Conditional symmetries and phase space reduction towards GCT invariant wave functions, for the class A Bianchi type VI and VII vacuum cosmologies

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 GCTs through the action of the automorphism group — especially through the action of the outer automorphism subgroup. Thus, a GCT invariant wave function is found, which depends on one combination of the two curvature invariants — which uniquely and irreducibly, characterize the hypersurfaces t=const.