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Path Integrals and Singular Potentials: A Novel Regularization for Heat Kernel Asymptotics and Brownian Motion

Revista, Zen, MATH, 10

Published Dec 5, 2025• 0 citations
Publisher

Abstract

We present a novel regularization scheme for path integrals applied to quantum mechanical systems with singular potentials. Our approach focuses on the heat kernel asymptotics and their connection to Brownian motion. We introduce a short-time expansion for the heat kernel, incorporating the effects of singular potentials through a carefully constructed effective potential. This effective potential is derived using a combination of distributional calculus and stochastic calculus, ensuring that the resulting path integral is well-defined and produces physically meaningful results. The regularization method is then applied to specific examples of singular potentials, such as the inverse square potential, demonstrating its effectiveness in resolving the ambiguities associated with these potentials. We further explore the implications of our regularization scheme for the behavior of Brownian motion in the presence of singular potentials, connecting the quantum mechanical description to the classical stochastic process.

DOI
10.5281/zenodo.17830464

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Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.
A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the inverse square potential — possibly combined with a delta-function interaction. The emergence of these singular potentials as low-energy nonrelativistic limits of quantum field theory is highlighted. Not surprisingly, the analog of ultraviolet regularization is required for the interpretation of these singular problems.
We present the path-integral solutions to the distributions in classical (Gibbs) and quantum (Wigner) statistical mechanics. The kernel of the distributions are derived in two ways - one by time slicing and defining the appropriate short-time interval phase space matrix element and second by making use of the kernel in the path-integral approach to quantum mechanics. We show that the two approaches are perturbatively identical. We also present another computation for the Wigner kernel, which is also the Liouville kernel, for the harmonic oscillator and free particle. These kernels may be used as the starting point in the perturbative expansion of the Wigner kernel for any potential. With the kernel solved, we essentially solve also the distributions in classical and quantum statistical mechanics.
CONTENTS: Preface Bosonic Path Integrals Nelson's Stochastic Mechanics by Means of Feynman Path Integrals Feymman path-integrals for Brownian Classical Motions (1+1)-dimensional Burger turbulence in the framework of Hopf-Cole Transformation and Feymman Trajectories path-integrals Feynman-Wiener Path Integral Representations for the Quantum Harmonic Oscillator with Time-Dependent Stochastic Frequency: Quantum Brownian Motions and Navier-Stokes Weakly Turbulence Some studies on functional integrals representations for fluid motion with random conditions Statistics turbulence suppression by a strong magnetic field: A Feynman path integral argument A non-linear diffusion equation, Some existence and uniqueness results on weak solutions for random initial conditions and its path integral representation -- a mathematical methods approach The Exact Evalution of the Response Function for Wave Propagation in Random Media: A Functional Integral Approach On the Abstract Method of Integration for Wave Equation and some Applications for Wave Physics Feynman Path-Integral Studies for Quantum Particles Propagators Interacting with Environments The Quantum Fractional Dimensional Brownian Motion: A Path-Integral Study Ohm's Law: A Langevin Path-Integral Study at Classical and Quantum W.K.B. Regime On the Statistical of An Ideal Gas with Mass Fluctuations and the Boltzman Ergodic Theorem: The Combined Boltzman-Tsallis Statistics The Suppression of the Quantum Phenomena by an External Strong White-Noise Magnetic Field: A Path Integral Argument Index.
We study the heat statistics of a quantum Brownian motion described by the Caldeira-Leggett model. By using the path integral approach, we introduce a concept of the quantum heat functional along every pair of Feynman paths. This approach has the advantage of improving our understanding about heat in quantum systems. First, we demonstrate the microscopic reversibility of the system by connecting the heat functional to the forward and time-reversed probabilities. Second, we analytically prove the quantum-classical correspondence of the heat functional and their statistics, which allows us to obtain better intuitions about the difference between classical and quantum heat.
By introducing novel concepts of work and heat functionals along individual pathpathpath, we reformulate quantum Jarzynski equality based on the path integral formulation of quantum mechanics. When applied to an open quantum system described by the quantum Brownian motion model, we establish a consistent framework of quantum thermodynamics in the strong coupling regime. Using the work and heat functionals, we derive a path-integral expression for the work and heat statistics. This formalism provides an effective way to calculate the work and heat in open quantum systems by utilizing various path integral techniques. By performing the ℏ\hbarℏ expansion, we analytically prove the quantum-classical correspondence of the work and heat statistics. In addition, we obtain the nnn-th order quantum correction to the classical work.