Hamiltonian solutions of the 3-body problem in (2+1)-gravity

TL;DR

This paper thoroughly analyzes the 3-body gravitational problem in (2+1) dimensions, deriving the Hamiltonian, exploring symmetries, and setting the stage for quantization, revealing a U(2) symmetry linked to braid group structures.

Contribution

It provides an explicit ADM Hamiltonian for the 3-body problem in (2+1)-gravity and highlights a U(2) symmetry related to braid groups, advancing the understanding of quantization in this context.

Findings

Explicit form of the ADM Hamiltonian in (2+1)-gravity

Identification of a U(2) symmetry related to braid groups

Discussion on single-valued energy eigenfunctions

Abstract

We present a full study of the 3-body problem in gravity in flat (2+1)-dimensional space-time, and in the nonrelativistic limit of small velocities. We provide an explicit form of the ADM Hamiltonian in a regular coordinate system and we set up all the ingredients for canonical quantization. We emphasize the role of a U(2) symmetry under which the Hamiltonian is invariant and which should generalize to a U(N-1) symmetry for N bodies. This symmetry seems to stem from a braid group structure in the operations of looping of particles around each other, and guarantees the single-valuedness of the Hamiltonian. Its role for the construction of single-valued energy eigenfunctions is also discussed.