Functional Integration on Constrained Function Spaces II: Applications
J. LaChapelle
TL;DR
The paper develops a conditional/conjugate functional-integral framework for constrained function spaces, introducing delta functionals on constraint spaces and gamma/Poisson-type integrators to enforce kinematical and dynamical constraints. It re-derives classical quantum-mechanical propagators for fixed end-points, quotient spaces, bounded and segmented configuration spaces, using path-decomposition ideas and holonomy considerations. As a novel application, it models average prime-counting functions as constrained gamma processes, deriving expressions that approximate the classical pi(x) and extending the approach to twin primes with gap-dependent normalizations, yielding Goldbach-type implications on average. Overall, the work links functional integration, constraint handling, and number-theoretic counting, suggesting new computational techniques and potential extensions to quantum fields and matrix-valued paths.
Abstract
Some well-known examples of constrained quantum systems commonly quantized via Feynman path integrals are re-examined using the notion of conditional integrators introduced in [1]. The examples yield some new perspectives on old results. As an interesting new application, the formalism is used to construct a physical model of average prime counting functions modeled as a constrained gamma process.