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

It is shown that the reduced particle dynamics of (2+1)-dimensional gravity in the maximally slicing gauge has a 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 accessory 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 function for the two-body problem is given.