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We have added a source term for the four components of AI', although A 3 vanishes and AO may be integrated explicitly. The introduction of a source term enables us to question the system. Some answers such as the Green functions may depend on the choice of gauge and are just an intermediary step to get physical information. It would be tempting, of course, to consider directly gauge-invariant quantities such as the S matrix. Unfortunately, the asymptotic states are unknown and any computation of the would-be S-matrix elements is plagued with severe-though interesting-infrared divergences. Moreover, the theory requires renormalization, and an algorithm for
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eliminating ultraviolet divergences which does not rely on the Green functions remains to be found. Since E appears only quadratically in the exponent of (12-62), the gaussian integration may be carried out:
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eGP)
(AI')
I) b(A 3) exp {;2
d 4 x [2' - 2g tr (A' J)] }
(12-63)
i tr (PVFl'v)
The subscript A of GA(J) stresses the fact that Green functions depend on the choice of gauge. In spite of the simplicity of this axial gauge, the Feynman rules that we may derive are not Lorentz covariant. It is therefore natural to consider more general conditions. We shall proceed in several steps. We first study another noncovariant gauge, the Coulomb gauge, defined by the auxiliary condition div A = 0 (12-64)
As stated above, it is always possible to find a local gauge transformation so as to satisfy (12-64). This condition was generally considered as determining uniquely the gauge transformation g. Phrased differently, if div A = 0, it was traditionally stated that the solution in g of div (g A) = reduces to the identity under suitable conditions at spatial infinity. This is what happens in the abelian case. If div A = 0, div A' = with A' = A + VA, the harmonic function A vanishes at infinity and hence everywhere. However, as pointed out recently by Gribov, this is no longer true in the nonabelian case. The equation for g involves A and admits solutions for A large enough. Given A such that div A = let us look for a time-independent infinitesimal gauge transformation A" = Ai + [oa, Ai] + [ioa such that div A' = 0:
(12-65) For A small enough (equivalently, for small enough coupling constant) we may expand
oa = oalD ] + goal!]
+ ...
etc.
alk ].
If a is assumed to vanish at infinity, so do all the There is no nontrivial solution. However, Eq. (12-65) may be regarded as a Schrodinger equation. For a potential A large enough, it can be checked that bound states do exist, i.e., solutions of
+ il;[a, A;J
with E < 0. Therefore, for intermediate magnitudes of A, there must exist zero-energy solutions of fast decrease at infinity. This objection seems embarrassing for our program, since the comparison of the quantization in different gauges relies on the assumption of uniqueness of the transformation which relates them. However, since this phenomenon appears for large values of the field, it does not affect the construction of the perturbative series which is by essence a small-field (small-fluctuation) expansion about a given classical configuration. We shall henceforth discard this Gribov phenomenon. Whenever we talk of uniqueness of the choice of gauge (12-64), we understand it in a perturbative sense.
The auxiliary conditions (12-64) have a nontrivial Poisson bracket with the constraint
NONABELIAN GAUGE FIELDS
{div Aa, rb(y)}XO=YO
OIX~(Y) div A'a(x)
OabL\x
= [ -
+ Cabc Vx Ac(X)]03(X y)
(12-66)
We introduce the operator vii:
vIIab(x, y)
= [ = [ -
L\xOab
+ Cabc V x Ac(x)J 03(X L\xOab + CabcAc(X) . V xJ 03(X -
(12-67)
where the last expression holds when vii is restricted to the constraint manifold. In the Coulomb gauge, we thus write the generating functional of Green functions as
eGc(J)
(A)
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