Selection rules for the Wheeler-DeWitt equation in quantum cosmology

The incompleteness of the Dirac quantization scheme leads to a redundant set of solutions of the Wheeler-DeWitt equation for the wave function in the superspace of quantum cosmology. The selection of physically meaningful solutions that match quantum initial data can be attained by a reduction of the theory to the sector of true physical degrees of freedom and their canonical quantization. The resulting physical wave function unitarily evolving in the time variable introduced within this reduction can then be raised to the level of the cosmological wave function in the superspace of 3-metrics to form a needed subset of all solutions of the Wheeler-DeWitt equation. We apply this technique in several simple but nonlinear minisuperspace models and discuss (at both the classical and quantum level) the physical reduction in extrinsic time—the time variable determined in terms of extrinsic curvature (or momentum conjugated to the cosmological scale factor). Only this extrinsic time gauge can be consistently used in the vicinity of turning points and bounces where the scale factor reaches extremum and cannot monotonically parametrize the evolution of the system. Since the 3-metric scale factor is canonically dual to the extrinsic time variable, the transition from the physical wave function to the wave function in superspace represents a kind of generalized Fourier transform. This transformation selects square-integrable solutions of the Wheeler- DeWitt equation, which guarantees the Hermiticity of canonical operators of the Dirac quantization scheme. This makes this scheme consistent, a property that is not guaranteed with general solutions of the Wheeler-DeWitt equation. Semiclassically this means that wave functions are represented by oscillating waves in classically allowed domains of superspace and exponentially fall off in classically forbidden (underbarrier) regions. This is explicitly demonstrated in a flat Friedmann-Robertson-Walker (FRW) model with a scalar field having a constant negative potential, and for the case of a phantom scalar field with a positive potential. The FRW model of a scalar field with a vanishing potential does not lead to selection rules for solutions of the Wheeler-DeWitt equation, but this does not violate Hermiticity properties, because all these solutions are plane-wave type and describe cosmological dynamics without turning points and bounces. In models with turning points the description of classically forbidden domains goes beyond the original principles of the unitary quantum reduction to the physical sector, because it includes the complexification of the physical time variable or the complex nature of the physical Hamiltonian. However, this does not alter the formalism of the Wheeler-DeWitt equation, which continues describing underbarrier quantum dynamics in terms of real superspace variables.