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QUANTUM FIELD THEORY
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eG~(J) = f ~(A) IJ b[%(A)] det vii
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{;2 f d4x [ff' - 2g tr
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(12-79)
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(12-80)
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(12-81)
where For instance, if we use the Lorentz gauge
the operator vii reads
vIIL,ab(X, y)
OI'Dl'ab b4(X - y)
= [Db ab + Cabco~Al'c(x)Jb4(x - y)
[Db ab
+ Cabc Ajlc(x)Ol'xJb4(x -
(12-82)
Returning to the axial gauges
%=n-A=O
we see that vii is independent of A, when restricted to the hence det vii may be absorbed into the normalization.
~onstraint
manifold;
Instead of relying on the canonical hamiltonian quantization and on the subsequent functional manipulations, we might decide to modify the iII-defined integral
9&(A)p(A)
through the insertion of the identity written as
f9&(g)~(A) I)
o[%(gA)]
The gauge invariance of p(A), of
~(A),
and of the measure 9&(A) would then enable us to factorize
NONABELIAN GAUGE FIELDS
an infinite group volume
!!&(g) ]
f!!&(A)fl(A)~(A) I]
b[ $'(A)]
This crude argument has the merit of exhibiting clearly the enormous degeneracy of the degrees of freedom as the origin of the problem.
The previous developments are readily extended to auxiliary conditions of the form
~(A) =
where ~ and the given function C(x) take their values in the Lie algebra. This modification does not affect the form of the operator At in (12-80).
The variations (bfbg) [$'(g A) - C] do not depend on C, and the only dependence of ~-'" on C comes from go, the gauge transformation such that $'(gO A) = C:
We may write
f9&(g)~(YoA)b[$'(gA)
- C] =
f!!&(g)~(gA)b[$'(gA)
- C]
where the second equality holds because the b function implies g = go. Any reference to C has disappeared.
Since gauge-invariant quantities should not be sensitive to changes of auxiliary conditions, it is possible to average over C with a gaussian weight, i.e., to substitute for b[~(A) - C] the quantity
f 22(C) exp G~ f d x tr (C2)Jb[~(A) 4
eG(J) '=
C] = exp
{~~ f d x tr [~2(A)]}
In its final form the generating function reads
22(A) det At.fl' exp
(;2 f
d4 x {2(x)
+ A tr [~2(A)]
- 2g tr (J . An)
(12-83) The perturbative expansion of det At leads to nonlocal interactions between gauge fields. It is more convenient to perform a final manipulation on this determinant, and to reexpress it as a local interaction of fictitious fields. The discussion of the integration on a Grassmann algebra has yielded the formula (9-76) :
where At denotes an n x n matrix. Changing At into iAt and extending this
582 QUANTUM FIELD THEORY
result to an infinite algebra, we may write the determinant det A!F as a functional integral
eG(J)
(A, 11, ij) exp
(;2 I
d4 x {!i'
+ A tr [ff2(A)]
- ijAI1 - 2g tr (J. An)
(12-84) The modified or effective lagrangian
!i'eff(A, 11, if) = !i'(A)
+ A tr ff2(A) -
ijAI1
(12-85)
involves the gauge field A and the new anticommuting auxiliary scalar fields 11 and ij, the so-called Faddeev-Popov ghosts. We emphasize that these fields are unphysical and only play an algebraic role. To insure global in variance, 11 and ij transform according to the adjoint representation. The ghost term in the lagrangian (12-85) reads
d x ijA!F11 =
II I
d x d Y ija(x)
bffa[gA (x)] blXb(Y) I1b(Y)
d x l1a(X) OAl'b D bcl1c(X)
offiA)
(12-86)
(DI'I1)a
Ol'l1a
+ Cabcl1bAl'c
In general, the kernel A !F is not hermitian; therefore the ghost lines in a Feynman diagram will have to be oriented. For instance, if ff is the Lorentz covariant condition ff = ol'AI', we find
!i'eff =
d x ijAI1 =
d x ijaol'Dl'abl1b
(12-87)
In this case, the lagrangian of Eq. (12-85) reads
tr (Fl'vPV)
+ A tr (0' A)2 -
ij0I'DI'I1
(12-88)
The parameter A (also denoted IX-lor C 1 in the literature) reminds us of the arbitrariness of the auxiliary condition. The choices A = 1 or A- 1 = 0 are referred to as Feynman or Landau gauges respectively.
12-2-3 Feynman Rules
Equation (12-88) gives the desired solution to the quantization problem, since it provides a local Lorentz covariant expression for the effective lagrangian. The quadratic part in A is invertible as a consequence of the fact that the condition ff = C has picked a single representative in each equivalence class. We are now in a position to write the Feynman rules. We rescale the fields A and 11, ij by g, the coupling constant. As the Feynman diagrams also include ghost fields, it is helpful to introduce (anticommuting) sources ~,~ coupled to
NONABELIAN GAUGE FIELDS 583
11, if We have thus eG(J.U)
9&(A, 11, if) exp {i
d4 x [.:i'eff(A, 11, if; g, A)
+ jI"aA/la + ~al1a + ifa~a]}
A .:i'eff(A, 11, if; g, A) = -!F/lvaP va - 2(O/lA/la
(12-89)
+ o/lifiD/lI1)a
(12-90)
(D/lI1)a = o/ll1a - gCabc A/lbl1c
The propagators of the gauge and ghost fields are respectively
/1"! a
ev {-i[O-(I-A)oC8>O]}-1= -ibab[kzg/lv. -(I-A- 1) zk/lk: ] b + Ie (k + Ie
(12-91)
and behave as k- z at large momentum. Therefore, both fields are given the dimension one for the ultraviolet power counting. The coupling constant g is dimensionless. There are three kinds of vertices. If we orient the ghost lines from the if to the 11 (as we do for genuine fermions) and include the factor i from the expansion of exp [i J d4 z .:i'int(Z)], we get
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