A Path Integral Treatment of Time-dependent Dunkl Quantum Mechanics

TL;DR

This work extends the Feynman path-integral formalism to time-dependent quantum systems within the Wigner-Dunkl framework by employing generalized canonical transformations that render the propagator stationary. The authors derive a general time-dependent propagator decomposed into parity sectors and apply it to the Dunkl harmonic oscillator, obtaining exact expressions in terms of modified Bessel functions and the deformed exponential $E_\nu$, with the evolution governed by an auxiliary equation for the scale function $\rho(t)$. Two analytically solvable models—the Dunkl Caldirola–Kanai oscillator and a strongly pulsating-mass oscillator—yield explicit propagators and parity-separated wavefunctions, illustrating the method's capacity to capture damping and periodic energy modulation. The results highlight the role of reflection symmetry in Dunkl mechanics and provide benchmarks for dissipative and non-Hermitian quantum dynamics, suggesting avenues for multi-dimensional and PT-symmetric generalizations.

Abstract

This paper presents an analytical treatment of the path integral formalism for time-dependent quantum systems within the framework of Wigner-Dunkl mechanics, emphasizing systems with varying masses and time-dependent potentials. 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.