Physics with Curved Momentum Space

Abstract

In this note I briefly describe motivations that lead to the concept of momentum space possessing a nontrivial geometry. Then I discuss some physical consequences of curved momentum space.The idea that momentum space might be curved is quite old. It seems that it was first spelled out by Max Born in the paper [1], where it is argued that some kind of 'reciprocity principle' should be adopted, stating that both curved spacetime and curved momentum space should be involved simultaneously in the description of (quantum) physics. About ten years later in the seminal paper [2] Snyder argued that curvature in momentum space might be necessary to handle ultraviolet divergencies of quantum field theory. This paper introduces, as a bi-product, a non-commutativity of spacetime coordinates and a minimal length, arguing that both do not need to be in conflict with Lorentz symmetry (for a recent review of minimal length scenarios see Ref.