From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 207

To illustrate the implementation of the renormalization mechanism, let us con-
sider the case of the Hamiltonian operator of the free field theory Eq. (3.26).
The regularization can be introduced in several different ways. Let us consider
two different methods:

(i) Sharp momentum cut-off

HˆΛ=^1

2

ω∑n<Λ

n∈Z^3

(

δ

δφ−n

δ

δφn

+ωn^2 |φn|^2

)

, (3.33)

(ii) Heat kernel regularization

Hˆ=^1

2

∑

n∈Z^3

(

δ

δφ−n

δ

δφn

+ωn^2 e−ω

(^2) n
|φn|^2

)

, (3.34)

which can be related by choosing =

√ 2
Λ^2.
There are other methods which include higher derivative terms or lattice dis-
cretization of the continuum space, but for simplicity we shall not discuss them in
this course.
The renormalization of the fundamental Hamiltonian is obtained by subtract-
ing anunobservableconstant quantityE 0 in such a way that the observable (renor-
malized) quantum Hamiltonian

Hˆren= lim

Λ→∞(

HˆΛ−E 0 (Λ)), (3.35)

is a well defined quantum operator with a finite energy spectrum.
Even if the renormalization of the Hamiltonian solves the divergence problem,
one might wonder about its physical meaning. First, let us analyze the structure
of the divergences of vacuum energy.
In the largeLlimit the vacuum energyE 0 becomes a good approximation to
the Riemann integral

E 0 (Λ,L)=^1

2

L^3

∫

|k|≤Λ

d^3 k

(2π)^3

√

k^2 +m^2 +O(LΛ).

Now it becomes clear that the infrared divergence is just due to the infinite volume
of the system and translation invariance. However the vacuum energy density

E 0 (Λ) = limL→∞

E 0 (Λ,L)

L^3

=

1

2

∫

|k|≤Λ

d^3 k

(2π)^3

√

k^2 +m^2 (3.36)

is free of IR divergences. However, the integral Eq. (3.36) is UV divergent. In the
sharp momentum cut-off regularization

E 0 (Λ) =

Λ^4

16 π^2

+

m^2 Λ^2

16 π^2

+

m^4

64 π^2

log

m^2

Λ^2

+

m^4 (1−log 16)

128 π^2

+O

(

1

Λ^2

)