Quantum Gravity

MINISUPERSPACE MODELS 245The usual symmetry reduction proceeds as follows; see, for example, Torre(1999). One starts from a classical field theory and specifies the action of agroup with respect to which the fields are supposed to be invariant. A prominentexample is the rotation group. One then constructs the invariant (‘reduced’) fieldsand evaluates the field equations for them. An important question is whetherthere is a shortcut in the following way. Instead of reducing the field equationsone might wish to reduce first the Lagrangian and then derive from it directly thereduced field equations. (Alternatively, this can be attempted at the Hamiltonianlevel.) This would greatly simplify the procedure, but in general it is not possible:reduction of the Lagrangian is equivalent to reduction of the field equations onlyin special situations. When do such situations occur? In other words, when docritical points of the reduced action define critical points of the full action?Criteria for this symmetric criticality principle were developed by Palais (1979).If restriction is made to local Lagrangian theories, one can specify such criteriamore explicitly (Torre 1999; Fels and Torre 2002).Instead of spelling out the general conditions, we focus on three cases thatare relevant for us:1. The conditions are always satisfied for a compact symmetry group, that is,the important case of spherical symmetry obeys the symmetric criticalityprinciple.2. In the case of homogeneous cosmological models, the conditions are satisfiedif the structure constants cab of the isometry group satisfy cab =0.cbTherefore, Bianchi-type-A cosmological models and the Kantowski–Sachsuniverse can be treated via a reduced Lagrangian. For Bianchi-type-B models,the situation is more subtle (cf. MacCallum (1979) and Ryan andWaller (1997)).3. The symmetric criticality principle also applies to cylindrical or toroidalsymmetry reductions (which are characterized by two commuting Killingvector fields). The reduced theories can be identified with parametrizedfield theories on a flat background. With such a formal identification it iseasy to find solutions. Quantization can then be understood as quantizationon a fixed background with arbitrary foliation into Cauchy surfaces. In twospace–time dimensions, where these reduced models are effectively defined,time evolution is unitarily implementable along arbitrary foliations. Thisceases to hold in higher dimensions; cf. Giulini and Kiefer (1995), Helfer(1996), and Torre and Varadarajan (1999).In the case of homogeneous models, the wave function is of the form ψ(q i ),i =1,...,n, that is, it is of a ‘quantum-mechanical’ type. In this section, wefollow a pragmatic approach and discuss the differential equations for the wavefunctions, together with appropriate boundary conditions. A general discussionof boundary conditions will be presented in Section 8.3 below. Extensive reviewsof quantum cosmology include Halliwell (1991), Wiltshire (1996), and Coule(2005).