Principle of relative locality

We propose a deepening of the relativity principle according to which the invariant arena for nonquantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes 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 energy-momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of energy-momentum space geometry, such as its curvature, torsion and nonmetricity, 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 energy-momentum space with a metric compatible connection and constant curvature.

Figure 1

Curvature of the connection on momentum space produces nonassociativity of composition rule.

Figure 2

Torsion implies that only one order of (deformed) summation of momenta actually is allowed for the process. In figure the atom initially in internal state $I$ with momentum $k_{\mu }$ could absorb the photon with momentum $q_{\mu }$ to go in an another internal state $II$ with momentum $q_{\mu }$, with $q_{\mu }=k_{\mu }\oplus p_{\mu }\ne p_{\mu }\oplus k_{\mu }$.

Figure 3

Schematic description of a measurement procedure aimed at exposing the implications of a nontrivial connection on momentum space. The route A and route B described in the text are here shown both in momentum space [panel (a)] and as spacetime trajectories of the atoms [panel (b)].