Quantum Gravity

ARROW OF TIME 319Kiefer et al. (1998), as well as section 4.2.4 of Joos et al. (2003) for a review. Ithappens when the wavelength of the primordial quantum fluctuations becomesmuch bigger than the Hubble scale H −1Iduring the inflationary regime, whereH 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 verybroad. Such a state is highly sensitive to any interaction, albeit small, with other(‘environmental’ or ‘irrelevant’) fields. It thereby decoheres into an ensemble ofnarrow wave packets that are approximately eigenstates of the field amplitude.A prerequisite is the classical nature of the background variables discussed in thelast 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 tensorperturbations of the metric. They correspond to gravitons (Chapter 2). Likefor the scalar part the tensor part evolves into a highly squeezed state duringinflation, and decoherence happens for it, too. The primordial gravitons wouldmanifest themselves in a stochastic background of gravitational waves, whichcould probably be observed with the space-borne interferometer LISA to belaunched in a couple of years. Its observation would constitute a direct test oflinearized quantum gravity.The decoherence time turns out to be of the ordert d ∼ H Ig , (10.41)where g is a dimensionless coupling constant of the interaction with other ‘irrelevant’fields causing decoherence. The ensuing coarse-graining brought about bythe 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, His approximately constant and the entropy increases linear with t. Inthepostinflationaryphases (radiation- and matter-dominated universe), H ∝ t −1 andthe entropy increases only logarithmically in time. The main part of the entropyfor the fluctuations is thus created during inflation. Incidentally, this behaviourresembles the behaviour for chaotic systems, although no chaos is involved here.The role of the Lyapunov coefficient is played by the Hubble parameter, and theKolmogorov entropy corresponds to the entropy production mentioned here.Decoherence also plays an important role for quantum black holes and in thecontext of wormholes and string theory; see section 4.2.5 of Joos et al. (2003).10.2 Arrow of timeOne of the most intriguing open problems is the origin of irreversibility in ouruniverse, also called the problem of the arrow of time. Since quantum gravitymay provide the key for its solution, this topic will be briefly reviewed here. Moredetails and references can be found in Zeh (2001).