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


SIGMA 4 (2008), 028, 9 pages      hep-th/0610061      http://dx.doi.org/10.3842/SIGMA.2008.028
Contribution to the Proceedings of the 3-rd Microconference Analytic and Algebraic Methods III

Noncommutative Lagrange Mechanics

Denis Kochan a, b
a) Dept. of Theoretical Physics, FMFI UK, Mlynská dolina F2, 842 48 Bratislava, Slovakia
b) Dept. of Theoretical Physics, Nuclear Physics Institute AS CR, 250 68 Rez, Czech Republic

Received November 26, 2007, in final form January 29, 2008; Published online February 25, 2008

Abstract
It is proposed how to impose a general type of ''noncommutativity'' within classical mechanics from first principles. Formulation is performed in completely alternative way, i.e. without any resort to fuzzy and/or star product philosophy, which are extensively applied within noncommutative quantum theories. Newton-Lagrange noncommutative equations of motion are formulated and their properties are analyzed from the pure geometrical point of view. It is argued that the dynamical quintessence of the system consists in its kinetic energy (Riemannian metric) specifying Riemann-Levi-Civita connection and thus the inertia geodesics of the free motion. Throughout the paper, ''noncommutativity'' is considered as an internal geometric structure of the configuration space, which can not be ''observed'' per se. Manifestation of the noncommutative phenomena is mediated by the interaction of the system with noncommutative background under the consideration. The simplest model of the interaction (minimal coupling) is proposed and it is shown that guiding affine connection is modified by the quadratic analog of the Lorentz electromagnetic force (contortion term).

Key words: noncommutative mechanics; affine connection; contortion.

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References

  1. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  2. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  3. Connes A., Noncommutative geometry, Academic Press, 1994.
    Madore J., An introduction to noncommutative differential geometry and its physical applications, Cambridge University Press, Cambridge, 1999.
    Gracia-Bondia J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser, Boston, 2001.
  4. Douglas M.R., Nekrasov N.A., Noncommutative field theory, Rev. Modern Phys. 73 (2001), 977-1029, hep-th/0106048.
    Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, hep-th/0109162.
  5. Nair V.P., Polychronakos A.P., Quantum mechanics on the noncommutative plane and sphere, Phys. Lett. B 505 (2001), 267-274, hep-th/0011172.
    Gamboa J., Loewe M., Rojas J.C., Noncommutative quantum mechanics, Phys. Rev. D 64 (2001), 067901, 3 pages, hep-th/0010220.
    Demetrian M., Kochan D., Quantum mechanics on non-commutative plane, Acta Phys. Slov. 52 (2002), 1-9, hep-th/0102050.
    Djemai A.E.F., Smail H., On quantum mechanics on noncommutative quantum phase space, Commun. Theor. Phys. (Beijing) 41 (2004), 837-844, hep-th/0309006.
    Ho P.M., Kao H.C., Noncommutative quantum mechanics from noncommutative quantum field theory, Phys. Rev. Lett. 88 (2002), 151602, 4 pages, hep-th/0110191.
    Dragovich B., Rakic Z., path integrals in noncommutative quantum mechanics, Theoret. and Math. Phys. 140 (2004), 1299-1308, hep-th/0309204.
  6. Duval C., Horváthy P.A., The exotic Galilei group and the "Peierls substitution", Phys. Lett. B 479 (2000), 284-290, hep-th/0002233.
    Horváthy P.A., Plyushchay M.S., Nonrelativistic anyons, noncommutative plane and exotic Galilean symmetry, J. High Energy Phys. 2002 (2002), no. 6, 033, 11 pages, hep-th/0201228.
    Banerjee R., A novel approach to noncommutativity in planar quantum mechanics, Modern Phys. Lett. A 17 (2002), 631-645, hep-th/0106280.
    Horváthy P.A., Plyushchay M.S., Anyon wave equations and the noncommutative plane, Phys. Lett. B 595 (2004), 547-555, hep-th/0404137.
  7. 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.
    Djemai A.E.F., On noncommutative classical mechanics, Internat. J. Theoret. Phys. 42 (2004), 299-314, hep-th/0309034.
  8. Arnol'd V.I., Mathematical methods of classical mechanics, 2nd ed., Springer-Verlag, New York, 1989.
    Abraham R., Marsden J.E., Foundation of mechanics, Addison-Wesley, 1978.
    Burke W.L., Applied differential geometry, Cambridge University Press, 1985.
    Fecko M., Differential geometry and Lie groups for physicists, Cambridge University Press, 2006.
  9. Kobayashi S., Nomizu K., Foundation of differential geometry, Vol. I, Wiley, New York, 1963.
    do Carmo M.P., Riemannian geometry, Birkhäuser, Boston, 1992.
    Frankel T., The geometry of physics, Cambridege University Press, Cambridge 1997.
  10. Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization, Ann. Phys. 111 (1978), 61-151.
    Fedosov B.V., A Simple Geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), 213-238.
    Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, q-alg/9709040.
  11. Frölicher A., Nijenhuis A., Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Nederl. Akad. Wetensch. Proc. Ser. A 59 (1956), 338-359.
  12. Shabanov S.V., Constrained systems and analytical mechanics in spases with torsion, J. Phys. A: Math. Gen. 31 (1998), 5177-5190, physics/9801023.
  13. Hehl F.W., von der Heyde P., Kerlick G.D., Nester J.M., General relativity with spin and torsion: foundations and prospects, Rev. Modern Phys. 48 (1976), 393-416.
    Hehl F.W., McCrea J.D., Mielke E.W., Ne'eman Y., Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rep. 258 (1995), 1-171.

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