Subgroups of the Group of Generalized Lorentz Transformations and Their Geometric Invariants

TL;DR

This paper explores the generalized Lorentz transformations as symmetries of a Finslerian event space, revealing subgroups and invariants that influence the dynamics of fundamental fields and lead to nonlinear Dirac equations.

Contribution

It identifies subgroups of the generalized Lorentz group and their invariants, providing new insights into the symmetry structure of Finslerian spacetime and nonlinear field equations.

Findings

Identified two noncompact subgroups of generalized Lorentz transformations.

Derived geometric invariants associated with these subgroups.

Constructed exact solutions to the nonlinear generalized Dirac equation.

Abstract

It is shown that the group of generalized Lorentz transformations serves as relativistic symmetry group of a flat Finslerian event space. Being the generalization of Minkowski space, the Finslerian event space arises from the spontaneous breaking of initial gauge symmetry and from the formation of anisotropic fermion-antifermion condensate. The principle of generalized Lorentz invariance enables exact taking into account the influence of condensate on the dynamics of fundamental fields. In particular, the corresponding generalized Dirac equation turns out to be nonlinear. We have found two noncompact subgroups of the group of generalized Lorentz symmetry and their geometric invariants. These subgroups play a key role in constructing exact solutions of such equation.