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

SIGMA 6 (2010), 016, 8 pages      arXiv:0911.2592      https://doi.org/10.3842/SIGMA.2010.016
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

From Noncommutative Sphere to Nonrelativistic Spin

Alexei A. Deriglazov
Dept. de Matematica, ICE, Universidade Federal de Juiz de Fora, MG, Brazil

Received November 12, 2009, in final form January 26, 2010; Published online February 04, 2010

Abstract
Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory. It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.

Key words: noncommutative geometry; nonrelativistic spin.

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References

  1. Berezin F.A., Marinov M.S., Particle spin dynamics as the Grassmann variant of classical mechanics, Ann. Physics 104 (1977), 336-362.
  2. Gitman D.M., Path integrals and pseudoclassical description for spinning particles in arbitrary dimensions, Nuclear Phys. B 488 (1997), 490-512, hep-th/9608180.
  3. Deriglazov A.A., Gitman D.M., Classical description of spinning degrees of freedom of relativistic particles by means of commuting spinors, Modern Phys. Lett. A 14 (1999), 709-720, hep-th/9811229.
  4. Frenkel J., Die Elektrodynamik des rotierenden Elektrons, Z. f. Physik 37 (1926), 243-262.
    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., Electrodynamics and classical theory of fields and particles, MacMillan, New York, 1964.
    Hanson A.J., Regge T., The relativistic spherical top, Ann. Physics 87 (1974), 498-566.
  5. Kuzenko S.M., Lyakhovich S.L., Segal A.Yu., A geometric model of the arbitrary spin massive particle, Internat. J. Modern Phys. A 10 (1995), 1529-1552, hep-th/9403196.
    Nersessian A., Ramos E., Massive spinning particles and the geometry of null curves, Phys. Lett. B 445 (1998), 123-128, hep-th/9807143.
    Barreto M.N., Ferreira F.J.S., Zlatev S.I., Strictly canonical quantization of a massless spinning particle and a quaternionic extension of pseudoclassical mechanics, hep-th/0510122.
    Casalbuoni R., Gomis J., Kamimura K., Longhi G., Space-time vector supersymmetry and massive spinning particle, J. High Energy Phys. 2008 (2008), no. 2, 094, 16 pages, arXiv:0801.2702.
    Das S., Ghosh S., Relativistic spinning particle in a nonnommutative extended spacetime, Phys. Rev. D 80 (2009), 085009, 8 pages, arXiv:0907.0290.
  6. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  7. Landau L.D., Lifshitz E.M., Quantum mechanics. Non-relativistic theory, Vol. 3, 3rd ed., Pergamon Press, Oxford - New York - Toronto, 1991, 1991.
  8. Lukierski J., Stichel P.C., Zakrzewski W.J., Galilean-invariant (2+1)-dimensional models with a Chern-Simons-like term and D=2 noncommutative geometry, Ann. Physics 260 (1997), 224-249, hep-th/9612017.
  9. Dunne G., Jackiw R., "Peireles substitution" and Chern-Simons quantum mechanics, Nuclear Phys. B Proc. Suppl. 33C (1993), 114-118, hep-th/9204057.
    Deriglazov A.A., Neves C., Oliveira W., Open string with a background B-field as the first order mechanics and noncommutativity, hep-th/0110183.
    Deriglazov A.A., Neves C., Oliveira W., Abreu E.M.C., Wotzasek C., Filgueiras C., Open string with a background B field as the first order mechanics, noncommutativity, and soldering formalism, Phys. Rev. D 76 (2007), 064007, 8 pages, arXiv:0707.1799.
  10. Deriglazov A.A., Noncommutative version of an arbitrary nondegenerated mechanics, hep-th/0208072.
    Deriglazov A.A., Noncommutative relativistic particle on the electromagnetic background, Phys. Lett. B 555 (2003), 83-88, hep-th/0208201.
  11. Bemfica F.S., Girotti H.O., Noncommutative quantum mechanics as a gauge theory, Phys. Rev. D 79 (2009), 125024, 8 pages, arXiv:0906.2161.
    Bemfica F.S., Girotti H.O., Noncommutative quantum mechanics: uniqueness of the functional description, Phys. Rev. D 78 (2008), 125009, 6 pages, arXiv:0810.1224.
  12. Baldiotti M.C., Gazeau J.-P., Gitman D.M., Semiclassical and quantum description of motion on noncommutative plane, Phys. Lett. A 373 (2009), 3937-3943, arXiv:0906.0388.
  13. Gomes M., Kupriyanov V.G., Position-dependent noncommutativity in quantum mechanics, Phys. Rev. D 79 (2009), 125011, 6 pages, arXiv:0902.3252.
    Gomes M., Kupriyanov V.G., da Silva A.J., Dynamical noncommutativity, arXiv:0908.2963.
  14. Amorim R., Tensor operators in noncommutative quantum mechanics, Phys. Rev. Lett. 101 (2008), 081602, 4 pages, arXiv:0804.4400.
    Amorim R., Tensor coordinates in noncommutative mechanics, J. Math. Phys. 50 (2009), 052103, 7 pages, arXiv:0804.4405.
  15. Bellucci S., Nersessian A., Phases in noncommutative quantum mechanics on (pseudo)sphere, Phys. Lett. B 542 (2002), 295-300, hep-th/0205024.
    Deriglazov A.A., Quantum mechanics on noncommutative plane and sphere from constrained systems, Phys. Lett. B 530 (2002), 235-243, hep-th/0201034.
    Daszkiewicz M., Lukierski J., Woronowicz M., κ-deformed statistics and classical four-momentum addition law, Modern Phys. Lett. A 23 (2008), 653-665, hep-th/0703200.
    Li K., Dulat S., The Aharonov-Bohm effect in noncommutative quantum mechanics, Eur. Phys. J. C 46 (2006), 825-828, hep-th/0508193.
    Dulat S., Li K., Commutator anomaly in noncommutative quantum mechanics, Modern Phys. Lett. A 21 (2006), 2971-2976, hep-th/0508060.
    Adorno T.C.M., Baldiotti C., Chaichian M., Gitman D.M., Tureanu A., Dirac equation in noncommutative space for hydrogen atom, Phys. Lett. B 682 (2009), 235-239, arXiv:0904.2836.
    Gitman D.M., Kupriyanov V.G., Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action, Eur. Phys. J. C 54 (2008), 325-332, arXiv:0707.0310.
    Miao Y.-G., Müller-Kirsten H.J.W., Park D.K., Chiral bosons in noncommutative spacetime, J. High Energy Phys. 2003 (2003), no. 8, 038, 21 pages, hep-th/0306034.
    Acatrinei C.S., A simple signal of noncommutative space, Modern Phys. Lett. A 20 (2005), 1437-1441, hep-th/0311134.
    Bérard A., Mohrbach H., Monopole and Berry phase in momentum space in noncommutative quantum mechanics, Phys. Rev. D 69 (2004), 127701, 4 pages, hep-th/0310167.
    Lukierski J., Stichel P.C., Zakrzewski W.J., Noncommutative planar particle dynamics with gauge interactions, Ann. Physics 306 (2003), 78-95, hep-th/0207149.
    Romero J.M., Santiago J.A., Cosmological constant and noncommutativity: a Newtonian point of view, Modern Phys. Lett. A 20 (2005), 781-790, hep-th/0310266.
  16. Aloisio R., Galante A., Grillo A., Luzio E., Méndez F., A note on DSR-like approach to space-time, Phys. Lett. B 610 (2005), 101-106, gr-qc/0501079.
    Aloisio R., Galante A., Grillo A.F., Liberati S., Luzio E., Méndez F., Deformed special relativity as an effective theory of measurements on quantum gravitational backgrounds, Phys. Rev. D 73 (2006), 045020, 11 pages, gr-qc/0511031.
    Aloisio R., Galante A., Grillo A.F., Liberati S., Luzio E., Méndez F., Modified special relativity on a fluctuating spacetime, Phys. Rev. D 74 (2006), 085017, 7 pages, gr-qc/0607024.
    Deriglazov A.A., Doubly special relativity in position space starting from the conformal group, Phys. Lett. B 603 (2004), 124-129, hep-th/0409232.
    Deriglazov A.A., Interpretation of Lorentz boosts in conformally deformed special relativity theory, hep-th/0510015.
    Mignemi S., Doubly special relativity and translation invariance, Phys. Lett. B 672 (2009), 186-189, arXiv:0808.1628.
  17. Girotti H.O., Gomes M., Rivelles V.O., da Silva A.J., A consistent noncommutative field theory: the Wess-Zumino model, Nuclear Phys. B 587 (2000), 299-310, hep-th/0005272.
    Girotti H.O., Noncommutative quantum field theories, Amer. J. Phys. 72 (2004), 608-612, hep-th/0301237.
    Amorim R., Dynamical symmetries in noncommutative theories, Phys. Rev. D 78 (2008), 105003, 7 pages, arXiv:0808.3062.
    Amorim R., Fermions and noncommutative theories, J. Math. Phys. 50 (2009), 022303, 7 pages, arXiv:0808.3903.
    Daszkiewicz M., Lukierski J., Woronowicz M., Towards quantum noncommutative κ-deformed field theory, Phys. Rev. D 77 (2008), 105007, 10 pages, arXiv:0708.1561.
    Gonera C., Kosinski P., Maslanka P., Giller S., Global symmetries of noncommutative space-time, Phys. Rev. D 72 (2005), 067702, 3 pages, hep-th/0507054.
    Amorim R., Abreu E.M.C., Quantum complex scalar fields and noncommutativity, Phys. Rev. D, to appear, arXiv:0909.0465.
  18. Szabo R.J., Quantum gravity, field theory and signatures of noncommutative spacetime, Gen. Relativity Gravitation 42 (2010), 1-29, arXiv:0906.2913.
    Banerjee R., Chakraborty B., Ghosh S., Mukherjee P., Samanta S., Topics in noncommutative geometry inspired physics, Found. Phys. 39 (2009), 1297-1345, arXiv:0909.1000.
  19. Snyder H.S., Quantized space-time, Phys. Rev. 71 (1947), 38-41.
  20. Deriglazov A.A., Poincaré covariant mechanics on noncommutative space, J. High Energy Phys. 2003 (2003), no. 3, 021, 9 pages, hep-th/0211105.
  21. Gitman D.M., Kupriyanov V.G., Gauge invariance and classical dynamics of noncommutative particle theory, arXiv:0910.1341.
  22. Bjorken J.D., Drell S.D., Relativistic quantum fields, McGraw-Hill Book Co., New York - Toronto - London - Sydney, 1965.
  23. Belich H., Colatto L.P., Costa-Soares T., Helayël-Neto J.A., Orlando M.T.D., Magnetic moment generation from non-minimal couplings in a scenario with Lorentz-symmetry violation, Eur. Phys. J. C 62 (2009), 425-432, arXiv:0806.1253.

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