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 si x 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-dim ensional configuration space to a 4-dimensional one has been achieved by th e usage of the information furnished by the quantum form of the linear cons traints. Further reduction of the space in which the wave function - obeyin g the Wheeler-DeWitt equation - lives, is accomplished by unrevealing the e xtra symmetries of the Hamiltonian. These symmetries appear in the form of linear in momenta first integrals of motion. Most of these symmetries, corr espond to GCT through the action of the automorphism group. Thus, a GCT inv ariant wave function is found, which depends on the only true degree of fre edom, i.e., the unique curvature invariant, characterizing the hypersurface s t = const. (C) 2001 Elsevier Science B.V. All rights reserved.