Feynman Path Integrals on Compact Lie Groups with Bi-invariant Riemannian Metrics

In this work, we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan development map, the notion of oscillatory integral and the Chernoff approximation theorem. We prove that, for a class of functions of a dense subspace of the relevant Hilbert space, the Feynman map produces the solution of the Schrödinger equation, where the Laplace–Beltrami operator coincides with the second-order Casimir operator of the group.