From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 37

{|α〉}form an overcomplete set of states that generateH. More explicitly, we have
the following structures:

• law of transformation

D(α)|β〉=eıIm (αβ

∗)

|α+β〉; (1.173)

• non-orthogonality condition

〈α|β〉=eα

∗β

; (1.174)

• resolution of identity

I=

∫

(dReα)(dImα)

π

e−|α|

2

|α〉〈α|≡

∫

dμα|α〉〈α|. (1.175)

From the latter, it is immediate to see that any|ψ〉∈Hcan be written as:

|ψ〉=

∫

dμα|α〉〈α|ψ〉=

∫

dμαψ(α), (1.176)

where∫ ψ(α)=〈α|ψ〉is called the symbol of the state|ψ〉. Clearly: 〈ψ|ψ〉=
dμα|ψ(α)|^2.
Such a construction can be easily extended to a finite or an infinite number of
creation/annihilation operators ofbosonic type, i.e. to a set of operators{(ai,a†i)}i
satisfying the canonical commutation relations:

[ai,aj]=[a†i,a†j]=0, (1.177)

[ai,a†j]=δijI, (1.178)

acting on the bosonic Fock Hilbert spaceHFwhich is generatedby the orthonormal
basis|n 1 ···nk···〉=√Π^1 jnj!(a† 1 )n^1 ···(a†k)nk···| 0 〉, withnk∈N. For instance,

this is the framework in which to discuss the quantization of the electromagnet
field: in vacuum^20 , each component of the latter satisfies d’Alembert equation
and hence can be described by using itsFourier modes, each of which behaves
independently as a 1D harmonic oscillator. Thus, each of these modes is described
quantum mechanically by a couple of creation/annihilation operators, which in all
satisfy Eqs. (1.177, 1.178).
Thanks to (1.177), a coherent state can be defined as the common eigenvector
|φ〉≡|φ 1 φ 2 ···〉of all annihilation operators:

aj|φ〉=φj|φ〉 withφj∈C. (1.179)

(^20) For the definition of Fock space in a more general QFT see sect. 3.5.1 of the third part of this
volume.