Kiefer C. Quantum gravity

MINISUPERSPACE MODELS 245
The usual symmetry reduction proceeds as follows; see, for example, Torre
(1999). One starts from a classical field theory and specifies the action of a
group with respect to which the fields are supposed to be invariant. A prominent
example is the rotation group. One then constructs the invariant (‘reduced’) fields
and evaluates the field equations for them. An important question is whether
there is a shortcut in the following way. Instead of reducing the field equations
one might wish to reduce first the Lagrangian and then derive from it directly the
reduced field equations. (Alternatively, this can be attempted at the Hamiltonian
level.) 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 only
in special situations. When do such situations occur? In other words, when do
critical 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 criteria
more explicitly (Torre 1999; Fels and Torre 2002).
Instead of spelling out the general conditions, we focus on three cases that
are 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 criticality
principle.
2. In the case of homogeneous cosmological models, the conditions are satisfied
if the structure constants cab of the isometry group satisfy cab =0.
c
b
Therefore, Bianchi-type-A cosmological models and the Kantowski–Sachs
universe can be treated via a reduced Lagrangian. For Bianchi-type-B models,
the situation is more subtle (cf. MacCallum (1979) and Ryan and
Waller (1997)).
3. The symmetric criticality principle also applies to cylindrical or toroidal
symmetry reductions (which are characterized by two commuting Killing
vector fields). The reduced theories can be identified with parametrized
field theories on a flat background. With such a formal identification it is
easy to find solutions. Quantization can then be understood as quantization
on a fixed background with arbitrary foliation into Cauchy surfaces. In two
space–time dimensions, where these reduced models are effectively defined,
time evolution is unitarily implementable along arbitrary foliations. This
ceases 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, we
follow a pragmatic approach and discuss the differential equations for the wave
functions, together with appropriate boundary conditions. A general discussion
of boundary conditions will be presented in Section 8.3 below. Extensive reviews
of quantum cosmology include Halliwell (1991), Wiltshire (1996), and Coule
(2005).