  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.

High Energy Physics - Theory;General Relativity and Quantum Cosmology