Towards Canonical Quantum Gravity for G1 Geometries in 2+1 Dimensions with a Lambda--Term

TL;DR

This paper develops a canonical quantization approach for (2+1)-dimensional pure gravity with a cosmological constant, reducing the state space to three unique scalar functionals and solving the Wheeler-DeWitt equation explicitly.

Contribution

It introduces a method to impose quantum constraints in (2+1) gravity with a Lambda-term, leading to a complete integration of the Wheeler-DeWitt equation for G1 geometries.

Findings

Reduced state space to three scalar functionals

Derived and integrated the Wheeler-DeWitt equation explicitly

Demonstrated the role of re-normalization in constraint implementation

Abstract

The canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term is considered, under the assumption of the existence of one spacelike Killing vector field. 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) smooth 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.