Conditional symmetries, the true degree of freedom and GCT invariant wave functions for the general Bianchi type II vacuum cosmology

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 unrevealing 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 GCT through the action of the automorphism group. Thus, a GCT 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.