Calculation of Spin Group Elements Revisited

TL;DR

This paper introduces a comprehensive method for calculating spin group elements from pseudo-orthogonal group elements across various dimensions and signatures, utilizing Clifford algebra, matrices, quaternions, and split-quaternions, with applications in multiple fields.

Contribution

It provides a unified approach for computing spin group elements using multiple formalisms, applicable to arbitrary dimensions and signatures, with explicit examples up to dimension three.

Findings

Method applicable to any dimension and signature

Explicit calculations for dimensions up to three

Multiple formalisms suited for different applications

Abstract

In this paper, we present a method for calculation of spin groups elements for known pseudo-orthogonal group elements with respect to the corresponding two-sheeted coverings. We present our results using the Clifford algebra formalism in the case of arbitrary dimension and signature, and then explicitly using matrices, quaternions, and split-quaternions in the cases of all possible signatures (p,q) of space up to dimension n=p+q=3. The different formalisms are convenient for different possible applications in physics, engeneering, and computer science.