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

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 precision 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. 1 Finite Minkowski spaces and statement of the theorem The purpose of this paper is to establish a relationship of approximation between Lorentz transformations of Minkowski space R and Lorentz transformations of 4-dimensional spaces over very large finite fields, in the precise sense of Theorem 1 below. In connection with particle theory Lorentz transformations over finite fields were already considered by Coish in [C]. The approximation described in Theorem 1 relies on modifying a relationship of approximation between positive real numbers and quadratic residues modulo a large prime established by Kustaanheimo [JK, K1, K2], and modifying Järnefelt’s approximation of collinear ranges of points in Euclidean space by collinear ranges in finite geometries [J]. The algebraic background and terminology used in this Theorem are as follows.