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

It is shown that the reduced particle dynamics of (2 + 1)-dimensional gravi ty in the maximally slicing gauge has a Hamiltonian form. This is proved di rectly for the two-body problem and for the three-body problem by using the Garnier equations for isomonodromic transformations. For a number of parti cles greater than three the existence of the Hamiltonian is shown to be a c onsequence of a conjecture by Polyakov which connects the accessory paramet ers of the Fuchsian differential equation which solves the SU(1, 1) Riemann -Hilbert problem, to the Liouville action of the conformal factor which des cribes the space metric. We give the exact diffeomorphism which transforms the expression of the spi nning cone geometry in the Deser-Jackiw-'t Hooft gauge to the maximally sli cing gauge. It is explicitly shown that the boundary term in the action, wr itten in Hamiltonian form gives the Hamiltonian for the reduced particle dy namics. The quantum mechanical translation of the two-particle Hamiltonian gives ri se to the logarithm of the Laplace-Beltrami operator on a cone whose angula r deficit is given by the total energy of the system irrespective of the ma sses of the particles thus proving at the quantum level a conjecture by 't Hooft on the two-particle dynamics. The quantum mechanical Green function f or the two-body problem is given.