Dirac spinors for Doubly Special Relativity and $\kappa$-Minkowski noncommutative spacetime

TL;DR

This paper develops a Dirac equation compatible with doubly-special relativity, incorporating an observer-independent length/momentum scale, and explores its natural emergence in kappa-Minkowski noncommutative spacetime, highlighting the importance of differential calculus choices.

Contribution

It constructs a consistent Dirac equation for doubly-special relativity and clarifies its relation to kappa-Minkowski spacetime, emphasizing the role of differential calculus in maintaining observer-independence.

Findings

Mild deformation of Dirac spinors due to the second invariant scale

Derivation of the Dirac equation within kappa-Minkowski spacetime framework

Differential calculus choice critically affects the form of the Dirac equation

Abstract

We construct a Dirac equation that is consistent with one of the recently-proposed schemes for a "doubly-special relativity", a relativity with both an observer-independent velocity scale (still naturally identified with the speed-of-light constant) and an observer-independent length/momentum scale (possibly given by the Planck length/momentum). We find that the introduction of the second observer-independent scale only induces a mild deformation of the structure of Dirac spinors. We also show that our modified Dirac equation naturally arises in constructing a Dirac equation in the kappa-Minkowski noncommutative spacetime. Previous, more heuristic, studies had already argued for a possible role of doubly-special relativity in kappa-Minkowski, but remained vague on the nature of the consistency requirements that should be implemented in order to assure the observer-independence of the two scales. We find that a key role is played by the choice of a differential calculus in kappa-Minkowski. A much-studied choice of the differential calculus does lead to our doubly-special relativity Dirac equation, but a different scenario is encountered for another popular choice of differential calculus.