Integrable Minisuperspace Models with Liouville Field: Energy Density Self-Adjointness and Semiclassical Wave Packets

TL;DR

This paper explores quantum cosmological models with Liouville fields, analyzing energy operator self-adjointness, spectrum properties, and wave packet evolution, revealing ambiguities and conditions for classical-quantum correspondence.

Contribution

It introduces a detailed analysis of energy density operators in Liouville field cosmologies, highlighting self-adjointness issues and the use of pseudo-Hermitian quantum mechanics techniques.

Findings

Energy density operators lack essential self-adjointness in certain models.

Discrete energy spectrum observed for phantom fields.

Localized wave packets evolve similarly under different norms.

Abstract

The homogeneous cosmological models with a Liouville scalar field are investigated in classical and quantum context of Wheeler-DeWitt geometrodynamics. In the quantum case of quintessence field with potential unbounded from below and phantom field, the energy density operators are not essentially self-adjoint and self-adjoint extensions contain ambiguities. Therefore the same classical actions correspond to a family of distinct quantum models. For the phantom field the energy spectrum happens to be discrete. The probability conservation and appropriate classical limit can be achieved with a certain restriction of the functional class. The appropriately localized wave packets are studied numerically using the Schrodinger's norm and a conserved Mostafazadeh's norm introduced from techniques of pseudo-Hermitian quantum mechanics. These norms give a similar packet evolution that is confronted with analytical classical solutions.