From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 43

Recalling that (i) the classical action is given by:

S=

∫t

0

dt′

m

2

(

dx

dt

) 2

(1.208)

and (ii) the solution of the classical equation of motiond^2 x(t)/dt^2 = 0, satisfying
the boundary conditionsx(0) =x′,x(t)=x, yields the classical trajectory:

xcl(t′)=

t′(x−x′)

t

+x′, x ̇cl(t′)=

x−x′

t

, (1.209)

one can easily verify that:

Sclas(x, x′)≡S|xcl=

∫t

0

dt′

m

2

(

x−x′

t′

) 2

. (1.210)

Comparing with (1.207), we see that the quantum kernel is given by the exponen-
tial of the classical action:

〈x|e−ıtH|x′〉=F(t)exp

[ı



Sclas(x, x′)

]

,F(t)=

√

m

2 πit

. (1.211)

up to a pre-factorF(t) which depends only on time. It is an interesting fact that
it can be calculated exactly by also makinguse of a semi-classical stationary phase
approximation[ 35 ].

Example 1.3.3. The 1D harmonic oscillator.
Similar calculations can be done for the Hamiltonian of the 1D harmonic oscillator,
to get:

K(x, x′;t)=Fho(t)exp

[ı



Sclas(x, x′)

]

, (1.212)

with

Scl(x, x′)=

ωm

2 sin(ωt)

[

(x+x′)^2 cos(ωt)− 2 xx′

]

, (1.213)

Fho=

√

mω

2 πısin(ωt). (1.214)

As in the previous example,Fhocan be calculated exactly by means of the sta-
tionary phase approximation[ 35 ].

1.3.2.2 Feynman integral in imaginary time and partition function

For a quantum system, the (canonical) partition function can be calculated as:

Z≡Tr

[

e−βH

]

=

∫

dx〈x|e−βH|x〉. (1.215)