Supersymmetric Quantum Mechanics on a noncommutative plane through the lens of deformation quantization

A gauge invariant mathematical formalism based on deformation quantization is outlined to model an N = 2 supersymmetric system of a spin 1/2 charged particle placed in a nocommutative plane under the influence of a vertical uniform magnetic field. The noncommutative involutive algebra ( C infinity ( R 2 )[[ # ]] , * r ) of formal power series in # with coefficients in the commutative ring C infinity ( R 2 ) was employed to construct the relevant observables, viz., SUSY Hamiltonian H, supercharge operator Q and its adjoint Q dagger all belonging to the 2 x 2 matrix algebra M 2 ( C infinity ( R 2 )[[ # ]] , * r ) with the help of a family of gauge -equivalent star products * r . The energy eigenvalues of the SUSY Hamiltonian all turned out to be independent of not only the gauge parameter r but also the noncommutativity parameter #. The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of #. The Witten index for the noncommutative SUSY Landau problem turns out to be - 1 corroborating the fact that there is no broken supersymmetry for the model we are considering.