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

TL;DR

This paper explores Dirac's quantization of (2+1)-dimensional Ashtekar gravity, presenting new solutions to the Hamiltonian constraint that involve spin network states and quantum effects on geometry.

Contribution

It introduces a generalized set of solutions to the Hamiltonian constraint in (2+1)-dimensional Ashtekar gravity, extending previous work by Jacobson and Smolin, with implications for quantum geometry.

Findings

Solutions are linear combinations of spin network states.

Quantum effects lead to nonvanishing area operator action.

Classical solutions have degenerate metrics, but quantum effects modify this.

Abstract

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 degenerate metric, due to a \lq quantum effect' the area operator has nonvanishing action on these states. We also discuss how to extend our results to (3+1)-dimensions.