Physical velocity of particles in relativistic curved momentum space

TL;DR

This paper proves that in relativistic theories with curved momentum space, the physical velocity of particles equals their group velocity, resolving a long-standing discrepancy and extending previous linear-order results to all orders.

Contribution

It provides a general proof that particle velocity matches group velocity in curved momentum space theories at all orders, clarifying the velocity concept in deformed relativistic symmetries.

Findings

Velocity equals group velocity in curved momentum space theories.

Result holds at all orders in deformation parameter.

Valid even when deformation depends on coordinates and momenta.

Abstract

We show in general that for a relativistic theory with curved momentum space, i.e.~a theory with deformed relativistic symmetries, the physical velocity of particles coincides with their group velocity. This clarifies a long-standing question about the discrepancy between coordinate and group velocity for this kind of theories. The first evidence that this was the case had been obtained at linear order in the deformation parameter in Phys.Lett.B700(2011)150 for the specific case of $\kappa$-momentum space. The proof was based on the recent understanding of how relative locality affects these scenarios. We here rely again on a careful implementation of relative locality effects, and obtain our result for a generic (relativistic) curved momentum space framework at all orders in the deformation/curvature parameter. We also discuss the validity of this result when the deformation depends on the coordinates as well as on the momenta.