# Fractional Laplacian Spinning Particle in External Electromagnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Fractional Laplacian

## 3. Fractional Spinning Particle in Electromagnetic Field

#### 3.1. Classical Relativistic Particle

#### 3.2. FLSP Model

#### 3.3. Equations of Motion

## 4. Examples of FLSP in Simple External Fields

#### 4.1. FLSP in Constant Magnetic Field

#### 4.2. FLSP in Constant Electric Field

#### 4.3. FLSP in Quadratic Electromagnetic Potential

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Beghin, L.; Mainardi, F.; Garrappa, R. (Eds.) Nonlocal and Fractional Operators; Springer: Cham, Switzerland, 2021. [Google Scholar]
- Treeby, B.E.; Cox, B.T. Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian. J. Acoust. Soc.
**2014**, 136, 1499–1510. [Google Scholar] [CrossRef] [PubMed] - Ciaurri, O.; Roncal, L.; Stinga, P.; Torrea, J.; Varona, J. Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications. Adv. Math.
**2018**, 330, 688–738. [Google Scholar] [CrossRef] - Giusti, A. MOND-like fractional Laplacian theory. Phys. Rev.
**2020**, D101, 124029. [Google Scholar] [CrossRef] - Giusti, A.; Garrappa, R.; Vachon, G. On the Kuzmin model in fractional Newtonian gravity. Eur. Phys. J. Plus
**2020**, 135, 798. [Google Scholar] [CrossRef] - Sorensen, T.O. The large-Z behavior of pseudorelativistic atoms. J. Math. Phys.
**2005**, 46, 052307. [Google Scholar] [CrossRef] - Fournais, S.; Lewin, M.; Triay, A. The Scott Correction in Dirac–Fock Theory. Commun. Math. Phys.
**2020**, 378, 569–600. [Google Scholar] [CrossRef] - La Nave, G.; Limtragool, K.; Phillips, P.W. Fractional Electromagnetism in Quantum Matter and High-Energy Physics. Rev. Mod. Phys.
**2019**, 91, 021003. [Google Scholar] [CrossRef] - Li, Z.; Tang, H.; Yuan, S.; Zhang, H.; Kong, L.; Sun, H. Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains. Fractal Fract.
**2023**, 7, 823. [Google Scholar] [CrossRef] - Heydeman, M.; Jepsen, C.B.; Ji, Z.; Yarom, A. Polyakov’s confinement mechanism for generalized Maxwell theory. JHEP
**2023**, 4, 119. [Google Scholar] [CrossRef] - Stephanovich, V.A.; Olchawa, W. Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity. Sci. Rep.
**2022**, 12, 384. [Google Scholar] [CrossRef] - El-Nabulsi, R.A.; Anukool, W. A family of nonlinear Schrodinger equations and their solitons solutions. Chaos Solitons Fractals
**2023**, 166, 112907. [Google Scholar] [CrossRef] - Frassino, A.M.; Panella, O. Quantization of nonlocal fractional field theories via the extension problem. Phys. Rev.
**2019**, D100, 116008. [Google Scholar] [CrossRef] - Paulos, M.F.; Rychkov, S.; van Rees, B.C.; Zan, B. Conformal Invariance in the Long-Range Ising Model. Nucl. Phys.
**2016**, B902, 246–291. [Google Scholar] [CrossRef] - Rajabpour, M.A. Conformal symmetry in non-local field theories. JHEP
**2011**, 6, 076. [Google Scholar] [CrossRef] - Vancea, I.V. Fractional Particle and Sigma Model. arXiv
**2023**, arXiv:2309.03054. [Google Scholar] - Samiee, M.; Akhavan-Safaei, A.; Zayernouri, M. A fractional subgrid-scale model for turbulent flows: Theoretical formulation and a priori study. Phys. Fluids
**2020**, 32, 055102. [Google Scholar] [CrossRef] - Suzuki, J.L.; Gulian, M.; Zayernouri, M.; D’Elia, M. Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials. J. Peridyn. Nonlocal Model
**2023**, 5, 392–459. [Google Scholar] [CrossRef] - Lin, Z.; Liu, F.; Wang, D.; Gu, Y. Reproducing kernel particle method for two-dimensional time-space fractional diffusion equations in irregular domains. Eng. Anal. Bound. Elem.
**2018**, 97, 131–143. [Google Scholar] [CrossRef] - Laskin, N. Fractional quantum mechanics and Lévy path integrals. Phys. Lett.
**2000**, A268, 298–305. [Google Scholar] [CrossRef] - Bertoin, J. Lévy Processes; Cambridge University Press: Cambridge, MA, USA, 1996. [Google Scholar]
- Frydryszak, A. Lagrangian Models of Particles with Spin: The First Seventy Years. In From Field Theory to Quantum Groups; Jancewicz, B., Sobczyk, J., Eds.; World Scientific: Singapore, 1996; pp. 151–172. [Google Scholar]
- Corben, H.C. Classical and Quantum Theories of Spinning Particles; Holden-Day: San Francisco, CA, USA, 1968. [Google Scholar]
- Ellis, J. Motion of a classical particle with spin. Math. Proc. Camb. Philos. Soc.
**1975**, 78, 145–156. [Google Scholar] [CrossRef] - Deriglazov, A. Classical Mechanics: Hamiltonian and Lagrangian Formalism; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Brink, L.; Di Vecchia, P.; Howe, P. A Lagrangian formulation of the classical and quantum dynamics of spinning particles. Nuc. Phys.
**1997**, B118, 76–94. [Google Scholar] [CrossRef] - Chen, W.; Li, Y.; Ma, P. The Fractional Laplacian; World Scientific: Singapore, 2020. [Google Scholar]
- D’Ovidio, M.; Garra, R. Fractional gradient and its application to the fractional advection equation. arXiv
**2013**, arXiv:1305.4400. [Google Scholar] - Case, J.S.; Chang, S.Y. On Fractional GJMS Operators. Commun. Pur. Appl. Math.
**2016**, 69, 1017–1061. [Google Scholar] [CrossRef] - Tarasov, V.E. Geometric Interpretation of Fractional-Order Derivative. FCAA
**2016**, 19, 1200–1221. [Google Scholar] [CrossRef] - Pisarski, R.D. Field theory of paths with a curvature-dependent term. Phys. Rev.
**1986**, D34, 670. [Google Scholar] [CrossRef] - Caffarelli, L.; Silvestre, L. An Extension Problem Related to the Fractional Laplacian. Commun. Partial. Differ. Equations
**2007**, 32, 1245–1260. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Porto, C.M.; Godinho, C.F.d.L.; Vancea, I.V.
Fractional Laplacian Spinning Particle in External Electromagnetic Field. *Dynamics* **2023**, *3*, 855-870.
https://doi.org/10.3390/dynamics3040046

**AMA Style**

Porto CM, Godinho CFdL, Vancea IV.
Fractional Laplacian Spinning Particle in External Electromagnetic Field. *Dynamics*. 2023; 3(4):855-870.
https://doi.org/10.3390/dynamics3040046

**Chicago/Turabian Style**

Porto, Claudio Maia, Cresus Fonseca de Lima Godinho, and Ion Vasile Vancea.
2023. "Fractional Laplacian Spinning Particle in External Electromagnetic Field" *Dynamics* 3, no. 4: 855-870.
https://doi.org/10.3390/dynamics3040046