Triply special relativity

We describe an extension of special relativity characterized by three invariant scales, the speed of light $c$, a mass $\kappa$, and a length $R$. This is defined by a nonlinear extension of the Poincaré algebra $\mathcal{A}$, which we describe here. For $R\to \infty$, $\mathcal{A}$ becomes the Snyder presentation of the $\kappa$-Poincaré algebra, while for $\kappa \to \infty$ it becomes the phase space algebra of a particle in de Sitter 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.