Many-body path integrals 471g 0 =[
m
2 πβ ̄h^2] 1 / 2
, gn=[
mβω^2 n
2 π] 1 / 2
. (12.4.35)
Whenω= 0, the product is exactly 1 for this choice ofg 0 andgn, which leaves the
overall free particle prefactor in eqn. (12.2.30).
Forω 6 = 0, the infinite product is
∏∞n=1[
ωn^2
ω^2 +ω^2 n] 1 / 2
=
∏∞
n=1[
π^2 n^2 /β^2 ̄h^2
ω^2 +π^2 n^2 /β^2 ̄h^2] 1 / 2
=
[∞
∏
n=1(
1 +
β^2 ̄h^2 ω^2
π^2 n^2)]−^1 /^2
. (12.4.36)
The product in the square brackets is one of many infinite product formulas for simple
functions.^2 In this case, the product formula of interest is
sinh(x)
x=
∏∞
n=1[
1 +
x^2
π^2 n^2]
. (12.4.37)
Using this formula, eqn. (12.4.36) becomes
I 0 =g 0[
β ̄hω
sinh(β ̄hω)] 1 / 2
=
[
mω
2 π ̄hsinh(β ̄hω)] 1 / 2
. (12.4.38)
Thus, the density matrix for a harmonic oscillator is finally given by
ρ(x,x′;β) =[
mω
2 π ̄hsinh(β ̄hω)] 1 / 2
×exp[
−
mω
2 ̄hsinh(β ̄hω)(
(x^2 +x′ 2
)cosh(β ̄hω)− 2 xx′)]
. (12.4.39)
Note that in the free-particle limit, we take the limitω→0, set sinh(β ̄hω)≈β ̄hωand
cosh(β ̄hω)≈1, so that eqn. (12.4.39) reduces to eqn. (12.2.30).
12.5 Many-body path integrals
Construction of a path integral for a system ofNindistinguishable particles is nontriv-
ial because we must take into account the symmetry of the physical states. Consider,
for example, the case of two identical particles described by a HamiltonianHˆ. If we
wish to compute the partition functionQ= Tr[exp(−βHˆ)] by performing the trace in
the coordinate basis, how we write down the proper coordinate eigenvectors depends
on whether the overall state is symmetric or antisymmetric. If thecoordinate labels
(^2) See, for example, Weber and Arfken’sMethods of Mathematical Physics(2005).