Automorphism Inducing Diffeomorphisms, Invariant Characterization of Homogeneous 3-Spaces and Hamiltonian Dynamics of Bianchi Cosmologies

TL;DR

This paper provides an invariant geometric framework for Bianchi homogeneous 3-spaces by utilizing automorphism groups, clarifies the relation between classical invariants and quantum dynamics, and addresses discrepancies in Bianchi Type A models.

Contribution

It introduces an automorphism-based invariant description of Bianchi 3-spaces and links classical integrals of motion to quantum symmetries, resolving discrepancies in quantum cosmology.

Findings

Invariant characterization of Bianchi 3-spaces via automorphism groups

Identification of classical integrals of motion as quantum symmetries

Resolution of discrepancies between kinematics and quantum dynamics in Bianchi Types

Abstract

An invariant description of Bianchi Homogeneous (B.H.) 3-spaces is presented, by considering the action of the Automorphism Group on the configuration space of the real, symmetric, positive definite, $3\times 3$ matrices. Thus, the gauge degrees of freedom are removed and the remaining (gauge invariant) degrees, are the --up to 3-- curvature invariants. An apparent discrepancy between this Kinematics and the Quantum Hamiltonian Dynamics of the lower Class A Bianchi Types, occurs due to the existence of the Outer Automorphism Subgroup. This discrepancy is satisfactorily removed by exploiting the quantum version of some classical integrals of motion (conditional symmetries) which are recognized as corresponding to the Outer Automorphisms.