Triply Special Relativity

TL;DR

This paper introduces a novel extension of special relativity with three invariant scales, connecting it to quantum gravity and de Sitter spacetime, and explores its implications for particle dynamics.

Contribution

It presents a non-linear algebra extending Poincare symmetry with three invariants, linking it to quantum gravity and de Sitter space, and analyzes particle motion modifications.

Findings

The algebra reduces to known structures in specific limits.

Modified particle dynamics are derived from the algebra.

Potential relevance for quantum gravity at low energies.

Abstract

We describe an extension of special relativity characterized by {\it three} invariant scales, the speed of light, $c$, a mass, $\kappa$ and a length $R$. This is defined by a non-linear extension of the Poincare algerbra, $\cal A$, which we describe here. For $R\to \infty$, $\cal A$ becomes the Snyder presentation of the $\kappa$-Poincare algebra, while for $\kappa \to \infty$ it becomes the phase space algebra of a particle in deSitter spacetime. We conjecture that the algebra is relevant for the low energy behavior of quantum gravity, with $\kappa$ taken to be the Planck mass, for the case of a nonzero cosmological constant $\Lambda = R^{-2}$. We study the modifications of particle motion which follow if the algebra is taken to define the Poisson structure of the phase space of a relativistic particle.