Probabilities in Quantum Cosmological Models: A Decoherent Histories Analysis Using a Complex Potential

TL;DR

This paper develops a rigorous operator-based framework using decoherent histories and complex potentials to assign probabilities in quantum cosmological models, improving upon heuristic WKB methods.

Contribution

It introduces a formalism for constructing class operators in quantum cosmology using complex potentials, linking decoherent histories with the Wheeler-DeWitt equation.

Findings

Class operators commute with the Hamiltonian.

Probabilities match heuristic methods semiclassically.

Decoherence occurs for large configuration space regions.

Abstract

In the quantization of simple cosmological models (minisuperspace models) described by the Wheeler-DeWitt equation, an important step is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation. This gives sensible semiclassical results but lacks an underlying operator formalism. In this paper, we address the issue of constructing probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory. We show that the appropriate class operators (the generalization of strings of projectors) in quantum cosmology are readily constructed using a complex potential. We derive the class operator for entering or not entering one or more regions in configuration space. They commute with the Hamiltonian, have a sensible classical limit and are closely related to intersection number operators. We show that oscillatory WKB solutions to the Wheeler-DeWitt equation give approximate decoherence of histories, as do superpositions of WKB solutions, as long as the regions of configuration space are sufficiently large. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures. In brief, we exhibit the well-defined operator formalism underlying the usual heuristic interpretational methods in quantum cosmology.