Quantum cosmology in the energy representation

TL;DR

This paper investigates the Wheeler-DeWitt equation in quantum cosmology, revealing a basis of solutions distinguished by analyticity that splits into positive and negative definite Hilbert spaces, with implications for understanding universe dynamics.

Contribution

It introduces a new basis of solutions for the Wheeler-DeWitt equation based on analyticity properties, suggesting a deeper structure in quantum cosmology models.

Findings

Solutions form two Hilbert spaces with positive and negative definite scalar products.

Preferred basis describes contracting and expanding universes with specific scalar field energies.

The framework may aid in analyzing the Wheeler-DeWitt equation in more complex models.

Abstract

The Hawking minisuperspace model (closed FRW geometry with a homogeneous massive scalar field) provides a fairly non-trivial testing ground for fundamental problems in quantum cosmology. We provide evidence that the Wheeler-DeWitt equation admits a basis of solutions that is distinguished by analyticity properities in a large scale factor expansion. As a consequence, the space of solutions decomposes in a preferred way into two Hilbert spaces with positive and negative definite scalar product, respectively. These results may be viewed as a hint for a deeper significance of analyticity. If a similar structure exists in full (non-minisuperspace) models as well, severe implications on the foundations of quantum cosmology are to be expected. Semiclassically, the elements of the preferred basis describe contracting and expanding universes with a prescribed value of the matter (scalar field) energy. Half of the basis elements have previously been constructed by Hawking and Page in a wormhole context, and they appear in a new light here. The technical tools to arrive at these conclusions are transformations of the wave function into several alternative representations that are based on the harmonic oscillator form of the matter energy operator, and that are called oscillator, energy and Fock representation. The framework defined by these may be of some help in analyzing the Wheeler-DeWitt equation for other purposes as well.