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arXiv:2602.11394

Coherent states for the exotic Landau problem and related properties

Isiaka Aremua

quant-phSubmitted Feb 11, 2026• Announced Feb 13, 2026• 0 citations

Abstract

This work presents a comprehensive study of the exotic Landau model in a two-dimensional noncommutative plane. Beginning with the classical formulation where two conserved quantities $\mathcal{P}_i$ and $\mathcal{K}_i$ are derived, we proceed to the quantum level where these lead to two independent oscillator representations generating bosonic Fock spaces $Γ_{\mathcal{P}}$ and $Γ_{\mathcal{K}}$. Coherent states satisfying all Klauder's criteria are explicitly constructed, and their physical properties including normalization, continuity, resolution of the identity, temporal stability, and action identity are rigorously proven. We further develop matrix vector coherent states and quaternionic vector coherent states, examining their mathematical structure and physical implications. Detailed calculations of the free particle propagator via path integrals, uncertainty relations, and time evolution of probability densities are provided.

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This work presents a comprehensive study of the exotic Landau model in a two-dimensional noncommutative plane. Beginning with the classical formulation where two conserved quantities

\mathcal{P}_i

and

\mathcal{K}_i

are derived, we proceed to the quantum level where these lead to two independent oscillator representations generating bosonic Fock spaces

Γ_{\mathcal{P}}

and

Γ_{\mathcal{K}}

. Coherent states satisfying all Klauder's criteria are explicitly constructed, and their physical properties including normalization, continuity, resolution of the identity, temporal stability, and action identity are rigorously proven. We further develop matrix vector coherent states and quaternionic vector coherent states, examining their mathematical structure and physical implications. Detailed calculations of the free particle propagator via path integrals, uncertainty relations, and time evolution of probability densities are provided.

In this paper, we study the exotic Landau problem at the classical level where two conserved quantities are derived. At the quantum level, the corresponding quantum operators of the conserved quantities provide two oscillator representations from which we derive two Boson Fock spaces. Using the normalized coherent states which are the minimum uncertainty states on non-commutative configuration space isomorphic to each of the boson Fock space, we form entangled coherent states which are Bell-like states labeled quasi-Bell states. The effect of non-maximality of a quasi-Bell state based quantum channel is investigated in the context of a teleportation of a qubit

Inspired by a recent work that proposes using coherent states to evaluate the Feynman kernel in noncommutative space, we provide an independent formulation of the path-integral approach for quantum mechanics on the Moyal plane, with the transition amplitude defined between two coherent states of mean position coordinates. In our approach, we invoke solely a representation of the noncommutative algebra in terms of commutative variables. The kernel expression for a general Hamiltonian was found to contain Gaussian-like damping terms, and it is non-perturbative in the sense that it does not reduce to the commutative theory in the limit of vanishing θ—the noncommutative parameter. As an example, we studied the free particle's propagator which turned out to be oscillating with period being the product of its mass and θ. Further, it satisfies the Pauli equation for a charged particle with its spin aligned to a constant, orthogonal B field in the ordinary Landau problem, thus providing an interesting evidence of how noncommutativity can induce spin-like effects at the quantum mechanical level.

Abstract In this paper, we study the exotic Landau problem at the classical level where two conserved quantities are derived. At the quantum level, the corresponding quantum operators of the conserved quantities provide two oscillator representations from which we derive two Boson Fock spaces. Using the normalized coherent states which are the minimum uncertainty states on noncommutative configuration space isomorphic to each of the boson Fock space, we form entangled coherent states which are Bell- like states labeled quasi-Bell states. The effect of non-maximality of a quasi-Bell state based quantum channel is investigated in the context of a teleportation of a qubit.

The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.