Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 099, 21 pages      arXiv:1403.0255      https://doi.org/10.3842/SIGMA.2014.099
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Wong's Equations and Charged Relativistic Particles in Non-Commutative Space

Herbert Balasin a, Daniel N. Blaschke b, François Gieres c and Manfred Schweda a
a) Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
b) Los Alamos National Laboratory, Theory Division, Los Alamos, NM, 87545, USA
c) Université de Lyon, Université Claude Bernard Lyon 1 and CNRS/IN2P3, Institut de Physique Nucléaire, Bat. P. Dirac, 4 rue Enrico Fermi, F-69622-Villeurbanne, France

Received March 02, 2014, in final form October 17, 2014; Published online October 24, 2014

Abstract
In analogy to Wong's equations describing the motion of a charged relativistic point particle in the presence of an external Yang-Mills field, we discuss the motion of such a particle in non-commutative space subject to an external $U_\star(1)$ gauge field. We conclude that the latter equations are only consistent in the case of a constant field strength. This formulation, which is based on an action written in Moyal space, provides a coarser level of description than full QED on non-commutative space. The results are compared with those obtained from the different Hamiltonian approaches. Furthermore, a continuum version for Wong's equations and for the motion of a particle in non-commutative space is derived.

Key words: non-commutative geometry; gauge field theories; Lagrangian and Hamiltonian formalism; symmetries and conservation laws.

pdf (457 kb)   tex (33 kb)

References

  1. Acatrinei C., Comments on noncommutative particle dynamics, hep-th/0106141.
  2. Acatrinei C., Lagrangian versus quantization, J. Phys. A: Math. Gen. 37 (2004), 1225-1230, hep-th/0212321.
  3. Acatrinei C.S., Surprises in noncommutative dynamics, Ann. U. Craiova Phys. 21 (2011), 107-116.
  4. Adorno T.C., Gitman D.M., Shabad A.E., Vassilevich D.V., Classical noncommutative electrodynamics with external source, Phys. Rev. D 84 (2011), 065003, 13 pages, arXiv:1106.0639.
  5. Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., On the unitarity problem in space/time noncommutative theories, Phys. Lett. B 533 (2002), 178-181, hep-th/0201222.
  6. Balachandran A.P., Borchardt S., Stern A., Lagrangian and Hamiltonian descriptions of Yang-Mills particles, Phys. Rev. D 17 (1978), 3247-3256.
  7. Balachandran A.P., Marmo G., Skagerstam B.S., Stern A., Gauge symmetries and fibre bundles. Applications to particle dynamics, Lecture Notes in Physics, Vol. 188, Springer-Verlag, Berlin, 1983.
  8. Blaschke D.N., Kronberger E., Sedmik R.I.P., Wohlgenannt M., Gauge theories on deformed spaces, SIGMA 6 (2010), 062, 70 pages, arXiv:1004.2127.
  9. Brown L.S., Weisberger W.I., Vacuum polarization in uniform nonabelian gauge fields, Nuclear Phys. B 157 (1979), 285-326, Errata, Nuclear Phys. B, 172 (1980), 544.
  10. Chamseddine A.H., Connes A., Marcolli M., Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007), 991-1089, hep-th/0610241.
  11. Delduc F., Duret Q., Gieres F., Lefrançois M., Magnetic fields in noncommutative quantum mechanics, J. Phys. Conf. Ser. 103 (2008), 012020, 26 pages, arXiv:0710.2239.
  12. Deriglazov A.A., Poincare covariant mechanics on noncommutative space, J. High Energy Phys. 2003 (2003), no. 3, 021, 9 pages, hep-th/0211105.
  13. Dixon W.G., Dynamics of extended bodies in general relativity. I. Momentum and angular momentum, Proc. Roy. Soc. London Ser. A 314 (1970), 499-527.
  14. Dubois-Violette M., Kerner R., Madore J., Noncommutative differential geometry and new models of gauge theory, J. Math. Phys. 31 (1990), 323-330.
  15. Duval C., On the prequantum description of spinning particles in an external gauge field, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., Vol. 836, Springer, Berlin, 1980, 49-66.
  16. Duval C., Horváthy P., Particles with internal structure: the geometry of classical motions and conservation laws, Ann. Physics 142 (1982), 10-33.
  17. Duval C., Horváthy P.A., The `Peierls substitution' and the exotic Galilei group, Phys. Lett. B 479 (2000), 284-290, hep-th/0002233.
  18. Duval C., Horváthy P.A., Exotic Galilean symmetry in the non-commutative plane and the Hall effect, J. Phys. A: Math. Gen. 34 (2001), 10097-10107, hep-th/0106089.
  19. Estienne B., Haaker S.M., Schoutens K., Particles in non-Abelian gauge potentials: Landau problem and insertion of non-Abelian flux, New J. Phys. 13 (2011), 045012, 24 pages, arXiv:1102.0176.
  20. Fatollahi A.H., Mohammadzadeh H., On the classical dynamics of charges in non-commutative QED, Eur. Phys. J. C 36 (2004), 113-116, hep-th/0404209.
  21. Gracia-Bondía J.M., Várilly J.C., Algebras of distributions suitable for phase-space quantum mechanics. I, J. Math. Phys. 29 (1988), 869-879.
  22. Horváthy P.A., The non-commutative Landau problem, Ann. Physics 299 (2002), 128-140, hep-th/0201007.
  23. Horváthy P.A., Exotic Galilean symmetry and non-commutative mechanics in mathematical and in condensed matter physics, Talk given at International Conference on Noncommutative Geometry and Quantum Physics (January 4-10, 2006, Kolkata, India), hep-th/0602133.
  24. Horváthy P.A., Martina L., Stichel P.C., Exotic Galilean symmetry and non-commutative mechanics, SIGMA 6 (2010), 060, 26 pages, arXiv:1002.4772.
  25. Kerner R., Generalization of the Kaluza-Klein theory for an arbitrary non-abelian gauge group, Ann. Inst. H. Poincaré Sect. A (N.S.) 9 (1968), 143-152.
  26. Kosyakov B.P., Exact solutions in the Yang-Mills-Wong theory, Phys. Rev. D 57 (1998), 5032-5048, hep-th/9902039.
  27. Kosyakov B.P., Introduction to the classical theory of particles and fields, Springer-Verlag, Berlin, 2007.
  28. Linden N., Macfarlane A.J., van Holten J.W., Particle motion in a Yang-Mills field: Wong's equations and spin-$\frac 12$ analogues, Czechoslovak J. Phys. 46 (1996), 209-215, hep-th/9512071.
  29. Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts in Applied Mathematics, Vol. 17, 2nd ed., Springer-Verlag, New York, 1999.
  30. Mathisson M., Neue Mechanik materieller Systeme, Acta Phys. Polon. 6 (1937), 163-200.
  31. Mezincescu L., Star operation in quantum mechanics, hep-th/0007046.
  32. Nair V.P., Polychronakos A.P., Quantum mechanics on the noncommutative plane and sphere, Phys. Lett. B 505 (2001), 267-274, hep-th/0011172.
  33. Papapetrou A., Spinning test-particles in general relativity. I, Proc. Roy. Soc. London. Ser. A. 209 (1951), 248-258.
  34. Ramond P., Field theory: a modern primer, Frontiers in Physics, Vol. 51, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1981.
  35. Rivasseau V., Non-commutative renormalization, in Quantum Spaces, Prog. Math. Phys., Vol. 53, Editors B. Duplantier, V. Rivasseau, Birkhäuser, Basel, 2007, 19-107, arXiv:0705.0705.
  36. Sikivie P., Weiss N., Classical Yang-Mills theory in the presence of external sources, Phys. Rev. D 18 (1978), 3809-3821.
  37. Souriau J.M., Structure of dynamical systems. A symplectic view of physics, Progress in Mathematics, Vol. 149, Birkhäuser Boston, Inc., Boston, MA, 1997.
  38. Stern A., Noncommutative point sources, Phys. Rev. Lett. 100 (2008), 061601, 4 pages, arXiv:0709.3831.
  39. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, hep-th/0109162.
  40. Thirring W., Classical mathematical physics. Dynamical systems and field theories, 3rd ed., Springer-Verlag, New York, 1997.
  41. Vanhecke F.J., Sigaud C., da Silva A.R., Noncommutative configuration space. Classical and quantum mechanical aspects, Braz. J. Phys. 36 (2006), 194-207, math-ph/0502003.
  42. Várilly J.C., Gracia-Bondía J.M., Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988), 880-887.
  43. Vignes-Tourneret F., Renormalisation des theories de champs non commutatives, Ph.D. Thesis, Université Paris 11, 2006, math-ph/0612014.
  44. Wipf A.W., Nonrelativistic Yang-Mills particles in a spherically symmetric monopole field, J. Phys. A: Math. Gen. 18 (1985), 2379-2384.
  45. Wong S.K., Field and particle equations for the classical Yang-Mills field and particles with isotopic spin, Nuovo Cimento A 65 (1970), 689-694.

Previous article  Next article   Contents of Volume 10 (2014)