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 relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrödinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrödinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space.

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.

A method of Foldy-Wouthuysen transformation for relativistic spin-1/2 particles in external fields is proposed; in the present work the basic properties of the Dirac hamiltonian in the FW representation in the noncommutative phase-space are investigated and the Schrödinger-Pauli equation is found, knowing that the used methods for extracting the full phase-space noncommutative Dirac equation are, the Bopp-shift linear translation method, and the Moyal-Weyl product (*-product).