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 s ix scale factors, as well as the lapse function and the shift vector presen t - is considered. A first reduction of the initial 6-dimensional configura tion space is achieved by the usage of the information furnished by the qua ntum form of the linear constraints. Further reduction of the space in whic h the wave function - obeying the Wheeler-DeWitt equation - lives, is accom plished by revealing a classical integral of motion, tantamount to an extra symmetry of the corresponding classical Hamiltonian. This symmetry generat or - member of a larger group - is linear in momenta and corresponds to GCT s through the action of the automorphism group - especially through the act ion 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 . (C) 2001 Elsevier Science BN. All rights reserved.