16 4
GAUGE FIELDS AND STRINGS
constant factor in front of (9.46). After that we obtain a finite amplitude
in the limit a -► 0.
In view of the future generalizations we have to develop the same
formalism as above for closed paths.
9.3 Closed Paths
In this case there are some technical differences in the integration over
the metric field. Let us begin with the integration over x(t), provided
that the Lagrange multiplier is replaced by its average, which we shall
choose to be ^ by a choice of scale. We have:
jTCMt)]
= i
9x(x) exp( - -
x (0) = x(T)
d t
T= h^'\T)dx
(9.49)
First of all we have to remove the trivial divergence in (9.49) connected
with the fact that the integrand has translation invariance -►
and so we have to fix some point of our loop. Such fixing is most
conveniently performed by inserting the relation
del Y (5(x ( t ) — c) d i I = 1
(9.50)
and omitting the j dc = K After that we have (with the choice c = 0):
I
T
= I
¿ ( x ( t ))
e x p ^ -
(9.51)
The ¿-function in (9.51) will exclude the dangerous zero mode of the
operator —d^/dt^. If we expand
■«(t) = ^ « 0 + Z
^
(9 .5 2 )