Kiefer C. Quantum gravity

DECOHERENCE AND THE QUANTUM UNIVERSE 313
Note that ∆ n (φ) is just the (constant) Wronskian corresponding to (10.12). 3 We
must emphasize that ∆ n is a non-trivial function of the background variable φ,
since it is defined on the full configuration space and not only along semiclassical
trajectories. In a sense, it gives the weights in the ‘Everett branches’. It is
therefore not possible to normalize the ψ n artificially to one, since this would be
inconsistent with respect to the full Wheeler–DeWheeler equation (Barvinsky et
al. 1999a).
The solution (10.9) forms the basis for our discussion of decoherence. Since
the {f n } are interpreted as the environmental degrees of freedom, they have to
be integrated out to get the reduced density matrix (cf. (10.5)) for φ or a (a
and φ can be used interchangeably, since they are connected by t). The reduced
density matrix thus reads here
∫
ρ(t|φ, φ ′ )= df Ψ(t|φ, f)Ψ ∗ (t|φ ′ ,f) , (10.16)
where Ψ is given by (10.9), and it is understood that df = ∏ n df n.Afterthe
integration, one finds
ρ(t|φ, φ ′ 1
[
)=C
exp − √v 1 φ ∗(t)v′ φ (t) 2 I − 1 ]
2 I′ +i(S 0 − S 0 ′ )
× ∏ n
[
v ∗ nv ′ n(Ω n +Ω ′∗ n )] −1/2
, (10.17)
where C is a numerical constant. The diagonal elements ρ(t|φ, φ) describe the
probabilities for certain values of the inflaton field to occur.
It is convenient to rewrite the expression for the density matrix (10.17) in
the form
ρ(t|φ, φ ′ )=C ∆1/4 φ
∆′1/4 φ
exp
(− √v 1 φ ∗(t)v′ φ (t) 2 Γ − 1 )
2 Γ′ +i(S 0 − S 0 ′ )
where
×D(t|φ, φ ′ ) , (10.18)
Γ = I(φ)+Γ 1−loop (φ) (10.19)
is the full Euclidean effective action including the classical part and the one-loop
part (cf. Section 2.2.4). The latter comes from the next-order WKB approximation
and is important for the normalizability of the wave function with respect
to φ. The last factor in (10.18) is the decoherence factor
3 The corresponding Wronskian for the homogeneous mode φ is ∆ φ ≡ ia 3 (vφ ∗v˙
φ − ˙v φ ∗v φ).