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Calculation of Spin Group Elements Revisited

D. S. Shirokov

TL;DR

This work addresses how to compute the two-sheeted spin lift for a given $P \in {\rm SO}_+(p,q)$ by developing a universal Clifford-algebra framework valid for arbitrary dimension $n=p+q$, and by providing explicit matrix, quaternion, and split-quaternion representations for all signatures with $n \le 3$. The main approach centers on constructing rotors $S$ via $S=\pm \frac{M_F}{\sqrt{\widetilde{M_F}M_F}}$ with $M_F:=p_A^B e_B e_F e^A$ for a suitable even $e_F$ ensuring $M_F\neq 0$, enabling $Se_aS^{-1}=p_a^b e_b$. The paper also delivers concrete, low-dimensional isomorphisms (e.g., Spin$(2)\simeq U(1)$, Spin$(3)\simeq SU(2)$, Spin$_+(2,1)\simeq SU(1,1)$) and explicit rotor formulas using unit quaternions and split-quaternions, with theorems and constructions laid out for the general case and $n=3$ specifically. The results broaden the toolkit for applications in physics, engineering, and computer science and set the stage for higher-dimensional extensions in future work.

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.

Calculation of Spin Group Elements Revisited

TL;DR

This work addresses how to compute the two-sheeted spin lift for a given by developing a universal Clifford-algebra framework valid for arbitrary dimension , and by providing explicit matrix, quaternion, and split-quaternion representations for all signatures with . The main approach centers on constructing rotors via with for a suitable even ensuring , enabling . The paper also delivers concrete, low-dimensional isomorphisms (e.g., Spin, Spin, Spin) and explicit rotor formulas using unit quaternions and split-quaternions, with theorems and constructions laid out for the general case and specifically. The results broaden the toolkit for applications in physics, engineering, and computer science and set the stage for higher-dimensional extensions in future work.

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.

Paper Structure

This paper contains 4 sections, 5 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Clifford Algebra Formalism
  3. Formalism of Matrices, Quaternions, and Split-quaternions
  4. Conclusions

Key Result

Theorem 1

Let $P\in{\rm SO}_+(p,q)$, $p+q=n$. We can always choose a basis element $e_F\in\{e_D \, | \, |D|=0\mod 2\}\in{C}\!\ell_{p,q}$ such that Next, we can find elements $S=\pm T\in{\rm Spin}_+(p,q)$ that correspond to $P=(p_a^b)\in{\rm SO}_+(p,q)$ as the two-sheeted covering $Se_aS^{-1}=p_a^b e_b$ in the following way:

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 2
  • Theorem 3: $n=3$
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 3 more