# Six-Dimensional Space with Symmetric Signature and Some Properties of Elementary Particles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Pseudo-Euclidean Space ${\mathbb{E}}_{\mathbf{3},\mathbf{3}}$ as an Image of the Spinor Space

## 3. Hidden Groups of Proper Motions of the Metric

## 4. Group $\mathit{SU}\left(\mathbf{4}\right)$ and the Conserved Quantum Characteristics It Generates

## 5. Simplest Irreducible Representation of $\mathit{SU}\left(\mathbf{4}\right)$

## 6. Representation of the Metric of the ${\mathbb{E}}_{\mathbf{3},\mathbf{3}}$ Space Using Hyperbolic Complex Numbers

## 7. Hyperbolic Groups of Unitary Symmetry and Their Representations

## 8. Reduction to Unisotropic Space

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Commutators of operators (14).

$\langle \xb7,\mathit{b}\rangle $ | ${\mathit{I}}_{\pm}$ | ${\mathit{V}}_{\pm}$ | ${\mathit{U}}_{\pm}$ | ${\mathit{N}}_{\pm}$ | ${\mathit{M}}_{\pm}$ | ${\mathit{W}}_{\pm}$ | |
---|---|---|---|---|---|---|---|

$\langle \mathit{a},\xb7\rangle $ | |||||||

${I}_{3}$ | $\pm {I}_{\pm}$ | $\pm \frac{1}{2}{V}_{\pm}$ | $\mp \frac{1}{2}{U}_{\pm}$ | $\pm \frac{1}{2}{N}_{\pm}$ | $\mp \frac{1}{2}{M}_{\pm}$ | 0 | |

Y | 0 | $\pm {V}_{\pm}$ | $\pm {U}_{\pm}$ | $\pm \frac{1}{3}{N}_{\pm}$ | $\pm \frac{1}{3}{M}_{\pm}$ | $\mp \frac{2}{3}{W}_{\pm}$ | |

Q | $\pm {I}_{\pm}$ | $\pm {V}_{\pm}$ | 0 | $\pm \frac{2}{3}{N}_{\pm}$ | $\mp \frac{1}{3}{M}_{\pm}$ | $\mp \frac{1}{3}{W}_{\pm}$ | |

B | 0 | 0 | 0 | $\pm \frac{4}{3}{N}_{\pm}$ | $\pm \frac{4}{3}{M}_{\pm}$ | $\pm \frac{4}{3}{W}_{\pm}$ |

**Table 2.**Commutators of operators (14), continued.

$\langle \xb7,\mathit{b}\rangle $ | ${\mathit{V}}_{+}$ | ${\mathit{V}}_{-}$ | ${\mathit{U}}_{+}$ | ${\mathit{U}}_{-}$ | ${\mathit{N}}_{+}$ | ${\mathit{N}}_{-}$ | ${\mathit{M}}_{+}$ | ${\mathit{M}}_{-}$ | ${\mathit{W}}_{+}$ | ${\mathit{W}}_{-}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

$\langle \mathit{a},\xb7\rangle $ | |||||||||||

${I}_{+}$ | 0 | $-{U}_{-}$ | ${V}_{+}$ | 0 | 0 | $-{M}_{-}$ | ${N}_{+}$ | 0 | 0 | 0 | |

${I}_{-}$ | ${U}_{+}$ | 0 | 0 | $-{V}_{-}$ | ${M}_{+}$ | 0 | 0 | $-{N}_{-}$ | 0 | 0 | |

${V}_{+}$ | 0 | (18) | 0 | ${I}_{+}$ | 0 | $-{W}_{-}$ | 0 | 0 | ${N}_{+}$ | 0 | |

${V}_{-}$ | (18) | 0 | $-{I}_{-}$ | 0 | ${W}_{+}$ | 0 | 0 | 0 | 0 | $-{N}_{-}$ | |

${U}_{+}$ | 0 | ${I}_{-}$ | 0 | (18) | 0 | 0 | 0 | $-{W}_{-}$ | ${M}_{+}$ | 0 | |

${U}_{-}$ | $-{I}_{+}$ | 0 | (18) | 0 | 0 | 0 | ${W}_{+}$ | 0 | 0 | $-{M}_{-}$ | |

${N}_{+}$ | 0 | $-{W}_{+}$ | 0 | 0 | 0 | (18) | 0 | ${I}_{+}$ | 0 | ${V}_{+}$ | |

${N}_{-}$ | ${W}_{-}$ | 0 | 0 | 0 | (18) | 0 | $-{I}_{-}$ | 0 | $-{V}_{-}$ | 0 | |

${M}_{+}$ | 0 | 0 | 0 | $-{W}_{+}$ | 0 | ${I}_{-}$ | 0 | (18) | 0 | ${U}_{+}$ | |

${M}_{-}$ | 0 | 0 | ${W}_{-}$ | 0 | $-{I}_{+}$ | 0 | (18) | 0 | $-{U}_{-}$ | 0 | |

${W}_{+}$ | $-{N}_{+}$ | 0 | $-{M}_{+}$ | 0 | 0 | ${V}_{-}$ | 0 | ${U}_{-}$ | 0 | (18) | |

${W}_{-}$ | 0 | ${N}_{-}$ | 0 | ${M}_{-}$ | $-{V}_{+}$ | 0 | $-{U}_{+}$ | 0 | (18) | 0 |

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Popov, N.; Matveev, I.
Six-Dimensional Space with Symmetric Signature and Some Properties of Elementary Particles. *Axioms* **2022**, *11*, 650.
https://doi.org/10.3390/axioms11110650

**AMA Style**

Popov N, Matveev I.
Six-Dimensional Space with Symmetric Signature and Some Properties of Elementary Particles. *Axioms*. 2022; 11(11):650.
https://doi.org/10.3390/axioms11110650

**Chicago/Turabian Style**

Popov, Nikolay, and Ivan Matveev.
2022. "Six-Dimensional Space with Symmetric Signature and Some Properties of Elementary Particles" *Axioms* 11, no. 11: 650.
https://doi.org/10.3390/axioms11110650