Generalized composition law from 2x2 matrices

TL;DR

This paper introduces a generalized composition law based on 2x2 matrices that simplifies the addition of bounded quantities, with applications to phase factors in relativity and matrix operations.

Contribution

It presents a novel generalization of Einstein's velocity addition law using 2x2 matrices, linking it to phase factors and matrix composition in a unified framework.

Findings

Provides a new mathematical framework for bounded quantity addition.

Connects the generalized law to phase factors like the Wigner angle.

Shows the relation to 2x2 S matrix composition.

Abstract

Many results that are difficult can be found more easily by using a generalization in the complex plane of Einstein's addition law of parallel velocities. Such a generalization is a natural way to add quantities that are limited to bounded values. We show how this generalization directly provides phase factors such as the Wigner angle in special relativity and how this generalization is connected in the simplest case with the composition of 2x2 S matrices.