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

SIGMA 10 (2014), 011, 15 pages      arXiv:1304.7293
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane

Carles Batlle a, Joaquim Gomis b and Kiyoshi Kamimura c
a) Departament de Matemàtica Aplicada 4 and Institut d'Organització i Control, Universitat Politècnica de Catalunya - BarcelonaTech, EPSEVG, Av. V. Balaguer 1, 08800 Vilanova i la Geltrú, Spain
b) Departament d'Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
c) Department of Physics, Toho University, Funabashi, Chiba 274-8510, Japan

Received August 29, 2013, in final form January 29, 2014; Published online February 08, 2014

We study all the symmetries of the free Schrödinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrödinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.

Key words: non-commutative plane; Schrödinger equation; Schrödinger symmetries; higher spin symmetries.

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  1. 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.
  2. Ballesteros A., Gadella M., del Olmo M.A., Moyal quantization of (2+1)-dimensional Galilean systems, J. Math. Phys. 33 (1992), 3379-3386.
  3. Banerjee R., Deformed Schrödinger symmetry on noncommutative space, Eur. Phys. J. C Part. Fields 47 (2006), 541-545, hep-th/0508224.
  4. Bekaert X., Meunier E., Moroz S., Symmetries and currents of the ideal and unitary Fermi gases, J. High Energy Phys. 2012 (2012), no. 2, 113, 59 pages, arXiv:1111.3656.
  5. Brihaye Y., Gonera C., Giller S., Kosiński P., Galilean invariance in 2+1 dimensions, hep-th/9503046.
  6. Callan C.G., Coleman S.R., Wess J., Zumino B., Structure of phenomenological Lagrangians. II, Phys. Rev. 177 (1969), 2247-2250.
  7. Coleman S.R., Wess J., Zumino B., Structure of phenomenological Lagrangians. I, Phys. Rev. 177 (1969), 2239-2247.
  8. 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.
  9. 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.
  10. Eastwood M., Higher symmetries of the Laplacian, Ann. of Math. 161 (2005), 1645-1665, hep-th/0206233.
  11. Gomis J., Kamimura K., Schrödinger equations for higher order non-relativistic particles and N-Galilean conformal symmetry, Phys. Rev. D 85 (2012), 045023, 6 pages, arXiv:1109.3773.
  12. Hagen C.R., Scale and conformal transformations in Galilean-covariant field theory, Phys. Rev. D 5 (1972), 377-388.
  13. Hagen C.R., Galilean-invariant gauge theory, Phys. Rev. D 31 (1985), 848-855.
  14. Horváthy P.A., Martina L., Stichel P.C., Galilean symmetry in noncommutative field theory, Phys. Lett. B 564 (2003), 149-154, hep-th/0304215.
  15. Horváthy P.A., Martina L., Stichel P.C., Exotic Galilean symmetry and non-commutative mechanics, SIGMA 6 (2010), 060, 26 pages, arXiv:1002.4772.
  16. Horváthy P.A., Plyushchay M.S., Anyon wave equations and the noncommutative plane, Phys. Lett. B 595 (2004), 547-555, hep-th/0404137.
  17. Horváthy P.A., Plyushchay M.S., Nonrelativistic anyons in external electromagnetic field, Nuclear Phys. B 714 (2005), 269-291, hep-th/0502040.
  18. Ivanov E.A., Ogieveckii V.I., Inverse Higgs effect in nonlinear realizations, Theoret. and Math. Phys. 25 (1975), 1050-1059.
  19. Jackiw R., Introducing scale symmetry, Phys. Today 25 (1972), no. 1, 23-27.
  20. Kastrup H.A., Gauge properties of the Galilei space, Nuclear Phys. B B7 (1968), 545-558.
  21. Lévy-Leblond J.M., Galilei group and Galilean invariance, in Group Theory and its Applications, Vol. II, Academic Press, New York, 1971, 221-299.
  22. 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.
  23. Niederer U., The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972), 802-810.
  24. Valenzuela M., Higher-spin symmetries of the free Schrödinger equation, arXiv:0912.0789.
  25. Vasiliev M.A., Higher spin gauge theories in any dimension, C. R. Phys. 5 (2004), 1101-1109, hep-th/0409260.

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