From Classical Mechanics to Quantum Field Theory

216 From Classical Mechanics to Quantum Field Theory. A Tutorial

M. However, as we have seen in the case of free fields, the theory needs a renor-
malization.
Using the plane wave basis on a finite torus, we have

Hˆ=^1

2

∑

n∈Z^3

⎛

⎝ δ

δφ−n

δ

δφn

+ωn^2 |φn|^2 + λ

12

∑

n 1 ,n 2 ∈Z^3

φnφn 1 φn 2 φ−n−n 1 −n 2

⎞

⎠.

Again the regularization of UV divergences requires the introduction of a regular-
ization, e.g.

HˆΛ=^1

2

ω∑n<Λ

n∈Z^3

⎛

⎝ δ

δφ−n

δ

δφn

+ω^2 n|φn|^2 +

λ

12

ωn 1 ,ω∑n 2 <Λ

n 1 ,n 2 ∈Z^3

φnφn 1 φn 2 φ−n−n 1 −n 2

⎞

⎠.(3.46)

The Hamiltonian Eq. (3.46) can be split into two terms

HˆΛ=H 0 +HˆΛint.

The first term

Hˆ 0 =^1

2

ω∑n<Λ

n∈Z^3

(

δ

δφ−n

δ

δφn

+ω^2 n|φn|^2

)

is just the Hamiltonian of the free bosonic theory, whereas the second term

HˆΛint=λ

4!

ωn 1 ,ωn∑ 2 ,ωn 3 <Λ

n 1 ,n 2 ,n 3 ∈Z^3

φn 1 φn 2 φn 3 φ−n 1 −n 2 −n 3.

contains the interaction terms. The renormalization ofH 0 can be performed as in
previous section by subtracting the vacuum energy of the free theory,

HˆΛren=H 0 ren+HˆintΛ.
But there are new divergences generated by the interacting terms which require
an extra renormalization.
The easiest way of dealing with the interacting theory is to consider the in-
teracting termHˆΛintas a perturbation. In first order of perturbation theory, the
vacuum energy gets an additional contribution

ΔE 0 =〈 0 |Hˆint| 0 〉,

which by Wick’s theorem

ΔE 0 =

λ

8

ωn 1 ,ω∑n 2 <Λ

n 1 ,n 2 ∈Z^3

〈 0 |φ−n 1 φn 1 | 0 〉〈 0 |φ−n 2 φn 2 | 0 〉,