As a result, in the limit e 0 the formula (9.17) gives:
156
GAUGE FIELDS AND STRINGS
jT(x, x', T) = const.
exp -
(X' - x)2 + T
(9.19)
In the process of this derivation it became clear that we could do much
better than performing the dumb discretization (9.16). If we look at
(9.14), we discover that the fluctuations of a(i) are short ranged. Indeed,
if we write
a(0 = + m
(9.20)
then the bilinear term in jS in the effective action, 14^,, obtained after
Gaussian integration has the representation:
. C ^ k k (q - k)k (q - k)
2n
2 n
k \ q - k ) \ ( x y
(9.21)
(Here we have the vertex of the jSxx-interaction from the term
j
dt in the Lagrangian.) This computation shows that the field
P does not acquire any kinetic energy, and therefore has a correlation in
t of the order of e. That in turn means that in all expressions containing
a “large number” of a, as in j (x(t)x^(t) dt (provided x(t) is smooth) one
can boldly replace a(i) by its mean value . This is an important
lesson.
So far we have solved only one part of the problem with the result:
jT(x, x', h(x))
-
i
x(0) = x
x(l) = x'
^ x ( t )^ (x ( t ) — h{x))
= const. T
exp( -
^
+ r), ~ ^
(9.22)
1
T = j dx(h(x)yi^
0
Now we have to complete the integration in (9.7). Let us notice, first of
all, that (9.22) depends on h(r) only through T. This of course is not an
accident since T is the only invariant quantity in one-dimensional
Riemannian geometry (things like scalar curvatures etc. are zero in this
case). It is natural to expect that the integration over ^ /i(t)/^ /(t) in