arXiv subjects

S. Hasibul Hassan Chowdhury

Publications and source records attributed to S. Hasibul Hassan Chowdhury.

16 recordsLinked to original sources

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.

math-ph↗

arXiv2026

From Bopp Shifts to Toroidal Shadows: K-Theoretic Gap Labels in Noncommutative Quantum Mechanics

We study Bopp shifts in two-dimensional noncommutative quantum mechanics (NCQM) through a functorial lens. A nondegenerate NCQM sector with central character $(\hbar,\vartheta,B_{\rm in})$ determines a self-adjoint infinitesimal representation of the NCQM Lie algebra $\mathfrak{g}_{\rm NC}$. A Darboux normalization of its represented phase-space operators produces a self-adjoint infinitesimal representation of the Weyl-Heisenberg Lie algebra $\mathfrak{g}_{\rm WH}$ with central parameter $\hbar$, and hence defines a Bopp-shift functor collapsing $(\hbar,\vartheta,B_{\rm in}) \mapsto \hbar$. In particular, a generic NCQM sector is not equivalent, as a $\mathfrak{g}_{\rm NC}$-sector, to the ordinary QM sector $(\hbar,0,0)$, even though their Bopp-shift images have the same Weyl-Heisenberg parameter. To measure what this collapse forgets, we construct a toroidal shadow functor assigning to each periodic datum $L=(a_x,a_y)$ and each NCQM sector $χ$ a phase-space noncommutative four-torus $A^4_{Θ_{χ,L}}$. Its $K_0$-trace pairing yields sector-sensitive gap labels whose top-degree coefficient is $(\vartheta B_{\rm in}-\hbar)/((2π)^2\hbar)$. This coefficient is independent of the spatial cell area $a_xa_y$ and equals the Pfaffian ${\rm Pf}(Θ_{χ,L})$, the top-degree generator of the $K_0$-trace range. Since strong Morita equivalence preserves this trace range up to positive scaling, non-proportionality of the trace ranges obstructs Morita equivalence of the shadows, separating equal-$\hbar$ sectors that the Bopp-shift functor identifies. In the arithmetic subfamily where $\vartheta/A$ and $B_{\rm in}A/\hbar$ are algebraic, the trace-range scale is forced to be trivial and $|{\rm Pf}|$ becomes a computable separation criterion: $|\vartheta B_{\rm in}-\hbar| \neq |\vartheta' B'_{\rm in}-\hbar|$ already implies inequivalence.

math-ph↗

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.

math-ph↗

arXiv2023

Quantum simulation of quantum mechanical system with spatial noncommutativity

Quantum simulation has become a promising avenue of research that allows one to simulate and gain insight into the models of High Energy Physics whose experimental realizations are either complicated or inaccessible with current technology. We demonstrate the quantum simulation of such a model, a quantum mechanical system with spatial noncommutativity, which is inspired by the works in Noncommutative Geometry and Noncommutative Field theory for a universal quantum computer. We use the novel group theoretical formalism to map the Hamiltonian of such a noncommutative quantum system into the ordinary quantum mechanical Hamiltonian and then carry out the quantum simulation using the Trotter-Suzuki product formula. Furthermore, we distinguish the impact of the noncommutativity parameter on the quantum simulation, especially on the Trotter error, and point out how its sizable value affects the simulation.

hep-ph↗

arXiv2023

On a charged spinless point particle minimally coupled to a constant magnetic field in a noncommutative plane

In this paper, we provide a mathematically and physically consistent minimal prescription for a charged spinless point particle coupled to a constant magnetic field in a 2-dimensional noncommutative plane. It turns out to be a gauge invariant prescription in contrast to the widely and carelessly used naive minimal prescription in the context of 2-dimensional quantum mechanics in a noncommutative plane. Besides, we explore the noncommutative U(1) gauge theoretic structure of the underlying noncommutative system by explicitly computing the 1-parameter family of Seiberg-Witten maps.

math-ph↗

arXiv2021

Gauge invariant energy spectra in 2-dimensional noncommutative quantum mechanics

In this paper we consider an electron moving on a two dimensional noncommutative plane immersed in a constant magnetic field under the influence of an anisotropic harmonic potential. We work out the gauge invariant energy spectra of this system using a consistent gauge prescription proposed in \cite{nctori,gauge-wigfunc}. We also show subsequently that straightforward generalization of the Landau problem in the noncommutative setting using minimal coupling prescription as is done naively on many occasions in the literature violates gauge invariance of the energy spectra.

math-ph↗

arXiv2017

Wigner functions for gauge equivalence classes of unitary irreducible representations of noncommutative quantum mechanics

While Wigner functions forming phase space representation of quantum states is a well-known fact, their construction for noncommutative quantum mechanics (NCQM) remains relatively lesser known, in particular with respect to gauge dependencies. This paper deals with the construction of Wigner functions of NCQM for a system of 2-degrees of freedom using 2-parameter families of gauge equivalence classes of unitary irreducible representations (UIRs) of the Lie group $\g$ which has been identified as the kinematical symmetry group of NCQM in an earlier paper. This general construction of Wigner functions for NCQM, in turn, yields the special cases of Landau and symmetric gauges of NCQM.

math-ph↗

arXiv2017

On the Plethora of Representations Arising in Noncommutative Quantum Mechanics and An Explicit Construction of Noncommutative 4-tori

We construct a 2-parameter family of unitarily equivalent irreducible representations of the triply extended group $\g$ of translations of $\mathbb{R}^{4}$ associated with a family of its 4-dimensional coadjoint orbits and show how a continuous 2-parameter family of gauge potentials emerges from these unitarly equivalent representations. We show that the Landau and the symmetric gauges of noncommutative quantum mechanics, widely used in the literature, in fact, belong to this 2-parameter family of gauges. We also provide an explicit construction of noncommutative 4-tori and compute the associated star products using the unitary dual of the group $\g$ that was studied at length in an earlier paper (\cite{ncqmjpa}). Finally, we construct projective modules over such noncommutative 4-tori and compute constant curvature connections on them using Rieffel's method.

math-ph↗

arXiv2016

Noncommutative coherent states and related aspects of Berezin-Toeplitz quantization

In~this paper, we construct noncommutative coherent states using various families of unitary irreducible representations (UIRs) of $\g$, a connected, simply connected nilpotent Lie group, that was identified as the kinematical symmetry group of noncommutative quantum mechanics for a system of 2-degrees of freedom in an earlier paper. Likewise described are the degenerate noncommutative coherent states arising from the degenerate UIRs of $\g$. We~then compute the reproducing kernels associated with both these families of coherent states and study Berezin-Toeplitz quantization of the observables on the underlying 4-dimensional phase space, analyzing in particular the semi-classical asymptotics for both these cases.

math-ph↗

arXiv2016

Deformation of Noncommutative Quantum Mechanics

In this paper, the Lie group $G_{NC}^{α,β,γ}$, of which the kinematical symmetry group $G_{NC}$ of noncommutative quantum mechanics (NCQM) is a special case due to fixed nonzero $α$, $β$ and $γ$, is three-parameter deformation quantized using the method suggested by Ballesteros and Musso in "Quantum Algebras as Quantizations of Dual Poisson-Lie Groups" [J. Phys. A: Math. Theor., 46 (2013), 195203]. A certain family of QUE algebras, corresponding to $G_{NC}^{α,β,γ}$ with two of the deformation parameters approaching zero, is found to be in agreement with the existing results of the literature on quantum Heisenberg group. Finally, we dualize the underlying QUE algebra to obtain an expression for the underlying star-product between smooth functions on $G_{NC}^{α,β,γ}$.

math-ph↗

arXiv2016

On Derivation of Goldman Bracket

In this paper, we obtain an infinite dimensional Lie algebra of exotic gauge invariant observables that is closed under Goldman-type bracket associated with monodromy matrices of flat connections on a compact Riemann surface for $G_{2}$ gauge group. As a by-product, we give an alternative derivation of known Goldman bracket for classical gauge groups $GL(n,\mathbb{R})$, $SL(n,\mathbb{R})$, $U(n)$, $SU(n)$, $Sp(2n,\mathbb{R})$ and $SO(n)$.

math-ph↗

arXiv2015

Wigner Functions for Noncommutative Quantum Mechanics: a group representation based construction

This paper is devoted to the construction and analysis of the Wigner functions for noncommutative quantum mechanics, their marginal distributions and star-products, following a technique developed earlier, {\it viz\/,} using the unitary irreducible representations of the group $\g$, which is the three fold central extension of the abelian group of $\mathbb R^4$. These representations have been exhaustively studied in earlier papers. The group $\g$ is identified with the kinematical symmetry group of noncommutative quantum mechanics of a system with two degrees of freedom. The Wigner functions studied here reflect different levels of non-commutativity -- both the operators of position and those of momentum not commuting, the position operators not commuting and finally, the case of standard quantum mechanics, obeying the canonical commutation relations only.

math-ph↗

arXiv2014

Coadjoint Orbits and Wigner functions of (1+1)-Extended Affine Galilei Group and Galilei Group

We study the coadjoint orbits of the noncentrally extended (1+1)-affine Galilei group and compute the relevant Wigner functions built on them explicitly. We consider the centrally extended (1+1)-Galilei group and study its coadjoint orbits in the second half of the paper. We also compute the Wigner functions built on the corresponding coadjoint orbits subsequently. Finally, a comparative study of the structure of the coadjoint orbits and corresponding Wigner functions between the extended (1+1)-affine Galilei group and the centrally extended (1+1)-Galilei group is presented along with possible physical interpretations.

math-ph↗

arXiv2014

Triply Extended Group of Translations of $\mathbb{R}^{4}$ as Defining Group of NCQM: relation to various gauges

The role of the triply extended group of translations of $\mathbb{R}^{4}$, as the defining group of two dimensional noncommutative quantum mechanics (NCQM), has been studied in \cite{ncqmjmp}. In this paper, we revisit the coadjoint orbit structure and various irreducible representations of the group associated with them. The two irreducible representations corresponding to the Landau and symmetric gauges are found to arise from the underlying defining group. The group structure of the transformations, preserving the commutation relations of NCQM, has been studied along with specific examples. Finally, the relationship of a certain family of UIRs of the underlying defining group with a family of deformed complex Hermite polynomials has been explored .

math-ph↗

arXiv2012

The Symmetry Groups of Noncommutative Quantum Mechanics and Coherent State Quantization

We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in $(2+1)$-space-time dimensions and the two-fold extension of the group of translations of $\mathbb R^4$. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.

math-ph↗

arXiv2011

All the Groups of Signal Analysis from the (1+1)-affine Galilei Group

We study the relationship between the (1+1)-affine Galilei group and four groups of interest in signal analysis and image processing, viz., the wavelet or the affine group of the line, the Weyl-Heisenberg, the shearlet and the Stockwell groups. We show how all these groups can be obtained either directly as subgroups, or as subgroups of central extensions of the affine Galilei group. We also study this at the level of unitary representations of the groups on Hilbert spaces.

math-ph↗