Geometry of deformed special relativity

Abstract

We undertake a study of the classical regime in which Planck's constant and Newton's gravitational constant are negligible, but not their ratio, the Planck mass, in hopes that this could possibly lead to testable quantum gravity (QG) effects in a classical regime. In this quest for QG phenomenology we consider modifications of the standard dispersion relation of a free particle known as deformed special relativity (DSR). We try to geometrize DSR to find the geometric origin of the spacetime and momentum space. In particular, we adopt the framework of Hamilton geometry which is set up on phase space, as the cotangent bundle of configuration space in order to derive a purely phase space formulation of DSR. This is necessary when one wants to understand potential links of DSR with modifications of quantum mechanics such as Generalised Uncertainty Principles. It is subsequently observed that space-time and momentum space emerge naturally as curved and intertwined spaces. In conclusion we mention examples and applications of this framework as well as potential future developments.