Relativistic deformed kinematics from momentum space geometry

Abstract

We present a way to derive a deformation of special relativistic kinematics (possible low energy signal of a quantum theory of gravity) from the geometry of a maximally symmetric curved momentum space. The deformed kinematics is fixed (up to change of coordinates in the momentum variables) by the algebra of isometries of the metric in momentum space. In particular, the wellknown example of κ-Poincaré kinematics is obtained when one considers an isotropic metric in de Sitter momentum space such that translations are a subgroup of the isometry group, and for a Lorentz covariant algebra one gets the also well-known case of Snyder kinematics. We prove that our construction gives generically a relativistic kinematics and explain how it relates to previous attempts of connecting a deformed kinematics with a geometry in momentum space.

Phenomenological consequences of a geometry in the cotangent bundle

“…Also, we consider that the momentum space tetrad do not change under such transformation. In the following, we will prove that with the definition of the new momentum metric given in this work, starting with a momentum space metric, we can still define momentum transformations for a fixed space-time point x that leave invariant the form of the metric, taking into account the curvature of the spacetime (the generalization to curved spacetime of the results obtained in [23]): there are still 10 momentum isometries of the metric that correspond to the four translations and six transformations that leave the origin invariant (the point in phase space (x, 0)), and we can identify the squared distance from a point in the momentum space to the origin as the deformed dispersion relation. In Appendix A it is shown that when the starting momentum space metric is of constant curvature (maximally symmetric space), the momentum scalar of curvature given by the contraction of Eq.…”

“…In this section we will first review the main results of [23]. We will expose how a deformed relativistic kinematics can be understood through a maximally symmetric momentum space, characterized by a metric g µν k (k).…”

“…In [23] a proposal to derive a (relativistic) deformed kinematics from a geometry in maximally symmetric momentum space is given, defining a deformed composition and transformation laws from the isometries of the momentum metric associated to translations and Lorentz respectively, and the deformed dispersion relation as the (square of the) distance from the origin to a point in momentum space. In particular, it was shown in that paper that, when the momentum space is de Sitter with the choice of coordinates in which the metric is 2…”

“…Since these transformations leave the tetrad invariant (and then the metric), they are isometries of the momentum metric. With this prescription to obtain the composition law, it can be shown [23] that the only compatible kinematics is κ-Poincaré (note that the composition law is associative by construction).…”

“…This means that we can identify, for a fixed x, the isometries of this metric that leaves the form of the tetrad of the whole metric invariant as the deformed composition law when a curvature of spacetime is present, in the same way it was done in [23]. By construction, the deformed composition law generators of Eq.…”