From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 33

Given any|ψ〉∈H, we can use the resolution of the identity (1.131) to write:

|ψ〉=

(∫

(dRez)(dImz)

π e

−|z|^2 |z〉〈z|

)

|ψ〉=

∫

(dRez)(dImz)

π e

−|z|^2 |z〉ψ(z∗),

(1.151)

whereψ(z∗) ≡〈z|ψ〉is the wave functions associated to the vectorψ in the
coherent state basis.
It is not difficult to verify that, for any|ψ〉,|φ〉∈H:

〈ψ|φ〉=

∫

(dRez)(dImz)

π

e−|z|

2

ψ(z∗)∗φ(z∗), (1.152)

which shows that we are working in the Hilbert spaceHBFof all anti-holomorphic
functions inzsuch that:

‖ψ‖^2 BF≡

∫

(dRez)(dImz)

π

e−|z|

2

|ψ(z∗)|^2 <∞, (1.153)

thus obtaining the so-called Bargmann-Fock representation. This Hilbert space
might be thought of as the completion of the linear space of polynomials in the
variablez∗:P={P(z∗)=a 0 +a 1 z∗+···+an(z∗)n}with respect to the scalar
product defined by the measure:

dμ(z)≡

(dRez)(dImz)

π e

−|z|^2. (1.154)

In this representation, the vectors of the Fock basis are given by the monomials in
z∗. Indeed, from (1.129), one has:

Φn(z∗)=〈z|n〉=

∑∞

m=0

(z∗)m

√

m!

〈m|n〉=

(z∗)n

√

n!

. (1.155)

It is interesting to see how the creation/annihilation operatorsa†/aare repre-
sented onHBF. From the definition of a coherent state, we know thata|z〉=z|z〉,
i.e.:〈z|a†=z∗〈z|. Therefore we can write:

a†|z〉=

∑∞

m=0

(√z∗)m

m!

a†|m〉=

∑∞

m=0

(√z∗)m

m!

√

m+1|m+1〉

=

∑∞

n=1

(z√∗)n−^1

n!

n|n〉=

∂

∂z

∑∞

n=0

(√z∗)n

n!

|n〉.

This means thata†|z〉=∂z∂|z〉and〈z|a=∂z∂∗〈z|.Inotherwords,inHBF:

(i)aacts as the derivative with respect toz∗:

a:f(z∗)=〈z|f〉→〈z|a|f〉= ∂

∂z∗

〈z|f〉= ∂

∂z∗

f(z∗) ; (1.156)