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not lead to a covariant quantity.
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(5-93)
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We define a covariant T product as (5-95) with
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iTJ,!)(x - y) =
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L~ dM 2 P(M2{(gpV -
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O;:;)e(x O- yO)WM(x - y)
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2 dM P(M2{gpV - O;::-)GF(X - y)
f'" dM2 p(M2)
4 d k
e-ik'(x-y)
(21lt .
gpv - k pk v/M2 k 2 - M2 + iE
(5-96)
Here i!iTpv(x, y) is a local (i.e., concentrated on x = y) covariant term which we leave undetermined for the time being. We shall see that it can be adjusted to take care of current conservation. Let us study the difference between the two time-ordered quantities. To do this we rewrite the term under square brackets in Eq. (5-96), using the fact that WM(x - y) depends only on (x - yf.
From canonical quantization it is clear that
Therefore
iT)'!)(x - y) - i<O! Tjp(x)Uy)!O)
= -gpogvo04(x -
f'" dM 2 p(M2 ) Mi M
(5-97)
The difference is a contact term concentrated on x = y but not covariant. Let us use this expression to study an interesting property of equal-time commutators, or rather of their vacuum expectation value in the present case. It follows from (5-96) that
oPiT(1)(x - y) pv
dM 2 p(M2)
4 2 d k - - e-ik.(x-y) k 1 - k /M2 (2n)4 v k 2 _ M2 + iE
-ov04(x - y)
dM 2 p<;2)
According to Eq. (5-97) this must be equal to
The current is conserved so that the divergence of the naive time-ordered symbol must equal the
QUANTUM FIELD THEORY
equal-time commutator; hence
which means (5-98)
The fact that a gradient of a delta function appears in the equal-time commutator of a time and space component of the current, as a consequence of covariance and positivity, was first noticed by Schwinger~hence the denomination of the c-number Schwinger term to recall that it even occurs in the vacuum expectation value. The vanishing of the equal-time commutator of two time components of the currents reflects the fact that integrated over space they generate the conserved electric charge. If the theory allows for a larger invariance group than U(I), the currents would carry internal indices corresponding to the various generators of the symmetry, and the discussion of equal-time commutators would involve the Lie algebra of the group (see Chap. 11). We can now return to the term i!J. Tpv(x, y) occurring in Eq. (5-95) and chosen to satisfy
iOf(OI Tjp(x)jv(Y) 10) = 0
(5-99)
The choice
z!J.Tpix,y)
gpyD (x - y)
p(M2) -2~ M
(5-100)
satisfies all the requirements, so that the complete expression of the covariant T product is
(5-101)
5-2 APPLICATIONS
Before presenting in the following chapter the covariant perturbation theory, let us illustrate the methods developed so far on some simple examples involving photons and charged fermions.
5-2-1 Compton Effect
We computed in Chap. 1 the classical elastic Thomson scattering of light off a charged center. In 1923 A. H. Compton, in his study of X-rays, discovered the scattering process with the frequency shift which carries his name. To be specific the target is taken as a free electron. We shall base our reasoning on formula (5-88) to compute the amplitude of the effect to lowest order in e, the electron charge. To this order Z3 is equal to one and the current can be identified with the free electron current (5-102) Henceforth we drop the index "in" and identify the in and out electronic states which means that the amplitude is evaluated to order e 2 The process itself is depicted on Fig. 5-2 where wiggly lines represent photons and solid ones electrons.
ELEMENTARY PROCESSES
Pi' CX;
Figure 5-2 General kinematics of the Compton effect.
The connected S-matrix element (c stands for connected) is
SJ<-i = _e 2
d4 x d4 y ei(kj x-k, y) <PI IT: li/(x)hljJ(x): : li/(y)tiljJ(y): IpY
(5-103)
The time-ordered operator is expressed by Wick's theorem in terms of normal products involving as coefficients the contractions
r--1
r--1
ljJ~(x)ljJ~,(y)
= li/~(x)li/~,(y) = 0
Let us apply Eq. (4-75); we omit contractions between operators referring to the same point because of the normal ordering of the current. Thus
T: li/(x)ypljJ(x): : li/(Y)YvljJ(y):
i :li/(X)YpSF(X - y)yvljJ(y):
+ i :li/(Y)YvSF(y -
x)ypljJ(x):
+ ...
The remaining terms do not contribute to the connected matrix element. We expand the free fields ljJ and li/ in terms of creation and annihilation operators [Eq. (3-157)]' If !Xi and !XI denote the initial and final electron polarizations
Ipi) = b t(Pi' !Xi) 10) IpI) = bt(PI' !XI) 10)
we have typically to evaluate
<Olb(pI'!XI):li/~(x)ljJq(y): bt(pi'!Xi)IO) = f(~:~~ ~:~; q :~ ei(q!'x-q,.y)
L ii~(ql' !XdU q(q2' !X2)<01 b(PI'!XI)b t(ql,!Xdb(q2,!X2)b t (qi'!Xi) 10)
= ei(Pj' x- p,' y) ii~(PI' !XI)Uq(Pi, !Xi)
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