The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space

Haouam, Ilyas

doi:10.3390/sym11020223

Cited by 25 publications

(13 citation statements)

References 38 publications

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“…The definition of Moyal-Weyl product between two arbitrary functions f (x, p) and g(x, p) in phase-space is given by [37]…”

Section: Review Of Noncommutative Algebramentioning

confidence: 99%

“…In other words, it is the nonrelativistic limit of the Dirac equation. Furthermore, the Pauli equation could be extracted from other relativistic higher spin equations such as the DKP equation considering the particle interacting with an electromagnetic field [37]. The nonrelativistic Schrödinger equation that describes an electron in interaction with an electromagnetic potential…”

Section: Formulation Of Noncommutative Pauli Equationmentioning

confidence: 99%

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On the Three-Dimensional Pauli Equation in Noncommutative Phase-Space

Haouam

2021

Acta Polytech

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In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in a noncommutative phase-space as well as the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic fields are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.

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“…The definition of Moyal-Weyl product between two arbitrary functions f (x, p) and g(x, p) in phase-space is given by [37]…”

Section: Review Of Noncommutative Algebramentioning

confidence: 99%

Section: Formulation Of Noncommutative Pauli Equationmentioning

confidence: 99%

On the Three-Dimensional Pauli Equation in Noncommutative Phase-Space

Haouam

2021

Acta Polytech

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“…(6) Now, using Bopp-shift transformation, one can express the NC variables in terms of the standard commutative variables [31] x nc a x s ¢ P~p s y ; p nc x a p s x ; y nc a y s C ¢ P~p s y ; p nc y a p s y ; (7) where the index s refers to the standard commutative space. The interesting point is that in the DNC space there is a minimum length for X in a simultaneous X, Y measurement [16]:…”

Section: Review Of Position-dependent Noncommutativitymentioning

confidence: 99%

Dirac Oscillator in Dynamical Noncommutative Space

Haouam¹

2021

Preprint

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In this paper, we address the energy eigenvalues of two-dimensional Dirac oscillator perturbed by dynamical noncommutative space. We derived the relativistic Hamiltonian of Dirac oscillator in dynamical noncommutative space ( τ -space), in which the space-space Heisenberg–like commutation relations and noncommutative parameter are position-dependent. Then used this Hamiltonian to calculate the first-order correction to the eigenvalues and eigenvectors, based on the second quantization and using the perturbation theory. It is shown that the energy shift depends on the dynamical noncommutative parameter τ . Knowing that with a set of two-dimensional Bopp-shift transformation, we mapped the noncommutative problem to the standard commutative one.

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“…We would like to note that, it would be quite well to approach the problem via the Seiberg-Witten map if we have an electromagnetic field interaction. But while we have a magnetic field interaction, we will follow a quite standard approach that has been widely used in the literature on noncommutative quantum mechanics (NCQM); which depends on obtaining a noncommutative version of a given field theory, and based on replacing the product of the fields by the Moyal-Weyl product (⋆-product) defined as [31][32][33]…”

Section: A Fisk-tait Equationmentioning

confidence: 99%

On the Fisk–Tait equation for spin-3/2 fermions interacting with an external magnetic field in noncommutative space-time

Haouam¹

2020

J. Phys. Stud.

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The Fisk-Tait equation in interaction with an external magnetic field in noncommutative spacetime is investigated, consequently, we studied the continuity equation in both commutative and noncommutative space-time; there we examined the influence of the space-time noncommutativity on the current density quadri-vector. And we also find that the total charge obtained from the probability density still indefinite even when space does not commute. Furthermore we found the spin current density in the two different spin directions. We also investigated the linking between the fermions and the bosons in the Fock space using the Holstein-Primakoff transformation.Over the years the particle equations for an arbitrary spin was considered the subject for careful investigations, that's why today we are interested in the relativistic equations that describe the motion of spin-3/2 particles. Such as the relativistic Rarita-Schwinger equation (1940) [1], the Fisk-Tait equation (1973) [2], Hurley's field equation [3], Bhabha-Gupta equation (Bhabha, Gupta 1952, 1954, 1974, the approach for arbitrary spin equation by V. Bargman, E.P. Wigner (1948) [6]. Also the Heisenberg equations of motion for the spin-3/2 field (1977) [7], in which, it is shown for dynamical systems with constraints depending upon external fields, the Lagrange and Heisenberg equations of motion are the same for the quantized charged spin-3/2 field in the presence of a minimal external electromagnetic interaction. The Rarita-Schwinger spin-3/2 equation in the weak-field limit is obtained to satisfy the Heisenberg equations of motion. This is similar to the case of spin-3/2 field minimally coupled with an external electromagnetic field by Mainland and Sudarshan (1973) [8].As well recently, we have the link between the relativistic canonical quantum mechanics of arbitrary spin and the covariant local field theory by V.M. Simulik (2017) [9] (where the found equations are without redundant components). Where it has been confirmed that, the synthesis of the relativistic canonical quantum mechanics of the spins-3/2 particle and antiparticle doublet is completely similar to the synthesis of the Dirac equation from the relativistic canonical quantum mechanics of the spin-1/2 particle-antiparticle doublet. On the basis of the investigation of solutions and transformation properties with respect to the Poincare group the obtained new 8-component equation is suggested to be well defined for the description of spin s=3/2 fermions. Despite the Rarita-Schwinger equation which has 16 components and needs the additional condition.But the equation for a particle of spin-3/2 originally was given by Fierz and Pauli (I939) in spinor form [10]. Knowing that the Klein-Gordon, Dirac, and Proca equations provide a relativistic description of the particles that have the lowest spins cases (S=0, 1/2, 1 respectively).For instance, the Rarita-Schwinger equation was formulated for the first time by William Rarita and Julian Schwinger, it was the most famous equation that describes the m...

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The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space

Cited by 25 publications

References 38 publications

On the Three-Dimensional Pauli Equation in Noncommutative Phase-Space

On the Three-Dimensional Pauli Equation in Noncommutative Phase-Space

Dirac Oscillator in Dynamical Noncommutative Space

On the Fisk–Tait equation for spin-3/2 fermions interacting with an external magnetic field in noncommutative space-time

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