Relative-locality geometry for the Snyder model

Abstract

We investigate the geometry of the energy–momentum space of the Snyder model and of its generalizations according to the definitions proposed in [G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Phys. Rev. D 84 (2011) 084010], in connection with the theory of relative locality. In this setting, the geometric structures of the energy–momentum space are defined in terms of the deformed composition law of momenta, and we show that in the Snyder case they describe a maximally symmetric space, with vanishing torsion and nonmetricity. However, one cannot apply straightforwardly the phenomenological relations between the geometry and the dynamics postulated in [G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Phys. Rev. D 84 (2011) 084010], because they were obtained assuming that the leading corrections to the composition law of momenta are quadratic, which is not the case with the Snyder model and its generalizations.