Consistent Probabilities in Wheeler-DeWitt Quantum Cosmology

TL;DR

This paper develops a rigorous framework for calculating quantum probabilities in Wheeler-DeWitt quantum cosmology, revealing that such models are generically singular with probability one, and clarifies the relation between histories and Dirac observables.

Contribution

It provides the first complete decoherent histories formulation for a Wheeler-DeWitt cosmological model with explicit probability calculations.

Findings

Probabilities for singularities are unity in this model.

The framework replaces heuristic interpretations with explicit formulas.

The relation between histories and Dirac observables is clarified.

Abstract

We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the model's Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to study the singular behavior of quantum Wheeler-DeWitt universes. Within this framework, rigorous formulae are given for calculating the corresponding probabilities from the wave function when those probabilities may be consistently defined, thus replacing earlier heuristics for interpreting the wave function of the universe with explicit constructions. It is shown according to a rigorously formulated standard, and in a quantum-mechanically consistent way, that in this quantization these models are generically singular. Independent of the choice of state we show that the probability for these Wheeler-DeWitt quantum universes to ever encounter a singularity is unity. In addition, the relation between histories formulations of quantum theory and relational Dirac observables is clarified.