Quantum Mechanics of Klein-Gordon-Type Fields and Quantum Cosmology

TL;DR

This paper develops a framework for the quantum mechanics of Klein-Gordon-type fields and applies it to quantum cosmology, resolving the Hilbert space and time evolution issues in the Wheeler-DeWitt equation.

Contribution

It introduces a method to construct positive-definite inner products and observables for Klein-Gordon-type fields, and applies these to formulate a consistent quantum cosmology with a well-defined notion of time.

Findings

Constructed all positive-definite inner products for Klein-Gordon solutions

Identified a unique Hilbert space structure leading to a probabilistic interpretation

Developed a formulation of quantum cosmology with unitary evolution

Abstract

We formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (\partial_t^2+D)\psi(t)=0, where D is a positive-definite operator acting in a Hilbert space \tilde H. We determine all the positive-definite inner products on the space H of the solutions of such an equation and establish their physical equivalence. We use a simple realization of the unique Hilbert space structure on H to construct the observables of the theory explicitly. In general, there are infinitely many choices for the Hamiltonian each leading to a different notion of time-evolution in H. Among these is a particular choice that generates t-translations in H and identifies t with time whenever D is t-independent. For a t-dependent D, we show that t-translations never correspond to unitary evolutions in H, 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. We solve the Hilbert space problem, construct the observables, introduce a new set of the wave functions of the universe, reformulate the quantum theory in terms of the latter, reduce the problem of time to the determination of a Hamiltonian operator acting in L^2(\R)\oplus 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 allows for a genuine probabilistic interpretation and a unitary Schreodinger time-evolution.