12 February 1998 Ž . Physics Letters B 419 1998 30–36 Isolation of the true degree of freedom and normalizable wave functions for the general type V cosmology T. Christodoulakis, G. Kofinas, E. Korfiatis, A. Paschos Nuclear and Particle Physics Section , Physics Department, UniÕersity of Athens, Panepistimiopolis, Athens 15771, Greece Received 28 April 1997; revised 23 October 1997 Editor: R. Gatto Abstract Ž . The quantization of the most general Type V geometry with all six scale factors as well as the shift vector present is Ž considered. The information carried by the linear constraints is used to reduce the Wheeler–DeWitt equation arising from a . valid Hamiltonian found earlier , which initially included six variables, to a final PDE in three variables, getting rid of three Ž . redundant variables gauge degrees of freedom . The full space of solutions to this equation is presented. In trying to interpret these wave functions, we are led through further consideration of the action of the automorphism group on the configuration space, to a final reduction to the one and only true degree of freedom, i.e. the only independent curvature invariant of the slice t sconstant. Thus, a normalizable wave function in terms of the true degree of freedom is obtained. q 1998 Elsevier Science B.V. 1. It is well known that in order to quantize gravity in a non-perturbative way one has to realize wx the following steps 1 : Ž. Ž . mn Ž . i define the basic operators g x, t , p x, t ˆ ˆ mn and the canonical commutation relations they satisfy; ˆ Ž. Ž . ii define quantum operators H x, t whose clas- m sical counterparts are the constraint functions Ž . H x, t ; m Ž . w x iii define the quantum states C g as the com- ˆ Ž . mon null eigenvectors of H x, t , i.e. these satisfy- m ˆ w x ing H C g s 0. As a consequence one has to m ˆ Ž . check that H x, t form a closed algebra under the m Ž . basic Canonical Commutation Relations CCR ; Ž . iv find the states and define the inner product in the space of these states. It is fair to say that the full program has not yet been carried out, although partial steps have been wx made 2 . In this work we apply these steps to the Bianchi Type V homogeneous cosmological model and not to the full quantum gravity. Since this way we only quantize a finite number of degrees of freedom we are in the quantum cosmology approximation. The difference of the present work from previous investi- gations on quantum cosmology is twofold: on one hand, usually only up to three gravitational degrees Ž . of freedom chosen in some sense arbitrarily were Ž considered at the classical level say, the three scale . factors of some anisotropic Bianchi type model and Ž.Ž thus quantized, while we allow for all six g t the ab time components of the spatial metric with respect to . the invariant basis one-forms , as well as the shift 0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž . PII S0370-2693 97 01419-6