)
a
(242)
58
18 Interactions and Collision States
[1] S. Weinberg, The Quantum Theory of Fields, Vol.I, Foundations, Cambridge Univer-
sity Press (1995).
[2] C. Itzykson and J. Zuber, Quantum Field Theory, MacGraw-Hill Int. Eds.(1980).
[3] M. Peskin and D. Schroder, An Introduction to Quantum Field Theory, Perseus 2008.
[4] M. Srednicki, Quantum field Theory, Cambridge, 2007 [5] J. D. Bjorken and S. D. Drell,
Relativistic Quantum Fields, McGraw-Hill (1965).
Up to now we encountered field equations that were only linear partial differential equa-
tions, corresponding to Lagrangians which were quadratic functions of the fields and their
first order derivatives. Such field equations could not describe interactions and this was
reflected into the spectrum of the observables, whose eigenvalues on multiparticle states
were sums over single particle eigenvalues, without terms involving more than one particle
at a time. For example, acting on a state (some notation: each qj is a triplet (pj , σj , sj )
referring to a particle of type sj having four-momentum pj and z-component of the spin
equal to σj )
59
written as functions of suitable asymptotic free fields, for which a prescription exist to give
them sense, namely the Wick’s normal ordering. The operator Lagrangian is thus a rather
formal device, but one with a great euristic value. One can read off from it the interaction
operators to be used in perturbation theory and, even more, a Lagrangian is indispensable
for the construction of covariant theories and in the path integral method, especially when
complicated symmetries like non abelian gauge symmetry or supersymmetry are included
in the theory.
It is interesting to compare with the situation in ordinary non relativistic quantum dy-
namics. A two-body interaction is described there as a potential
X
V (x1 , . . . , xn ) = U(|xi − xj |)
1≤i<j≤n
Its elements are sequences Ψ = (ψ0 , ψ1 , ψ2 , . . . ) with ψ0 just a complex number and
ψn (x1 , . . . , xn ) ∈ H n a normalizable symmetric wave function (we speak about spin-0
particles for simplicity) and with finite norm:
∞
X
2
kΨk = kψn k2 < ∞
n=0
The vector Φ0 = (1, 0, 0, . . . ) is the vacuum state. Then one introduces annihilation and
creation operators, acting on N-particle wave functions: for f ∈ H 1 (omitting vector
notation for simplicity)
Z
[a(f )ψN ](x2 , . . . , xN ) = N 1/2
f¯(x)ψN (x, x2 , . . . , xN )dx (251)
N
X
∗ −1/2
[a (f )ψN ](x1 , . . . , xN +1 ) = (N + 1) f (xk )ψN (x1 , . . . , x̂k , . . . , xN ) (252)
k=1
States of the form a∗ (g1 ) . . . a∗ (gn )Φ0 generate an everywhere dense subset in H. One
proceeds to define quantum field (at time zero), φ(x), by writing
Z
a(f ) = φ(x)f¯(x)d3 x
60
The CCR for fields follow from (253)
XN
∇2
H0 ψ(x1 , . . . , xN ) = − ψ(x1 , . . . , xN )
k=1
2m
The interaction can be computed similarly. For the two-body potential we obtain
Z
1
H1 = d3 xd3 yφ∗(x)φ∗ (y)U(|x − y|)φ(x)φ(y) (255)
2
E 42: Prove the formula giving H1 , namely show that
X
H1 ψ(x1 , . . . , xN ) = U(|xi − xj |)ψ(x1 , . . . , xN ) (256)
1≤i<j≤n
We see that a two-body potential appears as a quartic interaction in this formalism. Two
aspects of this Hamiltonian do not find a place in the relativistic theory. The first is
that H1 Φ0 = 0: it follows that the Fock vacuum is an eigenstate of the full Hamiltonian.
The second is the non local character of H1 , which is given by a double integral with a
potential which extend over spacelike regions.
61
defined as eigenvalues of the Casimir operator −P 2 = −Pa P 2 of the theory. The bound
states are to be included into H0 as new elementary systems produced by the interaction.
Consider now the operator
V (0, t) = exp(itH) exp(−itH0 ) (257)
If Φ is an element of Fock space with sufficiently regular wave functions then we have
Z 0
0 0 0
V (0, t)Φ = exp(itH) exp(−itH0 )Φ = Φ − i exp(it H)H1 exp(−it H0 )Φ dt
t
0 0
The state exp(−it H0 )Φ rappresents free particles and if |t | is very large the probability
that their distance be less than the potential range is very small.
0 0
Then kH1 exp(−it H0 )Φk → 0 as t → ±∞ and a quantitative evaluation show that the
integral written above will converge in the limit t → ±∞. In conclusion
Z 0
0 0 0
Φin = lim V (0, t)Φ = Φ − i exp(it H)H1 exp(−it H0 )Φ dt
t→−∞ −∞
exists. In the Heisenberg representation we are using, the vectors Φout and Φin represent
states which relative to measurements performed at t → ±∞ behave as the free particle
state Φ.
The operators Ω± = limt→±∞ V (0, t) are called Møller scattering operators: being strong
limits of unitary operators they are isometrics and in the absence of bound states also uni-
tary. If α and β label orthonormal basis of incoming and outgoing states which represent
collections of incoming and outgoing free particles respectively, the probability amplitude
of a scattering process α → β, that is the probability amplitude that a test performed on
the incoming vector24 Ψin out
α will find the system in the outgoing state Ψβ is
Sβα = Ψout in
β , Ψα (258)
This is called an S-matrix element. When the Møller scattering operators exist there is a
formula which gives these amplitudes as matrix elements of an operator S0 acting in the
free particle Fock space, since
Sβα = Ω+ Φβ , Ω− Φα = Φβ , (Ω+ )∗ Ω− Φα (259)
Then
where
62
In general, if α = (q1 , q2 , . . . , qn ), with qk = (pk , σk , sk ) listing four-momentum, z com-
0 0 0
ponent of the spin or elicity and particle specie sk , and similarly β = (q1 , q2 , . . . , qn ) for
final configurations, then Ψout in
β and Ψα are two complete orthonormal basis in the Hilbert
space and thus are connected by a unitary operator
SΨout in
β = Ψβ (262)
This operator acts on the full Hilbert space of the interacting theory and is different from
S0 .
63
for a wave packet f ∈ H1 and any Ψ1 , Ψ2 in the domain of φ(x), would be time independent
and equal to the smeared creation operator a∗ (f ) if the field were a free field. For the
interacting field it is time-dependent so we define the ingoing/outgoing creation operators
by the weak limits (first proposed by Lehmann, Symanzik e Zimmermann (LSZ) in a 1965
paper, and called LSZ conditions)
Z
Φ, −i dx (f ∂0 φ − φ∂0 f ) Ψ −−−−→ (Φ, a∗ (f, in)Ψ) (264)
t t→−∞
Z
Φ, −i dx (f ∂0 φ − φ∂0 f ) Ψ −−−→ (Φ, a∗ (f, out)Ψ) (265)
t t→∞
where
a
fk (x) = (2π)−3/2 (2Ek )−1/2 eika x
are the usual plane waves associated to a scalar field. We may now define the asymptotic
free fields
Z
φin (x) = d3 k a(k, in)fk (x) + a∗ (k, in)f¯k (x) (267)
Z
φout (x) = d3 k a(k, out)fk (x) + a∗ (k, out)f¯k (x) (268)
where a(k, in) = a(fk , in) and so on. We are going to use the above asymptotic operators
to define multi-particle states in the same way that we used the free fields to create the
free particle Fock space out of the vacuum. But to be sure that the operators a∗ (f, in)
really create one-particle states out of the vacuum some conditions must be met: it is
necessary first of all that
(Ω, φ(x)Ω) = 0
since otherwise a∗ (f, in)Ω will have a vacuum component, but also that for multi-particle
states with sufficiently regular wave functions
since otherwise a∗ (f, in)Ω will also have components along the multi-particle states.
By translational invariance v = (Ω, φ(x)Ω) = 0 = (Ω, φ(0)Ω) = 0; if this is not zero then
0
we have to shift φ = φ + v so that the new field will have zero vacuum expectation value.
To study the secondR property let us denote with Ψ(P, n) the wave function of the state
ΨM , so that ΨM = dn d3P |P, n >, where P label the total momentum and n the other
variables needed to specify the state (the integral over n may then include also discrete
64
sums) and |P, n > is an improper eigenstate of all these variables. Then with a simple
computation
XZ Z
0 0
∗
(ΦM , a (f, t)Ω) = −i d P Ψ (P, n) d3 kf (k)δ(k − P)(P 0 + k 0 )ei(k −P )t A(P, n)
3 ∗
whence
a
(Φk , φ(x)Ω) = A(k)eika x , A(k) = (Φk , φ(0)Ω)
We have just seen that φ(x) will create from the vacuum some multi-particle state so one
expects only that
Z 1/2
A(k) =
(2π)3/2 (2Ek )1/2
for some 0 ≤ Z ≤ 1. Thus√we need also to scale the field by the so called wave function
renormalization constant, Z, in order to apply the LSZ formalism. A field for which
Eq. (269) is satisfied is called a renormalized field. In perturbation theory this constant is
ultra-violet divergent, so a field theory typically needs a proper regularization procedure
before it can be said to be meaningful. But then futher conditions are needed to insure
that physical predictions be insensitive of the regularization scheme chosen.
IMPORTANT REMARK: Eq. (269) also assumes the one-particle states are stable,
since otherwise there should be a small imaginary part in the energy of the wave function.J
In general there are no reasons for the fields in the original Lagrangian (to fix ideas we
consider a quartic interaction)
1 λ
L=− (∂φ)2 + m2 φ2 − φ4 (270)
2 4!
to satisfy the normalization conditions just discussed. Even more, parameters like m or λ
are very likely to be affected by the interactions, as it happens for example to an electron
in a crystal lattice. Scaling and shifting fields, masses and coupling constants changes the
Lagrangian into
Z 1 Zλ λ 4
L = − (∂φ)2 + Zm m2 φ2 − φ (271)
2 2 4!
65
which can be rewritten as
1 Zλ λ 4
L=− (∂φ)2 + m2 φ2 − φ + Lc = L0 + L1 + Lc (272)
2 4!
where L0 is the free part, L1 the quartic interaction and
1 1
Lc = − (Z − 1)(∂φ)2 + (Zm − 1)m2 φ2 + wφ (273)
2 2
the so called counterterm lagrangian. Were λ = 0 the renormalization constants would
all be vanishing and so, in general, one expects for them an expansion in power of λ.
The counterterm Lagrangian has then to be treated as an additional interaction whose
purpose will be to mantain order by order in perturbation theory the physical definition
of the basic parameters like masses and coupling constants. Examples of this will be seen
after the rules of perturbation theory will be presented.
Asymptotic states representing free particles with wave packets f1 , . . . , fn are defined
either by
Ψin ∗ ∗ ∗
f1 ,...,fm = a (f1 , in)a (f2 , in) · · · a (fm , in)Ω (274)
or by
Ψout ∗ ∗ ∗
f1 ,...,fn = a (f1 , out)a (f2 , out) · · · a (fn , out)Ω (275)
where
eiq1 ·x
f1 (x) = p
(2π)3/2 2q10
66
is the wave function of the incoming particle and φ(x) the interacting field. Assuming
that pi 6= qj for all values i ∈ {1, . . . , n} and j ∈ {1, . . . , m} we can write, using Eq. (266),
Z
out in
lim Ψp1 ,...,pn , i dx (f1 (x)∂0 φ(x) − φ(x)∂0 f1 (x)) Ψq2 ,...,qm
t→−∞
Z t
out in
= lim Ψp1 ,...,pn , i dx (f1 (x)∂0 φ(x) − φ(x)∂0 f1 (x)) Ψq2 ,...,qm
t→∞
Z t
out 4 in
− Ψp1 ,...,pn , i d x ∂0 (f1 (x)∂0 φ(x) − φ(x)∂0 f1 (x)) Ψq2 ,...,qm
using (2 − m2 )f1 = 0 and integrating by parts25 . The risult of the first reduction is now
Ψout in
p1 ,...,pn , Ψq1 ,...,qm
Z
out 4 2
in
= −i Ψp1 ,...,pn , d x f1 (x)(2 − m )φ(x) Ψq2 ,...,qm (277)
Now we proceed, for example by reducing a second incoming particle. Using as before
Ψin ∗ in
q2 ,...,qm = ain (q2 )Ψq3 ,...,qm , we have
Z
out 4 2
∗ in
Ψp1 ,...,pn , d x f1 (x)(2 − m )φ(x) ain (q2 )Ψq3,...,qm
Z
out
= lim Ψp1 ,...,pn , d4 x f1 (x)(2 − m2 )φ(x)
t→−∞
Z
in
×i dy (f2 (y)∂0 φ(y) − φ(y)∂0f2 (y)) Ψq3 ,...,qm
t
as the ordinary product of the operators ordered in decreasing time sequence from left to
right26 . For example we can write
67
t = −∞ (one should here work with a test function in order to fully justify the procedure),
then change the order of the fields (because under T -product this is legitimate), use again
Eq. (266) to exchange the two limits, pull the contributions at t = +∞ outside the T -
prodotto and note that the out operator remaining will annihilate the final state because
it destroys a particle not present in the final state.
E 43: Try to execute the described operations.
What remains is
Z
out 4 2
∗ in
Ψp1 ,...,pn , d x f1 (x)(2 − m )φ(x) ain (q2 )Ψq3,...,qm
Z
out
= − Ψp1 ,...,pn , d4 x f1 (x)(2 − m2 )φ(x)
Z
4 in
×i d y ∂0 (f2 (y)∂0 φ(y) − φ(y)∂0 f2 (y)) Ψq3 ,...,qm
Z Z
= −i d x d4 y f1 (x)f2 (y)(2x − m2 )(2y − m2 ) Ψout
4
p1 ,...,pn ,
× T (φ(x)φ(y))Ψin q3,...,qm
so that
Z Z
Ψout in
p1 ,...,pn , Ψq1 ,...,qm = (−i) 2
d x d4 y f1 (x)f2 (y)(2x − m2 )(2y − m2 )
4
× Ψout in
p1 ,...,pn , T (φ(x)φ(y))Ψq3,...,qm (279)
It should now be clear how successive reductions will proceed: each initial particle is to
be substituted by its wave function, a KG operator and the insertion of its field inside the
T -product until all particles have gone leaving only the vacuum, and each final particle
is to be substituted with the complex coniugate wave function , a KG operator and a
field in the T -product since out the final state only remains the vacuum. One obtains in
this way the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, one of the key
equations of QFT
Z
out in
1 (−i)−(n+m)/2
Ψp1 ,...,pn , Ψq1,...,qm = p d4 y1 · · · d4 xm (280)
(2π)3(n+m)/2 2p01 . . . 2qm
0
!
X X
× exp i qi · xi − i pj · yj (2y1 − m2 ) · · · (2xm − m2 ) (Ω, T [φ(y1 ) · · · φ(xm )]Ω)
i j
This formula implies that the four-dimensional Fourier transform of the distributions
have poles at the physical momenta p2i = −m2 of the final particles and poles at qj2 =
−m2 of the initial particles and that the scattering amplitudes are the residues at the
poles times the product of the corresponding wave functions. The reduction formula also
connect the amplitudes to the vacuum expectation values of the chronological product
of the interacting fields. In that sense, knowledge of the n-point functions τ (x1 , . . . , xn )
is equivalent to solve a quantum field theory: the infinite hierarchy of the chronological
functions is equivalent to the non linear field equations of the theory.
68
21 Perturbation theory
[1] C. Itzykson and J. Zuber, Quantum Field Theory, McGraw-Hill (1980), Chaper 5.
[2] J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, McGraw-Hill (1965)
[3] M. Srednicki, Quantum Field Theory, Chapter 1.
There is a magic formula, due to Gell-mann and Low, which permits the computation of
the chronological functions in terms of free fields and which looks like
R
Ω, T φin (x1 ) · · · φin (xn ) exp −i H1in (z)d4 z Ω
(Ω, T [φ(x1 ) · · · φ(xn )]Ω) = R (281)
Ω, T exp −i H1in (z)d4 z Ω
where H1in (z) is the interaction Hamiltonian density expressed as a function of the incom-
ing free fields. For example in the φ4 theory
λ
H1in (z) = : φin(z)4 :
4!
and in QED
The formula can be (formally) obtained in few steps: first by searching the unitary oper-
ator such that
φin (x) = U(t)φ(x)U(t)−1 , t = x0 (282)
then using this to replace φ(x) with the incoming field into the chronological vacuum
expectation value, Eq. (281), then finally taking care of various limits.
To find U(t) one uses the equations of motion
φ̇in (x) = i[φin (x), H0 (φin , Πin )], φ̇ = i[φ(x), H(φ, Π)] (283)
to obtain the equation and the appropriate initial condition
iU̇ (t) = H1 (t)U(t), U(−∞) = 1 (284)
where
H1 (t) = U(t)H1 (φ, Π)U(t)−1 = H1 (φin, Πin ) (285)
Converting to an integral equation
Z t
U(t) = 1 − i H1 (s)U(s)ds (286)
−∞
69
usually succinctly abbreviated as
Z t
U(t) = T exp −i H1 (s)ds (289)
−∞
E 44: Show that eq. (288) really is the solution of the integral equation.
More generally, for t > t1 the operator
where T denotes anti-chronological order, i.e. increasing time ordering from left to right.
Of course
and
and everything in S appears to be Lorentz invariant except for the time ordering.
To justify (281) we insert φ(x) = U(t)−1 φin(x)U(t) into < T φ(1) . . . φ(n) >V AC and use
(293) to write
(Ω, T [φ(x1 ) · · · φ(xn )]Ω) = Ω, T [U(t1 )−1 φin (x1 )U(t1 , t2 ) · · · U(tn−1 , tn )φin(xn )U(tn )]Ω
00 0
Then choose t > t1 , . . . , tn > t and write
00 00 0 0
U(t1 )−1 = U(t )−1 U(t , t1 ), U(tn ) = U(tn , t )U(t )
70
Upon substitution and reordering within the T -product we get
00
h 00 0
i 0
(Ω, T [φ(x1 ) · · · φ(xn )]Ω) = U(t )Ω, T φin (x1 ) · · · φin (xn )U(t , t ) U(t )Ω
0 0
Now while U(t )Ω → Ω as t → −∞, the stability of the vacuum would imply
00
lim U(t )Ω = CΩ,
00
|C| = 1
t →∞
00
so that C = (Ω, SΩ) and C̄ = (Ω, SΩ)−1 ; hence taking the limits t → ∞, t → −∞ we
finally get Eq. (281). J
Eq. (281) is the basis of the covariant perturbative expansion: simply expand the expo-
nential in powers of the interaction and evaluate it term by term. For example in the φ4
theory one would have
τ (x1 , . . . , xn ) = (295)
∞
X (−i)k λk Z
4 4 4 4
= Ω, T φin (x1 ) · · · φin (xn )] d z1 · · · d zk : φin(z1 ) : · · · : φin (zk ) : Ω
k=1
k!(4!)k
and the next problem is to learn how to evaluate the cumbersome vacuum expectation
value of the T -products so displayed.
T (φ(x)φ(y)) = : φ(x)φ(y) :
+ θ(x0 − y 0)[φ+ (x), φ− (y)] + θ(y 0 − x0 )[φ+ (y), φ− (x)] (296)
−i∆F (x − y) = θ(x0 − y 0)[φ+ (x), φ− (y)] + θ(y 0 − x0 )[φ+ (y), φ− (x)] (297)
71
We start with the familiar formula
Z
+ − d3 k 1 −iεk (t−t0 ) ik·(x−y)
[φ (x), φ (y)] = e e
(2π)3 2k 0
0 √
where t = x0 , t = y 0 and εk = k2 + m2 . The theta function has the integral represen-
tation
Z ∞ iE(t−t0 )
0 1 e
θ(t − t ) = dE
2πi −∞ E − iε
hence we have
Z
0 0 + − dk 0 d3 k 1 1
θ(x − y )[φ (x), φ (y)] = i 4 0
eik·(x−y)
(2π) 2εk k − εk + iε
Similarly
Z
0 0 + − dk 0 d3 k 1 1
θ(y − x )[φ (y), φ (x)] = −i 4 0
eik·(x−y)
(2π) 2εk k + εk − iε
Adding the two equations and noting that
1 1 2εk
− 0 =− 2
k0 − (εk − iε) k + (εk − iε) k + m2 − iε
one obtains the expression27
Z
1 eip·x
∆F (x) = d4 p (298)
(2π)4 p2 + m2 − iε
The so called Feynman iε-prescription
p tell us how to evade the poles of p2 + m2 , the
standard energy values p0 = ± p2 + m2 , and integration in dp0 is to done first. Obvioulsy
one has
(2x − m2 )∆F (x − y) = −δ 4 (x − y)
Then formally
Z ∞
∆F (x − y) = i Ks (x, y)ds (300)
0
27
To be sure, in place of k 2 + m2 − iε one would obtain k 2 + m2 − 2iεk ε, but since εk is positive and ε
is infinitesimal, the two forms are equivalent.
72
This Cauchy problem has a unique solution
−i 2
Ks (x, y) = 2 2
e−i(m s−σ/4s) (301)
(4π) s
where σ = (x − y)2 . Hence
Z ∞
1 2
∆F (x − y) = 2
s−2 e−i(m s−σ/4s) ds (302)
(4π) 0
The integral converge at infinity but in the origin it converges only if σ is in the upper
half-plane, H+ , of the complex plane, and then it defines an analytic function in H+ .
Hence we think about σ as it were σ + iε and we take the limit for ε → 0+ . An integral
representation of Hankel functions of the second kind gives finally the formula
m (2) √
∆F (x − y) = − √ H1 (m −σ − iε) (303)
8π −σ − iε
The function has a cut along the negative real axis, and the Feynman propagator is
obtained as the limiting value on the lower side of the cut, if σ < 0, or on the real axis if
σ > 0, of a analytic function in H+ .
Eq. (296) is the theorem of Wick in the case of two operators; its generalization is expressed
by the following identity
Z Z
4 4
T exp −i j(x)φ(x)d x = : exp −i j(x)φ(x)d x :
Z Z
i 4 4
× exp j(x)∆F (x − y)j(y)d xd y (304)
2
where j(x) is a classical external source (not an operator). The identity can be obtained
in the following way.
Let us integrate everything between x0 = t0 and x0 = t: the T -product satisfies then the
Dyson’s equation with H1 (x) = j(x)φ(x). Using the identity
valid if [A, [A, B]] = [B, [A, B]], one sees that the operator
Z t Z Z
4 i t 4 0 0 4
exp −i j(x)φ(x)d x exp − d x θ(x − y )j(x)∆m (x − y)j(y)d y
t0 2 t0
obey the same equation with the same initial data, where ∆m (x − y) = −i[φ(x), φ(y)]
E 43: Prove this fact. Identity (305) is used to compute the derivative of the first factor.
Hence we have
Z t
4
T exp −i j(x)φ(x)d x (306)
t0
Z t Z Z
4 i t 4 0 0 4
= exp −i j(x)φ(x)d x exp − d x θ(x − y )j(x)∆m (x − y)j(y)d y
t0 2 t0
73
R −
R
Using again
R Eq. (305) with A = −i j(x)φ (x) e B = −i j(x)φ+ (x) to put the operator
exp(−i j(x)φ(x)dx) into normal form one finally get
R
Z Z
−i φ(x)j(x)d4 x 4 i (+) 4 4
e =: exp −i φ(x)j(x)d x : exp j(x)∆m (x − y)j(y)d xd y
2
Then substituting into (306) and taking the limits t0 → −∞ and t → ∞ Wick’s theorem
(304) is finally obtained. In Eq. (306) it appears the retarded Green function
so there is a Wick’s theorem for T -products in terms of ordinary products and retarded
Green functions. Expanding Eq. (304) in power series of j(x) we obtain the Wick’s
identities (with an effort we do not advise students to undertake)
k X l
! " n #
X m2k,l Y Y
T (φ(x1 ) · · · φ(xn )) = i∆F (xπ(2i−1) − xπ(2i) ) : φ(xπ(j) ) :
l=0 π∈S
n! i=1 j=2l+1
n
where the dots indicate terms which can be obtained from the second term of the equation
by cyclically permuting the variables x1 , x2 , x3 . In practice, calling field contraction the
substitution of φ(xi )φ(xk ) with the distribution −i∆F (xi − xk ), i.e. with the symbol
Z
d4 k eik(xi −xk )
φ(xi )φ(xk ) = −i
(2π)4 k 2 + m2 − iε
Wick’s identity can be written symbolically as
XY
T (φ(x1 ) . . . φ(xn )) =: φ(x1 ) . . . φ(xn ) : + (contrazioni)× : φ(xk1 ) . . . φ(xkj ) :
where the sum over contractions includes all possible field contractions, putting everything
else into a normal product.
Inside matrix elements there are also operators within normal products, like
(φ(x1 ) · · · φ(xm ) : φ(y1 )4 : · · · : φ(yl )4 :
The Wick theorem works exactly as before, except that contractions of operators within
the normal product are to be avoided since (looking at Eq. (296)) there is no need to
correct with commutators the product of any two such operators. For example
T φ(x) : φ(y)4 : = −4i∆F (x − y) : φ3 (y) :
74
Hence we see that the propagator is the central element for the computation of the S-
matrix elements.
REMARK: The propagator is undefined at coincident points x = y due to θ(x0 − y 0).
The ambiguity is anyway a distribution with support at x = y and so it is proportional
to a δ 4 (x − y) and its derivatives. The higher order chronological function are determined
from causality and covariance only up to distributions with support on coincidents points.
Being local, they can be considered as the effect of local operators added to the initial
Lagrangian. This observation is at the root of renormalization theory: it means that
the interactions can be so defined, or adjusted, so as to absorb into masses and coupling
parameters the ultra-violet divergences one encounters in perturbative evaluation of the
chronological functions. Theories for which this is really possible by adjusting a finite
number of parameters are called renormalizable theories.
The vacuum expectation values of the T -products of the fields φin (z), which are free
fields, are exactly computable in terms of Feynman propagators and in Eq. (295), after
all contractions have been done by the use of wick’s theorem, only all possible such
contractions remain.
If to each contraction, to which it correspond the term −i∆F (x1 − x2 ), a line is associated
connecting the points x1 con x2 , then to each product of such contractions a graph is
associated with a certain number of vertex and external legs. These are the famous
Feynman diagrams. As an example let us consider the two-point function τ (x1 , x2 ) in the
theory φ4 to second order in λ. We must then compute
Z Z
(−iλ)2
−i∆F (x1 − x2 ) + 2
d z1 d4 z2 Ω, T φin (x1 )φin (x2 ) : φ4in(z1 ) :: φ4in(z2 ) : Ω
4
2(4!)
and dividing the result for (Ω, SΩ), again to second order in λ. The first term is the free
field propagator. The radiative corrections (si chiamano in questo modo le correzioni al
risultato di ordine zero) are contained in the second term. One obtains
Z Z
(−iλ)2 4 4
4 4
d z1 d z2 Ω, T φ in (x1 )φ in (x2 ) : φ in (z1 ) :: φ in (z2 ) : Ω
2(4!)2
Z Z
(−iλ)2
= d z1 d4 z2 (−i∆F (x1 − z1 ))(−i∆F (x2 − z2 )(−i∆F (z1 − z2 ))3
4
3!
Z Z
(−iλ)2
+ (−i∆F (x1 − x2 )) d z1 d4 z2 (−i∆F (z1 − z2 ))4
4
(307)
4!2
In fact there are 192 distinct contraction giving the second line and 4·3·2 ways to contract
φin (x1 ) with φin (x2 ) and each field φin(z1 ) with a field φin (z2 ).
The corresponding diagrams are shown in Fig. (1) and Fig. (2), this one being topologically
disconnected. It represents a vacuum to vacuum transition.
We may note in the connected graph the following elements:
75
Figure 1: Connected Feynman diagram for the radiative correction to the propagator.
z z
1 2
x x
1 2
z1 z
2
x1 x2
one can cut two lines without disconnecting the diagram: one says it is a 2-loop
graph,
there are no other graphs with such properties except those obtained by a permu-
tation of the vertices and internal lines of the given diagrams.
after division of Eq. (307) by (Ω, S Ω) the diagram with a vacuum subdiagram is
cancelled by the sum of all graphs.
The properties listed are general and give rise to the so called Feynman’s rules for the
theory φ4 , a set of sistematic prescriptions for associating to chronological functions sums
over diagrams (and viceversa) in such a way that to each diagram there corresponds
a particular terms of the Wick expansion of the chronological products. To calculate
τ (x1 , . . . , x2n ) the rules are (see Itzykson/Zuber, Chap.[6], where also QED is considered)
1. draw all distinct Feynman diagrams with 2n external points x1 , . . . , x2n and m vertices
y1 , . . . , ym , taking into account the incident rule of lines at the vertices (three lines
at each vertex for φ3 interaction, four for φ4 , etc.), and omitting all diagrams which
contain vacuum subdiagrams.
76
4. divide the diagram by a symmetry factor S, equal to the order of the group of all
permutations of internal lines (those connecting vertices with vertices) with fixed
vertices
5. sum the contribution of all such diagrams.
Notice that it is not necessary to add the contribution of the m! permutations of the
vertices since the factor 1/m! coming from the expansion of the exponential takes care of
this automatically.
Looking back at τ (x1 , x2 ) and passing to Fourier transform (it is quite common to use the
0
notation −i∆ (x1 − x2 ) to indicate the exact propagator τ (x1 − x2 ))
Z
0 0
∆ (p) = d4 x ∆ (x) e−ip·x
where the sum is over the incoming momenta at the vertex, at each internal line with
momentum k the term
−i d4 k
k 2 + m2 − iε (2π)4
and at each external line carrying incoming momen tum p the factor
−i
p2 + m2 − iε
and if one integrates over all internal momenta, dividing then the result by a simmetry
factor. Finally one sums all topologically distinct graphs, the vertices being fixed. This
can be systematized by giving Feynman rules in momentum space (see Itzykson-Zuber
book for a detailed treatment).
77
If the integral in Eq. (309) were convergent (it is not) there would be a remarkable fact:
0
the pole of the function ∆ (p) would not be at p2 = −m2 (that is what we call mass would
not be constant m2 ) and the residue would not be 1, but (always to second order in λ)
dΠ∗ (p2 )
Z =1+ (311)
dp2 |p2 =−m2
Unfortunately Π(−m2 ) diverges for large momentum variables or, as it is custom to say,
in the ultraviolet region, and one cannot simply says that the mass becomes m2 + Π(−m2 )
due to self-interaction effects. It is therefore necessary, in the first instance, to regularize
the theory, for example bounding the integration region with a cutoff integrating over
momenta k 2 < Λ2 . We shall assume, thereafter, that such regularization have been done
once and for all, even if not indicated explicitely.
Thus, for example, our Π(p2 ) really depends also on the cutoff Λ. Then what is the mass
in the regularized theory? A key idea of all of physics is that measurable parameters like
masses and coupling constants should be defined operationally, i.e. by giving a detailed
description about how they are to be measured; clearly the constants appearing in the
Lagrangian have nothing to a pripri with measurement prescriptions. Perhaps this is
particularly clear in the case of the coupling parameter λ: one defines a coupling parameter
by measuring forces between particles in a long distance limit, for instance, of by studying
scattering at some prescribed energy. This means that the physical coupling really is a
function of distance or energy, not the constant appearing in the Lagrangian. In the case
of mass it is of course possible to give analogous definitions, but is much more convenient
mathematically to define the physical mass parameter as the pole of the propagator with
respect to p2 even if this is less intuitive physically. For a free field this is the same thing
as the Lagrangian parameter m, but not for the interacting theory as we have just seen
(that is Π(−m2 ) 6= 0). At the end of Sec. 19 we saw that a counterterm Lagrangian (273)
had to be added to the initial Lagrangian in order to keep the fields in accord with the
LSZ conditions: now we shall see that the same counterterm Lagrangian can be used to
eliminate the infinities which appears when one tries to remove the cutoff.
It is useful to note that Π(−m2 ) = 0 should hold true also by particle stability, besides
being required by the definition of m: in fact the LSZ formula gives the particle-particle
amplitude as
iπΠ∗ (−m2 ) 4
< p, out|in, q >= δ(p − q) + p δ (p − q)
q2 + m2
Stability of one-particle states28 requires < p, out|in, q >= δ(p − q), hence the condition.
As we repeatedly said this would not hold even if the integral (309) were convergent,
so a renormalization procedure of some sort would be necessary even if everything were
finite from the start. For the same reasons Eq. (311) in general does not give the wanted
condition Z = 1.
Noi vediamo dunque che per calcolare gli elementi di matrice S è necessario sottrarre a
Π∗ (p2 ) un polinomio, che scriviamo nella forma
0
Π∗ (p2 ) → Π (p2 ) = Π∗ (p2 ) − (Z − 1)(p2 + m2 ) − Zδm2
28
Decay of a particle into several of the same mass is also forbidden by energy-momentum conservation.
78
dove (gli integrali che definiscono Π∗ (p2 ) si intendono regolarizzati)
dΠ∗ (p2 )
δm2 = Π∗ (−m2 ), Z −1=
dp2 |p2 =−m2
0 0
in modo tale che Π (−m2 ) = 0 e dΠ (p2 )/dp2|p2=−m2 = 0. Il limite di cutoff infinito in
0
Π (p2 ) ora esiste e soddisfa le nostre richieste. La trasformata di Fourier della sottrazione
è la distribuzione
(Z − 1)(2 − m2 )δ(x − y) + Zδm2 δ(x − y)
Si può notare che se avessimo calcolato la funzione a due punti usando la nuova La-
grangiana
1 1 λ 1 1
L[φ; m2 ] = − ∂a φ∂ a φ − m2 φ2 − φ4 − (Z − 1) ∂a φ∂ a φ + m2 φ2 − Zδm2 φ2
2 2 24 2 2
invece della Lagrangiana originale, ma trattando gli ultimi due termini come una cor-
0
rezione di ordine λ2 , avremmo trovato proprio il risultato corretto per Π (p2 ). Alla fine
della Sez. 19 abbiamo notato che l’uso delle formule di riduzione e la costruzione stessa
degli stati asintotici richiedeva proprio la Lagrangiana scritta sopra (con lievi cambiamenti
di notazioni). Vediamo dunque che si possono usare i parametri Z, m e λ per assorbire
le divergenze ultraviolette. Nella presente teoria l’integrale (309) è divergente, ma si pos-
sono usare le sottrazioni δm2 e Z − 1 per rendere finito il risultato (ossia per cancellare le
divergenze). Il fatto che ordine dopo ordine nella teoria perturbativa si possano scegliere
i parametri Z − 1, δm2 e λ per cancellare tutte le divergenze è un aspetto non banale
della teoria φ4 : ci dice che è una teoria rinormalizzabile. Si osservi che alla base della
cancellazione delle divergenze c’è la definizione dei parametri della Lagrangiana: per la
teoria φ4 questa è
Z e m2 sono definiti dalla condizione che il propagatore del campo abbia un polo
per p2 = −m2 con residuo 1
λ è definito come il valore della trasformata di Fourier della funzione a quattro punti
79
Figure 3: Rappresentazione grafica del vertice.
p p
1 4
λ =?
p p
2 3
25 Scattering
Consideriamo la diffusione elastica di due particelle al più basso ordine in λ nella teoria
φ4 . Dalla formula di riduzione si vede che occorre calcolare la funzione a quattro punti, e
dalla formula di Gell-mann/Low il termine del primo ordine è
Z
λ
τ (x1 , x2 , x3 , x4 ) = −i d4 x < 0|T φ(1)φ(2)φ(3)φ(4) : φ4 (x) : |0 >
4!
Qui abbiamo scritto φ(x) al posto di φin(x), e omesso l’ampiezza vuoto-vuoto perché di
ordine λ2 , φ(1) sta per φ(x1 ), e cosı̀ via. Il grafico di Feynman che corrisponde al processo
è mostrato in Fig. [4] e contribuisce 4! volte all’elemento di matrice scritto sopra; dalla
formula di riduzione si ottiene allora l’elemento di matrice S
0 0 0 0
S12→10 20 = −iλ(2π)−6 (16E1 E2 E1 E2 )−1/2 (2π)4 δ(p1 + p2 − p1 − p2 ) (313)
0
dove le Ei , Ej sono le energie iniziali e finali delle particelle partecipanti. Per dedurre
dalla formula scritta qualcosa che abbia senso occorre adesso considerare il modo corretto
di usarla.
Innanzitutto la probabilità di transizione, cioè |S12→10 20 |2 , risulta finita solo se si lavora
in un volume di universo finito, diciamo V T ; infatti in caso contrario il quadrato della
distribuzione δ diverge, mentre in un volume finito (V T = volume spaziale × durata)
si ha29
0 0 0 0
[(2π)4 δ(p1 + p2 − p1 − p2 )]2 = (2π)4 δ(p1 + p2 − p1 − p2 )V T
In un volume spaziale finito la norma degli stati iniziali e finali è k|p > |k2 = δ(p − p) =
δ(0) = V /(2π)3, e quindi occorre ancora dividere l’elemento di matrice per quattro fattori
29
La formula non pretende di definire δ(x)2 , che non esiste, ma solo di far vedere in che modo diverge.
80
Figure 4: Primo contributo alla diffusione elastica
1 3
2 4
(V /(2π)3)1/2 , uno per ogni particella. Ciò equivale a usare le funzioni d’onda normalizzate
nel volume V
1
fp = √ eip·x
2V E
√
al posto delle precedenti fp = (2π)−3/2 eip·x / 2E. Infine, trattandosi di una probabilità di
transizione nello spettro continuo, occorre ancora moltiplicare il risultato per “il numero”
di stati disponibili nei volumi V d3 p dello spazio delle fasi: con le funzioni d’onda scritte
questo numero è la misura
0 0
1 V d 3 p1 V d 3 p2
#=
2 (2π)3 (2π)3
dove il fattore 1/2 risulta dal fatto che le particelle finali sono indistinguibili. Tenuto
conto di questi fattori, e dividendo il risultato per T , si ottiene la probabilità di transizione
per unità di tempo nell’elemento di volume indicato dello spazio delle fasi (o “rate” di
transizione, visto che si misura in sec.−1 )
0 0
λ2 1 d 3 p1 d 3 p2
dΓ = δ(P i − P f ) 0 0 (314)
64π 2 2E1 E2 V E1 E2
0 0
dove Pi = p1 + p2 , Pf = p1 + p2 . Si noti la presenza delle misure d3 p/E, invarianti
di Lorentz, e il fattore V −1 che non lo è; infatti un ”rate” non può essere invariante di
Lorentz a causa della dilatazione relativistica del tempo. Invariante di Lorentz è invece la
sezione d’urto, che misura il rate per flusso incidente di particelle. Consideriamo il caso in
cui la prima particella incide con momento p1 sul “target” in quiete: allora p1 = (E1 , p1 ),
p2 = (m, 0) e il flusso incidente (densità di particelle (1/V) per velocità relativa (|p1 |/E1 ))
è
1 |p1 | 1
I= = vrel
V E1 V
Dividendo il rate per I il volume di normalizzazione si cancella, e notiamo adesso che il
fattore restante, E1 E2 vrel , può essere scritto nella forma invariante di Lorentz
1/2
E1 E2 vrel = (p1 · p2 )2 − m4 (315)
81
Figure 5: diffusione nel centro di massa
P’
p −p
−P’
Consideriamo dettagliamente la sezione d’urto nel sistema del centro di massa, dove per
definizione p1 + p2 = 0.
0 0
Integrando dΓ/I sul momento p2 e sull’energia finale E1 si ottiene facilmente la sezione
d’urto per la diffusione nella direzione θ
λ2 1
dσ = 2 2
dΩ (316)
64π 4ECM
0
where dΩ = 2π d(cos θ) è l’angolo solido attorno alla direzione di p1 entro il quale si
2
ha la diffusione. ECM è l’energia di ciascuna particella incidente, cosicché 4ECM = s è
il quadrato dell’energia totale nel centro di massa (center-of-mass squared energy), cioè
s = −(p1 + p2 )2 . Un altro invariante importante è il momento trasferito alla prima
particella
0 0 0
t = −(p1 − p1 )2 = 2m2 − 2E1 E1 + 2p1 p1 cos θ1
e alla seconda
0 0 0
u = −(p1 − p2 )2 = 2m2 − 2E1 E2 + 2p1 p2 cos θ2
0 0 0
Nel centro di massa θ2 = π − θ1 , E1 = E2 ≡ E e E1 = E2 ≡ E . Dalla conservazione
0
dell’energia segue anche che |p1 | = |p1 | ≡ p0 . Le grandezze s, t, u si chiamano variabili
di Mandelstam, e soddisfano la relazione
s + t + u = 4m2
82
26 Decay and Resonances
Consider the following scalar-scalar interaction
g
L1 (x) = : φ(x)χ2 (x) :
2
Let m denote the mass of the φ-particle but χ be massless. To first order the vertex
function of this theory is given by
Z
g
−i d4 y < 0|T (φ(x1 )χ(x2 )χ(x3 ) : φ(y)χ2(y) :)|0 >
2
This matrix element has to with the decay process φ → χχ, which would be formally
forbidden within the LSZ formalism since we assumed the stability of one-particle states.
Ignoring this difficulty for the moment, the amplitude is
−ig
Sp,p1,p2 = 3/2
√ (2π)4 δ(p − p1 − p2 ) (319)
V 8EE1 E2
The decay rate is obtained by squaring this amplitude, dividing by T and multiplying
with the density of final states,
g 2E −1 4 d 3 p1 d 3 p2 p
dΓ = δ (p − p 1 − p 2 ) , E= |p|2 + m2 (320)
64π 2 E1 E2
The total rate is obtained by integration over the final momenta (τ = Γ−1 is then the
lifetime of the state)
g2
Γ=
32πE
Notice that since E = m(1 − v 2 )−1/2 and v = p/E, we have
g2 √ √
Γ= 1 − v 2 = Γ0 1 − v 2
32πm
in agreement with the formula of relativistic time dilatation.
E 46: Show that if the χ-particle has mass mχ the rate in the center of mass system is
1/2
g2 4m2χ
ΓCM = 1− 2
32πm m
The meaning of the decay rate computed above is that the φ-propagator has a self-energy
correction with a small imaginary part. In fact un unstable particle should not really
exists in the spectrum of the theory but only in intermediate states where it shows us as
resonance. To confirm this expectation we now compute this to one-loop order. The only
Feynman graph is drown below.
83
Figure 6: Self-energy
k
p p
p−k
The dotted line is the propagator of the χ-particle; from the Feynman rules (or sim-
0
ply computing the two-point function directly) we get ∆ (p), the Fourier transform of
0
∆ (x1 , x2 ) = −i < 0|T (φ(x1 )φ(x2 ))|0 >, up to the one-loop order as follows
0 1 1 1
i∆ (p) = + 2 Π(p2 ) 2
p2 2 2
+ m − iε p + m − iε p + m2 − iε
where
Z
2 −ig 2 1 1
Π(p ) = d4 k
32π 4 2 2 2 2
k + mχ − iε (p − k) + mχ − iε
0
∆ (x1 , x2 ) is then reconstructed by inverse Fourier transform
Z
0 d4 p 0
∆ (x1 , x2 ) = ∆ (p)eip·(x1 −x2 )
(2π)4
Π(−m2 ) = 0
84
ensuring that m is the physical observed mass, and
dΠ(p2 ) 0
2
≡ Π (−m2 ) = 0
dp p2 =−m2
ensuring that the field itself is properly normalized (remember the discussion in connection
with the LSZ formalism). But a look at Eq. (322) reveals that the argument of the
logarithm is negative for x− < x < x+ , with
s
1 1 4m2χ
x± = ± 1+ 2
2 2 p
when p2 < −4m2χ . Hence Π(p2 ) has an imaginary part. We have been fool: we are trying
to compute a quantity which is not in the spectrum of the Hamiltonian (the metastable
0
φ-particle mass). Hence we only require Re Π(−m2 ) = 0, Re Π (−m2 ) = 0, so that finally
the renormalized self-energy looks like
Z 1
2 g2 |m2χ + p2 x(1 − x)|
Π(p ) = dx ln − A(p2 + m2 ) (323)
32π 2 0 m2χ + p2 x(1 − x) − iε
0
with A fixed by the condition Re Π (−m2 ) = 0. Once more the result is finite when
expressed in terms of physical masses and renormalized fields.
Now A is real being the coefficient of an operator in the counterterm Lagrangian. Hence
the imaginary part is still coming from the logarithm in the integral (323): in fact
Im ln(m2χ + p2 x(1 − x) − iε) = −iπ
throughout the region x− < x < x+ and we see that, as a result,
Z x+ 1/2
2 g2 g2 4m2χ
Im Π(p ) = dx = 1+ 2 (324)
32π x− 32π p
Thus
1/2
2 g2 4m2χ
Im Π(−m ) = 1− 2 = mΓCM (325)
32π m
The fact that the imaginary part gives the decay rate of the unstable state is general and
known in field theory as the Cutkosky rule. It means that the φ-particle appears as a
resonance in χχ → χχ scattering: the Feynman graph is
T = jf i
i j
where dotted lines represent the χ-particles. If s ∼ m2 then the matrix element is domi-
nated by the pole of the φ-propagator
g2
T '
−s + m2 − Π(−s)
But close to m2 we have just seen that Π(−s) ∼ imΓCM so writing s = (m + ε)2 '
m2 + 2mε, ε m, we obtain
−g 2 /2m
T =
ε − iΓCM /2
i.e. the usual Breit-Wigner formula for a near resonance transition.
85
27 Diffusione di un fermione da un campo esterno
Consideriamo la diffusione di un fermione (per semplicità parleremo di elettroni) nel
campo coulombiano di un centro diffusore fisso. Il potenziale di interazione è
Z
Ze
H1 = e A0 : ψ̄γ 0 ψ : d4 x A0 =
r
Poiché non abbiamo derivato le formule di riduzione LSZ per i fermioni, calcoleremo
l’ampiezza di diffusione nel formalismo di interazione, dove i campi che figurano in H1
vanno considerati come campi liberi che si potranno scrivere in termini di operatori di
creazione e annichilazione, come segue
Z r
1 X 3 m ik·x ∗ −ik·x
ψ(x) = d k u(k, σ)a(k, σ)e + v(k, σ)b (k, σ)e
(2π)3/2 σ Ek
Figure 7: Grafico di Feynman per la diffusione in un campo esterno, al primo ordine nella
teoria delle perturbazioni.
k σ
Calcoleremo tale ampiezza al primo ordine della teoria delle perturbazioni, in approssi-
mazione di Born. Secondo la formula di Dyson, si ha
Z
0 0
A(k, σ; p, σ ) = −e dtdx < p, σ |A0 (x) : ψ ∗ (x)ψ(x) : |k, σ > (326)
86
dove compare la trasformata di Fourier del potenziale coulombiano
Z
4πZe
Ã0 (p − k) = A0 (x)e−i(p−k)·x dx =
|p − k|2
La distribuzione δ(Ep −Ek ) è dovuta alla stazionarietà del potenziale, cosicchè la diffusione
conserva l’energia, mentre si riconosce che l’argomento della trasformata di Fourier del
potenziale è il momento trasferito dall’elettrone al centro diffusore, del quale si trascura
il rinculo.
L’ampiezza di diffusione appare con le dimensioni fisiche di un volume, mentre dovrebbe
essere adimensionale. Ciò è dovuto all’uso degli stati impropri |k, σ >, che non sono
normalizzabili30 . Come chiarito precedentemente, in volume finito la norma è (2π)−3/2 V 1/2
e dunque occorre dividere la (327) per due di questi fattori. Si ottiene cosı̀ la corretta
ampiezza adimensionale
0 me 0
A(k, σ; p, σ ) = −2πδ(Ep − Ek ) p u∗ (p, σ )u(k, σ)Ã0 (p − k) (328)
V Ep Ek
Particelle non polarizzate
La probabilità di transizione è data dal quadrato del modulo dell’ampiezza e se siamo
interessati solamente a particelle di dato impulso ed energia, ma non allo stato di polariz-
zazione, allora dovremo fare la media sulle polarizzazioni iniziali (perché non sono note)
e sommare su quelle finali (perché contribuiscono entrambe). Più esattamente, se lo stato
iniziale è una miscela statistica dei due stati di polarizzazione nelle proporzioni 1 : 1, la
corrispondente matrice statistica è
1X
ρ(k) = |k, σ >< σ, k|
2 σ
0
e la probabilità della transizione i = (k, σ) → f = (p, σ ) sarà
0 0
Wif =< p, σ |H1 ρ(k)H1 |σ , p > (329)
R
dove H1 = e A0 : ψ̄γ 0 ψ : d4 x. Ciò corrisponde appunto a fare la media sulle polariz-
zazioni iniziali. Wif può anche scriversi nella forma
h 0
i
Wif = Tr ρ(p, σ )H1 ρ(k)H1
0 0 0
dove ρ(p, σ ) = |p, σ >< σ , p| è il proiettore sullo stato finale.
NOTA 2: nel resto della discussione non faremo uso, come si dovrebbe, di stati iniziali
e finali normalizzabili. Questo causa una difficoltà apparente, e cioè il problema di inter-
pretare il quadrato della delta di Dirac nell’Eq. (327) che, a rigore, non esiste. Useremo
dunque una procedura più formale ma, a posteriori, corretta, dato che se ne può dare la
giustificazione descrivendo la diffusione con pacchetti d’onda normalizzabili.
Per chiarire la natura della divergenza nel quadrato della delta di Dirac, si può scrivere31
Z T /2
2 1
δ(Ep − Ek ) = δ(Ep − Ek ) lim ei(Ep −Ek )T dT = δ(Ep − Ek ) lim T
T →∞ 2π −T /2 T →∞
30
Con le dimensioni corrette risulteranno invece le grandezze misurabili, quali le sezioni d’urto e le vite
medie, che si ottengono dalle ampiezze tenendo conto della densità e della normalizzazione degli stati.
31
Non stiamo cercando una definizione per δ(x)2 , che non esiste, ma semplicemente di descrivere in che
modo δ(x)2 diverge.
87
Da qui si vede che la divergenza è lineare in T , e dunque la grandezza ben definita è, in
realtà, la probabilità di transizione per unità di tempo. La definizione di quest’ultima,
richiede ancora che si moltiplichi per la densità degli stati finali. Con la normalizzazione
nel volume finito V questa è 1/(2π)3, e quindi il numero di stati nell’elemento V dp dello
spazio delle fasi sarà dN = V dp/(2π)3. Se scriviamo dp = p2 dp dΩ(p), allora dΩ(p)
determina l’elemento di angolo solido intorno alla direzione di p, e si avrà la probabilità
di transizione per unità di tempo nella forma
1 X 0
dW = |A(k, σ; p, σ )|2 p2 V dp dΩ(p) (330)
2T 0
σ,σ
Poichè il numero medio di particelle diffuse entro l’angolo solido dΩ(p) è proporzionale
al numero medio di particelle incidenti per unità di tempo e superficie, cioè al flusso, gli
sperimentali riportano il rapporto della probabilità al flusso, grandezza che si chiama la
sezione d’urto differenziale del processo di diffusione. La sua definizione è
Z
dσ dW
= dEp I −1 (332)
dΩ(p) dEp
dove I rappresenta il flusso incidente. Per calcolare l’integrale sui valori dell’energia
è necessario determinare la dipendenza di dW da Ep , e per calcolare dσ è necessario
determinare I. Il primo problema si risolve con le due formule di facile verifica
Tr (p/ + m)γ 0 (k/ + m)γ 0 = 8E 2 1 − v 2 sin2 (θ/2) (333)
θ
|p − k|4 = 16 |k|4 sin4 (334)
2
dove θ è l’angolo tra la direzione di p e k, v = |p|/Ep = |k|/Ek e, in entrambe le formule,
abbiamo posto |p| = |k| ed E = Ep = Ek , come impone la conservazione dell’energia. Si
noti anche che p dp = Ep dEp , come segue differenziando l’espressione Ep2 = |p|2 + m2 .
Per calcolare I, troviamo prima il valore medio dell’operatore corrente sugli stati iniziali.
Usando ancora lo sviluppo del campo libero in onde piane e le relazioni di commutazione,
si ha
1 kµ
< j µ >=< k, σ| : ψ̄γ µ ψ : |k, σ >= (335)
(2π)3 Ek
Occorre ancora dividere l’elemento di matrice per V /(2π)3 (a causa del volume finito); il
flusso risultante è
1 |k|
I =< j > ·n =
V Ek
88
dove n = k/|k| rappresenta la direzione incidente. Questi risultati danno infine la sezione
d’urto
(Ze2 )2 E 2 1
dσ = 1 − v 2 sin2 (θ/2) dΩ(p) (336)
4|k|4 sin4 θ
2
89
Appendix
A Polarization vectors
From eq.s(212), (213), (214) it follows trivially that
Then e(0) · k + e(3) · k = 0. Also l+ = e(0) + e(3) is lightlike and orthogonal to k, e(1) ed
e(2) , so that l+ = α(k) k.
An explicit description of the vectors ea(λ) (k) satisfying these conditions is the following.
Take e(0) = n to be any timelike vector, ea(λ) (k), λ = 1, 2, orthogonal to the plane [n, k],
with unit norm and orthogonal to each other. Let ea(3) (k) in the plane [n, k] such that
e2(3) = 1, e(3) · n = 0. This determines e(3)
k a + (n · k)na
ea(3) (k) = (A.1)
|n · k|
In the frame where n is the time axis (1, 0, 0, 0), one has
This determines uniquely the matrix L(k): in fact its columns are the components ea(λ) (k).
If the first condition is not evident, note that
L(k) · q = κ (n + e(3) )
where W (Λ, k) = L(Λk)−1 ΛL(k) is the usual Wigner matrix in the stability group of q.
But
so that
90
B Dirac theory: non relativistic limit and the fine
structure of hydrogen
We want to study the NR limit of Dirac’s equation coupled with an electromagnetic field.
We will find a remarkable prediction of Dirac theory, namely that the particles described
by the Dirac equations are fermions, with a giromagnetic ratio equal to 2. So actually,
what Dirac found was a relativistic theory of electrons (and positrons), or more generally
of any spin-1/2 particle. To start with we recall the equation to be studied
(∂/ −ieA
/ + m) ψ(x) = 0 (B.4)
where σ k are the well known Pauli matrices, the Dirac equation takes the form (reintro-
ducing ~ and c)
Zα e
i ∂0 − i φ − λ−1
c φ + iσ · ∇ + σ · A η=0
r ~c
Zα e
i ∂0 − i η + λ−1
c η − iσ · ∇ + σ · A φ=0
r ~c
where λc = ~/mc. We need to extract the high frequency part of the wave functions, so
we set
0 /λ 0 /λ
φ = e−ix c
u η = e−ix c
v
91
terms unless Zα ∼ 1 (which means Z = 137). Neglecting (∂0 − iZα/r)v and solving for
v gives
~ e
v= σ · i∇ + A u
2mc ~c
Substituting this into the first equation we obtain the Schrödinger like equation for the
two-component spinor u
Zα ~ e e
i ∂0 − i u+ iσ · ∇ + σ · A iσ · ∇ + σ·A u= 0
r 2mc ~c ~c
The spinor
φ
ψ=
η
(∂/ − ieA
/)ψ + mψ = 0
92
which in a Coulomb field reads
Zα
i ∂0 − i φ − λ−1
c φ + iσ · ∇η = 0
r
Zα
i ∂0 − i η + λ−1
c η + iσ · ∇φ = 0
r
For stationary states, ∂0 → −iE/~c = −iε, so
−1 Zα
iσ · ∇φ + λc + ε + η=0
r
−1 Zα
iσ · ∇η − λc − ε − φ=0
r
Second, we are going to exploit the spherical symmetry of the problem and reduce the
equations to ordinary differential equations in one radial variable. But before this, let us
recall how the spinor harmonics Θjlm , are constructed. They are eigenvectors of the spin
operator
1 σ 0
J = ` + σ, σ=
2 0 σ
This commutes with the Hamiltonian and with the parity operator, P ψ(t, x) = βψ(t, −x).
First in the hierarchy we have the familiar spherical harmonics
s
m+|m| 2l + 1 (l − |m|)! imφ
Ylm = (−1) 2 e Pl|m| (cos θ), −l ≤ m ≤ l
4π (l + |m|)!
q
j−m+1
2j+2
Yl,m−1/2 j = l − 1/2
Θ̂jlm = q ,
− j+m+1 Y −j ≤ m ≤ j
2j+2 l,m+1/2
Both have definite parity, (−1)l , and are eigenvectors of the operator ` · σ with eigenvalue
j(j + 1) − l(l + 1) − 3/4, as can be easily proved using
lz l− p
`·σ = , l± Yl,m = l(l + 1) − m(m ± 1) Yl,m±1 , l± = lx ± ily
l+ −lz
93
Hence they are also eigenvectors of j2 = `2 + ` · σ + 3/4,
and moreover,
r
σ · nΘjlm = Θ̂jl+1m , n=
r
We are ready to separate the angular variables from the radial one. The orbital angular
momentum is not a separately conserved quantity, but can serve to specify the parity
of states, which is conserved. Hence we look for eigenstates of H, j, jz and the parity
operator P .
Since under parity φ(x) → φ(−x) and η(x) → −η(−x), it is clear that either φ ∼ Θjlm
and η ∼ Θ̂jl+1m , or φ ∼ Θ̂jl+1m and η ∼ Θjlm . So we separate variables by setting32 either
or
Then both pairs (φ, η) will be parity eigenvectors, and there will be two states with
opposite parity, for given values of j, m and the energy eigenvalue. The Dirac equations
become
0 j − 1/2 −1 Zα
f− − f− + ε + λc + g− = 0
r r
0 j + 3/2 −1 Zα
g− + g − − ε − λc + f− = 0 (B.8)
r r
0 j + 3/2 Zα
f+ + f+ + ε + λ−1
c + g+ = 0
r r
0 j − 1/2 Zα
g+ − g+ − ε − λ−1
c + f+ = 0 (B.9)
r r
These four equations can be rewritten as a single pair if we define a parameter δ such that
−(j + 1/2) for l = j − 1/2
δ=
j + 1/2 for l = j + 1/2
Then
0 1+δ −1 Zα
f + f + ε + λc + g=0
r r
0 1−δ −1 Zα
g + g − ε − λc + f =0 (B.10)
r r
where it is understood that for δ < 0 f is f− and g is g− , respectively, and for δ > 0 f is
f+ and g is g+ , respectively.
32
The factor i in the second equation has been set for convenience.
94
Our problem has been reduced to the simpler one of solving (B.10) with the asymptotic
boundary conditions which are pertinent to the bound states.
The asymptotics for all is
f ∼ rν 1/2
ν ν = −1 + δ 2 − (Zα)2
g∼r
near r = 0, and
p −χ r
f ∼ pλ−1c + εe
−χ r
g ∼∼ λ−1 c −εe
p
near r = ∞, with χ = λ−2 2
c − ε . We look for solutions of the form
p
f = p λ−1
c + εr
γ−1 −χr
e (H1 + H2 ) 1/2
−1 γ−1 −χr , γ = δ 2 − (Zα)2
g = λc − ε r e (H1 − H2 )
Substituting this into (B.8), taking the sum and then the difference of the two resulting
equations and defining the dimensionless variable ρ = 2χ r, gives
0 Zαε Zα
ρH1 + γ − H1 + δ − H2 = 0 (B.11)
χ λc χ
0 Zαε Zα
ρH2 + γ + − ρ H2 + δ + H1 = 0 (B.12)
χ λc χ
Finally, eliminating H1 or H2 gives
00 Zαε
ρH1 + (2γ + 1 − ρ)H1 − γ − H1 = 0
χ
00 Zαε
ρH2 + (2γ + 1 − ρ)H2 − γ + 1 − H2 = 0
χ
which can be solved in terms of hypergeometric functions. For definiteness, the solutions
are
Zαε
H1 = AF (γ − , 2γ + 1; ρ)
χ
Zαε
H2 = BF (γ + 1 − , 2γ + 1; ρ)
χ
where
∞
X Γ(a + n)Γ(b) z n
F (a, b; z) =
n=0
Γ(b + n)Γ(a) n!
95
The series F (a, b; z) diverges at infinity like ez , unless a = −nr for some positive natural33
nr , in which case it reduces to a polynomial of degree nr . If nr = 0, then H1 still is a
polynomial but H2 is not, unless B = 0. This happens precisely when δ < 0, since for
δ > 0 one has A = B. Hence the possible values of nr are
0, 1, . . . for δ<0
nr =
1, 2, . . . for δ>0
The condition
Zαε
γ− = −nr
χ
fixes the energy spectrum of the hydrogen atom. Solving for ε = E/~c, gives the exact
result
p
2 nr + δ 2 − (Zα)2
Enr ,j = mc r 2 (B.13)
p
2 2
(Zα) + nr + δ − (Zα) 2
We note that for Z > 137, Enr j becomes complex, and the Dirac theory cannot be applied.
To six order34 in Zα, the eigenvalues are
2 (Zα)2 mc2 (Zα)4 mc2 1 3
Enr j − mc = − − −
2(nr + j + 1/2)2 2(nr + j + 1/2)3 j + 1/2 4(nr + j + 1/2
The first term is the NR hydrogen spectrum, the next one is the fine structure term due to
spin-orbit coupling. It splits each level with principal quantum number n = nr + j + 1/2
into n components of fine structure, pairwise degenerate (not counting m), except that
for odd n the terms with the highest j remain non degenerate. These correspond to the
two possible values of l = j ± 1/2, for a given j.
The degeneracy is present also in the exact formula for the energy eigenvalues: for a given
nr and j, the eigenfunctions with l = j − 1/2 are those with δ < 0 (the pair {f− , g− }),
those with l = j + 1/2 have δ > 0 (the pair {f+ , g+ }). An exception is the case nr = 0, for
which the eigenfunctions came only from the set with δ < 0. Thus the ground p state has
nr = 0, j = 1/2 and therefore also l = 0. The parity is positive and E0 = mc2 1 − (Zα)2,
so E0 is complex when Zα > 1, as anticipated.
References
[1] A. Messiah, Mécanique Quantique, Vol. II, Dunod, Paris (1969).
[2] I.S. Gradshteyn and I.M Ryzhik, Table of Integrals, Series and Products, Academic
Press, New York, (1980).
[3] J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley,
Cambridge, MA, 1955): Appendix A2.
33
We use the notation nr to remind that this is a radial quantum number.
34
These are the only meaningfull values, since the higher order terms mix with the radiative corrections
of quantum electrodynamics.
96