After substitution of (1.5) into (1.4) we obtain (1.2). Notice also that
without the potential v the formula (1.5) is exact for any value
— tj
and describes the propagation of a free particle.
In order to establish the analogy with classical statistical mechanics
one has to consider the propagation for imaginary time T. Namely, let
us look at
STATISTICAL MECHANICS AND QUANTUM FIELD THEORY
3
Z(x, x', T) = = F(x, x', — iT)
( 1.6)
We can repeat the splitting procedure again with the only difference
that the tj in (1.4) will acquire an extra factor — i. In this way we obtain:
Z(x, x', T) =
^x(i) exp) —
x(0 ) = x
x{T) = x'
1
I
+ o(x(i))^ d t|
(1.7)
which is to be understood in the same way as (1.1). The mnemonic rule
for passing from (1.1) to (1.7) is very simple: consider the expression:
-<;(x(i))jdi
( 1.8)
and introduce t = —it. We obtain:
(1-8) =
m /dx
2 \ di
+ v{ x ( t )) i dr
(1.9)
The derivation (1.9) shows also that we have even more freedom in
computing the functional integral. Namely, we can chose the splitting
points {tj} to lie on an arbitrary contour in the complex plane, and
therefore time not only can be imaginary but also can go along some
complex path (provided that the convergence condition for (1.5), Im
At < 0, is satisfied). For some problems this freedom is very useful. At
the moment, however, we are interested in a different aspect of all this.
Namely, that formula (1.9) has an important physical interpretation.
Let us consider an elastic string of length T and tension m with the ends
fixed at x and x'. Suppose that this string stays in an external potential
v(x). The potential energy of such a string will be given by:
► t[.^(0]
m (dx^ ^
2
I + <^(.<(^))| dT
( 1. 10)