Non-Abelian elastic collisions, associated difference systems of equations and discrete analytic functions

TL;DR

This paper generalizes elastic collision equations to arbitrary associative algebras, unifies relativistic cases, and connects these to discrete analytic functions through difference systems on a grid.

Contribution

It introduces a broad algebraic framework for elastic collisions and links it to discrete analytic functions, unifying linear and nonlinear approaches.

Findings

Elastic collision equations extended to arbitrary associative algebras.

Relativistic elastic collisions are special cases of the generalized equations.

Equations can be interpreted as difference systems on a two-dimensional grid.

Abstract

We extend the equations of motion that describe non-relativistic elastic collision of two particles in one dimension to an arbitrary associative algebra. Relativistic elastic collision equations turn out to be a particular case of these generic equations. Furthermore, we show that these equations can be reinterpreted as difference systems defined on the ${\mathbb Z}^2$ graph and this reinterpretation relates (unifies) the linear and the non-linear approach of discrete analytic functions.