From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 213

Using the basis of plane waves Eq. (3.20) we have

Hˆren=^1

4

∑

n∈Z^3

(

a†nan+ana†n

)

−E 0 =

1

2

∑

n∈Z^3

a†nan

wherean=a(fn),

Pˆren=^1

4

∑

n∈Z^3

n

(

a†nan+ana†n

)

,

and we can define the number operator as

Nˆ=^1

2

∑

n∈Z^3

1

ωn

a†nan.

The main property of the number operator are its commutation relations with the
creation and annihilation operators,

[N,aˆ n]=−an, [N,aˆ †n]=a†n.
The creation operatorsa(f) can generate by iterative actions on the vacuum a
basis of physical states. In particular, the state^8

|f〉=a(f)†| 0 〉

can be considered as a one-particle state with wave packetf. Indeed, it easy to
check that

Nˆ|f〉=|f〉.

The relativistic invariant normalization of the creation operators simplifies the
identification of the norm of one-particle states,

〈f|f〉r=‖f‖^2 r=

1

2

∫

d^3 x

∫

d^3 yf∗(x)(−Δ+m^2 )−^1 (x,y)f(y).

The completion of the space of one-particle states with this norm is then

H=L^2 (R^3 ,C)={f,‖f‖^2 r<∞},

which again can be identified with the dual space of the configuration space of
classical fields.
The next step is the identification of two particle states. They are of the form

|f 1 ,f 2 〉=√^1

2

a(f 1 )†a(f 2 )†| 0 〉=√^1

2

a(f 2 )†|f 1 〉,

(^8) We use the Dirac notation, where| 0 〉is the vacuum state, and the ket|f〉denotes the state
|f〉=a(f)†| 0 〉.