Conditional Symmetries and the Quantization of Bianchi Type I Vacuum Cosmologies with and without Cosmological Constant

TL;DR

This paper explores the quantization of Bianchi Type I vacuum cosmologies, employing conditional symmetries to reduce degrees of freedom and identify a wave function dependent on a single variable, with a proposed invariant measure for the Hilbert space.

Contribution

It introduces a method to reduce the quantum cosmology model using conditional symmetries, leading to a simplified wave function and a symmetry-respecting measure for the Hilbert space.

Findings

Wave function depends on the determinant of the scale factor matrix.

Reduction of the configuration space via quantum integrals of motion.

Proposed measure for the Hilbert space invariant under symmetries and scalings.

Abstract

In this work, the quantization of the most general Bianchi Type I geometry, with and without a cosmological constant, is considered. In the spirit of identifying and subsequently removing as many gauge degrees of freedom as possible, a reduction of the initial 6--dimensional configuration space is presented. This reduction is achieved by imposing as additional conditions on the wave function, the quantum version of the --linear in momenta-- classical integrals of motion (conditional symmetries). The vector fields inferred from these integrals induce, through their integral curves, motions in the configuration space which can be identified to the action of the automorphism group of Type I, i.e. $GL(3,\Re)$. Thus, a wave function depending on one degree of freedom, namely the determinant of the scale factor matrix, is found. A measure for constructing the Hilbert space is proposed. This measure respects the above mentioned symmetries, and is also invariant under the classical property of covariance under arbitrary scalings of the Hamiltonian (quadratic constraint).