From Classical Mechanics to Quantum Field Theory

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

⎫

⎬

⎭ (1.225)