Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation

TL;DR

This paper generalizes the Dirac equation for Finslerian space-time, deriving a new symmetry group and providing explicit forms and solutions, extending relativistic quantum mechanics to anisotropic geometries.

Contribution

It introduces a generalized Lorentz symmetry and Dirac equation for Finslerian space-time, expanding the framework of relativistic quantum theory.

Findings

Derived the metric and symmetry group for Finslerian space-time.

Formulated the generalized Dirac equation respecting the new symmetry.

Presented an exact solution to the nonlinear generalized Dirac equation.

Abstract

The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e., Finslerian space-time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented.