|
SIGMA 10 (2014), 012, 13 pages arXiv:1311.7005
https://doi.org/10.3842/SIGMA.2014.012
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
Abstract
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.
pdf (402 kb)
tex (46 kb)
References
- 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.
- Barut A.O., Bracken A.J., Zitterbewegung and the internal geometry of the
electron, Phys. Rev. D 23 (1981), 2454-2463.
- 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.
- Berezin F.A., Marinov M.S., Particle spin dynamics as the Grassmann variant of
classical mechanics, Ann. Physics 104 (1977), 336-362.
- 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.
- Corben H.C., Classical and quantum theories of spinning particles, Holden-Day,
San Francisco, 1968.
- Deriglazov A.A., Classical mechanics: Hamiltonian and Lagrangian formalism,
Springer, Heidelberg, 2010.
- Deriglazov A.A., Nonrelativistic spin: à la Berezin-Marinov quantization
on a sphere, Modern Phys. Lett. A 25 (2010), 2769-2777.
- 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.
- Deriglazov A.A., Classical-mechanical models without observable trajectories
and the Dirac electron, Phys. Lett. A 377 (2012), 13-17,
arXiv:1203.5697.
- Deriglazov A.A., Spinning-particle model for the Dirac equation and the
relativistic Zitterbewegung, Phys. Lett. A 376 (2012),
309-313, arXiv:1106.5228.
- 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.
- Deriglazov A.A., Variational problem for Hamiltonian system on SO(k,m)
Lie-Poisson manifold and dynamics of semiclassical spin,
arXiv:1211.1219.
- 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.
- Deriglazov A.A., Nersessian A., Rigid particle revisited: extrinsic curvature
yields the Dirac equation, arXiv:1303.0483.
- Deriglazov A.A., Pupasov-Maksimov A.M., Lagrangian for Frenkel electron and
position's non-commutativity due to spin, arXiv:1312.6247.
- 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.
- 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.
- 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.
- Frenkel J., Die Elektrodynamik des rotierenden Elektrons, Z. Phys.
37 (1926), 243-262.
- Frenkel J., Spinning electrons, Nature 117 (1926), 653-654.
- 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.
- Gitman D.M., Tyutin I.V., Quantization of fields with constraints, Springer
Series in Nuclear and Particle Physics, Springer-Verlag, Berlin, 1990.
- Grassberger P., Classical charged particles with spin, J. Phys. A:
Math. Gen. 11 (1978), 1221-1226.
- Hanson A.J., Regge T., The relativistic spherical top, Ann. Physics
87 (1974), 498-566.
- 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.
- 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.
- Ramirez W.G., Deriglazov A.A., Pupasov-Maksimov A.M., Frenkel electron and a
spinning body in a curved background, arXiv:1311.5743.
|
|