The Lorentz Extension as Consequence of the Family Symmetry

TL;DR

This paper proposes an algebraic model linking family symmetry of leptons to an extended Lorentz group, revealing new invariance properties and implications for weak interactions and neutrino oscillations.

Contribution

It introduces a novel mapping between SL(3,C) and a generalized Lorentz group, extending the Dirac equation and Lorentz symmetry to incorporate family symmetry effects.

Findings

Extended Lorentz group requires an additional generator.

Generalized vertex matrix explains weak interaction features.

Model provides insights into neutrino oscillations.

Abstract

In this paper we postulate an algebraic model to explain how the symmetry of three lepton species plays its role in the Lorentz extension. Inspired by the two-to-one mapping between the group SL (2, C) and the Lorentz group, we design a mapping between SL (3, C) group, which displays the family symmetry, and a generalized Lorentz group. Following the conventional method, we apply the mapping results to Dirac equation to discuss its transformation invariance, and it turns out that only when the vertex matrix is extended to the combination can the Dirac-equation-form be reserved. At the same time we find that the Lorentz group has to be extended with an additional generator . The generalized vertex matrix is helpful in understanding the axial-like form of weak interaction and the neutrino oscillations.