Abstract
Noncommutative geometry and deformation quantization provide a powerful framework for extending classical geometry to quantum settings. In this talk, I will talk about a gauge invariant mathematical formalism based on deformation quantization to model an (\mathcal{N}=2) supersymmetric system of a spin (1/2) charged particle placed in a noncommutative plane under the influence of a vertical uniform magnetic field. The talk is based on a research paper where the noncommutative involutive algebra ((C^{\infty}(\mathbb{R}^{2})[[\vartheta]],^r)) of formal power series in (\vartheta) with coefficients in the commutative ring (C^{\infty}(\mathbb{R}^{2})) was employed to construct the relevant observables, viz., SUSY Hamiltonian (H), supercharge operator (Q) and its adjoint (Q^{\dag}) all belonging to the (2\times 2) matrix algebra (\mathcal{M}_{2}(C^{\infty}(\mathbb{R}^{2})[[\vartheta]],^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 (\vartheta). The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of (\vartheta). 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.