46 From Classical Mechanics to Quantum Field Theory. A Tutorial
Thus, in a formal way we can write:
〈φf|e−ı
tH
|φi〉=
∫
φ(0)=φi,φ(t)=φf
[Dφ∗Dφ]eφ
∗(t)φ(t)
e
ı∫ 0 tdt[iφ∗∂t∂φ′−H(φ∗,φ)]
,
(1.219)
where the integral in the exponential represents the so-called Schr ̈odinger La-
grangian of the classical system, which contains a kinetic term which is linear in
the first derivatives with respect to time.
We can now proceed to evaluate the (grancanonical) partition function of the
system, by recalling that[ 31 ]:
Z=Tr
[
e−β(Hˆ−μNˆ)
]
=
∫
dφ ̃∗dφ ̃
N
e−φ ̃
∗φ ̃
〈ζφ ̃|e−β(Hˆ−μNˆ)|φ ̃〉, (1.220)
with
ζ=
{
+1 for bosons
−1forfermions.
From (1.219), puttingτ′=ıt′,β=it,=1,wehave:
〈ζφ ̃|e−β(Hˆ−μNˆ)|φ ̃〉= lim→ 0
∫ M∏− 1
n=1
dφ∗ndφn
N
e−i
∑M− 1
n=1φ∗nφne
∑M
n=1φ∗nφn
×
∏M
n=1
exp{− [H(φ∗n,φn− 1 )−μφ∗nφn− 1 ]}, (1.221)
with the boundary conditions:
φ 0 =φ, φ ̃ ∗M=ζφ ̃∗=ζφ∗ 0. (1.222)
Thus we get (as beforeζ=±1 for bosons/fermions):
Z= lim→ 0
∫(∏M
n=1
dφ∗ndφn
N
)
exp
{
−
∑M
n=1
φ∗n
φn−φn− 1
}
×exp
{
−
∑M
n=1
[H(φ∗n,φn− 1 )−μφ∗nφn− 1 ]
}
. (1.223)
Example 1.3.4. The 1D bosonic/fermionic harmonic oscillator.
We consider the Hamiltonian
H=Ωa†a, (1.224)
so thatH(φ∗n,φn− 1 )=Ωφ∗nφn− 1. The argument in the exponential of (1.223) is
then given by:
exp
⎧
⎨
⎩−
∑M
i,j=1
φ∗iMijφj