From Classical Mechanics to Quantum Field Theory

206 From Classical Mechanics to Quantum Field Theory. A Tutorial

3.4 TheQuantumVacuum........................

The advantage of the diagonal structure of the quantum Hamiltonian in the plane
wave basis is that it facilitates the analysis of its spectrum.
In particular the ground state, known in QFT as vacuum state, is given by

Ψ 0 =〈φ| 0 〉=

∏

n∈Z^3

√

ωn

π

e−

(^12) ωn|φn| 2
=exp

{

−

1

2

∑

n∈Z^3

(

ωn|φn|^2 +log

π

ωn

)}

(3.30)

Indeed,

HˆΨ 0 =E 0 Ψ 0 , (3.31)

where the vacuum energy

E 0 =^1

2

∑

n∈Z^3

ωn (3.32)

is half the divergent sum of all normal modes frequencies. This appearance of
this divergence is a genuine quantum effect which is induced by the fact that
the lowest energy (zero-point energy) of each quantum oscillator is non-vanishing.
The divergence is generated by the large momentumn(ultraviolet)modes. The
vacuum energy is the simplest quantity of the quantum theory which presents UV
divergences.

3.4.1 Renormalization of vacuum energy

The appearance of UV divergences postponed the formulation of quantum field
theories for two decades. The solution of the UV puzzle came from the renor-
malization program. The main idea behind the renormalization program is the
disassociation of the fundamental observables like the quantum Hamiltonian and
the observed quantities.
To implement the renormalization program, we need a previous step which is
known as regularization. This requires to introduce a modification of all funda-
mental (bare) operators depending on a UV scale parameter Λ in a way that they
become well defined operators with a finite spectrum, e.g. by cutting the infinite
sum in Eq. (3.32) to a finite sum. The modification must disappear in the limit
Λ→∞to recover the original divergent expressions. The second step consists of
aphysicalmodification of the fundamental observables by absorbing the sources of
divergences into the physical parameterslike particle masses and charges, coupling
constants or energy scales of the theory in a way that the they remain finite in the
Λ→∞.