QUANTUM STRINGS AND RANDOM SURFACES
167
In order to compute these integrals it is convenient to go to the
momentum representation:
00
T
T
F{q
r dT
1» • • •» Q n) —
^ x(t) expl
dr
(9.59)
The integrals in (9.59) are Gaussian. According to the general rules, to
find them we have first to solve the classical equations:
■id = - ' I 9j ¿(i - ty)
J
and then to compute the classical action. We have:
(9.60)
(9.61)
where ^ ( t|t') is a Green function for the operator —d^/dt^. The
substitution of (9.61) into the classical action gives:
1 r
Sci =
xi, dt - i X qjXjTj)
(i,J>
(9.62)
The Green function on a circle contains a zero mode contribution:
(9.63)
(here X„ = 4n^n^lT^, and C is the zero mode contribution which is
arbitrary). However if momentum is conserved (Lj qj = 0) this arbitrariness disappears from (9.62):
cZ?,?y = q ^ i ; ) = 0
(9.64)
The necessity of momentum conservation can be seen also from (9.60)
by integrating it over t. Then periodicity of
requires
= 0.
We obtained the following expression:
00
T
dT
F(.qi,-.,qN) =
Y~^/2 Q-m^T/l j n * ,
(9.65)