#
Construction of Discrete Symmetries Using the Pauli Algebra Form of the Dirac Equation^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Pauli Algebra Form of the Dirac Equation

**Proposition 1.**

**Proof.**

## 3. The Short Exact Sequence Connecting the Pauli Group with the Group of Symmetries

**Proposition 2.**

**Proof.**

## 4. The Charge Conjugation and the Mass Inversion Symmetries

## 5. Identifying Pauli Algebra Expressions for Parity and Time-Reversal Symmetries

**Proposition 3.**

**Proof.**

**Proposition 4.**

**Proof.**

## 6. The Proof of CPTM Invariance

**Proposition 5.**

**Proof.**

**Proof.**

## 7. Relation to Other Work

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Peng, J.C. Physics 570 Lecture Notes. Available online: https://courses.physics.illinois.edu/phys570/fa2018/note.html (accessed on 30 November 2022).
- Schwartz, M.D. Quantum Field Theory and the Standard Model; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Nofech, A. Biquaternionic Dirac equation predicts zero mass for Majorana fermions. Symmetry
**2020**, 12, 1144. [Google Scholar] [CrossRef] - Debergh, N.; Petit, J.P.; D’Agostini, G. On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator. J. Phys. Commun.
**2018**, 2, 115012. [Google Scholar] [CrossRef] - Marsch, E.; Narita, Y. CPTM Symmetry for the Dirac Equation and Its Extended Version Based on the Vector Representation of the Lorentz Group. Front. Phys.
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© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Nofech, A.
Construction of Discrete Symmetries Using the Pauli Algebra Form of the Dirac Equation. *Phys. Sci. Forum* **2023**, *7*, 41.
https://doi.org/10.3390/ECU2023-14054

**AMA Style**

Nofech A.
Construction of Discrete Symmetries Using the Pauli Algebra Form of the Dirac Equation. *Physical Sciences Forum*. 2023; 7(1):41.
https://doi.org/10.3390/ECU2023-14054

**Chicago/Turabian Style**

Nofech, Avraham.
2023. "Construction of Discrete Symmetries Using the Pauli Algebra Form of the Dirac Equation" *Physical Sciences Forum* 7, no. 1: 41.
https://doi.org/10.3390/ECU2023-14054