Many-body path integrals 473x(τ)(^0) ℏ
x
x
τ
1
2
x(τ)
(^0) ℏ
x
x
τ
1
2
β β
Fig. 12.9Representative paths in the direct (left) and exchange (right) terms in the path
integral of eqn. (12.5.3).
andx(1) 2 ,...,x( 2 P)for particles 1 and 2, respectively. Note that the path index is now a
superscript. The partition functions can be written as
Q(L,T) = lim
P→∞
(
mP
2 πβ ̄h^2)P∫
dx(1) 1 ···dx( 1 P)dx(1) 2 ···dx( 2 P)×
[
e−βφ(
x(1) 1 ,...,x( 1 P),x(1) 2 ,...,x( 2 P))
±e−β
φ ̃(
x(1) 1 ,...,x( 1 P),x(1) 2 ,...,x( 2 P))]
, (12.5.4)
where + and−are used for bosons and fermions, respectively, and
φ(
x(1) 1 ,...,x( 1 P),x(1) 2 ,...,x( 2 P))
=
∑P
k=1{
1
2
mω^2 P[(
x( 1 k)−x( 1 k+1)) 2
+
(
x( 2 k)−x( 2 k+1)) 2 ]
+
1
P
U(x( 1 k),x( 2 k))}
, (12.5.5)
withx( 1 P+1)=x(1) 1 andx( 2 P+1)=x(1) 2. The definition of the functionφ ̃has the same
mathematical form as eqn. (12.5.5) but with the endpoint conditionsx( 1 P+1)=x(1) 2
andx( 2 P+1)=x(1) 1. If the term exp(−βφ) is factored out of the brackets in eqn. (12.5.4),
then the partition functions for fermions and bosons can be shownto be
Q(L,T) = lim
P→∞(
mP
2 πβ ̄h^2)P∫
dx(1) 1 ···dx( 1 P)dx(1) 2 ···dx( 2 P)×e
−βφ(
x(1) 1 ,...,x( 1 P),x(1) 2 ,...,x( 2 P))[
det(
A ̃
)]
(12.5.6)
for fermions and
Q(L,T) = lim
P→∞(
mP
2 πβ ̄h^2)P∫
dx(1) 1 ···dx( 1 P)dx(1) 2 ···dx( 2 P)