# The Quantum Nature of Lorentz Invariance

## Abstract

**:**

## 1. Introduction

## 2. From Pauli to Lorentz

## 3. Elementary Properties of Quarks

## 4. Ternary ${Z}_{3}$-Commutation

## 5. Two-Generator Algebra and Its Invariance Group

## 6. Color Dirac Equation

## 7. The Lagrangian

## 8. Gauge Fields of the Standard Model

## 9. The ${Z}_{3}$ Lorentz Invariance

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Born, M.; Jordan, P. Zur quantenmechanik. Z. Phys.
**1925**, 34, 858–878. [Google Scholar] [CrossRef] - Von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1996. [Google Scholar]
- Dubois-Violette, M.; Kerner, R.; Madore, J. Noncommutative differential geometry of matrix algebras. J. Math. Phys.
**1990**, 31, 316–322. [Google Scholar] [CrossRef] - Dubois-Violette, M.; Kerner, R.; Madore, J. Noncommutative geometry and new models of gauge theories. J. Math. Phys.
**1990**, 31, 323–331. [Google Scholar] [CrossRef] - Froggatt, C.D.; Nielsen, H.B. Origin of Symmetries; World Scientific: Singapore, 1991. [Google Scholar]
- Pauli, W. The Connection Between Spin and Statistics. Phys. Rev.
**1940**, 58, 716–722. [Google Scholar] [CrossRef] - Dyson, F.J. Ground State Energy of a Finite System of Charged Particles. J. Math. Phys.
**1967**, 8, 1538–1545. [Google Scholar] [CrossRef] - Dyson, F.J.; Lenard, A. Stability of matter. II. J. Math. Phys.
**1968**, 9, 698–711. [Google Scholar] - Streater, R.F.; Wightman, A.S. PCT, Spin ans Statistics and All That; W.A. Benjamin, Inc.: New York, NY, USA; Amsterdam, The Netherlands, 1964. [Google Scholar]
- Dirac, P.A.M. The Principles of Quantim Mechanics, 4th ed.; Oxford University Press: Oxford, UK, 1958. [Google Scholar]
- Bjorken, J.D.; Drell, D.S. Relativistic Quantum Mechanics; McGraw-Hill: London, UK; New York, NY, USA, 1964. [Google Scholar]
- Gell-Mann, M.; Ne’eman, Y. The Eightfold Way; Benjamin: New York, NY, USA, 1964. [Google Scholar]
- Lipkin, H.J. Frontiers of the Quark Model; pr. WIS-87-47-PH; Weizmann Institute: Rehovot, Israel, 1987. [Google Scholar]
- Okubo, S. Triple products and Yang-Baxter equation. I. Octonionic and quaternionic triple systems. J. Math. Phys.
**1993**, 34, 3273–3291. [Google Scholar] [CrossRef] - Kerner, R. Graduation Z
_{3}et la racine cubique de l’opérateur de Dirac. C. R. Acad. Sci. Paris**1991**, 312, 191–196. [Google Scholar] - Kerner, R. Z
_{3}graded algebras and the cubic root of the supersymmetry translations. J. Math. Phys.**1992**, 33, 403–411. [Google Scholar] [CrossRef] - Kerner, R. Ternary Generalization of Pauli’s Principle and the Z6-Graded Algebras. Phys. At. Nucl.
**2017**, 80, 522–534. [Google Scholar] [CrossRef] - Kerner, R. Ternary Z
_{2}and Z_{3}graded algebras and color dynamics. In Mathematical Structures and Applications; Springer: New York, NY, USA, 2018; pp. 311–357. [Google Scholar] - Lee, T.D.; Wick, G.C. Negative metric and the unitarity of the S-matrix. Nucl. Phys. B
**1969**, 9, 209–243. [Google Scholar] [CrossRef] - Lee, T.D.; Wick, G.C. Finite theory of quantum electrodynamics. Phys. Rev. D
**1970**, 2, 1033. [Google Scholar] [CrossRef] - Anselmi, D.; Piva, M. Perturbative unitarity of Lee-Wick quantum field theory. Phys. Rev. D
**2017**, 96, 045009. [Google Scholar] [CrossRef] [Green Version]

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Kerner, R.
The Quantum Nature of Lorentz Invariance. *Universe* **2019**, *5*, 1.
https://doi.org/10.3390/universe5010001

**AMA Style**

Kerner R.
The Quantum Nature of Lorentz Invariance. *Universe*. 2019; 5(1):1.
https://doi.org/10.3390/universe5010001

**Chicago/Turabian Style**

Kerner, Richard.
2019. "The Quantum Nature of Lorentz Invariance" *Universe* 5, no. 1: 1.
https://doi.org/10.3390/universe5010001