General very special relativity is Finsler geometry

Abstract

We ask whether Cohen and Glashow’s very special relativity model for Lorentz violation might be modified, perhaps by quantum corrections, possibly producing a curved space-time with a cosmological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous deformations, but none of these give rise to noncommutative translations analogous to those of the de Sitter deformation of the Poincaré group: space-time remains flat. Only a 1-parameter family DISIM b ( 2 ) of deformations of SIM(2) is physically acceptable. Since this could arise through quantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be taken into account. The Lorentz-violating point-particle action invariant under DISIM b ( 2 ) is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive DISIM b ( 2 ) -invariant wave equations for particles of spins 0, 1 2 , and 1. The experimental bound, | b | < 10 − 26 , raises the question “Why is the dimensionless constant b so small in very special relativity?”