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Physica Scripta

Path integral formalism for the Klein–Gordon-Dunkl oscillator in (1+d)-dimensional space-time coordinates

H. Benzair, Tahar Boudjedaa, M. Merad

Published Aug 12, 2024• 10 citations
Publisher

Abstract

Abstract In this paper, we extend the path integral formalism for the Klein–Gordon oscillator by replacing the standard derivative with the Dunkl derivative in 1D, 2D, and even in d − dimensions. Using space-time Cartesian coordinates, we derive the propagators and solve the problems exactly. We determine the energy eigenvalues and corresponding wave functions for the spinless particle. In limiting cases, where the Dunkl derivative parameters are set to zero, our results converge appropriately to those found in the literature for these problems.

DOI
10.1088/1402-4896/ad6e2f

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Abstract In this paper, we extend the path integral formalism for the Klein–Gordon oscillator by replacing the standard derivative with the Dunkl derivative in 1D, 2D, and even in d − dimensions. Using space-time Cartesian coordinates, we derive the propagators and solve the problems exactly. We determine the energy eigenvalues and corresponding wave functions for the spinless particle. In limiting cases, where the Dunkl derivative parameters are set to zero, our results converge appropriately to those found in the literature for these problems.
Abstract In this paper, we extend the path integral formalism for the Dirac oscillator in (1+1) dimension by replacing the spatial derivative with the Dunkl derivative. Utilizing representations in position space-time coordinates, we precisely calculate the propagator, expressed in terms of generalized Hermite polynomials. The energy eigenvalues of the electron, along with their corresponding wave functions, are determined. In special cases, we can precisely evaluate the non-relativistic energy eigenvalues and wave functions, even in the absence of Dunkl parameters.
In this paper, we investigate a central coordinate system within the framework of the Wigner–Dunkl deformation of quantum mechanics in d spatial dimensions, focusing on the dynamics of a relativistic spinless particle described by the Klein–Gordon equation. The analysis is carried out using Feynman’s path integral formalism in hyperspherical coordinates, where the Lagrangian includes an effective potential depending explicitly only on radial variables. This approach allows for the exact separation of variables in d-dimensional Dunkl space, enabling an analytical evaluation of the path integral. Particular emphasis is placed on the attractive Coulomb-type potential, for which the exact propagator is obtained in closed form. As a result, we derive the energy spectrum and corresponding wave functions of the Dunkl–Klein–Gordon system, expressed in terms of special functions that reflect the modified symmetries induced by the Dunkl operators.
Abstract The Green’s functions for relativistic oscillators with spin-0 and spin-1/2, within the framework of a new generalization of the Dunkl derivative with three parameters, are calculated using the path integral approach. For simplicity, this derivative is reformulated to depend on only two parameters. As a result, the energy eigenvalues and their corresponding wave functions are determined in both cases as functions of the three deformed parameters. In the limiting cases of the deformed parameters, the standard Dunkl derivative system is recovered, even in the absence of the Dunkl parameters.
In this study, we replace the standard partial derivatives in the Klein–Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl–Klein–Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional hyperspherical coordinates, followed by an analysis of the d-dimensional Dunkl–Klein–Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl–Klein–Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.
This study addresses a significant gap in the literature by extending the path integral formalism to time-dependent systems within the Wigner–Dunkl framework, deriving exact propagator solutions for analytically solvable cases. By employing generalized canonical transformations, we reformulated the path integral to develop an explicit expression for the propagator. This formalism is applied to specific cases, including a Dunkl-harmonic oscillator with time-dependent mass and frequency. Solutions for the Dunkl–Caldirola-Kanai oscillator and a model with a strongly pulsating mass are derived, providing exact propagator expressions and corresponding wave functions. These findings extend the utility of Dunkl operators in quantum mechanics, offering new insights into the dynamics of time-dependent quantum systems and possibly find application in quantum optics, plasma physics, and other fields.