Skeletonizations of Phase Space Paths

Construction of skeletonized path integrals for a particle moving on a curved spatial manifold is considered. As shown by DeWitt, Kuchař and others, while the skeletonized configuration space action can be written unambiguously as a sum of Hamilton principal functions, different choices of the measure will lead to different Schrödinger equations. On the other hand, the Liouville measure provides a unique measure for a skeletonized phase space path integral, but there is a corresponding ambiguity in the skeletonization of a path through phase space. A family of skeletonization rules described by Kuchař and referred to here as geodesic interpolation is discussed, and shown to behave poorly under the involution process, wherein intermediate points are removed by extremization of the skeletonized action. A new skeletonization rule, tangent interpolation, is defined and shown to possess the desired involution properties. 1. SKELETONIZED PATH INTEGRALS The path integral method seems to provide a generic recipe for producing a quantum theory given a classical action. This recipe is based on the principle that amplitudes are given by sums, over the relevant histories, of the complex exponential of the action. For example, a wavefunction ψ(xi, ti) is propagated to a later time tf > ti by ψ(xf , tf) = ∫ xf Dx eψ(xi, ti), (1) e-mail: [email protected] where the integral is over all paths ending at the argument xf . However, in order to provide a constructive definition for an expression like (1) one needs to give a meaning to formal concepts like an integral over paths, and it is in this step that decisions need to be made in the implementation of the quantum theory. The system considered in this work is a free non-relativistic particle of unit mass moving on a curved n-dimensional spatial manifold, which is described by the action