Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.
We consider spin system defined on the coadjoint orbit with noncompact symmetry and investigate the quantization. Classical spin with noncompact SU ( N , 1) symmetry is first formulated as a dynamical system and the constraint analysis is performed to reduce the system from the group space to the coadjoint orbit which is a symplectic manifold with Kähler structure. We achieve this by solving the constraint directly. It is shown that the dynamical variables describing the noncompact spins can be written as functions of canonically conjugate variables and canonical quantization is possible on the reduced phase space. With the quantum mechanical Hamiltonian acting on the holomorphic coherent state in Hilbert space, we obtain the exact propagator by solving the time-dependent Schrödinger equation.