NONPERTURBATIVE CANONICAL QUANTIZATION OF MINISUPERSPACE MODELS - BIANCHI TYPE-I AND TYPE-II

We carry out the quantization of the full type I and II Bianchi models following the nonperturbative canonical quantization program. These h omogeneous minisuperspaces are completely soluble; i.e., it is possibl e to obtain the general solution to their classical equations of motio n in an explicit form. We determine the sectors of solutions that corr espond to different spacetime geometries, and prove that the parameter s employed to describe the different physical solutions define a good set of coordinates in the phase space of these models. Performing a tr ansformation from the Ashtekar variables to this set of phase-space co ordinates, we endow the reduced phase space of each of these systems w ith a symplectic structure. The symplectic forms obtained for the type I and II Bianchi models are then identified as those of the cotangent bundles over L(+,+)xS2XS1 (modulo some identification of points) and L(+,+)xS1, respectively, with L(+,+)+ the positive quadrant of the fut ure light cone. We construct a closed algebra of Dirac observables i n each of these reduced phase spaces, and complete the quantization pr ogram by finding unitary irreducible representations of these algebras . The real Dirac observables are represented in this way by self-adjoi nt operators, and the spaces of quantum physical states are provided w ith a Hilbert structure.