Non-Perturbative Canonical Quantization of Minisuperspace Models: Bianchi Types I and II

We carry out the quantization of the full type I and II Bianchi models following the non-perturbative canonical quantization program. These homogeneous minisuperspaces are completely soluble, i.e., it is possible to obtain the general solution to their classical equations of motion in an explicit form. We determine the sectors of solutions that correspond to different spacetime geometries, and prove that the parameters employed to describe the different physical solutions define a good set of coordinates in the phase space of these models. Performing a transformation from the Ashtekar variables to this set of phase space coordinates, we endow the reduced phase space of each of these systems with 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 ${\cal L}^+_{(+,+)}\times S^2\times S^1$ and ${\cal L}^+_{(+,+)}\times S^1$, respectively. We construct a closed *-algebra of Dirac observables in each of these reduced phase spaces, and complete the quantization program by finding unitary irreducible representations of these algebras. The real Dirac observables are represented in this way by self-adjoint operators, and the spaces of quantum physical states are provided with a Hilbert structure.