C•1 Ct ti arXiv:submit/0174271 [hep-th] The principle of relative locality Giovanni Amelino-Camelit, Laurent Freider, Jerzy Kowalski-Glikmanb, Lee Smolin` °Dipartimento di Fisica, University "La Sapienza" and Sez. Romal INFN, P.le A. Moro 2, 00185 Roma, Italy bInstitute for Theoretical Physics, University of Wroclaw, Pl. Maxa Boma 9, 50-204 Wroclaw, Poland `Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2J 2Y5, Canada (Dated: January 5. 2011) We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in space- times are constructed by observers, but different observers, separated from each other by translations. construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry. such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature. I. INTRODUCTION How do we know we live in a spacetime? And if so how do we know we all share the same spacetime? These are the fundamental questions we are investigating in this note. We local observers do not directly observe any events macroscopically displaced from our measuring instruments. As naive observer