From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 205

and it is evident that the system Eq. (3.22) describes an infinite series of harmonic
oscillators with frequencies

ωn=

√∣

∣∣

∣

2 πn

L

∣∣

∣∣

2

+m^2. (3.23)

Canonical quantization maps classical fields into operators in a Hilbert space
Hsatisfying the commutation relations obtained by replacing Poisson brackets by
operator commutators

{·,·}=⇒−i[·,·], (3.24)

which can be realized in the Schr ̈odinger representation on space of functionals of
Mby

ˆπ(x)=−i

δ

δφ(x)

; φˆ(x)=φ(x). (3.25)

The corresponding quantum Hamiltonian is

Hˆ=^1

2

(

‖ˆπ‖^2 +‖∇φ‖^2 +m^2 ‖φ‖^2

)

, (3.26)

In terms of smeared functions, the Schr ̈odinger representation of the momen-
tum operator

ˆπ(f)=−i

∫

d^3 xf(x) δ

δφ(x)

,

becomes just a Gateaux derivative operator

πˆ(f)F(φ)=−islim→ 0

1

s

(F(φ+sf)−F(φ)). (3.27)
For any orthonormal basis of test functionsfninL^2 (R^3 ) we have the expansion
of classical fieldsφ∈L^2 (R^3 )

φ(x)=

∑∞

n=0

φnfn(x),

whereφn=φ(fn). Moreover, by linearityφ(f)=

∑∞

n=0φn(fn,f)

πˆ(x)F(φ)=−i

δ

δφ(x)

F

(∞

∑

n=0

φnfn(x)

)

(3.28)

and from Eq. (3.27) it follows that

πˆ(fn)=−i δ

δφn

.

In the plane wave basis, the quantum Hamiltonian

Hˆ=^1

2

∑

n∈Z^3

(

δ

δφ−n

δ

δφn

+ω^2 n|φn|^2

)

, (3.29)

again corresponds to an infinite series of harmonic oscillators with frequenciesωn.