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

SIGMA 10 (2014), 011, 15 pages      arXiv:1304.7293      https://doi.org/10.3842/SIGMA.2014.011
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

Abstract
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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