Weyl Quantization of Fractional Derivatives

TL;DR

This paper develops a framework for defining quantum analogs of fractional derivatives, extending classical fractional calculus into the quantum domain using Weyl quantization and Fourier analysis.

Contribution

It introduces a method to construct quantum analogs of fractional Riemann-Liouville and Liouville derivatives using Weyl quantization and analytic function representations.

Findings

Quantum analogs of fractional derivatives are formulated.

The approach uses Fourier transforms and analytic function representations.

Potential applications in quantum physics and fractional calculus are suggested.

Abstract

The quantum analogs of the derivatives with respect to coordinates q_k and momenta p_k are commutators with operators P_k and $Q_k. We consider quantum analogs of fractional Riemann-Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann-Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for analytic functions. To define a quantum analog of the fractional Liouville derivative, which is defined on the real axis, we can use the representation of the Weyl quantization by the Fourier transformation.