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

TL;DR

This paper develops a gauge-invariant deformation quantization formalism to model supersymmetric quantum mechanics on a noncommutative plane, revealing unbroken supersymmetry and gauge-independent energy spectra.

Contribution

It introduces a novel gauge-invariant deformation quantization approach for noncommutative supersymmetric systems, analyzing their spectral properties and supersymmetry status.

Findings

Energy eigenvalues are gauge and noncommutativity independent.

Supersymmetry remains unbroken at all orders of noncommutativity.

Witten index is -1, indicating unbroken supersymmetry.

Abstract

A gauge invariant mathematical formalism based on deformation quantization is outlined to model an $\mathcal{N}=2$ 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 $(C^{\infty}(\mathbb{R}^{2})[[\vartheta]],*^r)$ 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.