Quantum Gravity

244 QUANTUM COSMOLOGYconcerns in particular a comparison between reduced and Dirac quantization.In quantum geometrodynamics, the wave functional Ψ is defined—apart fromnon-gravitational degrees of freedom—on Riem Σ, the space of all three-metrics.The presence of the diffeomorphism constraints guarantees that the true configurationspace is ‘superspace’, that is, the space Riem Σ/Diff Σ. Restricting theinfinitely many degrees of freedom of superspace to only a finite number, onearrives at a finite-dimensional configuration space called minisuperspace. Ifthenumber of the restricted variables is still infinite, the term ‘midisuperspace’ hasbeen coined. The example of spherical symmetry discussed in the last chapter isan example of midisuperspace. Since the most important example in the case offinitely many degrees of freedom is cosmology, the minisuperspace examples areusually applied to quantum cosmology, that is, the application of quantum theoryto the Universe as a whole. The present chapter deals with quantum cosmology.For simplicity, we shall restrict our attention mostly to quantum geometrodynamics;the application of loop quantum gravity (cf. Chapter 6) to cosmology iscalled loop quantum cosmology and is discussed in Section 8.4 below.The importance of quantum theory for an understanding of the origin of theuniverse was already emphasized by Georges Lemaître in the context of his atomeprimitif; seeLemaître (1958). However, he did not consider the quantization ofspace–time itself. The idea that the Universe as a whole is the result of a ‘vacuumfluctuation’ in quantum field theory can be traced back at least to Tryon (1973).Independent of any quantum theory of gravity, one can give general argumentsthat demand for reasons of consistency the application of quantum theoryto the Universe as a whole. Namely, macroscopic quantum systems are stronglycoupled to their natural environment; cf. the discussion in Chapter 10. Since theenvironment is again coupled to its environment, and so on, the only strictlyclosed system in the quantum theoretical sense is the Universe as a whole. Thisleads to quantum cosmology, independent of any particular interaction. However,since gravity is the dominating interaction on cosmic scales, a quantum theoryof gravity is needed as the formal framework for quantum cosmology.The first quantum-cosmological model based on quantum gravity was presented,together with its semiclassical approximation, in DeWitt (1967a). It dealtwith the homogeneous and isotropic case. The extension to anisotropic models(in particular, Bianchi models) was performed by Misner; cf. Misner (1972) andRyan (1972). Kuchař (1971) made the extension to the midisuperspace case anddiscussed the quantization of cylindrical gravitational waves; see also Ashtekarand Pierri (1996). Classically, in the general case of inhomogeneous models, differentspatial points seem to decouple near a big-bang singularity for generalsolutions of the Einstein equations (Belinskii et al. 1982). Such solutions consistof a collection of homogeneous spaces described, for example, by a ‘mixmasteruniverse’ (in which the universe behaves like a particle in a time-dependent potentialwall, with an infinite sequence of bounces). For this reason the use ofminisuperspace models may even provide a realistic description of the universenear its classical singularity.