560 Quantum time-dependent statistical mechanics
GAB(t) =1
Q(N,V,T)
∫
dx〈x|Aˆ(ˆx)ei
Hˆτc∗/ ̄hˆ
B(ˆx)e−i
Hˆτc/ ̄h
|x〉 (14.6.7)=
1
Q(N,V,T)
∫
dxdx′〈x|ei
Hˆτc∗/ ̄h
|x′〉b(x′)〈x′|e−i
Hˆτc/ ̄h
|x〉a(x).If the two matrix elements〈x|ei
Hˆτ∗c/h ̄
|x′〉and〈x′|e−i
Hˆτc/ ̄h
|x〉are represented as path
integrals, the interpretation of eqn. (14.6.8) is clear. We start atx, calculate the eigen-
valuea(x) ofAˆ, propagate along a complex time path tox′, calculate the eigenvalue
b(x′) ofBˆ, and then propagate back toxalong a complex conjugate time path. This
process is represented schematically in Fig. 14.6(b). Since the two matrix elements in
eqn. (14.6.8) are complex conjugates, and sincea(x) andb(x′) are both real,GAB(t)
is, itself, a real object. More importantly, in contrast toCAB(t), the complex time
paths needed to represent the two matrix elements in eqn. (14.6.8)have oscillatory
phases, but they also have positive-definite weights, which tends to make them some-
what better behaved numerically. If each matrix element in eqn. (14.6.8) is discretized
into paths ofPpoints, thenGAB(t) can be written as the limit of a discretized path
integral of the form
GAB,P(t) =1
Q(N,V,T)
∫
dx 1 ···dx 2 Pa(x 1 )b(xP+1)ρ(x 1 ,...,x 2 P)eiΦ(x^1 ,...,x^2 P) (14.6.8)(Krilovet al., 2001), whereρ(x 1 ,...,x 2 P) is a positive-definite distribution given by
ρ(x 1 ,...,x 2 P) =(
mP
2 π|τc| ̄h)P
exp[
−
mPβ
4 |τc|^2 ̄h^2∑^2 P
k=1(xk+1−xk)^2 −β
2 P∑^2 P
k=1U(xk)]
. (14.6.9)
Here,x 2 P+1 =x 1 due to the trace condition, and Φ(x 1 ,...,x 2 P) is a phase factor
defined by
Φ(x 1 ,...,x 2 P) =mPt
2 ̄h|τc|^2[P
∑
k=1(xk+1−xk)^2 −∑^2 P
k=P+1(xk+1−xk)^2]
−
t
̄hP[P
∑
k=2U(xk)−∑^2 P
k=P+2U(xk)]
. (14.6.10)
In the limitP→∞,GAB,P(t) =GAB(t). Note that the same path variables defineρ
and Φ, demonstrating explicitly that the paths have a positive-definite weight as well
as a phase factor. Moreover, becauseGAB,P(t) is real, the imaginary part of exp(iΦ)
must vanish. The fact that the paths inGAB(t) have positive-definite weights allows