Path integrals on homogeneous manifolds

Abstract

Quantum kinematics are considered for two classes of systems: (i) the coordinate space is a homogeneous Riemannian manifold and the kinetic energy is the Laplace–Beltrami operator, (ii) the phase-space is a homogeneous Kählerian manifold where the algebra of observables is the universal enveloping algebra of the symmetry group. In both the cases, the path integrals are derived from more fundamental principles. The derivation enables a definite description of the classes of trajectories which constitute the domain of integration in the path integrals. The specification of the classes of functions is important for consistency of the approach. The standard action functional and the variational principle appear in the steepest descent method corresponding to the semiclassical approximation.