Combinatorial solutions to the Hamiltonian constraint in (2+1)-dimensional Ashtekar gravity

Dirac's quantization of the (2+1)-dimensional analog of Ashtekar's approach to quantum gravity is investigated. After providing a diffeomorphism-invariant regularization of the Hamiltonian constraint, we find a set of solutions to this Hamiltonian constraint which is a generalization of the solution discovered by Jacobson and Smolin. These solutions are given by particular linear combinations of the spin network states. While the classical counterparts of these solutions have a degenerate metric, due to a 'quantum effect', the area operator has a non-vanishing action on these states. For computational simplicity, we restricted the analysis to piecewise analytic graphs with four-point vertices and to finite-dimensional representations of SL(2,R). It is considered, however, that the analysis will have to be extended to more generic cases in order to obtain geometrodynamical states. We also discuss how to extend our results to (3+1) dimensions.