The Phase Space Structure of Multi Particle Models in 2+1 Gravity

TL;DR

This paper explores the phase space structure of a simple 2+1 dimensional gravity model with point particles, revealing insights into quantum gravity and non-commutative spacetime features without full quantization.

Contribution

It derives the reduced phase space of a multi-particle 2+1 gravity model directly from Einstein gravity as a topological field theory, without prior assumptions.

Findings

Reduced phase space has a non-commutative spacetime structure upon quantization.

The model provides a simplified setting to study quantum gravity features.

The phase space is characterized by relative positions and conjugate momenta of particles.

Abstract

What can we learn about quantum gravity from a simple toy model, without actually quantizing it? The toy model consists of a finite number of point particles, coupled to three dimensional Einstein gravity. It has finitely many physical degrees of freedom. These are basically the relative positions of the particles in spacetime and the conjugate momenta. The resulting reduced phase space is derived from Einstein gravity as a topological field theory. The crucial point is thereby that we do not make any a priori assumptions about this phase space, except that the dynamics of the gravitational field is defined by the Einstein Hilbert action. This already leads to some interesting features of the reduced phase space, such as a non-commutative structure of spacetime when the model is quantized.