Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry

Abstract

We generalize the formulation of non-commutative quantum mechanics to three dimensional non-commutative space. Particular attention is paid to the identification of the quantum Hilbert space in which the physical states of the system are to be represented, the construction of the representation of the rotation group on this space, the deformation of the Leibnitz rule accompanying this representation and the implied necessity of deforming the co-product to restore the rotation symmetry automorphism. This also implies the breaking of rotational invariance on the level of the Schroedinger action and equation as well as the Hamiltonian, even for rotational invariant potentials. For rotational invariant potentials the symmetry breaking results purely from the deformation in the sense that the commutator of the Hamiltonian and angular momentum is proportional to the deformation.

21 pages

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Cited by in corpus (8)

• Hydrogen atom in rotationally invariant noncommutative space
• The thermodynamics of Fermi gases in three dimensional fuzzy space
• On the role of Schwinger's SU(2) generators for Simple Harmonic Oscillator in 2D Moyal plane
• UV/IR mixing as a twisted Poincaré anomaly
• The entropy of dense non-commutative fermion gases
• Voros product and noncommutative inspired black holes
• Effect of Noncommutativity of Space-time on Zitterbewegung
• Noncommutativity in Configuration Space Induced by A Conjugate Magnetic Field in Phase Space