Md. Rafsanjany Jim | Searcharxiv

arXiv2026

Affine Left--Right Dirac Data on the Moyal Plane, with an Application to $U(1)_\star$ Gauge Coupling in Noncommutative Quantum Mechanics

We study even non-unital Dirac data on the two-dimensional Moyal plane with fixed deformation parameter $\vartheta\neq0$, whose Dirac operator is affine in the left and right regular coordinate multipliers, with coefficient matrices $T_L$ and $T_R$. Two invariants organize this class. The right block controls algebra--Dirac transversality: local compactness holds if and only if $\det T_R\neq0$, with $|\det T_R|$ determining the local volume coefficient and $\operatorname{sgn}\det T_R$ the oriented Gaussian $K$-homology pairing. Independently, $Γ=\vartheta(\det T_L-\det T_R)$ controls the spectral regime and, together with $|\det T_R|$, the exact localized heat trace: $Γ\neq0$ gives infinitely degenerate Landau levels, while for $Γ=0$ in the transverse regime the spectrum is $\mathbb R$ with empty point spectrum. Applied to Wallet's Landau--Moyal family, the criterion shows that local compactness survives at the metric-degenerate point $ξ_{\mathrm W}=1$ but fails at $ξ_{\mathrm W}=-1$, thereby resolving the local-compactness behaviour at both exceptional one-sided points. We then derive a distinguished affine family from a fixed nondegenerate sector $(\hbar_0,\vartheta_0,B_0)$ of the kinematical symmetry group of two-dimensional noncommutative quantum mechanics. The sector fixes the effective Moyal scale $\vartheta_{\mathrm{eff}}=\vartheta_0/(1-\vartheta_0B_0/\hbar_0)$ and $T_R$, while an affine $U(1)_\star$ background coupling deforms only the left affine part of the Dirac operator. Consequently, local compactness, noncommutative volume and $K$-homology class remain unchanged, whereas $Γ=\tfrac12(1-eF_{xy}/B_0)$ drives a transition at $eF_{xy}=B_0$ from Landau levels to spectrum $\mathbb R$ with empty point spectrum.

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arXiv2024

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 $\mathcal{N}=2$ 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 $Q^†$ 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.

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