Physical Foundations of Cosmology

166 The very early universe

following equation for ̄χ(t):

χ ̄;α;α+V′(χ ̄)+

1

2

V′′′(χ ̄)

〈

φ^2

〉

= 0 , (4.94)

where the higher-order terms∼

〈

φ^3

〉

,etc. have been neglected. In quantum field
theory this corresponds to the so called one-loop approximation. We will now show
that in a hot universe the last term in (4.94) can be combined withV′(χ ̄)and
rewritten as the derivative of an effective potentialVeff(χ, ̄ T).To this purpose, let
us calculate the average

〈

φ^2

〉

.

Scalar field quantizationIn the lowest (linear) order, the inhomogeneous modesφ
obey the equation

φ;α;α+V′′(χ ̄)φ= 0 , (4.95)

obtained by linearizing (4.92). Assuming that the mass

m^2 φ(χ ̄)≡V′′(χ ̄)≥ 0

does not depend on time, and neglecting the expansion of the universe (this is a
good approximation for our purposes), the solution of (4.95) is

φ(x,t)=

∫

1

√

2 ωk

(

e−iωkt+ikxa−k+eiωkt−ikxak+

) d^3 k

( 2 π)^3 /^2

, (4.96)

where

ωk=

√

k^2 +V′′(χ ̄)=

√

k^2 +m^2 φ,

k≡|k|,andak−,a+k =

(

ak−

)∗

are the integration constants. Our task is to calculate
both quantum and thermal contributions to

〈

φ^2

〉

.

In quantum theory, the fieldφ(x,t)≡φx(t)becomes a “position” operatorφˆx(t)
and the spatial coordinatesxcan be considered simply as enumerating the degrees of
freedom of the physical system. That is, at each point in space, we have one degree of
freedom – a field strength – which plays the role of position in a configuration space.
Hence, a quantum field is a quantum mechanical system with an infinite number of
degrees of freedom. As in usual quantum mechanics, the position operatorsφˆx(t)
and their conjugated momenta

πˆy≡∂L/∂φ ̇=∂φˆy/∂t

should satisfy the Heisenberg commutation relations:

[

φˆx(t),πˆy(t)

]

=

[

φˆx(t),∂

φˆy(t)

∂t

]

=iδ(x−y), (4.97)