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

SIGMA 10 (2014), 012, 13 pages      arXiv:1311.7005

Geometric Constructions Underlying Relativistic Description of Spin on the Base of Non-Grassmann Vector-Like Variable

Alexei A. Deriglazov and Andrey M. Pupasov-Maksimov
Departamento de Matemática, ICE, Universidade Federal de Juiz de Fora, MG, Brasil

Received December 17, 2013, in final form February 04, 2014; Published online February 08, 2014

Basic notions of Dirac theory of constrained systems have their analogs in differential geometry. Combination of the two approaches gives more clear understanding of both classical and quantum mechanics, when we deal with a model with complicated structure of constraints. In this work we describe and discuss the spin fiber bundle which appeared in various mechanical models where spin is described by vector-like variable.

Key words: semiclassical description of relativistic spin; Dirac equation; theories with constraints.

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  1. Bargmann V., Michel L., Telegdi V.L., Precession of the polarization of particles moving in a homogeneous electromagnetic field, Phys. Rev. Lett. 2 (1959), 435-436.
  2. Barut A.O., Bracken A.J., Zitterbewegung and the internal geometry of the electron, Phys. Rev. D 23 (1981), 2454-2463.
  3. Barut A.O., Thacker W., Covariant generalization of the Zitterbewegung of the electron and its SO(4,2) and SO(3,2) internal algebras, Phys. Rev. D 31 (1985), 1386-1392.
  4. Berezin F.A., Marinov M.S., Particle spin dynamics as the Grassmann variant of classical mechanics, Ann. Physics 104 (1977), 336-362.
  5. Cognola G., Vanzo L., Zerbini S., Soldati R., On the Lagrangian formulation of a charged spinning particle in an external electromagnetic field, Phys. Lett. B 104 (1981), 67-69.
  6. Corben H.C., Classical and quantum theories of spinning particles, Holden-Day, San Francisco, 1968.
  7. Deriglazov A.A., Classical mechanics: Hamiltonian and Lagrangian formalism, Springer, Heidelberg, 2010.
  8. Deriglazov A.A., Nonrelativistic spin: à la Berezin-Marinov quantization on a sphere, Modern Phys. Lett. A 25 (2010), 2769-2777.
  9. Deriglazov A.A., A semiclassical description of relativistic spin without the use of Grassmann variables and the Dirac equation, Ann. Physics 327 (2012), 398-406, arXiv:1107.0273.
  10. Deriglazov A.A., Classical-mechanical models without observable trajectories and the Dirac electron, Phys. Lett. A 377 (2012), 13-17, arXiv:1203.5697.
  11. Deriglazov A.A., Spinning-particle model for the Dirac equation and the relativistic Zitterbewegung, Phys. Lett. A 376 (2012), 309-313, arXiv:1106.5228.
  12. Deriglazov A.A., Variational problem for the Frenkel and the Bargmann-Michel-Telegdi (BMT) equations, Modern Phys. Lett. A 28 (2013), 1250234, 9 pages, arXiv:1204.2494.
  13. Deriglazov A.A., Variational problem for Hamiltonian system on SO(k,m) Lie-Poisson manifold and dynamics of semiclassical spin, arXiv:1211.1219.
  14. Deriglazov A.A., Evdokimov K.E., Local symmetries and the Noether identities in the Hamiltonian framework, Internat. J. Modern Phys. A 15 (2000), 4045-4067, hep-th/9912179.
  15. Deriglazov A.A., Nersessian A., Rigid particle revisited: extrinsic curvature yields the Dirac equation, arXiv:1303.0483.
  16. Deriglazov A.A., Pupasov-Maksimov A.M., Lagrangian for Frenkel electron and position's non-commutativity due to spin, arXiv:1312.6247.
  17. Deriglazov A.A., Rizzuti B.F., Zamudio G.P., Castro P.S., Non-Grassmann mechanical model of the Dirac equation, J. Math. Phys. 53 (2012), 122303, 31 pages, arXiv:1202.5757.
  18. Dirac P.A.M., Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series, Vol. 2, Belfer Graduate School of Science, New York, 1967.
  19. Foldy L.L., Wouthuysen S.A., On the Dirac theory of spin 1/2 particles and its non-relativistic limit, Phys. Rev. 78 (1950), 29-36.
  20. Frenkel J., Die Elektrodynamik des rotierenden Elektrons, Z. Phys. 37 (1926), 243-262.
  21. Frenkel J., Spinning electrons, Nature 117 (1926), 653-654.
  22. Gavrilov S.P., Gitman D.M., Quantization of pointlike particles and consistent relativistic quantum mechanics, Internat. J. Modern Phys. A 15 (2000), 4499-4538, hep-th/0003112.
  23. Gitman D.M., Tyutin I.V., Quantization of fields with constraints, Springer Series in Nuclear and Particle Physics, Springer-Verlag, Berlin, 1990.
  24. Grassberger P., Classical charged particles with spin, J. Phys. A: Math. Gen. 11 (1978), 1221-1226.
  25. Hanson A.J., Regge T., The relativistic spherical top, Ann. Physics 87 (1974), 498-566.
  26. Laroze D., Gutiérrez G., Rivera R., Yáñez J.M., Dynamics of a rotating particle under a time-dependent potential: exact quantum solution from the classical action, Phys. Scr. 78 (2008), 015009, 5 pages.
  27. Peletminskii A., Peletminskii S., Lagrangian and Hamiltonian formalisms for relativistic dynamics of a charged particle with dipole moment, Eur. Phys. J. C Part. Fields 42 (2005), 505-517.
  28. Ramirez W.G., Deriglazov A.A., Pupasov-Maksimov A.M., Frenkel electron and a spinning body in a curved background, arXiv:1311.5743.

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