Three dimensional loop quantum gravity: coupling to point particles

TL;DR

This paper develops a quantum gravity framework in three dimensions that includes point particles, extending loop quantum gravity techniques to describe their quantum states and interactions.

Contribution

It introduces a complete quantum description of 3D gravity coupled with spinning particles, including the kinematical and physical Hilbert spaces and explicit distance operators.

Findings

Constructed the kinematical Hilbert space for the coupled system

Defined the physical Hilbert space using spin foam amplitudes

Derived an explicit expression for the distance operator between particles

Abstract

We consider the coupling between three dimensional gravity with zero cosmological constant and massive spinning point particles. First, we study the classical canonical analysis of the coupled system. Then, we go to the Hamiltonian quantization generalizing loop quantum gravity techniques. We give a complete description of the kinematical Hilbert space of the coupled system. Finally, we define the physical Hilbert space of the system of self-gravitating massive spinning point particles using Rovelli's generalized projection operator which can be represented as a sum over spin foam amplitudes. In addition we provide an explicit expression of the (physical) distance operator between two particles which is defined as a Dirac observable.