Supersymmetry breaking in noncommutative quantum mechanics

Abstract

Supersymmetric quantum mechanics is formulated on a two dimensional noncommutative plane and applied to the supersymmetric harmonic oscillator. We find that the ordinary commutative supersymmetry is partially broken and only half of the number of supercharges are conserved. It is argued that this breaking is closely related to the breaking of time reversal symmetry arising from noncommutativity.

The {\cal N}=1 supersymmetric Landau problem and its supersymmetric Landau level projections: the {\cal N}=1 supersymmetric Moyal–Voros superplane

“…Deformations of the algebraic structures of even a Grassmann graded Landau problem have so far not led to a projection onto Landau levels on which supersymmetry acts nontrivially [9,10]. We also note that the supersymmetry inherent to the Landau problem with a spin-1/2 charged particle of gyromagnetic factor g = 2 does not survive a noncommutative geometry on the Euclidean plane [11]. More specifically, all previous approaches, whether those of [8] (and references therein) or even [12], have considered a massless limit of the Landau problem or some procedure equivalent to that limit.…”

“…Consequently, in order to gain some new structure, one needs to consider a projection involving at least two 11 Landau sectors separated by the quantum energy gap hω c .…”

“…Deformations of the algebraic structures of even a Grassmann graded Landau problem have so far not led to a projection onto Landau levels on which supersymmetry acts non trivially [9,10]. We note also that the supersymmetry inherent to the Landau problem with a spin 1/2 charged particle of gyromagnetic factor g = 2 does not survive a noncommutative geometry on the Euclidean plane [11].…”

Dirac equation with a magnetic field in 3D non-commutative phase space

“…Discoveries in string theory and Mtheory that the effects of noncommutative (NC) spaces may appear near the string scale and at higher energies [2][3][4] motivate studies in these areas greatly. Recently, a lot of problems have been investigated on the theory of NC spaces such as the quantum Hall effects [5][6][7][8][9], the harmonic oscillator [10][11][12][13][14], the coherent states [15], the thermodynamics [16], the classical-quantum relationship [17], the motion of the spin-1/2 particle under a uniform magnetic field [18], various kinds of relativistic oscillators [19,20,23,24], etc. In [18], the particle is confined to the plane the applied magnetic field is perpendicular to.…”