Hierarchical order of Galilei and Lorentz invariance in the structure of matter

Abstract

The structure of matter shows a hierarchical order: (1) from Lorentz invariance in high-energy physics; (2) to Galilei invariance in the low-energy nonrelativistic limit of high-energy physics; and (3) again to Lorentz invariance in condensed matter physics (where the velocity of sound takes the place of the velocity of light). The hierarchical order can be continued downward further to: (4) non-relativistic (velocity small compared to the velocity of sound) condensed matter excitons, obeying Galilei invariance; and (5) to excitonic matter obeying Lorentz invariance with an excitonic matter sound velocity. It was previously conjectured that Lorentz invariance of high-energy physics is preceded by Galilei invariance at the Planck scale. Still further, the conjectured Galilei invariance at the Planck scale may be the result of an underlying five-dimensional non-Euclidean conform invariant metric structure, with three spatial and two time dimensions, compactified onto three spatial and one time dimension.