Conditional Symmetries, the True Degree of Freedom and G.C.T. Invariant Wave functions for the general Bianchi Type II Vacuum Cosmology

TL;DR

This paper quantizes the general Bianchi Type II vacuum cosmology, identifies symmetries to reduce the wave function's configuration space, and constructs a G.C.T.-invariant wave function depending on a single curvature invariant.

Contribution

It introduces a method to reduce the Wheeler-DeWitt equation's solution space by revealing symmetries, leading to a G.C.T.-invariant wave function with a single true degree of freedom.

Findings

Wave function depends on the unique curvature invariant.

Symmetries simplify the quantum cosmology model.

G.C.T. invariance is achieved in the wave function.

Abstract

The quantization of the most general Bianchi Type II geometry -with all six scale factors, as well as the lapse function and the shift vector, present- is considered. In an earlier work, a first reduction of the initial 6-dimensional configuration space, to a 4-dimensional one, has been 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 unrevealling the extra symmetries of the Hamiltonian. These symmetries appear in the form of -linear in momenta- first integrals of motion. Most of these symmetries, correspond to G.C.T.s through the action of the automorphism group. Thus, a G.C.T. invariant wave function is found, which depends on the only true degree of freedom, i.e. the unique curvature invariant, characterizing the hypersurfaces t=const.