Conditional Symmetries and Phase Space Reduction towards G.C.T. Invariant Wave Functions, for the Class A Bianchi Type VI & VII Vacuum Cosmologies

TL;DR

This paper develops a method to reduce the quantum state space of certain cosmological models by exploiting classical symmetries related to G.C.T.s, leading to invariant wave functions that depend on key curvature invariants.

Contribution

It introduces a novel reduction technique using classical integrals of motion linked to G.C.T. symmetries, enabling the construction of invariant wave functions in Bianchi Type VI & VII models.

Findings

Reduced the configuration space using linear constraints.

Identified a classical integral of motion related to G.C.T. symmetries.

Derived G.C.T.-invariant wave functions depending on curvature invariants.

Abstract

The quantization of Class A Bianchi Type VI and VII geometries -with all six scale factors, as well as the lapse function and the shift vector present- is considered. A first reduction of the initial 6-dimensional configuration space is achieved by the usage of the information furnished by the quantum form of the linear constraints. Further reduction of the space in which the wave function -obeying the Wheeler-DeWitt equation- lives, is accomplished by revealing a classical integral of motion, tantamount to an extra symmetry of the corresponding classical Hamiltonian. This symmetry generator -member of a larger group- is linear in momenta and corresponds to G.C.T.s through the action of the automorphism group -especially through the action of the outer automorphism subgroup. Thus, a G.C.T. invariant wave function is found, which depends on one combination of the two curvature invariants --which uniquely and irreducibly characterizes the hypersurfaces t=const.