General very special relativity is Finsler geometry

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 ${\mathrm{DISIM}}_b(2)$ 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 ${\mathrm{DISIM}}_b(2)$ is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive ${\mathrm{DISIM}}_b(2)$-invariant wave equations for particles of spins 0, $\frac{1}{2}$, and 1. The experimental bound, $|b|<{10}^{-26}$, raises the question “Why is the dimensionless constant $b$ so small in very special relativity?”