# Noether’s Theorem in Non-Local Field Theories

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## Abstract

**:**

## 1. Introduction

## 2. Symmetries and the Conserved Currents

## 3. Non-Local Charged Scalar Field

#### 3.1. Time-Like Components

#### 3.2. Vector Current

#### 3.3. Energy-Momentum Tensor

#### 3.4. Angular Momentum Tensor

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CPT symmetry | Charge, parity, and time reversal symmetry |

QCD | quantum chromodynamics |

T product | time-ordered product |

## Appendix A. Field Derivatives

## Appendix B. Series Summation

## Appendix C. Vector Current from the Minimal Substitution

## References

- Noether, E. Invariante Variationsprobleme. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse
**1918**, 235–257, (English translation: Noether, E. Invariant Variation Problems. Transp. Theory Stat. Phys.**1971**, 1, 183–207, arXiv:physics/0503066.). [Google Scholar] - Faddeev, L.D.; Slavnov, A.A. Gauge Fields: An Introduction to Quantum Theory, 2nd ed.; Westview Press: Boulder, CO, USA, 1993. [Google Scholar]
- Gasser, J.; Leutwyler, H. Chiral perturbation theory: Expansions in the mass of the strange quark. Nucl. Phys. B
**1985**, 250, 465. [Google Scholar] [CrossRef][Green Version] - Scherer, S. Introduction to chiral perturbation theory. Adv. Nucl. Phys.
**2003**, 27, 277. [Google Scholar] - Cremaschini, C.; Tessarotto, M. Statistical treatment of the electromagnetic radiation-reaction problem: Evaluation of the relativistic Boltzmann-Shannon entropy. Phys. Rev. E
**2013**, 87, 032107. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Hamiltonian formulation for the classical EM radiation-reaction problem: Application to the kinetic theory for relativistic collisionless plasmas. Eur. Phys. J. Plus
**2011**, 126, 63. [Google Scholar] [CrossRef][Green Version] - Szabados, L.B. Quasi-Local Energy-Momentum and Angular Momentum in General Relativity. Living Rev. Rel.
**2009**, 12, 4. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tessarotto, M.; Cremaschini, C. Theory of Nonlocal Point Transformations in General Relativity. Adv. Math. Phys.
**2016**, 2016, 9619326. [Google Scholar] [CrossRef][Green Version] - Eringen, A.C. Nonlocal Continuum Field Theories; Springer: New York, NY, USA, 2002. [Google Scholar]
- Lunev, F.A. Analog of Noether’s theorem for non-Noether and nonlocal symmetries. Theor. Math. Phys.
**1990**, 84, 816. [Google Scholar] [CrossRef] - Okun, L.B. C, P, T are broken. Why not CPT? In Proceedings of the 14th Rencontres de Blois: Matter- Anti-Matter Asymmetry, Chateau de Blois, France, 17–22 June 2002. [Google Scholar]
- Barnaby, N.; Cline, J.M. Predictions for Nongaussianity from Nonlocal Inflation. JCAP
**2008**. [Google Scholar] [CrossRef][Green Version] - Weinberg, S. The Quantum Theory of Fields, Volume 1: Foundations; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]

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**MDPI and ACS Style**

Krivoruchenko, M.I.; Tursunov, A.
Noether’s Theorem in Non-Local Field Theories. *Symmetry* **2020**, *12*, 35.
https://doi.org/10.3390/sym12010035

**AMA Style**

Krivoruchenko MI, Tursunov A.
Noether’s Theorem in Non-Local Field Theories. *Symmetry*. 2020; 12(1):35.
https://doi.org/10.3390/sym12010035

**Chicago/Turabian Style**

Krivoruchenko, Mikhail I., and Arman Tursunov.
2020. "Noether’s Theorem in Non-Local Field Theories" *Symmetry* 12, no. 1: 35.
https://doi.org/10.3390/sym12010035