Quantum mechanics of Klein-Gordon-type fields and quantum cosmology

Abstract

With a view to address some of the basic problems of quantum cosmology, we formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (partial derivative(t)(2) + D) psi(t) = 0, where t is an element of R and D is a positive-definite operator acting in a Hilbert space X In particular, we determine all the positive-definite inner products on the space H of the solutions of such an equation and establish their physical equivalence. This specifies the Hilbert space structure of R uniquely. We use a simple realization of the latter to construct the observables of the theory explicitly. The field equation does not fix the choice of a Hamiltonian operator unless it is supplemented by an underlying classical system and a quantization scheme supported by a correspondence principle. In general, there are infinitely many choices for the Hamiltonian each leading to a different notion of time-evolution in R. Among these is a particular choice that generates t-translations in R and identifies t with time whenever D is t-independent. For a t-dependent D, we show that regardless of the choice of the inner product the t-translations do not correspond to unitary evolutions in R, and t cannot be identified with time. We apply these ideas to develop a formulation of quantum cosmology based on the Wheeler-DeWitt equation for a Friedman, Robertson-Walker model coupled to a real scalar field with an arbitrary positive confining potential. In particular, we offer a complete solution of the Hilbert space problem, construct the observables, use a position-like observable to introduce the wave functions of the universe (which differ from the Wheeler-DeWitt fields), reformulate the corresponding quantum theory in terms of the latter, reduce the problem of the identification of time to the determination of a Hamiltonian operator acting in L-2(R) circle plus L-2(R), show that the factor-ordering problem is irrelevant for the kinematics of the quantum theory, and propose a formulation of the dynamics. Our method is based on the central postulates of nonrelativistic quantum mechanics, especially the quest for a genuine probabilistic interpretation and a unitary Schrodinger time-evolution. It generalizes to arbitrary minisuperspace (spatially homogeneous) models and provides a way of unifying the two main approaches to the canonical quantum cosmology based on these models, namely quantization before and after imposing the Hamiltonian constraint. (C) 2003 Elsevier Inc. All rights reserved.