Physical Foundations of Cosmology

4.4 “Symmetry restoration” and phase transitions 167

where

[ˆ

φx(t),πˆy(t)

]

≡φˆx(t)πˆy(t)−πˆy(t)φˆx(t)

and Planck’s constant is set to unity. The field operatorφˆx(t)obeys (4.95) and its
solution is given in (4.96), but now the integration constants should be considered
as time-independent operatorsaˆk−,aˆk+. Substituting (4.96) in (4.97), we find that
the operatorsaˆ+k,aˆ−ksatisfy the commutation relations

[

aˆk−,aˆ+k′

]

=δ

(

k−k′

)

,

[

aˆ−k,aˆ−k′

]

=

[

aˆ+k,aˆk+′

]

= 0. (4.98)

Except for the appearance of theδfunction, these behave like the creation and
annihilation operators of harmonic oscillators. The Hilbert space in which these
operators act then resembles the Hilbert space of a set of harmonic oscillators. The
vacuum state| 0 〉is defined via

aˆ−k| 0 〉= 0 (4.99)

for allk,and corresponds to the minimal energy state. The vectors

|nk〉=

(

aˆk+

)n

√

n!

| 0 〉 (4.100)

are interpreted as describingnkparticlesper single quantum statecharacterized by
the wave vectork.

Problem 4.16The operator Nˆk≡aˆ+kaˆk− corresponds to the total number of
particles with wave vectork.Using commutation relations (4.98), verify that
Nˆk|nk〉=δ( 0 )nk|nk〉.The appearance ofδ( 0 )can easily be understood if we
take into account that the total number of quantum states with a given momentum
is proportional to the volume. Because we consider an infinite volume, the factor
δ( 0 )simply reflects this infinity. The number of particlesper unit volumeis finite
and equal to the occupation numbernk.
Verify that

〈

aˆk+aˆ−k′

〉

Q≡

〈nk|aˆ+kaˆk−′|nk〉

〈nk|nk〉

=nkδ

(

k−k′

)

,

〈

aˆk+aˆ+k′

〉

Q=

〈

aˆ−kaˆk−′

〉

Q=^0.

(4.101)

Now we can proceed with the calculation of

〈

φ^2

〉

.Let us take a quantum state
with occupation numbersnkin every modek.In a homogeneous isotropic universe
the spatial average can be replaced by the quantum average. Squaring (4.96) and