Quantum Gravity

256 QUANTUM COSMOLOGY)(− ∂2 ∂2+∂ ¯φ2 ∂β 2 − λ2 −2 ¯φs V (β, ¯φ)e ψ(β, ¯φ) =0, (8.32)where V (β, ¯φ) denotes here the dilaton potential. Since no external time t existsin quantum cosmology, it does not make sense to talk of a transition betweenthe pre- and post-big-bang regime. Boundary conditions have to be imposedintrinsically, that is, with respect to the configuration-space variables β and¯φ. Constructing wave packet-solutions to (8.32), one recognizes that the preandpost-big-bang branches just correspond to different solutions (D¸abrowskiand Kiefer 1997). An intrinsic distinction between ‘expanding’ and ‘contracting’solution is not possible since the reference phase e −iωt is lacking (Zeh 1988).This can only be achieved if additional degrees of freedom are introduced and aboundary condition of low entropy is imposed; see Section 10.2.Quantum-cosmological models can, and have been, discussed in more generalsituations. Cavaglià and Moniz (2001), for example, have investigated theWheeler–DeWitt equation for effective actions inspired by ‘M-theory’ (Section9.1). Lidsey (1995) has discussed general scalar-tensor theories in which it turnsout that the duality symmetry of the classical action corresponds to hiddenN = 2 SUSY. Quantum cosmology for relativity in more than three space dimensionsalso possesses some interesting features (see e.g. Zhuk 1992).Instead of modifying the gravitational sector one can also introduce exoticmatter degrees of freedom. One example are phantom fields. They have beeninvoked as a possible explanation for the observed Dark Energy in the Universe,and are characterized by a negative kinetic term. Such phantom models exhibit,in the classical theory, some new types of singularities. Among them is the big ripwhere the matter density diverges at finite times for large a instead of small a.The corresponding quantum scenario is discussed in D¸abrowski et al. (2006). Itturns out that wave-packet solutions of the Wheeler–DeWitt equation dispersenear the region that corresponds to the big-rip singularity. One thus arrives ata genuine quantum region at large scales.Quantum cosmology can also be discussed using methods of connection orloop dynamics (Chapter 6). Paternoga and Graham (1998), for example, investigatedBianchi IX models with Λ ≠ 0 in the connection representation. Theystarted from the Chern–Simons state (6.8), which is a solution of the Euclideanquantum constraints for Barbero–Immirzi parameter β = 1. Through a generalizedFourier transform to the metric transformation, they were able—usinginequivalent contours in the transformation formula—to find various solutions tothe Wheeler–DeWitt equation. One can also address quantum cosmology directlyin the loop representation. The ensuing scenario of loop quantum cosmology isdiscussed in Section 8.4.8.1.3 (2+1)-dimensional quantum gravityGeneral relativity in 2+1 dimensions is ‘trivial’ in the sense that there are nolocal dynamical degrees of freedom. The Riemann tensor depends linearly onthe Ricci tensor and thus the vacuum solutions of Einstein’s equations are either