From Classical Mechanics to Quantum Field Theory

36 From Classical Mechanics to Quantum Field Theory. A Tutorial

(vi) The operator of the one-parameter group generated byN:A=exp[−τa†a]

is not written in the normal form and to calculate its kernel, we may

proceed as follows. Noticing that

Amk=〈m|exp[−τa†a]|k〉=δmkexp[−τm], (1.168)

one gets

A(z∗,z′)=

∑∞

m,k=0

δmkexp[−τm]

(√z′)k

k!

(√z∗)m

m!

=

∑∞

m=0

(e−τz∗z′)m

m! =exp[e

−τz∗z′]. (1.169)

(vii) These results can be used in the context of Quantum Statistical Mechanics

[ 30 ]to calculate the, say canonical, partition function of a gas ofNdis-

tinguishable harmonic oscillators, which is given byZN=(Z 1 )N,where

Z 1 ≡TrH[e−βH], withH=ω(a†a+1/2) andβ=1/kBTthe inverse of

the absolute temperature, up to Boltzmann constantkB. Thus one has:

Z 1 =e−βω/^2 TrH

[

e−βωa

†a]

. (1.170)
Settingτ=βωin the last example, using (1.160) and changing to polar
coordinates, we immediately see that:

Z=e−βω/^2

∫

(dRez)(dImz)

π

e−|z|

2

exp[e−βω|z|^2 ]

=e−βω/^2

∫ 2 π

0

dθ

π

∫ρ

0

ρdρe−ρ

(^2) (1−e−βω)

=

1

2sinh(βω/2)

. (1.171)

1.3.1.4 Generalized coherent states and comments

We will introduce here the notion of generalized coherent states[ 33 ], applied to
the Heisenberg-Weyl groupW 1.
Let us take any UIR ofW 1 ,T(g), and denote with|ψ 0 〉any (non-zero) vector in
the (necessarily infinite-dimensional) representation spaceH. The stability group
of|ψ 0 〉isgivenonlybythecenterofW 1 , i.e. by the elements of the formT((s,0)).
We define the set of generalized coherent states as:

|α〉≡T(g)|ψ 0 〉=D(α)|ψ 0 〉. (1.172)

The set of coherent states that we have studied in the previous section corresponds
to the choice: |ψ 0 〉=| 0 〉. Similarly to what was done before, one can show that