Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity

TL;DR

This paper explores deformed relativistic wave equations within Doubly Special Relativity using a geometric approach based on curved momentum space, addressing unresolved issues like symmetry relations and covariance.

Contribution

It introduces a geometric framework for Klein-Gordon and Dirac equations in DSR, providing new insights into their relations and symmetries.

Findings

Rederived algebraic expressions for wave equations

Analyzed discrete symmetries for Dirac particles

Discussed covariance and Hilbert space formalism

Abstract

In this work we discuss the deformed relativistic wave equations, namely the Klein--Gordon and Dirac equations in a Doubly Special Relativity scenario. We employ what we call a geometric approach, based on the geometry of a curved momentum space, which should be seen as complementary to the more spread algebraic one. In this frame we are able to rederive well-known algebraic expressions, as well as to treat yet unresolved issues, to wit, the explicit relation between both equations, the discrete symmetries for Dirac particles, the fate of covariance, and the formal definition of a Hilbert space for the Klein--Gordon case.