Dirac factorization and fractional calculus

TL;DR

This paper introduces a novel application of Dirac factorization to fractional calculus, extending Pauli matrices to handle fractional powers of operators and linearize complex evolution equations.

Contribution

It presents a new method using Dirac factorization and extended Pauli matrices for fractional operator powers and evolution equations, offering an alternative to existing transform techniques.

Findings

Successfully applies Dirac factorization to fractional powers of operators

Extends Pauli matrices for fractional calculus applications

Provides a comparison with other transform-based methods

Abstract

We show that the Dirac factorization method can be successfully employed to treat problems involving operators raised to a fractional power. The technique we adopt is based on an extension of the Pauli matrices and the properties of the roots of unity. We also comment about the possibility of using the method to linearize evolution equations containing the $n$-th root of differential operators and make a comparison with other techniques involving suitable transforms.