# Behaviour of Charged Spinning Massless Particles

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. The Spinless Relativistic Particle

#### 2.1. The Massive Case

#### Example

#### 2.2. The Massless Case

#### 2.2.1. Equations of Motion

#### 2.2.2. Energy Equation

#### 2.2.3. Example of a Constant Electromagnetic Field

## 3. The Spinning Charged and Massless Particle

#### 3.1. Time Parametrization

**N.B.**We observe that the einbein variable $e\left(t\right)$ is dynamical, its evolution being defined by the first of Equation (44). On the other hand, the constraint (43) fixes the absolute value of the velocity, which may thus be variable and different from that of the light. This feature is a peculiarity of the massless theory. We will check this for some concrete examples in Section 3.4.

#### 3.2. Conservation Laws

#### 3.3. Physical Interpretation of the Classical Theory

#### 3.4. Constant Electromagnetic Field

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Geim, A.K.; MacDonal, A.H. Graphene: Exploring carbon flatland. Phys. Today
**2007**, 60. [Google Scholar] [CrossRef] - Castro, A.H.; Guinea, N.F.; Peres, N.M.P.; Novoselov, K.S.; Geim, A.K. The electronic properties of graphene. Rev. Mod. Phys.
**2009**, 81, 109. [Google Scholar] [CrossRef] - Frenkel, J. Zur Elektrodynamik punktförmiger Elektronen. Z. Phys.
**1925**, 32, 518–534. [Google Scholar] [CrossRef] - Frenkel, J. Die Elektrodynamik des Rotierenden Electrons. Z. Phys.
**1926**, 37, 243–262. [Google Scholar] [CrossRef] - Landau, L.D.; Lifschitz, E.M. The Classical Theory of Fields. In Course of Theoretical Physics; Oxford-Pergamon Press: Oxford, UK, 1975; Volume 2. [Google Scholar]
- Martin, J.L. Generalized Classical Dynamics and the ‘Classical Analogue’ of a Fermi Oscillator. Proc. R. Soc. Lond. A
**1959**, 251, 536. [Google Scholar] - Berezin, F.A.; Marinov, M.S. Classical Spin and Grassmann Algebra. Pisma Zh. Eksp. Teor. Fiz.
**1975**, 21, 678–680. (In Russian) [Google Scholar] - Berezin, F.A.; Marinov, M.S. Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics. Ann. Phys.
**1977**, 104, 336–362. [Google Scholar] - Casalbuoni, R. Relativity and supersymmetries. Phys. Lett. B
**1976**, 62, 49. [Google Scholar] [CrossRef] - Casalbuoni, R. The classical mechanics for Bose-Fermi systems. Nuovo Cimento A
**1976**, 33, 389–431. [Google Scholar] [CrossRef] - Brink, L.; Deser, S.; Zumino, B.; Di Vecchia, P.; Howe, P. Local supersymmetry for spinning particles. Phys. Lett. B
**1976**, 64, 435–438, Errata. Phys. Lett. B**1977**, 64, 488. [Google Scholar] - Brink, L.; Di Vecchia, P.; Howe, P. A Lagrangian formulation of the classical and quantum dynamics of spinning particles. Nucl. Phys. B
**1977**, 118, 76–94. [Google Scholar] [CrossRef] - Balachandran, A.P.; Salomonson, P.; Skagerstam, B.-S.; Winnberg, J.-O. Classical description of a particle interacting with a non-Abelian gauge field. Phys. Rev. D
**1977**, 15, 2308. [Google Scholar] [CrossRef] - Salomonson, P. Sypersymmetric actions for spinning particles. Phys. Rev. D
**1978**, 18, 1868. [Google Scholar] [CrossRef] - Carlos, G.A.P.; Teitelboim, C. Classical supersymmetric particles. J. Math. Phys.
**1980**, 21, 1863–1880. [Google Scholar] - Van Holten, J.W. Quantum theory of a massless spinning particle. Z. Phys. C
**1988**, 41, 497–504. [Google Scholar] [CrossRef] - Van Holten, J.W. On the electrodynamics of spinning particles. Nucl. Phys. B
**1991**, 356, 3–26. [Google Scholar] [CrossRef] - Fainberg, V.Y.; Marshakov, A.V. Local Supersymmetry and Dirac Particle Propagator as a Path Integral. Nucl. Phys. B
**1988**, 306, 659–676. [Google Scholar] [CrossRef] - Aliev, T.M.; Fainberg, V.Y.; Pak, N.K. Path integral for spin: A New approach. Nucl. Phys. B
**1994**, 429, 321–343. [Google Scholar] [CrossRef] - Geyer, B.; Dmitry, G.; Shapiro, I.L. Path integral and pseudoclassical action for spinning particle in external electromagnetic and torsion fields. Int. J. Mod. Phys. A
**2000**, 15, 3861–3876. [Google Scholar] [CrossRef] - Bittencourt, J.A. Fundamentals of Plasma Physics, 3rd ed.; Springer: New York, NY, USA, 2004. [Google Scholar] [CrossRef]
- Rohrlich, F. Classical Charged Particles, 3rd ed.; World Scientific Publishing: Singapore, 2007. [Google Scholar]
- Kosyakov, B. Introduction to the Classical Theory of Particles and Fields; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Kosyakov, B.P. Massless interacting particles. J. Phys. A
**2008**, 41, 465401. [Google Scholar] [CrossRef] - Azzurli, F.; Lechner, K. The Lienard-Wiechert field of accelerated massless charges. Phys. Lett. A
**2013**, 377, 1025–1029. [Google Scholar] [CrossRef] - Azzurli, F.; Lechner, K. Electromagnetic fields and potentials generated by massless charged particles. Ann. Phys.
**2014**, 349, 1–32. [Google Scholar] [CrossRef] - Lechner, K. Electrodynamics of massless charged particles. J. Math. Phys.
**2015**, 56, 022901. [Google Scholar] [CrossRef] - Deriglazov, A.A.; Walberto, G.R. World-line geometry probed by fast spinning particle. Mod. Phys. Lett. A
**2015**, 30, 1550101. [Google Scholar] [CrossRef] - Deriglazov, A.A.; Walberto, G.R. Ultrarelativistic Spinning Particle and a Rotating Body in External Fields. Adv. High Energy Phys.
**2016**, 2016, 1376016. [Google Scholar] [CrossRef]

**Figure 1.**Particle trajectories in the $z=0$ plane for $0\le t\le 5$. Charge $q=1$, constant electric field E in the positive y direction, constant magnetic field B in the positive z direction. Energy $\mathcal{E}=0.2$, initial velocity ${\mathbf{v}}_{0}$ = $(0.1,0.995,0)$. Solid line: $B=1.6,\phantom{\rule{0.166667em}{0ex}}E=1$; dotted line: $B=E=1$; dashed line: $B=0.4,\phantom{\rule{0.166667em}{0ex}}E=1$; dotted-dashed line: $B=0,\phantom{\rule{0.166667em}{0ex}}E=1$. The $B=0$ trajectory would be on the upper vertical axis in the case of ${\mathbf{v}}_{0}$ = $(0,1,0)$.

**Figure 2.**(

**a**) particle trajectories in the $z=0$ plane for $0\le t\le 5$. Charge $q=1$, constant electric field $E=1$ in the positive y direction, constant magnetic field $B=0.4$ in the positive z direction. Initial velocity ${\mathbf{v}}_{0}$ = $(1,0,0)$. Dashed line: $\mathcal{E}=0.1$; dotted line: $\mathcal{E}=0.3$; solid line: $\mathcal{E}=0.7$; (

**b**) Particle trajectories in the $z=0$ plane for $0\le t\le 5$. Charge $q=1$, constant electric field $E=1$ in the positive y direction, constant magnetic field $B=1.6$ in the positive z direction. Initial velocity ${\mathbf{v}}_{0}$ = $(0.1,0.995,0)$. Dashed line: $\mathcal{E}=0.2$; dotted line: $\mathcal{E}=0.3$; solid line: $\mathcal{E}=0.5$.

**Figure 3.**(a) Particle trajectories in the $z=0$ plane, (b) spin ${s}_{z}\left(t\right)$ and (c) velocity $\left|\mathbf{v}\right|$ = $|\dot{\mathbf{x}}\left(t\right)|$ for a constant electric field E in the positive y direction and a constant magnetic field B in the positive z direction. Parameters’ values are chosen as: charge $q=1$, energy $\mathcal{E}=2$, initial spin $\mathbf{s}\left(0\right)$ = $(0,0,0.5)$ and initial velocity $\mathbf{v}\left(0\right)$ = $({v}_{0x},0.9,0)$, ${v}_{0x}$ being the largest of the solutions of the constraint (50). The following field configurations have been chosen: $B=3.2,\phantom{\rule{0.166667em}{0ex}}E=2$ (solid lines); $B=E=2$ (dotted lines); $B=0.8,\phantom{\rule{0.166667em}{0ex}}E=2$ (dashed lines); $B=0,\phantom{\rule{0.166667em}{0ex}}E=2$ (dotted-dashed lines). The $B=0$ trajectory would be on the upper vertical axis in the case of $\mathbf{v}\left(0\right)$ = $({v}_{0x},1,0)$.

© 2017 by the authors. 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Morales, I.; Neves, B.; Oporto, Z.; Piguet, O.
Behaviour of Charged Spinning Massless Particles. *Symmetry* **2018**, *10*, 2.
https://doi.org/10.3390/sym10010002

**AMA Style**

Morales I, Neves B, Oporto Z, Piguet O.
Behaviour of Charged Spinning Massless Particles. *Symmetry*. 2018; 10(1):2.
https://doi.org/10.3390/sym10010002

**Chicago/Turabian Style**

Morales, Ivan, Bruno Neves, Zui Oporto, and Olivier Piguet.
2018. "Behaviour of Charged Spinning Massless Particles" *Symmetry* 10, no. 1: 2.
https://doi.org/10.3390/sym10010002