Generalization of Hajicek and Kuchař’s canonical quantization scheme to the 2+1 geometries admitting one Killing vector field

Abstract

The Canonical analysis and subsequent Quantization of the 2+1 action of pure gravity plus a Cosmological constant term is considered, under the assumption of the existence of one manifest Killing field (axial symmetry). The proper imposition of the Quantum analogues of the two Linear (momentum) constraints reduces an initial collection of state vectors, consisting of all smooth functionals of the components(and/or their derivatives) of the spatial metric, to particular scalar smooth functionals. The demand that the midi‐superspace metric (inferred from the kinetic part of the Quadratic (Hamiltonian) constraint) must define on the space of these states an induced metric whose components are given in terms of the same states, which is made possible through an appropriate re‐normalization assumption, severely reduces the possible state vectors to three unique (up to general coordinate transformations) scalar functionals. The Quantum analogue of the Hamiltonian constraint produces a Wheeler‐DeWitt equation based on this reduced manifold of states, which is completely integrated.