From Classical Mechanics to Quantum Field Theory

204 From Classical Mechanics to Quantum Field Theory. A Tutorial

In this representation, it is clear that the system describes an infinite series
of coupled harmonic oscillators. The way of disentangling the coupling is to find
the normal modes, i.e. to choose a basis of test functionsfnwhere the interaction
operator Δ is diagonal. The normal modes are plane waves which do not belong to
L^2 (R^3 ). For such a reason it is convenient to introduce an infrared regulator, i.e.
to consider the system in a finite volume. There are many physical reasons why
this method is necessary. In the quantum case there are two types of divergences:

(i) ultraviolet (UV) divergences, which are due to short range singularities

associated to the local products of distributions; and

(ii) infrared (IR) divergences which are due to the infinite volume of space.

Both need to be regularized and renormalized as we will see later on.

From this perspective, the introduction of a finite volume can be considered as
a regulator of IR divergences. Poincar ́e invariance will be recovered in the limit of
infinite volume at the very end.
We shall consider mostly the torusT^3 compactification ofR^3 ,thatis,aboxwith
periodic boundary conditions. The normal modes in this case are normalizable
plane waves,

fn+(x)=

1

2

(fn(x)+fn(x)∗); fn−(x)=−

i

2

(fn(x)−fn(x)∗) n∈Z^3 +,

with

fn(x)=

1

√

L^3

ei^2 πn·x/L, (3.20)

wheren∈Z^3 ,n·x=n 1 x 1 +n 2 x 2 +n 3 x 3 andLis the length of each side of the
torus, i.e.x∈[0,L]^3. The normal modes diagonalize the Hamiltonian because

Δfn±(x)=−

(

2 π

L

) 2

(n·n)fn±(x). (3.21)

However, it is more convenient to use the complex modesfn, provided that in the
mode expansion the fields

φ(x)=

∑

n∈Z^3

φnfn(x),

the coefficientsφn =φ(f) satisfy the reality conditionsφ∗n=φ−nin order to
guarantee the reality of the fieldsφ∗=φ.
In terms of the complex modes the Hamiltonian is diagonal

H=^1

2

∑

n∈Z^3

(

|πn|^2 +

(∣

∣∣

∣

2 πn

L

∣∣

∣∣

2

+m^2

)

|φn|^2

)

, (3.22)