The Lorentz group and its finite field analogues: local isomorphism and approximation

TL;DR

This paper explores the relationship between the real Lorentz group and its finite field analogues, showing how finite Lorentz groups can approximate the real group through local isomorphisms and rational subgroups.

Contribution

It introduces a method to approximate the real Lorentz group using finite groups over large finite fields via local isomorphisms and rational subgroups.

Findings

Finite Lorentz groups are homomorphic images of rational subgroups of the real Lorentz group.

Bounded subsets of the real Lorentz group can be approximated by finite subsets of rational subgroups.

Finite approximations correspond to well-behaved subsets over finite fields, establishing a link between real and finite Lorentz groups.

Abstract

Finite Lorentz groups acting on 4-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real 4-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary accuracy to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields.