I <01 T {[jo(x), !/J(Xi)]<5(X O i=l in .NET framework

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x !/J(xd!ii(Yl) !/J(Xi)!ii(Yi) .. App( Z p)
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zJ) App(zp) 10 ) (8-75)
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+ I <01 T!/J(X1) !/J(Yn)A pl (Zl) [jo(x), ApJ(zj)] <5(XO j=l
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QUANTUM FIELD THEORY
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The term containing OpjP has dropped out, while the remaining terms just come from the Xo dependence implicit in the T product. We now appeal to the canonical commutation rules:
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[jo(x), ljI(x')] c5(xo - X'D) = - eljl(x)c5 4(x - x') [jo(x), il/(x')] c5(xo - X,D) = eil/(x)c5 4 (x - x') [jo(x), Ap(x')] c5(xO - X'D) = 0 /
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(8-76)
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which express that ljI, ill, and A create quanta of electric charge Q = Jjo(x, t) d 3 x equal to -e, e, and zero respectively. In contrast with the explicit form of the current, which follows from the minimal coupling prescription, the previous commutation rules, or at least their integrated versions, are crucial for the conservation of the charge. The Ward-Takahashi identity (8-75) reads
o~ <01 Tj p(x)ljI(xdil/(Yl)'" App(zp) 10
= e <01 TljI(xdil/(yd'" App(zp) 10) .L [c5 4(x - y;) - c5 4(x - Xi)] ,= 1
(8-77) We shall investigate further the following cases:
n=1 n=O n=O
P= 1
self-energy and vertex vacuum polarization photon-photon scattering
We remark that (8-77) holds also for the connected part of Green functions. The forthcoming discussion is formal, as we sidestep ultraviolet divergences. It will be justified in the next subsection, where we exhibit a regularization that preserves the identities.
1. Let Gpa be the complete photon propagator and G~Oj the free one (Fig. 8-14): Gpa(x)
= G; j(x) -
d4 x'
G~(~l(x -
<01 Tja,(x')Aa(x) 10)
(8-78)
It follows from Eq. (8-77) that
(8-79) Interpreted in momentum space, this is seen to imply the transversity of the
-<0-
Figure 8-14 The vacuum polarization in quantum electrodynamics.
RENORMALIZATION
vacuum polarization
kPwpa(k)
wpa(k) = -i(gpak2 - k pk a)w(k 2)
(8-80)
which generalizes the result derived in Eq. (7-6).
Indeed, in momentum space, Eq. (8-79) reads
where M2
j121A or, after multiplication by G;;pl kp =
ir up',
M2 j12(k 2 _ M2) kUrup(k)
(8-81)
If we parametrize r up in the form
rup(k)
[gpuk2 - k pk u(1 - A)]
+ gpuj12 + wpu(k)
(8-82)
= A(k2)(gpuk2 - kpk u) + B(k 2)k pk u the identity (8-81) telIs us that
In other words, B(e) is not affected by radiative correction and wpu is transverse.
2. The relation between the 'electron self-energy and the vertex functions is obtained by considering the complete (not necessarily proper but certainly connected) vertex function "Yp(p', p) defined as (Fig. 8-15)
-ie(21lf8 4 (p' - p - q)"Yp(p',p)
d4x d4x1 d4Y1
ei(p" Xt- p. Yt-q' x)
<01 T Ap(x)l/I(x r)i/f(y 1) 10) 0) <01 Tr(x)l/I(x l)i/f(y 1) 1
(8-83)
-iGlJJ(q)
d4xd 4X 1 d4Y1
ei(p"Xt-P Yt-q x)
Figure 8-15 The vertex function and its decomposition into a proper vertex dressed with complete propagators.
QUANTUM FIELD THEORY
In terms of the vertex function Ap(p', p) already encountered in Eq. (7-46) = yp] and of the complete electron propagator is(p) [S[Ol(p) = (J m)-l], we have (compare Fig. 8-15)
[A~Ol(p', p)
Yp(p', p)
= Gpp,(q) [is(p')AP' (p', p)iS(p)]
(8-84)
Contracting Eq. (8-83) with k p and using Eqs. (8-84) and (8-79), we get
e(2n)4J4(p' - p - q)qPG~~l(q)S(p')AG(p', p)S(p)
_qPG~~l(q) f
d4x d4xl d4Yl ei(p" x,- p' y,-q' x) <01 TjG(X)l/!(Xl)i/I(Yl) 1 ) 0
(8-85) Since qPG~~l(q) is proportional to qG, it follows that
e(2n)4J4(p' - p - q)S(p')qPAp(p', p)S(p)
= i f d4x d4xl d4Yl ei(P' x,-p.y,-q'X)of<01 TjP(X)l/!(Xl)i/I(Yl) 10 )
We may now use the general identity (8-77) for n = 1, p = 0:
e(2n)4J4(p' - p - q)S(P')qP Ap(p', p)S(p)
= ie f d4x d4xl d4Yl ei(p" x,- p' y,-q' x) <01 Tl/!(Xl)i/I(Yl) 10)
x [J 4(x - Yl) - J4(X - Xl)]
Therefore,
S(p')qPAP(p', p)S(p) = S(p) - S(p') qPAp(p', p) = S-l(p') - S-l(P) (8-86)
= [P' - m - 2)P')] - [pi - m - 2)P)]
(8-87)
Differentiating with respect to p'P at q = 0 yields
Ap(p, p) = opP S
(8-88)
in agreement with Eq. (7-47b). 3. Finally, the Ward identity for the photon-photon scattering amplitude enables us to factorize four powers of the external momenta and hence to improve the power counting. From Eq. (8-77) for n = 0, p = 3, we learn that the fourphoton Green function satisfies
and similar transversity conditions with respect to k2' k3, k 4 . It follows that
RENORMALIZATION
where r 1 is antisymmetric in the first pair of indices (1, Pl. This factorization of photon momenta may be pursued for k2 , k 3 , k4 , without introducing singularities. We end up with a four-point function whose effective superficial degree of divergence is minus four instead of zero.
It remains to show that these identities are preserved by the regularization and renormalization operations.
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