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10-1-1 Field Equations
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For definiteness we shall deal with electrodynamics, but the technique is general. We introduce as usual a generating functional with a source J,ix) for the electromagnetic potential and anticommuting I1(X) and ij(x) sources for the electronpositron field. A path integral expression for this functional is
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Z(J, 11, ij)
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EiJ(A, ljJ, lj/)
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J(A, ljJ, lj/)
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d x [J1l(X)AIl(X)
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+ ij(x)ljJ(x) + lj/(X)I1(X)]
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Here J is the action as a function of A, ljJ, ljJ given in (6-25), and a normalization factor is understood to insure that the connected functional, where we omit the index c for shortness, satisfies G(O) = O. To cope with renormalization, the action may be regularized and counterterms included. For the sake of simplicity these will not be explicitly spelled out until the end of the derivation. Field equations will follow from the observation that the integral of a derivative vanishes. We have, for instance,
M (1 (i (i ) [ (iAIl(X) i()J' i(iij' - i(i11
+ JIl(X) ]
Z(J, 11, if)
(10-2)
Write
(10-3)
Then Eq. (10-2) reads
J (x)
+ [OgIl
(iG (iG) - (1 - A)O 0] - - - e -(iG y -(iG - e -(i ( y II
i(iJv(x)
(i11
(iij
(i11
(iij
(10-4)
It is advantageous to perform the Legendre transformatIOn to irreducible functions
G(J, 11, ij) = ir(A, ljJ, lj/)
d4 x (J A
+ Ii1I1 + ijljJ)
1 (iG (i11(X)
(10-5)
satisfying
AIl(x) =
(iG ()J1l(X)
ljJ(x) =
1 (iG (iij(x)
ljJ(x)
(iT' JIl(X) = - (iAIl(X)
(iT' I1(X) = - (ilj/(x)
(iT' ij(x) = (iljJ(x)
(10-6)
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
(10-7) we may reexpress it when the fermion sources vanish as
~Ab:( ) I
"'= iJj=O
[Og!,v - (1 - A)8!,8 v] AV(x) - ie tr [Y!' (
bl/J 01/J
~r )-(~, X)]
(10-8)
The inverse of b2rjblj/(x)01/J(y) is the electron propagator including radiative corrections in the presence of an external field. We take one derivative with respect to A and set it equal to zero. In this process we encounter the irreducible vertex function (7-46)
A,,,,,iJj= 0
bA!'(x) bl/J(y)ol/J(z)
= eA!'(x; y, z)
(10-9)
the vacuum polarization amplitude
OA!'(~;~AV(y) I
A,,,, ,iJj = 0
= [Og!,v - (1 - A)8!,8 v]b (x - y) + (Og!,v - 8!,8 v)w(x - y)
(10-10)
and the complete electron propagator is (x, y), satisfying the relation (Fig. 10-1)
(g!'vO - 8!,8 v)w(x, y)
= ie 2 d4z1 d4z2 tr [y!,S(x, zl)A v(Y; Z1, Z2)S(Z2, x)]
(10-11)
The normalizations are such that to lowest order Av(Y; Z1, Z2) = yvb4(y - Zl)04(y - Z2) d4p e- ip (x-y) S(x, y) = (2n)4 p - m - Z)p)
+ ...
(10-12)
In other words, Eq. (10-4) is a sophisticated form of Maxwell's equations of which (10-11) is only a small part. Higher functional derivatives would lead to further relations. The graphical interpretation given in Fig. 10-1 could, of c,ourse, be used for a direct derivation. Similarly, Dirac's equation will follow from the
Figure 10-1 Integral equation for the vacuum polarization.
QUANTUM FIELD TIIEORY
identity 0=
22 (A,!iI,!/J)
[~I + I1(X)] e 15!/J(X)
(10-13)
It is convenient here to work with complete Green functions without disconnected photon amplitudes, obtained as derivatives of Z(J, 11, ij)jZ(J, 0,0). For shortness we write Z(1, 0, 0) == Z(J) and obtain
[11(X)
+ (i
- m - eyP- il5J:(X)) il5;(X)] Z(J, 11, ij) = 0
(10-14)
We take a derivative with respect to 11 and set 11
ij = 0
15 (x - y)Z(J) - [i - m - eyP-
iI5J~(X)] Z(J)S(x, y; J) =
where S(x, y; J) describes the propagation in the presence of the source J. If we denote
Ap-(x, J)
(1) il5JP-(x) Z(J)
= i 15JP-(x) G(J, 0, 0)
this equation may be rewritten
15 4 (x - y) - [i - m - eJ(x; J) - eyP-
il5J~(X)] S(x, y; J) =
(10-15)
with a Kronecker 15 function for spinor indices omitted. In (10-15) the differentiation with respect to J can be performed, the source set equal to zero, and as a consequence A(x; J) IJ = 0 = O. This yields
(i - m)S(x, y) - ie 2
d4 z d4 x' d4 y' yp-GP-V(x, z)S(x, x')
x Av(z; x', y')S(y', y)
15 4 (x - y)
(10-16)
since the complete three-point function involves the irreducible vertex convoluted with propagators for the two fermions and for the photon
GP-V(x y) =
d4 k e- ik (x- y) { -
k2 [1
+ w(k 2 )]
gp-v
+ longitudinal terms}
(10-17)
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