1 2 0
GAUGE FIELDS AND STRINGS
which picks up ¿-like terms in the second functional derivative. If the
quantum theory is regularized, then the first term in (7.34) does not
contain singularities as s ^ s'. From this we infer that:
= (rr(^PiV,F,Ms)) exp(^ j A,x,
(7.36)
For classical fields the r.h.s. of (7.36) would be zero. In the quantum
case it is finite and calculable. To find it we consider the following (noninvariant) functional integral:
tr
A, dx^
(7.37)
and change variables by
A
öA^. There are two effects coming
from this change, which must cancel each other. The first is the
variation of the action, proportional to
while the second is the
variation of the phase factor. The cancellation condition gives the
identities:
- ^ V ,F ;,(z m x , x))= (\) dy,^(z - yX'Fix, y)A^'F(y, x)>
0
1
\
%
(7.38)
p Tr(V,F,,(z)'F(x, x)) \ = (j) dy^z - y) x)>
c
Here the index a labels the generators A" of our Lie algebra and
y)
is a piece of the phase factor for the part of the contour C connecting x
and y(x, yeC). For the SU(N) group, (7.38) is further simplified
through the use of the identity:
1
Z Kß^ö — ^O L Ö ^ß y ~ ^O iß^
(7.39)
We have:
-^ T r (V ,F ,,(z )'F (x ,x ))
^0
= d)
- y)
X
P(x, y) Tr
x)> - - [. (7.40)