Conservation laws for linear equations on quantum Minkowski spaces

TL;DR

This paper develops a method to find conservation laws for linear equations on quantum Minkowski spaces, applying it to key equations like Klein-Gordon, Dirac, and wave equations, including their symmetries and quantum deformations.

Contribution

It introduces explicit formulas for conserved currents in quantum Minkowski spaces and applies them to important relativistic equations, revealing quantum deformations of classical symmetries.

Findings

Explicit conserved currents for quantum Minkowski space equations

Quantum deformations of classical symmetry operators

Additional operators related to non-commutative calculus

Abstract

The general, linear equations with constant coefficients on quantum Minkowski spaces are considered and the explicit formulae for their conserved currents are given. The proposed procedure can be simplified for *-invariant equations. The derived method is then applied to Klein-Gordon, Dirac and wave equations on different classes of Minkowski spaces. In the examples also symmetry operators for these equations are obtained. They include quantum deformations of classical symmetry operators as well as an additional operator connected with deformation of the Leibnitz rule in non-commutative differential calculus.