Wave functions for arbitrary operator ordering in the de Sitter minisuperspace approximation

TL;DR

This paper derives exact solutions for the Wheeler-DeWitt equation in a closed universe with a cosmological constant, addressing operator ordering ambiguities and their implications for wave function normalizability.

Contribution

It provides a comprehensive analysis of wave functions for arbitrary operator orderings in the de Sitter minisuperspace, extending previous models and examining their physical viability.

Findings

Wave functions are derived for arbitrary operator orderings.

Vilenkin's tunneling wave function remains non-normalizable under general conditions.

Inclusion of matter degrees of freedom does not resolve non-normalizability.

Abstract

We derive exact series solutions for the Wheeler-DeWitt equation corresponding to a spatially closed Friedmann-Robertson-Walker universe with cosmological constant for arbitrary operator ordering of the scale factor of the universe. The resulting wave functions are those relevant to the approximation which has been widely used in two-dimensional minisuperspace models with an inflationary scalar field for the purpose of predicting the period of inflation which results from competing boundary condition proposals for the wave function of the universe. The problem that Vilenkin's tunneling wave function is not normalizable for general operator orderings, is shown to persist for other values of the spatial curvature, and when additional matter degrees of freedom such as radiation are included.