QUANTIZATION OF A RELATIVISTIC PARTICLE ON THE SL(2,R) MANIFOLD BASEDON HAMILTONIAN REDUCTION

A quantum theory is constructed for the system of a relativistic parti cle with mass m moving freely on the SL(2,R) group manifold. Applied t o the cotangent bundle of SL(2,R), the method of Hamiltonian reduction allows us to split the reduced system into two coadjoint orbits of th e group. We find that the Hilbert space consists of states given by th e discrete series of the unitary irreducible representations of SL(2,R ), and with a positive-definite, discrete spectrum.