On the Three-Dimensional Pauli Equation in Noncommutative Phase-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 noncommutative phase-space, as well the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic field 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.


I. INTRODUCTION
It is known that the relativistic wave equation describing the fermions with spin-1/2 is the Dirac equation; on the other hand, the non-relativistic wave equation describing them, namely the Pauli equation, which is a topic of great interest in physics [1][2][3][4]. It is relative to the explanation of many experimental results, and its probability current density changed to including an additional spin-dependent term recognized as the spin current [5][6][7]. Pauli equation shown [8][9][10][11][12] as the non-relativistic limit of Dirac equation. Knowing that historically at first time Pauli in 1927 [13] presented his known spin matrices in modifying the non-relativistic Schrödinger equation to account for Goudsmit-Uhlenbeck's hypothesis (1925) [14,15]. Therefore, he applied an ansatz for adding a phenomenological term to the ordinary non-relativistic Hamiltonian in the presence of an electromagnetic field, the interaction energy of a magnetic field and electronic magnetic moment relative to the intrinsic spin angular momentum of the electron. Describing this spin angular momentum through the spin matrices requires replacing the complex scalar wave function by a two-component spinor wave function in the wave equation. Since then, the study of the Pauli equation became a matter of considerable attention.
In 1928 when Dirac presented his relativistic free wave equation in addition to the minimal coupling replacement to include electromagnetic interactions [16], he showed that his equation contained a term involving the electron magnetic moment interacting with a magnetic field, which was the same one inserted by hand in Pauli's equation. After that, it became common to account electron spin as a relativistic phenomenon, and the corresponding spin-1/2 term could be inserted into the spin-0 non-relativistic Schrodinger equation as will be discussed in the following to see how this is possible. However, motivated by attempts to understand string theory and describe quantum gravitation using noncommutative geometry and by trying to have drawn considerable attention to the phenomenological implications, we focus here on studying the problem of a non-relativistic spin-1/2 particle in the presence of an electromagnetic field within 3-dimensional noncommutative phase-space.
As a mathematical theory, noncommutative geometry is by now well established, although at first, its progress has been narrowly restricted to some branches of physics such as quantum mechanics. However, recently, the noncommutative geometry has become a topic of great interest. It has been finding applications in many sectors of physics and rapidly has become involved in them, continued to promote fruitful ideas and the search for a better understanding. Such as in the quantum gravity [17]; the standard model of fundamental interactions [18]; as well in the string theory [19]; and its implication in Hopf algebras [20] gives the Connes-Kreimer Hopf algebras [21][22][23] etc. There are many papers devoted to the study such various aspects especially in quantum field theory [24][25][26] and quantum mechanics [27][28][29].
This paper is organized as follows. In section 2, we present an analysis review of noncommutative geometry, in particular both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product. In section 3, we investigate the three-dimensional Pauli equation in the presence of an electromagnetic field and the corresponding continuity equation. Besides, we derived the current magnetization term in the deformed continuity equation. Section 4 is devoted to calculating the semi-classical noncommutative partition function of the Pauli system of the one-particle and N-particle systems. Consequently, we obtain the corresponding magnetization and the magnetic susceptibility through the Helmholtz free energy, all in both commutative and noncommutative phase-spaces and within a classical limit. Therefore, concluding with some remarks.

II. REVIEW OF NONCOMMUTATIVE ALGEBRA
Firstly, we present the most essential formulas of noncommutative algebra [29]. It is well known that at very tiny scales such as the string scale, the position coordinates do not commute with each other, neither do the momenta.
Let us accept in a d-dimensional noncommutative phase-space the operators of coordinates and momenta x nc i and p nc j , respectively. The noncommutative formulation of quantum mechanics corresponds to the following Heisenberglike commutation relations the effective Planck constant is the deformed Planck constant, which is given bỹ where Tr(Θη) 4αβ ≪ 1 is the condition of consistency in quantum mechanics. Θ µν , η µν are constant antisymmetric d × d matrices and δ µν is the identity matrix.
It is shown that x nc i and p nc j can be represented in terms of coordinates x i and momenta p j in usual quantum mechanics through the so-called generalized Bopp-shift as follows [27] x nc µ = αx µ − 1 2α Θ µν p ν , and p nc with α = 1 − Θη 8 2 and β = 1 α are scaling constants. To the 1rst order of Θ and η, in the calculations we take α = β = 1, so the Equations (3, 2) become If the system in which we study the effects of noncommutativity is three-dimensional, we limit ourselves to the following noncommutative algebra Θ l = (0, 0, Θ), η l = (0, 0, η) are the real-valued noncommutative parameters with the dimension of length 2 , momentum 2 respectively, they are assumed to be extremely small. And ǫ jkl is the Levi-Civita permutation tensor, with ǫ 123 = ǫ 231 = ǫ 312 = −ǫ 321 = −ǫ 132 = −ǫ 213 = 1, if j = k or k = l, ǫ jkl = 0. Therefor, we have In noncommutative quantum mechanics, it is quite possible that we replace the usual product with the Moyal-Weyl (⋆) product, then the quantum mechanical system will become simply the noncommutative quantum mechanical system. Let H (x, p) be the Hamiltonian operator of the usual quantum system, then the corresponding Schrödinger equation on noncommutative quantum mechanics is typically written as The definition of Moyal-Weyl product between two arbitrary functions f (x, p) and g(x, p) in phase-space is given by [30] with f (x, p) and g(x, p), assumed to be infinitely differentiable. If we consider the case of noncommutative space the definition of Moyal-Weyl product will be reduced to [31] ( Due to the nature of the ⋆product, the noncommutative field theories for low-energy fields (E 2 1/Θ) at classical level are completely reduced to their commutative versions. However, this is just the classical result and quantum corrections always reveal the effects of Θ even at low-energies.
On noncommutative phase-space the ⋆product can be replaced by a Bopp's shift, i.e. the ⋆product can be changed into the ordinary product by replacing H (x, p) with H (x nc , p nc ). Thus the corresponding noncommutative Schrödinger equation can be written as Note that Θ and η terms always can be treated as a perturbation in quantum mechanics. If Θ = η = 0, the noncommutative algebra reduces to the ordinary commutative one.

A. Formulation of noncommutative Pauli equation
The nonrelativistic Schrödinger equation that describes an electron in interaction with an electromagnetic potential where− → p = i − → ∇ is the momentum operator, m, e are the mass and charge of the electron, and c is the speed of light. ψ (r, t) is Schrödinger's scalar wave function. The appearance of real-valued electromagnetic Coulomb and vector potentials, φ ( − → r , t) and , is a consequence of using the gauge-invariant minimal coupling assumption to describe the interaction with the external magnetic and electric fields defined by However, the electron gains potential energy when the spin interacts with the magnetic field, therefore the Pauli equation of an electron with spin is given by where ψ (r, t) = ψ 1 ψ 2 T are the spinor wave function, which replaces the scalar wave function. With µ B = |e| 2mc = 9.27 × 10 −24 JT −1 is Bohr's magneton, − → B is the applied magnetic field vector, also µ B − → σ represents the magnetic moment. − → σ 's being the three Pauli matrices (Tr − → σ = 0), obey the following algebra − → a ,− → b are any two vector operators that commute with − → σ . It must be emphasized that the third term of equation (13) is the Zeeman term, which is generated automatically by using feature (16) with a correct g-factor of g = 2 as reduced in the Bohr's magneton rather than being introduced by hand as a phenomenological term, as is usually done.
The Pauli equation in noncommutative phase-space is Here we achieved the noncommutativity in space using Moyal ⋆product then the noncommutativity in phase through Bopp-shift. Using equation (9), we have (18) In case of a constant real magnetic field − → B = (0, 0, B) = B − → e 3 oriented along the axis (Oz), which is often referred to as the Landau system. We have the following symmetric gauge Therefore, the derivations in the equation (18) approximately shut down in the first-order of Θ, then the noncommutative Pauli equation in the presence of a uniform magnetic field can be written as follows with − → p nc , − → A = 0. We now make use of the Bopp-shift transformation (4), in the momentum operator to obtain we rewrite the above equation in a more compact form We restrict ourselves only to the first-order of the parameter η. The only reason behind this consideration is the balance with the noncommutativity in the space considered in the case of constant magnetic field. Thus we have now The existence of a Pauli equation for all orders of Θ parameter is explicitly relative to the magnetic field.
In the case of a non-constant magnetic field, we introduce a function depending on x in the Landau gauge as A 2 = xBf (x) which gives us a non-constant magnetic field. The magnetic field can be calculated easily using the second equation of equation (12) as follows [26] − → If we specify f (x) we obtain different classes of the non-constant magnetic field. If take f (x) = 1 in this case we get a constant magnetic field.
Having the equation (23) on hand, we calculate the probability density and the current density.

B. Deformed continuity equation
In the following we calculate the current density, which results from the Pauli equation (23 that describing a system of two coupled differential equations for ψ 1 and ψ 2 .
By putting the noncommutative Pauli equation in the presence of a uniform magnetic field simply reads Knowing that − → σ , − → L are Hermitian and the magnetic field is real, and Q * Θ is the adjoint of Q Θ . The adjoint equation of equation (26) reads Here * , † stand for the complex conjugation of the potentials, operators and for the wave-functions successively.
To find the continuity equation, we multiply equation (26) from left by ψ † and equation (27) from the right by ψ, then making the subtraction of these equations, yields after some minor simplefications, we have This will be recognized as the deformed continuity equation. The obtained equation (29) contains new quantity, which is the deformation due to the effect of the phase-space noncommutativity on the Pauli equation.
The third term on the left-hand side, which is the deformation quantity, can be simplified as follows , also we must pay attention to the order, ψ † is the first and ψ the second factor, we have Using the following identity also gives the same equation above [6] υ where υ, τ are arbitrary two-component spinor. Noting that − → A does not appear on the right-hand side of the identity; and that this identity is related to the fact that − → π is Hermitian.
It is evident that the noncommutativity affects the current density, and the deformation quantity may apear as a correction to it. The deformed current density satisfies the current conservation, which means, we have a conservation of the continuity equation in the noncommutative phase-space. Equation (29) may be contracted as where is the probability density and is the deformed current density of the electrons. The deformation quantity is The existence of a deformed continuity equation for all orders of Θ parameter also proportional to the magnetic field. Actually, one can explicitly calculate the conserved current to all orders of Θ. In the case of a non-constant magnetic field, and using equation (24), we have we calculate the n th order term in the general deformed continuity equation (37) as follows We note the absence of the magnetization current term in equation (35), as in commutative case when this was asserted by authors [1,4,8,13,16,20], where at first they attempted to cover this deficiency by explaining how to derive this additional term from the non-relativistic limit of the relativistic Dirac probability current density. Then, Nowakowski and others [6] provided a superb explanation of how to extract this term through the non-relativistic Pauli equation itself.
Knowing that, in commutative background the magnetization current − → j M from the probability current of Pauli equation is proportional to − → ∇ × ψ †− → σ ψ . However, the existence of such an additional term is important and it should be discussed when talking about the probability current of spin-1/2 particles. In following, we try to derive the current magnetization in noncommutative background without changing the continuity equation, and seek if such additional term is affected by noncommutativity or not.

C. Derivation of the magnetization current
At first it must be clarified that the authors Nowakowski and others (2011) in [4,6] derived the non-relativistic current density for a spin-1/2 particle using minimally coupled Pauli equation. In contrast, Wilkes, J. M (2020) in [32] derived the non-relativistic current density for a free spin-1/2 particle using directly free Pauli equation. However, we show here that the current density can be derived from the minimally coupled Pauli equation in noncommutative phase-space.
Starting with the noncommutative minimally coupled Pauli equation written in the form we multiply the above equation from left by ψ † and the adjoint equation of equation (39) from the right by ψ, the subtraction of these equations yields the following continuity equation noting that the noncommutativity of π nc has led us to express the two terms as follows While with only p i , we would have no reason for preferring p i p j ψ over p j p i ψ.
It is easy to verify that the identity (32) remains valid for − → π nc because the fact that − → π nc is Hermitian. Therefore, through identity (32), we have Knowing that the 2 nd sum in equation (42) gives zero by swapping i and j for one of the sums, then the probability current vector from the above continuity equation is Using the property (15), equation (44) becomes with ǫ jik = −ǫ ijk , and using one more time identity (32), we find (this is similar to investigation by [6] in the case of commutative phase-space) In the right-hand side of the above equation, the first term will be interpreted as the noncommutative current vector − → j nc given by equation (36), and the second term is the requested additional term, namely current magnetization Besides, − → j M can also be shown to be a part of the conserved Noether current [33], resulting from the invariance of the Pauli Lagrangian under the global phase transformation U(1).
What can be concluded here is that the magnetization current is not affected by the noncommutativity, perhaps because the spin operator could not be affected by noncommutativity. This is in contrast to what was previously found around the current density, which showed a great influence of noncommutativity.

IV. NONCOMMUTATIVE SEMI-CLASSICAL PARTITION FUNCTION
In this part of our work, we investigate the magnetization and the magnetic susceptibility quantities of our Pauli system using the partition function in noncommutative phase-space. We concentrate, at first, on the calculation of the semi-classical partition function Z. While our system is not completely classical but contains a quantum interaction concerning the spin, therefore, the noncommutative partition function is separable into two independent parts as follows Based on the proposal that noncommutative observables corresponding to the commutative one [34], and for noninteracting particles, the classical partition function in the canonical ensemble in noncommutative phase-space is given by the following formula [35,36] which is written for a N particle, 1 N ! is the Gibbs correction factor, considered due to accounting for indistinguishability, which means that there are N ! ways of arranging N particles at N sites.h ∼ △x nc △p nc , with 1 3 is a factor that makes the volume of the noncommutative phase-space dimensionless.
β defined as 1 KB T , K B is the Boltzmann constant, where K B = 1.38 × 10 −23 JK −1 . The Helmholtz free energy is we may derive the magnetization as follows For a single particle, The noncommutative classical partition function is then where d 3 is a shorthand notation serving as a reminder that the x and p are vectors in three-dimensional phase-space. The relation between equation (48) and (51) is given by the following formula Knowing that using equation (6), we have and according to uncertainty principle [36]h For an electron with spin in interaction with an electromagnetic potential, once the magnetic field − → B be in the zdirection, and by equation (19), bear in mind that − → p nc , − → A nc = 0, then for the sake of simplicity, the noncommutative Pauli Hamiltonian from equation (23) takes the form We split the noncommutative Pauli Hamiltonian as H nc P auli = H nc cla + H spin , with H spin = µ Bσz B . It is easy to verify that Using the three equations above, our noncommutative classical Hamiltonian becomes where L z = p y x − p x y = (x i × p i ) z , and m = m Now, following the definition given in equation (51) we express the single particle noncommutative classical partition function as It should be noted that once we want to factorize our Hamiltonian into a momentum and a position term. This is not always possible when there are matrices (or operators) in the exponent. However, within the classical limit, it is possible. Otherwise, to separate the operators in the exponent, we use the Baker-Campbell-Hausdorff (BCH) formula given by ( Finally, the Pauli partition function for a system of N particles in a three-dimensional noncommutative phase-space is In the limit of the noncommutativity, i.e. Θ → 0, η → 0, the above expression of Z nc tends to the result of Z in the usual commutative phase-space, which is Using formulae (49) and (50), we find the magnetization in noncommutative and commutative phase-space, thus and the noncommutative magnetization be The commutative magnetization is it is obvious that M nc | Θ=0 = M . We may derive the magnetic susceptibility of electrons χ = 1 V ∂ M ∂B in noncommutative phase-space using the magnetization (72) by where the commutative magnetic susceptibility χ = χ nc (Θ = 0) is Finally, we conclude with the following special cases. Let us first consider B = 0, then we have Armed with the partition function Z, we can compute other important thermal quantities, such as the average energy U = − ∂ ∂β lnz, the entropy S = lnz − β ∂ ∂β lnz and the specific heat C = β 2 ∂ 2 ∂ 2 β lnz.

V. CONCLUSION
In this work, we have exactly studied the three-dimensional Pauli equation and the corresponding continuity equation for a spin-1/2 particle in the presence of an electromagnetic field in noncommutative phase-space, considering constant and non-constant magnetic fields. 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. Given the known absence of the magnetization current term in the continuity equation, even in noncommutative phase-space as confirmed by our calculations, we extracted the magnetization current term from the Pauli equation itself without modifying the continuity equation. Furthermore, we found that the density current is conserved, which means, we have a conservation of the deformed continuity equation.
By using the classical treatment (within the classical limit), the magnetization and the magnetic susceptibility quantities are explicitly determined in both commutative and noncommutative phase-spaces through a semi-classical partition function of the Pauli system of the one-particle and N-particle systems in three dimensions. Besides, to see the behaviour of these deformed quantities, we carried out some special cases in commutative and noncommutative phase-spaces.
Finally, we can say that we successfully examined the influence of the noncommutativity on the problems in question, where the noncommutativity was introduced using both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product. Further, the noncommutative corrections to the nonrelativistic Pauli equation and the continuity equation are also valid up to all orders in the noncommutative parameter. Our results limits are in good agreement with those obtained by other authors as discussed and in the literature.