Hamiltonian structure of 2+1 dimensional gravity

TL;DR

This paper reviews the Hamiltonian ADM approach to 2+1 dimensional gravity, especially with point particles, showing how the gravitational field can be reduced to particle degrees of freedom and quantized.

Contribution

It demonstrates the reduction of 2+1 gravity to a Hamiltonian system involving only particle variables and provides a quantization framework for the two-particle case.

Findings

Hamiltonian structure established for 2+1 gravity with point particles

Reduced dynamics expressed solely in terms of particle positions and momenta

Canonical quantization achieved for the two-particle system

Abstract

A summary is given of some results and perspectives of the hamiltonian ADM approach to 2+1 dimensional gravity. After recalling the classical results for closed universes in absence of matter we go over the the case in which matter is present in the form of point spinless particles. Here the maximally slicing gauge proves most effective by relating 2+1 dimensional gravity to the Riemann- Hilbert problem. It is possible to solve the gravitational field in terms of the particle degrees of freedom thus reaching a reduced dynamics which involves only the particle positions and momenta. Such a dynamics is proven to be hamiltonian and the hamiltonian is given by the boundary term in the gravitational action. As an illustration the two body hamiltonian is used to provide the canonical quantization of the two particle system.