Hamiltonian structure and quantization of(2+1)-dimensional gravity coupled toparticles*

Abstract

It is shown that the reduced particle dynamics of(2+1)-dimensional gravity in the maximally slicinggauge has a Hamiltonian form. This is proved directly forthe two-body problem and for the three-body problem byusing the Garnier equations for isomonodromictransformations. For a number of particles greater thanthree the existence of the Hamiltonian is shown to be aconsequence of a conjecture by Polyakov which connectsthe accessory parameters of the Fuchsian differentialequation which solves theSU(1,1) Riemann-Hilbert problem, to the Liouvilleaction of the conformal factor which describes thespace metric.We give the exact diffeomorphism which transforms the expression of thespinning cone geometry in the Deser-Jackiw-'t Hooftgauge to the maximally slicing gauge. It is explicitlyshown that the boundary term in the action, written inHamiltonian form gives the Hamiltonian for the reducedparticle dynamics.The quantum mechanical translation of the two-particleHamiltonian gives rise to the logarithm of theLaplace-Beltrami operator on a cone whose angulardeficit is given by the total energy of the systemirrespective of the masses of the particles thus provingat the quantum level a conjecture by 't Hooft on thetwo-particle dynamics. The quantum mechanical Greenfunction for the two-body problem is given.