Anisotropic models are unitary: A rejuvenation of standard quantum cosmology

TL;DR

This paper demonstrates that quantum anisotropic cosmological models are fundamentally unitary by showing all operator orderings can produce self-adjoint Hamiltonians, challenging previous beliefs of non-conservation of probability.

Contribution

It proves that non-unitarity in quantum anisotropic models is a misconception and that all such models can have self-adjoint Hamiltonians, extending the understanding of quantum cosmology.

Findings

All operator orderings can lead to self-adjoint Hamiltonians in anisotropic models.

Non-uniqueness of self-adjoint extensions is present and not exclusive to anisotropic models.

Explicit calculations are provided for a Bianchi III cosmological model.

Abstract

The present work proves that the folk-lore of the pathology of non-conservation of probability in quantum anisotropic models is wrong. It is shown in full generality that all operator ordering can lead to a Hamiltonian with a self-adjoint extension as long as it is constructed to be a symmetric operator, thereby making the problem of non-unitarity in context of anisotropic homogeneous model a ghost. Moreover, it is indicated that the self-adjoint extension is not unique and this non-uniqueness is suspected not to be a feature of Anisotropic model only, in the sense that there exists operator orderings such that Hamiltonian for an isotropic homogeneous cosmological model does not have unique self-adjoint extension, albeit for isotropic model, there is a special unique extension associated with quadratic form of Hamiltonian i.e {\it Friedrichs extension}. Details of calculations are carried out for a Bianchi III model.