On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics

TL;DR

This paper extends Feynman's path integral formulation to Wigner-Dunkl quantum mechanics, analyzing Gaussian wave packets, solving the harmonic oscillator, and exploring Dunkl processes with jumps and Bessel process representations.

Contribution

It introduces a path integral approach within Wigner-Dunkl quantum mechanics and explicitly solves the harmonic oscillator, connecting Dunkl processes with quantum evolution.

Findings

Gaussian wave packet dispersion matches standard quantum mechanics

Explicit Feynman-Kac path integral for harmonic oscillator derived

Dunkl processes with jumps represented by Bessel processes

Abstract

Feynman's path integral approach is studied in the framework of the Wigner-Dunkl deformation of quantum mechanics. We start with reviewing some basics from Dunkl theory and investigate the time evolution of a Gaussian wave packet, which exhibits the same dispersion relation as observed in standard quantum mechanics. Feynman's path integral approach is then extended to Wigner-Dunkl quantum mechanics. The harmonic oscillator problem is solved explicitly. We then look at the Euclidean time evolution and the related Dunkl process. This process, which exhibit jumps, can be represented by two continuous Bessel processes, one with reflection and one with absorbtion at the origin. The Feynman-Kac path integral for the harmonic oscillator problem is explicitly calculated.