On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics

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.

arXiv commentTypos corrected and DOI added to references

References in corpus (20)

• Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
• Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions
• Deformed Heisenberg algebra with reflection
• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
• The functional-analytic versus the functional-integral approach to quantum Hamiltonians: The one-dimensional hydrogen atom.
• The Dunkl oscillator in three dimensions
• Deformed Heisenberg algebra and fractional spin field in 2+1 dimensions
• Hidden nonlinear supersymmetries in pure parabosonic systems
• Exact Solution of the Relativistic Dunkl Oscillator in $(2+1)$ Dimensions
• Bosons, fermions and anyons in the plane, and supersymmetry
• Minimal bosonization of supersymmetry
• Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential
• Asymptotic analysis for the Dunkl kernel
• A chaotic representation property of the multidimensional Dunkl processes
• Relativistic Solutions of Generalized-Dunkl Harmonic and Anharmonic Oscillators
• The Feynman Path Integral: An Historical Slice
• On the Hydrogen Atom via Wigner-Heisenberg Algebra
• Supersymmetry without Fermions
• Power-Duality in Path Integral Formulation of Quantum Mechanics

Cited by in corpus (12)

• Dunkl-Schrödinger equation with time-dependent harmonic oscillator potential
• The Dunkl-Fokker-Planck Equation in $1+1$ Dimensions
• Time-Dependent Dunkl-Schrödinger Equation with an Angular-Dependent Potential
• The Condensation of Ideal Dunkl-Bose Gas in Power-Law Traps
• A Path Integral Treatment of Time-dependent Dunkl Quantum Mechanics
• One-dimensional Dunkl Quantum Mechanics: A Path Integral Approach
• Dunkl-Klein-Gordon Equation in Higher Dimensions
• Time-dependent Dunkl-Pauli Oscillator
• Bounding the Wigner Deformation Parameter in Harmonically Trapped Bose Gases
• Spectral and Thermal Analysis of the Morse Potential within the Dunkl Formalism: Analytical Approximations and Applications
• Exact Solutions of the Schrödinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well
• A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties