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Square root of a multivector of Clifford algebras in 3D: A game with signs

A. Acus, A. Dargys

Abstract

An algorithm to extract the square root in radicals from a multivector (MV) in real Clifford algebras Cl(p,q) for n=p+q <=3 is presented. We show that in the algebras Cl(3,0), Cl(1,2) and Cl(0,3) there are up to four isolated roots in a case of the most general (generic) MV. The algebra Cl(2,1) makes up an exception and the MV here can have up to 16 isolated roots. In addition to isolated roots, a continuum of roots can appear in all algebras except p+q=1. A number of examples are provided to illustrate properties of various roots that may appear in n=3 Clifford algebras.

Square root of a multivector of Clifford algebras in 3D: A game with signs

Abstract

An algorithm to extract the square root in radicals from a multivector (MV) in real Clifford algebras Cl(p,q) for n=p+q <=3 is presented. We show that in the algebras Cl(3,0), Cl(1,2) and Cl(0,3) there are up to four isolated roots in a case of the most general (generic) MV. The algebra Cl(2,1) makes up an exception and the MV here can have up to 16 isolated roots. In addition to isolated roots, a continuum of roots can appear in all algebras except p+q=1. A number of examples are provided to illustrate properties of various roots that may appear in n=3 Clifford algebras.

Paper Structure

This paper contains 25 sections, 1 theorem, 79 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Square roots in $\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{1,2}$ algebras
  4. The generic case $s^2+S^2\neq 0$
  5. The special case $s^2+S^2= 0$
  6. $\mathit{Cl}_{1,2}\simeq\mathit{Cl}_{3,0}$ algebra
  7. Examples for $\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{1,2}$
  8. Square roots in $\mathit{Cl}_{0,3}$ algebra
  9. The generic case $s^2-S^2\neq 0$
  10. The special case $s^2 - S^2=0$
  11. The subcase $s=S\neq 0$
  12. The subcase $s=-S\neq 0$
  13. The subcase $s = S = 0$
  14. Examples for $\mathit{Cl}_{0,3}$
  15. Square roots in $\mathit{Cl}_{2,1}$ algebra
  16. ...and 10 more sections

Key Result

theorem \oldthetheorem

Figures (1)

  • Figure 1: text of caption

Theorems & Definitions (2)

  • theorem \oldthetheorem: Optional addition to theorem head
  • proof : Optional replacement proof heading