Hamiltonian structure and quantization of 2+1 dimensional gravity coupled to particles

TL;DR

This paper demonstrates the Hamiltonian formulation and quantization of 2+1 dimensional gravity coupled to particles, establishing explicit connections between classical dynamics, gauge transformations, and quantum properties, including Green's functions.

Contribution

It provides the first direct Hamiltonian formulation for multi-particle systems in 2+1 gravity and links classical solutions to quantum operators through the Liouville action and Riemann-Hilbert problems.

Findings

Hamiltonian form for 2+1 gravity with particles established

Quantum two-particle Hamiltonian related to Laplace-Beltrami operator on a cone

Green's function for two-body quantum problem derived

Abstract

It is shown that the reduced particle dynamics of 2+1 dimensional gravity in the maximally slicing gauge has hamiltonian form. This is proved directly for the two body problem and for the three body problem by using the Garnier equations for isomonodromic transformations. For a number of particles greater than three the existence of the hamiltonian is shown to be a consequence of a conjecture by Polyakov which connects the auxiliary parameters of the fuchsian differential equation which solves the SU(1,1) Riemann-Hilbert problem, to the Liouville action of the conformal factor which describes the space-metric. We give the exact diffeomorphism which transforms the expression of the spinning cone geometry in the Deser, Jackiw, 't Hooft gauge to the maximally slicing gauge. It is explicitly shown that the boundary term in the action, written in hamiltonian form gives the hamiltonian for the reduced particle dynamics. The quantum mechanical translation of the two particle hamiltonian gives rise to the logarithm of the Laplace-Beltrami operator on a cone whose angular deficit is given by the total energy of the system irrespective of the masses of the particles thus proving at the quantum level a conjecture by 't Hooft on the two particle dynamics. The quantum mechanical Green's function for the two body problem is given.