20
GAUGE FIELDS AND STRINGS
classical solutions must be included. Both of these questions can be
answered if we consider a finite size system and fix the value of g(x) at
the boundary F :
g(x)\,=giO
ie V
(2.4)
We are considering the functional
n g io ^ =
S>g{x)Q -d/ft)S[ÿ(x)]
(2.5)
g(x)\r = g(4)
(where S is the action). This functional is just the Euclidean analogue of
the Shrodinger wave function. The classical limit ft -► 0 would correspond to taking the minimum of the action S with the Dirichlet
boundary conditions (2.4). Integration over ft(x) in the decomposition
(2.2) corresponds to the inclusion of quantum fluctuations. It is clear
from the above discussion that we should fix the boundary condition
for ft:
ft(x)|r = /
( 2.6)
To sum up, we shall compute the integral over ft and obtain an eflective
action depending on
It must be understood, however, that ^ci(^)
is not an independent variable: in fact all classical solutions are
parametrized by their boundary values g(0
is ^ — 1-dimensional).
So, we are effectively computing the T-functional (2.5), which as we
already said is an analogue of the Shrodinger wave function and, on the
other hand is analogous to the on-shell amplitudes of the Minkowskian
theory (they also depend on ^ — 1-dimensional fields).
All this information about boundary conditions and T^-functionals
can be kept subconsciously so far as we are interested in infinite
systems. Actually, most often we need not bother to express ^ci(^)
through g(0- Substituting (2.2) into (2.1) we get:
=
= LI' + g;, '(h ' d^h)g^,
2el
+ \ Tr(R'>(;i - ' d^h)),
R f = (d^gjg:, '
^0
(2.7)