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

SIGMA 6 (2010), 060, 26 pages      arXiv:1002.4772
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Exotic Galilean Symmetry and Non-Commutative Mechanics

Peter A. Horváthy a, Luigi Martina b and Peter C. Stichel c
a) Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Parc de Grandmont, F-37200 Tours, France
b) Dipartimento di Fisica - Università del Salento and Sezione INFN di Lecce, via Arnesano, CP. 193, I-73100 Lecce, Italy
c) An der Krebskuhle 21 D-33 619 Bielefeld, Germany

Received March 23, 2010, in final form July 19, 2010; Published online July 26, 2010

Some aspects of the ''exotic'' particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.

Key words: noncommutative spaces; Galilean symmetry; dynamical systems; quantum field theory.

pdf (421 kb)   ps (228 kb)   tex (36 kb)


  1. Landau L., Lifschitz E., Physique théorique, Tome III, Mécanique quantique. Théorie non relativiste, Mir, Moscow, 1967.
  2. Simms D., Projective representations, symplectic manifolds and extensions of Lie algebras, Lectures given at Centre de Physique Théorique, CNRS, Marseille (1969), Preprint 69/P.300.
    Aldaya V., de Azcárraga J.A., Cohomology, central extensions, and (dynamical) groups, Internat. J. Theoret. Phys. 24 (1985), 141-154.
    Tuynman G.M., Wiegerinck W.A.J.J., Central extensions and physics, J. Geom. Phys. 4 (1987), 207-258.
    Marmo G., Morandi G., Simoni A., Sudarshan E.C.G., Quasi-invariance and central extensions, Phys. Rev. D 37 (1988), 2196-2205.
    de Saxcé G., Vallée C., Construction of a central extension of a Lie group from its class of symplectic cohomology, J. Geom. Phys. 60 (2010), 165-174.
  3. Souriau J.-M., Structure des systèmes dynamiques, Dunod, Paris, 1970.
    Souriau J.-M., Structure of dynamical systems. A symplectic view of physics, Progress in Mathematics, Vol. 149, Birkhäuser Boston, Inc., Boston, MA, 1997.
  4. Bargmann V., On unitary ray representations of continuous groups, Ann. of Math. (2) 59 (1954), 1-46.
  5. Lévy-Leblond J.-M., Galilei group and Galilean invariance, in Group Theory and Applications, Vol. II, Academic Press, New York, 1971, 221-299.
  6. Duval C., Exotic Galilei group, IQHE and Chern-Simons electrodynamics, Unpublished draft, 1995.
    Grigore D.R., Transitive symplectic manifolds in 1+2 dimensions, J. Math. Phys. 37 (1996), 240-253.
    Grigore D.R., The projective unitary irreducible representations of the Galilei group in 1+2 dimensions, J. Math. Phys. 37 (1996), 460-473, hep-th/9312048.
  7. Ballesteros A., Gadella M., del Olmo M.A., Moyal quantization of 2+1-dimensional Galilean systems, J. Math. Phys. 33 (1992), 3379-3386.
    Brihaye Y., Gonera C., Giller S., Kosinski P., Galilean invariance in 2+1 dimensions, hep-th/9503046.
  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. Duval C., Horváthy P.A., The exotic Galilei group and the "Peierls substitution", Phys. Lett. B 479 (2000), 284-290, hep-th/0002233.
    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.
    Horváthy P.A., The non-commutative Landau problem, Ann. Physics 299 (2002), 128-140, hep-th/0201007.
  10. Nair V.P., Polychronakos A.P., Quantum mechanics on the noncommutative plane and sphere, Phys. Lett. B 505 (2001), 267-274, hep-th/0011172.
  11. Sochichiu C., A note on noncommutative and false noncommutative spaces, Appl. Sciences 3 (2001), 48-51, hep-th/0010149.
    Bellucci S., Nersessian A., Sochichiu C., Two phases of the non-commutative quantum mechanics, Phys. Lett. B 522 (2001), 345-349, hep-th/0106138.
    Acatrinei C., Path integral formulation of noncommutative quantum mechanics, J. High Energy Phys. 2001 (2001), no. 9, 007, 7 pages, hep-th/0107078.
    Gamboa J., Loewe M., Méndez F., Rojas J.C., Noncommutative quantum mechanics, Phys. Rev. D 64 (2001), 067901, 3 pages.
    Banerjee R., A novel approach to noncommutativity in planar quantum mechanics, Modern Phys. Lett. A 17 (2002), 631-645, hep-th/0106280.
  12. Djemai A.E.F., Smail H., On quantum mechanics on noncommutative quantum phase space, Commun. Theor. Phys. 41 (2004), 837-844, hep-th/0309006.
    Banerjee R., Kumar K., Deformed relativistic and nonrelativistic symmetries on canonical noncommutative spaces, Phys. Rev. D 75 (2007), 045008, 5 pages, hep-th/0604162.
    Banerjee R., Deformed Schrödinger symmetry on noncommutative space, Eur. Phys. J. C 47 (2006), 541-545, hep-th/0508224.
    Papageorgiou G., Schroers B.J., A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions, J. High Energy Phys. 2009 (2009), no. 11, 009, 40 pages, arXiv:0907.2880.
    Scholtz F.G., Gouba L., Hafver A., Rohwer C.M., Formulation, interpretation and application of non-commutative quantum mechanics, J. Phys. A: Math. Theor. 42 (2009), 175303, 13 pages, arXiv:0812.2803.
    Gangopadhyay S., Scholtz F.G., Path integral action of a particle in the non commutative plane, arXiv:0904.0379.
    Baldiotti M.C., Gazeau J.P., Gitman D.M., Semiclassical and quantum description of motion on non-commutative plane, Phys. Lett. A 373 (2009), 3937-3943, arXiv:0906.0388.
    Gomes M., Kupriyanov V.G., Position-dependent noncommutativity in quantum mechanics, Phys. Rev. D 79 (2009), 125011, 6 pages, arXiv:0902.3252.
  13. Romero J.M., Santiago J.A., Vergara J.D., Newton's second law in a non-commutative space, Phys. Lett. A 310 (2003), 9-12, hep-th/0211165.
    Romero J.M., Vergara J.D., The Kepler problem and noncommutativity, Modern Phys. Lett. A 18 (2003), 1673-1680, hep-th/0303064.
  14. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, hep-th/0109162.
  15. Delduc F., Duret Q., Gieres F., Lefrancois M., Magnetic fields in noncommutative quantum mechanics, J. Phys. Conf. Ser. 103 (2008), 012020, 26 pages, arXiv:0710.2239.
  16. Jackiw R., Nair V.P., Anyon spin and the exotic central extension of the planar Galilei group, Phys. Lett. B 480 (2000), 237-238, hep-th/0003130.
    Duval C., Horváthy P.A., Spin and exotic Galilean symmetry, Phys. Lett. B 547 (2002), 306-312, hep-th/0209166.
  17. Fehér L., Equations of motion and dynamical symmetries of point particles, PhD thesis, University of Szeged, 1988 (in Hungarian).
    Skagerstam B.-S., Stern A., Topological quantum mechanics in 2+1 dimensions, Internat. J. Modern Phys. A 5 (1990), 1575-1595.
  18. Plyushchay M.S., Relativistic model of the anyon, Phys. Lett. B 248 (1990), 107-112.
    Plyushchay M.S., Fractional spin. Majorana-Dirac field, Phys. Lett. B 273 (1991), 250-254.
    Jackiw R., Nair V.P., Relativistic wave equation for anyons, Phys. Rev. D 43 (1991), 1933-1942.
  19. Horváthy P.A., Plyushchay M.S., Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane, J. High Energy Phys. 2002 (2002), no. 6, 033, 11 pages, hep-th/0201228.
    Horváthy P.A., Mathisson's spinning electron: noncommutative mechanics and exotic Galilean symmetry, 66 years ago, Acta Phys. Polon. B 34 (2003), 2611-2621, hep-th/0303099.
    Plyushchay M.S., Majorana equation and exotics: higher derivative models, anyons and noncommutative geometry, Electron. J. Theor. Phys. 3 (2006), no. 10, 17-31, math-ph/0604022.
  20. Horváthy P.A., Plyushchay M.S., Anyon wave equations and the noncommutative plane, Phys. Lett. B 595 (2004), 547-555, hep-th/0404137.
    Horváthy P.A., Plyushchay M.S., Valenzuela M., Bosons, fermions and anyons in the plane, and supersymmetry arXiv:1001.0274.
  21. Horváthy P.A., Martina L., Stichel P.C., Galilean symmetry in noncommutative field theory, Phys. Lett. B 564 (2003), 149-154, hep-th/0304215.
    Horváthy P.A., Martina L., Stichel P.C., Galilean noncommutative gauge theory: symmetries and vortices, Nuclear Phys. B 673 (2003), 301-318, hep-th/0306228.
  22. Chang M.C., Niu Q., Berry phase, hyperorbits, and the Hofstadter spectrum, Phys. Rev. Lett. 75 (1995), 1348-1351, cond-mat/9505021.
    Bohm A., Mostafazadeh A., Koizumi H., Niu Q., Zwanziger J., The geometric phase in quantum systems. Foundations, mathematical concepts, and applications in molecular and condensed matter physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2003, Chapter 12.
    Xiao D., Chang M.C., Niu Q., Berry phase effects on electronic properties, Rev. Modern Phys. 82 (2010), 1959-2007, arXiv:0907.2021.
  23. Jungwirth T., Niu Q., MacDonald A.H., Anomalous Hall effect in ferromagnetic semiconductors, Phys. Rev. Lett. 88 (2002), 207208, 4 pages, cond-mat/0110484.
    Culcer D., MacDonald A.H., Niu Q., Anomalous Hall effect in paramagnetic two-dimensional systems, Phys. Rev. B 68 (2003), 045327, 9 pages, cond-mat/0311147.
    Fang Z. et al., The anomalous Hall effect and magnetic monopoles in momentum space, Science 302 (2003), no. 5642, 92-95, cond-mat/0310232.
    Horváthy P.A., Anomalous Hall effect in noncommutative mechanics, Phys. Lett. A 359 (2006), 705-706, cond-mat/0606472.
  24. Murakami S., Nagaosa N., Zhang S.-C., Dissipationless quantum spin current at room temperature, Science 301 (2003), no. 5638, 1348-1351, cond-mat/0308167.
    Sinova J., Culcer D., Niu Q., Sinitsyn N.A., Jungwirth T., MacDonald A.H., Universal intrinsic spin-Hall effect, Phys. Rev. Lett. 92 (2004), 126603, 4 pages, cond-mat/0307663.
    Murakami S., Intrinsic spin Hall effect, Adv. Solid State Phys. 45 (2005), 197-209, cond-mat/0504353.
  25. Liberman V.S., Zeldovich B.Ya., Spin-orbit interaction of a photon in an inhomogeneous medium, Phys. Rev. A 46 (1992), 5199-5207.
    Bliokh K.Yu., Bliokh Yu.P., Optical Magnus effect as a consequence of Berry phase anisotropy, JETP Lett. 79 (2004), 519-522.
    Bliokh K.Yu., Bliokh Yu.P., Topological spin transport of photons: the optical Magnus effect and Berry phase, Phys. Lett. A 333 (2004), 181-186, physics/0402110.
    Sadykov N.R., Twistability of spin particle trajectories, Theoret. and Math. Phys. 135 (2003), 685-692.
    Bérard A., Mohrbach H., Spin Hall effect and Berry phase of spinning particles, Phys. Lett. A 352 (2006), 190-195, hep-th/0404165.
  26. Onoda M., Murakami S., Nagaosa N., Hall effect of light, Phys. Rev. Lett. 93 (2004), 083901, 4 pages, cond-mat/0405129.
    Bliokh K.Y., Niv A., Kleiner V., Hasman E., Geometrodynamics of spinning light, Nature Photon. 2 (2008), 748-753, arXiv:0810.2136.
  27. Duval C., Horváth Z., Horváthy P.A., Geometrical spinoptics and the optical Hall effect, J. Geom. Phys. 57 (2007), 925-941, math-ph/0509031.
    Duval C., Horváth Z., Horváthy P.A., Fermat principle for spinning light. Phys. Rev. D 74 (2006), 021701, 5 pages, cond-mat/0509636.
  28. Bliokh K.Y., Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium, J. Opt. A: Pure Appl. Opt. 11 (2009), 094009, 14 pages, arXiv:0903.1910.
  29. Horváthy P.A., Exotic galilean symmetry and non-commutative mechanics in mathematical and in condensed matter physics, hep-th/0602133.
  30. Martina L., Hamiltonian theory of anyons in crystals, J. Math. Sci. 151 (2008), 3159-3166.
  31. Stichel P.C., Dynamics and symmetries on the noncommutative plane, hep-th/0611172.
  32. Dunne G.V., Jackiw R., Trugenberger C.A., "Topological" (Chern-Simons) quantum mechanics, Phys. Rev. D 41 (1990), 661-666.
    Dunne G., Jackiw R., "Peierls substitution" and Chern-Simons quantum mechanics, Nuclear Phys. B Proc. Suppl. 33C (1993), 114-118.
  33. Laughlin R.B., Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50 (1983), 1395-1398.
    Stone M. (Editor), Quantum Hall effect, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.
  34. Chaichian M., Ghosh S., Langvik M., Tureanu A., Dirac quantization condition for monopole in noncommutative space-time, Phys. Rev. D 79 (2009), 125029, 5 pages, arXiv:0902.2453.
  35. Mendes A.C.R., Neves C., Oliveira W., Takakura F.I., A new approach to canonical quantization of the radiation damping, Eur. Phys. J. C 45 (2006), 257-261, hep-th/0503135.
  36. Snyder H.S., Quantized space-time, Phys. Rev. 71 (1947), 38-41.
  37. Ghosh S., A novel "magnetic" field and its dual non-commutative phase space, Phys. Lett. B 638 (2006), 350-355, hep-th/0511302.
  38. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., q-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  39. Amelino-Camelia G., Fundamental physics in space: a quantum-gravity perspective, Gen. Relativity Gravitation 36 (2004), 539-560, astro-ph/0309174.
  40. Giller S., Kosinski P., Majewski M., Maslanka P., Kunz J., More about the q-deformed Poincaré algebra, Phys. Lett. B 286 (1992), 57-62.
  41. de Azcárraga J.A., Pérez Bueno J.C., Relativistic and Newtonian κ-space-times, J. Math. Phys. 36 (1995), 6879-6896, q-alg/9505004.
  42. Plyushchay M.S., Relativistic particle with torsion, Majorana equation and fractional spin, Phys. Lett. B 262 (1991), 71-78.
    Plyushchay M.S., The model of relativistic particle with torsion, Nuclear Phys. B 362 (1991), 54-72.
  43. Alvarez P.D., Gomis J., Kamimura K., Plyushchay M.S., (2+1)D exotic Newton-Hooke symmetry, duality and projective phase, Ann. Physics 322 (2007), 1556-1586, hep-th/0702014.
    Alvarez P.D., Gomis J., Kamimura K., Plyushchay M.S., Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry, Phys. Lett. B 659 (2008), 906-912, arXiv:0711.2644.
    del Olmo M.A., Plyushchay M.S., Electric Chern-Simons term, enlarged exotic Galilei symmetry and noncommutative plane, Ann. Physics 321 (2006), 2830-2848, hep-th/0508020.
  44. Negro J., del Olmo M.A., Tosiek J., Anyons, group theory and planar physics, J. Math. Phys. 47 (2006), 033508, 19 pages, math-ph/0512007.
  45. Chou C., Nair V.P., Polychronakos A.P, On the electromagnetic interaction of anyons, Phys. Lett. B 304 (1993), 105-110, hep-th/9301037.
  46. Ghosh S., Spinning particles in 2+1 dimensions, Phys. Lett. B 338 (1994), 235-240, hep-th/9406089.
    Ghosh S., Anyons in an electromagnetic field and the Bargmann-Michel-Telegdi equation, Phys. Rev. D 51 (1995), 5827-5829, hep-th/9409169.
    Banerjee N., Ghosh S., Interacting anyons and the Darwin Lagrangian, Phys. Rev. D 52 (1995), 6130-6133, hep-th/9412022.
    Basu B., Ghosh S., Dhar S., Noncommutative geometry and geometric phases, Europhys. Lett. 76 (2006), 395-401, hep-th/0604068.
    Dhar S., Basu B., Ghosh S., Spin Hall effect for anyons, Phys. Lett. A 371 (2007), 406-409, cond-mat/0701096.
    Cortés J.L., Gamboa J., Velázquez L., Electromagnetic interaction of anyons in non-relativistic quantum field theory, Internat. J. Modern Phys. A 9 (1994), 953-968, hep-th/9211106.
    Cortés J.L., Plyushchay M.S., Anyons as spinning particles, Internat. J. Modern Phys. A 11 (1996), 3331-3362, hep-th/9505117.
  47. Maude D.K., Potemski M., Portal J.C., Henini M., Eaves L., Hill G., Pate M.A., Spin excitations of a two-dimensional electron gas in the limit of vanishing Landé g factor, Phys. Rev. Lett. 77 (1996), 4604-4607.
    Leadley D.R., Nicholas R.J., Maude D.K., Utjuzh A.N., Portal J.C., Harris J.J., Foxon C.T., Fractional quantum Hall effect measurements at zero g factor, Phys. Rev. Lett. 79 (1997), 4246-4249, cond-mat/9706157.
  48. Duval C., Horváthy P.A., Anyons with anomalous gyromagnetic ratio and the Hall effect, Phys. Lett. B 594 (2004), 402-409, hep-th/0402191.
    Duval C., Horváthy P.A., Noncommuting coordinates, exotic particles, and anomalous anyons in the Hall effect, Theoret. and Math. Phys. 144 (2005), 899-906, hep-th/0407010.
  49. Duval C., Fliche H.-H., Souriau J.-M., Un modèle de particule à spin dans le champ gravitationnel et électromagnétique, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1082-A1084.
    Duval C., Un modèle de particule à spin dans le champ électromagnétique et gravitationnel extérieure, Thèse de 3e cycle. Marseille, 1972.
  50. Souriau J.-M., Modèle de particule à spin dans le champ électromagnétique et gravitationnel, Ann. Inst. H. Poincaré Sect. A (N.S.) 20 (1974), 315-364.
    Duval C., The general relativistic Dirac-Pauli particle: an underlying classical model, Ann. Inst. H. Poincaré Sect. A (N.S.) 25 (1976), 345-362.
  51. Horváthy P.A., Plyushchay M.S., Nonrelativistic anyons in external electromagnetic field, Nuclear Phys. B 714 (2005), 269-291, hep-th/0502040.
  52. Lukierski J., Stichel P.C., Zakrzewski W.J., Noncommutative planar particle dynamics with gauge interactions, Ann. Physics 306 (2003), 78-95, hep-th/0207149.
  53. Fosco C.D., Torroba G., Noncommutative theories and general coordinate transformations, Phys. Rev. D 71 (2005), 065012, 9 pages, hep-th/0409240.
  54. Acatrinei C.S., A simple signal of noncommutative space, Modern Phys. Lett. A 20 (2005), 1437-1441, hep-th/0311134.
  55. Lukierski J., Stichel P.C., Zakrzewski W.J., N=2 supersymmetric planar particles and magnetic interactions from noncommutativity, Phys. Lett. B 602 (2004), 249-254, hep-th/0407247.
  56. Alvarez P.D., Cortés J.L., Horváthy P.A., Plyushchay M.S., Super-extended noncommutative Landau problem and conformal symmetry, J. High Energy Phys. 2009 (2009), no. 3, 034, 14 pages, arXiv:0901.1021.
  57. Horváthy P.A., Plyushchay M.S., Valenzuela M., Bosonized supersymmetry of anyons and supersymmetric exotic particle on the non-commutative plane, Nuclear Phys. B 768 (2007), 247-262, hep-th/0610317.
  58. Zhang S.C., Hansson T.H., Kivelson S., Effective-field-theory model for the fractional quantum Hall effect, Phys. Rev. Lett. 62 (1989), 82-85.
    Zhang S.C., The Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect, Internat. J. Modern Phys. B 6 (1992), 25-58.
  59. Lozano G.S., Moreno E.F., Schaposnik F.A., Self-dual Chern-Simons solitons in non-commutative space, J. High Energy Phys. 2001 (2001), no. 2, 036, 17 pages, hep-th/0012266.
    Bak D., Kim S.K., Soh K.-S., Yee J.H., Noncommutative Chern-Simons solitons, Phys. Rev. D 64 (2001), 025018, 9 pages, hep-th/0102137.
  60. Jackiw R., Pi S.-Y., Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett. 64 (1990), 2969-2972.
    Jackiw R., Pi S.-Y., Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D 42 (1990), 3500-3513.
    Jackiw R., Pi S.-Y., Self-dual Chern-Simons solitons, Prog. Theor. Phys. Suppl. (1992), no. 107, 1-40.
    Dunne G., Self-dual Chern-Simons theories, Springer Lecture Notes in Physics, Vol. 36, Springer, New York, 1995.
    Horváthy P.A., Zhang P., Vortices in (abelian) Chern-Simons gauge theory, Phys. Rep. 481 (2009), 83-142, arXiv:0811.2094.
  61. Hadasz L., Lindström U., Roçek M., von Unge R., Time dependent solitons of noncommutative Chern-Simons theory coupled to scalar fields, Phys. Rev. D 69 (2004), 105020, 12 pages, hep-th/0309015.
    Horváthy P.A., Stichel P.C., Moving vortices in noncommutative gauge theory, Phys. Lett. B 583 (2004), 353-356, hep-th/0311157.
  62. Ashcroft N.W., Mermin N.D., Solid state physics, Saunders, Philadelphia, 1976.
  63. Horváthy P.A., Martina L., Stichel P.C., Enlarged Galilean symmetry of anyons and the Hall effect, Phys. Lett. B 615 (2005), 87-92, hep-th/0412090.
  64. Duval C., Horváth Z., Horváthy P.A., Martina L., Stichel P.C., Berry phase correction to electron density in solids and exotic dynamics, Modern Phys. Lett. B 20 (2006), 373-378, cond-mat/0506051.
    Duval C., Horváth Z., Horváthy P.A., Martina L., Stichel P.C., Comment on "Berry phase correction to electron density in solids" by Xiao et al., Phys Rev. Lett. 96 (2006), 099701, 1 page.
    Stone M., Mathematical methods II, available at
    Bliokh K.Yu., On Hamiltonian nature of semiclassical motion equations in the presence of electromagnetic field and Berry curvature, Phys. Lett. A 351 (2006), 123-124, cond-mat/0507499.
    Gosselin P., Menas F., Bérard A., Mohrbach H., Semiclassical dynamics of electrons in magnetic Bloch bands: an Hamiltonian approach, Europhys. Lett. 76 (2006), 651-656, cond-mat/0601472.
  65. 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.
  66. Karplus R., Luttinger J.M., Hall effect in ferromagnetics, Phys. Rev. 95 (1954), 1154-1160.
  67. Kramer P., Saraceno M., Geometry of the time-dependent variational principle in quantum mechanics, Lecture Notes in Physics, Vol. 140, Springer-Verlag, Berlin - New York, 1981.
  68. Abraham R., Mardsen J., Foundations of mechanics, Addison-Wesley, Reading, 1978.
    Marmo G., Saletan E.J., Simoni A., Vitale B., Dynamical systems. A differential geometric approach to symmetry and reduction, John Wiley & Sons, Ltd., Chichester, 1985.
  69. Xiao D., Shi J.R., Niu Q., Berry phase correction to electron density of states in solids, Phys. Rev. Lett. 95 (2005), 137204, 4 pages, cond-mat/0502340.
  70. Zhang P.M., Horváthy P.A., Ngome J.P., Non-commutative oscillator with Kepler-type dynamical symmetry, arXiv:1006.1861.

Previous article   Next article   Contents of Volume 6 (2010)