Kiefer C. Quantum gravity

ARROW OF TIME 319
Kiefer et al. (1998), as well as section 4.2.4 of Joos et al. (2003) for a review. It
happens when the wavelength of the primordial quantum fluctuations becomes
much bigger than the Hubble scale H −1
I
during the inflationary regime, where
H I denotes the Hubble parameter of inflation (which is approximately constant);
cf. Fig. 7.4. The quantum state becomes strongly squeezed during this phase:
The squeezing is in the field momentum, while the field amplitude becomes very
broad. Such a state is highly sensitive to any interaction, albeit small, with other
(‘environmental’ or ‘irrelevant’) fields. It thereby decoheres into an ensemble of
narrow wave packets that are approximately eigenstates of the field amplitude.
A prerequisite is the classical nature of the background variables discussed in the
last subsection, which is why one could talk about a ‘hierarchy of classicality’.
Density fluctuations arise from the scalar part of the metric perturbations
(plus the corresponding matter part). In addition one has of course the tensor
perturbations of the metric. They correspond to gravitons (Chapter 2). Like
for the scalar part the tensor part evolves into a highly squeezed state during
inflation, and decoherence happens for it, too. The primordial gravitons would
manifest themselves in a stochastic background of gravitational waves, which
could probably be observed with the space-borne interferometer LISA to be
launched in a couple of years. Its observation would constitute a direct test of
linearized quantum gravity.
The decoherence time turns out to be of the order
t d ∼ H I
g , (10.41)
where g is a dimensionless coupling constant of the interaction with other ‘irrelevant’
fields causing decoherence. The ensuing coarse-graining brought about by
the decohering fields causes an entropy increase for the primordial fluctuations
(Kiefer et al. 2000). The entropy production rate turns out to be given by Ṡ = H,
where H is the Hubble parameter of a general expansion. During inflation, H
is approximately constant and the entropy increases linear with t. Inthepostinflationary
phases (radiation- and matter-dominated universe), H ∝ t −1 and
the entropy increases only logarithmically in time. The main part of the entropy
for the fluctuations is thus created during inflation. Incidentally, this behaviour
resembles the behaviour for chaotic systems, although no chaos is involved here.
The role of the Lyapunov coefficient is played by the Hubble parameter, and the
Kolmogorov entropy corresponds to the entropy production mentioned here.
Decoherence also plays an important role for quantum black holes and in the
context of wormholes and string theory; see section 4.2.5 of Joos et al. (2003).
10.2 Arrow of time
One of the most intriguing open problems is the origin of irreversibility in our
universe, also called the problem of the arrow of time. Since quantum gravity
may provide the key for its solution, this topic will be briefly reviewed here. More
details and references can be found in Zeh (2001).