# Relativistic Symmetries and Hamiltonian Formalism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Orbit Method

## 3. Poincaré Symmetry

## 4. The Conformal Group

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Loganayagam, R.; Surówka, P. Anomaly/Transport in an Ideal Weyl Gas. J. High Energy Phys.
**2012**, 1204, 97. [Google Scholar] [CrossRef] [Green Version] - Son, D.T.; Yamamoto, N. Berry Curvature, Triangle Anomalies, and the Chiral Magnetic Effect in Fermi Liquids. Phys. Rev. Lett.
**2012**, 109, 181602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hossenfelder, S. Lost in Math: How Beauty Leads Physics Astray; Basic Books: New York, NY, USA, 2018. [Google Scholar]
- Wigner, E. On Unitary Representations of the Inhomogeneous Lorentz Group. Ann. Math.
**1939**, 40, 149–204. [Google Scholar] [CrossRef] - Weinberg, S. The Quantum Theory of Fields I; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Lévy-Leblond, J.M. Nonrelativistic Particles and Wave Equations. Commun. Math. Phys.
**1967**, 6, 286–311. [Google Scholar] [CrossRef] - Kirillov, A.A. Elements of the Theory of Representations; Springer: Berlin/Heidelberg, Germany, 1976. [Google Scholar]
- Kirillov, A.A. Lectures on the Orbit Method. Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 2004; Volume 64. [Google Scholar]
- Souriau, J.M. Structure of Dynamical Systems: A Symplectic View of Physics; Birkhauser: Basel, Switzerland, 1997. [Google Scholar]
- Arnol’d, V.I. Mathematical Methods of Classical Mechanics; Springer: Berlin/Heidelberg, Germany, 1989. [Google Scholar]
- Marsden, J.E.; Ratiu, T.S. Introduction to Mechanics and Symmetry; Springer: Berlin/Heidelberg, Germany, 1999. [Google Scholar]
- Frenkel, J. Die Elektrodynamik des Rotierenden Elektrons. Z. Phys.
**1926**, 37, 243–262. [Google Scholar] [CrossRef] - Thomas, L.H. The Motion of the Spinning Electron. Nature
**1926**, 117, 514. [Google Scholar] [CrossRef] - Thomas, L.H. The Kinematics of an Electron with an Axis. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1927**, 3, 1–22. [Google Scholar] [CrossRef] - Kramers, H. On the Classical Theory of the Spinning Electron. Physica
**1934**, 1, 825–828. [Google Scholar] [CrossRef] - Mathisson, M. Neue Mechanik Materieller Systeme. Acta Phys. Pol.
**1937**, 6, 163–2900. [Google Scholar] - Papapetrou, A. Spinning Test-Particles in General Relativity. I. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1951**, 209, 248–258. [Google Scholar] - Corinaldesi, E.; Papapetrou, A. Spinning Test-Particles in General Relativity. II. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1951**, 209, 259–268. [Google Scholar] - Dixon, W. A Covariant Multipole Formalism for Extended Test Bodies in General Relativity. Il Nuovo Cim. (1955–1965)
**1964**, 34, 317–339. [Google Scholar] [CrossRef] - Dixon, W. Description of Extended Bodies by Multipole Moments in Special Relativity. J. Math. Phys.
**1967**, 8, 1591–1605. [Google Scholar] [CrossRef] - Dixon, W. Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1970**, 314, 499–527. [Google Scholar] - Bhabha, H.J.; Corben, H.C. General Classical Theory of Spinning Particles in a Maxwell Field. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1941**, 178, 273–314. [Google Scholar] - Corben, H. Spin in Classical and Quantum Theory. Phys. Rev.
**1961**, 121, 1833–1839. [Google Scholar] [CrossRef] - Corben, H. Spin Precession in Classical Relativistic Mechanics. Il Nuovo Cim. (1955–1965)
**1961**, 20, 529–541. [Google Scholar] [CrossRef] - Nyborg, P. On Classical Theories of Spinning Particles. Il Nuovo Cim. (1955–1965)
**1962**, 23, 47. [Google Scholar] [CrossRef] - Frydryszak, A. Lagrangian Models of Particles with Spin: The First Seventy Years. In From Field Theory to Quantum Groups; World Scientific: Singapore, 1996; pp. 151–172. [Google Scholar]
- Gaioli, F.H.; Alvarez, E.T.G. Classical and Quantum Theories of Spin. Found. Phys.
**1998**, 28, 1539–1550. [Google Scholar] [CrossRef] [Green Version] - Deriglazov, A.A. Variational Problem for the Frenkel and the Bargmann–Michel–Telegdi (BMT) Equations. Mod. Phys. Lett. A
**2013**, 28, 1250234. [Google Scholar] [CrossRef] [Green Version] - Deriglazov, A.A. Lagrangian for the Frenkel Electron. Phys. Lett. B
**2014**, 736, 278–282. [Google Scholar] [CrossRef] [Green Version] - Costa, L.F.; Herdeiro, C.; Natário, J.; Zilhao, M. Mathisson’s Helical Motions for a Spinning Particle: Are They Unphysical? Phys. Rev. D
**2012**, 85, 024001. [Google Scholar] [CrossRef] [Green Version] - Costa, L.F.; Natário, J. Center of Mass, Spin Supplementary Conditions, and the Momentum of Spinning Particles. In Equations of Motion in Relativistic Gravity; Springer: Berlin/Heidelberg, Germany, 2015; pp. 215–258. [Google Scholar]
- Fradkin, E. Application of Functional Methods in Quantum Field Theory and Quantum Statistics (II). Nucl. Phys.
**1966**, 76, 588–624. [Google Scholar] [CrossRef] - Berezin, F.; Marinov, M. Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics. Ann. Phys.
**1977**, 104, 336–362. [Google Scholar] [CrossRef] - Howe, P.; Penati, S.; Pernici, M.; Townsend, P. Wave Equations for Arbitrary Spin from Quantization of the Extended Supersymmetric Spinning Particle. Phys. Lett. B
**1988**, 215, 555–558. [Google Scholar] [CrossRef] - 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] - Wiegmann, P.B. Multivalued Functionals and Geometrical Approach for Quantization of Relativistic Particles and Strings. Nucl. Phys. B
**1989**, 323, 311–329. [Google Scholar] [CrossRef] - Carinena, J.F.; Garcia-Bondia, J.; Varilly, J.C. Relativistic Quantum Kinematics in the Moyal Representation. J. Phys. A Math. Gen.
**1990**, 23, 901–933. [Google Scholar] [CrossRef] - Andrzejewski, K.; Gonera, C.; Goner, J.; Kosiński, P.; Maślanka, P. Spinning Particles, Coadjoint Orbits and Hamiltonian Formalism. arXiv
**2020**, arXiv:2008.09478. [Google Scholar] - Novozhilov, Y.V. Introduction to Elementary Particle Theory; Pergamon Press: Oxford, UK, 1975. [Google Scholar]
- Duval, C.; Horvathy, P. Chiral Fermions as Classical Massless Spinning Particles. Phys. Rev. D
**2015**, 91, 045013. [Google Scholar] [CrossRef] [Green Version] - Duval, C.; Elbistan, M.; Horvathy, P.; Zhang, P.M. Wigner–Souriau Translations and Lorentz Symmetry of Chiral Fermions. Phys. Lett. B
**2015**, 742, 322–326. [Google Scholar] [CrossRef] - Andrzejewski, K.; Kijanka-Dec, A.; Kosiński, P.; Maślanka, P. Chiral Fermions, Massless Particles and Poincare Covariance. Phys. Lett. B
**2015**, 746, 417–423. [Google Scholar] [CrossRef] [Green Version] - Skagerstam, B. Localization of Massless Spinning Particles and the Berry Phase. arXiv
**1992**, arXiv:hep-th/9210054. [Google Scholar] - Kosiński, P.; Maślanka, P. Localizability, Gauge Symmetry and Newton–Wigner Operator for Massless Particles. Ann. Phys.
**2018**, 398, 203–213. [Google Scholar] [CrossRef] [Green Version] - Chen, J.Y.; Son, D.T.; Stephanov, M.A.; Yee, H.U.; Yin, Y. Lorentz Invariance in Chiral Kinetic Theory. Phys. Rev. Lett.
**2014**, 113, 182302. [Google Scholar] [CrossRef] [Green Version] - Andrzejewski, K.; Brihaye, Y.; Gonera, C.; Gonera, J.; Kosiński, P.; Maślanka, P. The Covariance of Chiral Fermions Theory. J. High Energy Phys.
**2019**, 8, 11. [Google Scholar] [CrossRef] [Green Version] - Berger, L. Side-Jump Mechanism for the Hall Effect of Ferromagnets. Phys. Rev. B
**1970**, 2, 4559. [Google Scholar] [CrossRef] - Bliokh, K.Y.; Bliokh, Y.P. Topological Spin Transport of Photons: The Optical Magnus Effect and Berry Phase. Phys. Lett. A
**2004**, 333, 181–186. [Google Scholar] [CrossRef] [Green Version] - Onoda, M.; Murakami, S.; Nagaosa, N. Hall Effect of Light. Phys. Rev. Lett.
**2004**, 93, 083901. [Google Scholar] [CrossRef] [Green Version] - Duval, C.; Horvath, Z.; Horváthy, P. Geometrical Spinoptics and the Optical Hall Effect. J. Geom. Phys.
**2007**, 57, 925–941. [Google Scholar] [CrossRef] [Green Version] - Duval, C.; Horváth, Z.; Horváthy, P. Fermat Principle for Spinning Light. Phys. Rev. D
**2006**, 74, 021701. [Google Scholar] [CrossRef] [Green Version] - Bliokh, K.Y.; Nori, F. Relativistic Hall Effect. Phys. Rev. Lett.
**2012**, 108, 120403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stone, M.; Dwivedi, V.; Zhou, T. Wigner Translations and the Observer Dependence of the Position of Massless Spinning Particles. Phys. Rev. Lett.
**2015**, 114, 210402. [Google Scholar] [CrossRef] [Green Version] - Bolonek-Lasoń, K.; Kosiński, P.; Maślanka, P. Lorentz Transformations, Sideways Shift and Massless Spinning Particles. Phys. Lett. B
**2017**, 769, 117–120. [Google Scholar] [CrossRef] - Todorov, I.T. Conformal Description of Spinning Particles; Springer: Berlin/Heidelberg, Germany, 1986. [Google Scholar]
- Mack, G.; Salam, A. Finite Component Field Representations of the Conformal Group. Ann. Phys.
**1969**, 53, 174–202. [Google Scholar] [CrossRef] - Mack, G. All Unitary Ray Representations of the Conformal Group SU (2,2) with positive energy. Commun. Math. Phys.
**1977**, 55, 1–28. [Google Scholar] [CrossRef] [Green Version] - Gonera, J.; Kosiński, P.; Maślanka, P. Conformal Symmetry, Chiral Fermions and Semiclassical Approximation. Phys. Lett. B
**2020**, 800, 135111. [Google Scholar] [CrossRef] - Kosiński, P.; Maślanka, P. Hamiltonian Description of Conformally Invariant Elementary Systems. Work in progress.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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**

Kosiński, P.; Maślanka, P.
Relativistic Symmetries and Hamiltonian Formalism. *Symmetry* **2020**, *12*, 1810.
https://doi.org/10.3390/sym12111810

**AMA Style**

Kosiński P, Maślanka P.
Relativistic Symmetries and Hamiltonian Formalism. *Symmetry*. 2020; 12(11):1810.
https://doi.org/10.3390/sym12111810

**Chicago/Turabian Style**

Kosiński, Piotr, and Paweł Maślanka.
2020. "Relativistic Symmetries and Hamiltonian Formalism" *Symmetry* 12, no. 11: 1810.
https://doi.org/10.3390/sym12111810