342 Inflation II: origin of the primordial inhomogeneities
Problem 8.11Considering only the physical (nonfictitious) mode of a long-
wavelength perturbatioun in the synchronous coordinate system, verify that the
second term in the second equality in (8.82) dominates over the first.
The first step in quantizing (8.79) is to define the momentumπcanonically
conjugated tov,
π≡∂L
∂v′=v′. (8.83)In quantum theory, the variablesvandπbecome operatorsvˆandπˆ, which at any
moment of timeηsatisfy the standard commutation relations:
[vˆ(η,x),vˆ(η,y)]=[πˆ(η,x),πˆ(η,y)]= 0 ,[vˆ(η,x),πˆ(η,y)]=[
vˆ(η,x),vˆ′(η,y)]
=iδ(x−y), (8.84)where we have set=1. The operatorvˆobeys the same equation as the corre-
sponding classical variablev,
vˆ′′−cs^2 vˆ−z′′
z
vˆ= 0 , (8.85)and its general solution can be written as
vˆ(η,x)=1
√
2
∫ [
vk∗(η)eikxaˆk−+vk(η)e−ikxaˆ+k] d^3 k
( 2 π)^3 /^2, (8.86)
where the temporal mode functionsvk(η)satisfy
v′′k+ω^2 k(η)vk= 0 ,ω^2 k(η)≡c^2 sk^2 −z′′/z. (8.87)We are free to impose the bosonic commutation relations for the creation and
annihilation operators on the conjugated operator-valued constants of integration
aˆk−andaˆ+k′:
[
aˆ−k,aˆ−k′
]
=
[
aˆk+,aˆk+′]
= 0 ,
[
aˆk−,aˆk+′]
=δ(
k−k′)
. (8.88)
Substituting (8.86) into (8.84), we find that they are consistent with commutation
relations (8.84) only if the mode functionsvk(η)obey the normalization condition
v′kv∗k−vkv∗′k = 2 i. (8.89)The expression on the left hand side is a Wronskian of (8.87) built from two inde-
pendent solutionsvkandv∗k; therefore it does not depend on time. It follows from
(8.89) thatvk(η)is acomplexsolution of the second order differential equation
(8.87). To specify it fully and thus determine the physical meaning of the opera-
torsaˆk±we need the initial conditions forvkandv′kat some initial timeη=ηi.