Self-Adjoint Wheeler-DeWitt Operators, the Problem of Time and the Wave Function of the Universe

TL;DR

This paper explores the self-adjointness of Wheeler-DeWitt operators in quantum cosmology, revealing how different parametrizations of time influence the mathematical domains and physical interpretations of wave functions of the universe.

Contribution

It analyzes the impact of self-adjointness conditions on Wheeler-DeWitt operators in minisuperspace models, linking mathematical domains to cosmological implications and different time parametrizations.

Findings

Two types of domains for self-adjoint WDW operators identified

Wave functions' domains depend on time parametrization and boundary conditions

New parameters emerge in the domain of Hartle-Hawking type wave functions

Abstract

We discuss minisuperspace aspects a non empty Robertson-Walker universe containing scalar matter field. The requirement that the Wheeler-DeWitt (WDW) operator be self adjoint is a key ingredient in constructing the physical Hilbert space and has non-trivial cosmological implications since it is related with the problem of time in quantum cosmology. Namely, if time is parametrized by matter fields we find two types of domains for the self adjoint WDW operator: a non trivial domain is comprised of zero current (Hartle-Hawking type) wave functions and is parametrized by two new parameters, whereas the domain of a self adjoint WDW operator acting on tunneling (Vilenkin type) wave functions is a {\em single} ray. On the other hand, if time is parametrized by the scale factor both types of wave functions give rise to non trivial domains for the self adjoint WDW operators, and no new parameters appear in them.