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

: 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.


Introduction
Lately, it has been very interesting to investigate the theoretical basis of the modern physics to explain the nature and the behavior of the matter and energy on the subatomic scale, sometimes referred to as quantum theories, such as the quantum gravity [1,2] or quantum general relativity (QGR) [3], quantum optics and information, the standard model and the gauge theories [4]. This investigation sometimes can be represented in terms of the low-energy regime through the examination of the non-relativistic properties in miscellaneous interactions such as the external electromagnetic fields (EMF), Dirac or DKP oscillator interaction [5,6], Lennard-Jones potential, Coulomb potential, square and step potential; the non-relativistic limit is about low speeds in front of the speed of the light, in more detail, it is for the regime of weak-energy in front of the mass-energy pc mc 2 1 [7], where the non-relativistic limit can be realized through numerous methods, among them the Foldy-Wouthuysen (FW) transformation [8], and Eriksen's method [9] proposed in 1958. Based on the methodical derivation of the unitary transformation that makes the Hamiltonian a diagonal operator, it can be also used when an electromagnetic field is present. There is also the Cayley transformation, and the Cini-Touschek transformation, and the method of development in power ofh [10] and the Douglas-Kroll-Hess (DKH) approach [11,12], which was used mostly as part of relativistic quantum chemistry, and it depends on separating (block-diagonalize) relativistic Hamiltonians into two parts. One part describes electrons in the case of Dirac Hamiltonian, for example, while the other gives rise to the negative energy states. The non-relativistic limit of the relativistic equations was essentially investigated by converting the Hamiltonian from an odd form to an even form.
In this work, we investigated the non-relativistic limit of the DKP equation according to the Foldy-Wouthuysen transformation which occupies an extraordinary position in quantum physics kinetic momentum in the (2+1)d non-commutative phase-space x nc i and p nc i , respectively, and the Heisenberg-like non-commutative commutation relations [22] appear as follows: x nc i , x nc j = iΘ ij , p nc i , p nc j = iη ij , x nc i , p nc j = ih e f f δ ij , (i, j = 1, 2), (1) the effective Plank constant can be written as h e f f =h(1 + Θη with Θ ij and η ij are antisymmetric constant tensors, Θ, η are real-valued non-commutativity parameters, they are supposed to be very small, with the dimension of lenght 2 , momentum 2 , respectively. For some investigations about non-commutative systems concerning the NC parameters, the experimental limit of about 100 nHz on possible sidereal variations (the highest energy variations supported by the experiment) gives estimated limits at about Θ 4.10 −40 m 2 , η 1.76 × 10 −61 Kg 2 m 2 s −2 , andh e f f 10 −67 (SI) [23]. These values agree with the higher limits on the basic scales of coordinate and momentum, and these bounds will be suppressed if the magnetic field used in the experiment is weak (B 5 mG).
In the (2+1)d commutative phase-space, the canonical variables x i and p i satisfy the following commutative algebra The non-commutative geometry Equation (1) in turn is described at the level of fields and functions, by the Gronewold-Moyal product ( -product) [24][25][26] defined as Because of the nature of the -product, the non-commutative field theories for the slowly varying fields or low energies (ΘE 2 < 1) completely reduce to their commutative version.
The NCPS operators are linked to the commutative operators through the Heisenberg-Weyl algebra in terms of the aka Bopp-shift translation which was introduced from Equation (5) [27,28], and it is given by When also Θ = η = 0, the NCPS algebra reduces to the commutative algebra.
, and Equation (8) has three irreducible representations: a 10d representation provides a description of spin-1 bosons, a 5d representation provides a description of spin-0 bosons (spinless particles), and a 1d representation which is a trivial representation.
The Kemmer equation for spin-0 is almost related to the Klein-Gordon equation [32], and for spin-1 is associated with the Proca equations [33].
We have multiplying Equation (7) by β 0 and getting the zero component of Equation (13), we can denote with Equation (7) can be also written in the form Substituting Equation (6) into Equation (14), the DKP Hamiltonian in a non-commutative phase becomes with η ij = 2η k kij , and according to the definition of the vectorial product (A × B) µ = µνλ A ν .B λ , we shall denote the DKP Hamiltonian in a non-commutative phase by The non-commutative phase parameter forms a tetrahedron with the position → x and the DKP vector → β .

The Foldy-Wouthuysen Transformation for a Free Boson in a Non-Commutative Phase
The Foldy-Wouthuysen transformation eliminates the odd part entirely from the wave equation Hamiltonian, and reduces it to an even part (diagonal form). The unitary FW transformation is presented by the following transformations: with U FW being a unitary operator, and S being the time-independent Hermitian operator where θ is a function, taking into account that tan(2| m . The transformed Hamiltonian should contain no odd operatorsH by applying the transformation (21) to Equation (18), knowing that S is the non-explicitly time-dependent operator,H as U FW U † FW = 1. In this case, Equation (22) is written as knowing that, from Equation (8), noting that  where the unitary operator is Equation (23) becomes using the property ( In order to eliminate the odd part, we chose one arrives atH Last but not least, this satisfies the Schrödinger equation, and Equation (32) is similar to the case of commutative phase and space, so that we find that the effect of the non-commutativity in phase on the non-relativistic limit of the DKP Hamiltonian vanished, due to the fact that the non-commutativity parameter entangled explicitly into the non-diagonal part of the non-commutative Hamiltonian. In another way, the non-commutativity in phase affects the odd part of the DKP Hamiltonian. This is for the case of no interaction with potentials.
For the transformed wave function, we merely take the case of the spin-0 representation. We choose the wave function From Equation (16), we rewrite the DKP equation in (2+1)d as follows: Substituting ψ into Equation (34) gives us It is clear that the five components ( ψ 1 , ψ 2 , ψ 3 , ψ 4 , ψ 5 ) are not independent from each , and combining the above equations, we get the dynamical equation of component ψ 1 This leads us to where see the Appendix A Then, we find so that, substituting Equation (40) into Equation (33), we obtain Then, our transformed wave function Equation (19) becomes unlike what happened with the Hamiltonian (18). Here, in the wavefunction Equation (42), the effect of the phase non-commutativity does not vanish because, in the calculations of the energy (Hamiltonian eigenvalue), in order to obtain the wavefunction, η was not entangled with an odd term for that it remains in the equation. In this part of the work, we say that FW transformation did not eliminate the effect of the phase non-commutativity from the Schrödinger equation as was expected.

Schrödinger-Pauli Equation from the DKP Equation in Non-Commutative Phase-Space
At first, defining the electromagnetic field A µ = (A 0 , → A) by inserting the following covariant derivative in the DKP equation satisfies the commutation relation then, the DKP equation in the presence of an electromagnetic interaction (EMI) is with / D = β µ D µ . Then, the suitable physical form for ψ (when there is EMI) can be written as We will explain the presence of an apparently abnormal term devoid of physical interpretation in the DKP Hamiltonian. This term is generated because of the consideration of the minimal coupling Equation (43) in the Kemmer equation, so that contracting Equation (45) on the left by D µ β µ β ν leads to After some algebraic considerations and simplifications, the above relation becomes Then, we multiply Equation (45) on the left by −iβ ρ and making ν = 0 in Equation (48), we obtain the Hamiltonian form of the Kemmer equation where Finally, the above equation becomes The term proportional to e 2m in Equations (48)-(51) is previously regarded by Kemmer himself in his own original research [34]. This term has no clear physical interpretation unlike the other terms in these equations which have physical interpretations similar to similar terms obtained in the Dirac equation interacting with the electromagnetic field.
Using the -product, we find the DKP equation in the non-commutative phase-space Firstly, using Equation (5), we link the non-commutative coordinates x nc i to the commutative one x i so that we achieve the non-commutativity in space x , with k being a real constant so that the derivation in Equation (5) turns off in the first order 0(Θ 2 ). Then, Equation (53) where Secondly, using Equation (6), we link the non-commutative kinetic momentum P nc i to the commutative one P i so that we achieve the the non-commutativity in phase into Equation (55), to find the DKP equation in the complete non-commutative phase-space with η ij = 2η k kij , and, according to the definition of the vectorial product (A × B) µ = µνλ A ν .B λ , and, after minor simplification

Foldy-Wouthuysen Transformation in Non-Commutative Phase-Space
We determine the Schrödinger-Pauli equation in NCPS, which means obtaining the non-relativistic limit from the DKP equation through the Foldy-Wouthuysen transformation, knowing that the FW transformation is suitable for weak fields. The DKP Hamiltonian in NCPS can be written in the form For performing the Foldy-Wouthuysen transformation, we split our non-commutative DKP Hamiltonian Equation (58) to a block diagonal part (even operator ξ) and an off-diagonal part (odd operator O ), (Odd operators (off-diagonal matrices): α i , β i , β 0 , ..., even operators (diagonal matrices): with These are defined to satisfy ξO = Oξ. We consider that, if we multiply two odd operators (or even operators), we find an even operator, and, if multiplying an even operator with an odd operator, we obtain an odd operator. Then, using the Foldy-Wouthuysen transformation, we remove all odd operators. We may successively eliminate these odd terms from the DKP Hamiltonian in NCPS. Later, we will obtain a Hamiltonian completely free of odd operators. We further assume that ξ and O can not be less in order than 1 while the Hermitian operator S may be considered small, and it is given by the terms containing large powers of 1 m (up of 1 m 4 ) in Equation (69) may be ignored. To be specific, we only consider terms of the order that we limit ourselves in the development. Hence, Equation (69) becomes Precisely, we consider only terms until the order of ( 1 m ) 2 . Therefore, terms up to or equal to ( 1 m ) 3 can be neglected, and, with Equation (A6), we obtain the following Hamiltonian: The above equation will be admitted basically as the non-commutative Schrödinger-Pauli Hamiltonian for a classical particle of zero or unity spin interacting with an EMF. The appearance of terms proportional to the explicit phase (even space) non-commutative terms involved in the Schrödinger-Pauli Hamiltonian because of the fact of the effect of the phase-space non-commutativity on the DKP equation, which means they appeared as terms containing the non-commutativity parameters (η, Θ). Then, after using the classical limit via the unitary Foldy-Wouthuysen transformation, those terms that appeared being responsible for generating new terms and correction terms containing the non-commutativity parameters.
In the above Equations (71) and (72), Σ stands for the spin operator of the bosons (with the eigenvalues of 0 or 1), and H, E are the magnetic and the electric fields, respectively.
We denote and interpret the separate terms in our non-commutative Schrödinger-Pauli Hamiltonian as follows. We can identify each term separately, starting with non-diagonal term mβ 0 as the rest energy (which can be eliminated simply from another FW transformation).
Then, e(A 0 + ((gradA 0 + div( Equation (16) through Equation (15), and, with (η = 0, → X → α ∼ 0), the term of kinetic energy is totally diagonalized and can be written as: The most important result we care about is the existence of the orbital angular momentum and the spin couplings with the external magnetic field, but they are modified and affected by the non-commutativity influence as it is obvious in the terms The following terms represent the diagonal spin-orbit coupling by the electric field, but they are affected and modified also by the NC influence The following terms can be explained by being analogous to the terms of Darwin for particles with spin-1/2 in interaction with an EMF The rest of the terms represent higher-order corrections: one of the FW transformation and of one of the PSNC influence. Under the condition η = Θ = 0, Equation (72) becomes Equation (77) is similar to the Schrödinger-Pauli Hamiltonian extracted from the Dirac equation in interaction with an external electromagnetic field.

Conclusions
In previous sections, we have studied the non-relativistic limit of the DKP equation which provides description of the zero or unity spin particles in the DKP representation using the FW unitary transformation in NCPS, where we introduced the phase-space non-commutativity influence. Then, subsequently applying the FW transformation to to take the system (in interaction with an EMF) to a non-relativistic regime, where we found the Schrödinger-Pauli equation (at least to the order of approximation we have considered), knowing that we investigated the non-relativistic limit of the DKP equation in two cases. In the first case, we considered only the non-commutativity in phase with the absence of the interaction with fields, but, for the second case, we considered the full NCPS in the presence of the external electromagnetic field.
In the first case, the concerned equation was the non-relativistic Schrödinger equation, knowing that the effect of the phase non-commutativity vanished in the DKP Hamiltonian but appeared in the corresponding wave-function. At the second case, the concerned equation was the phase-space non-commutative Schrödinger-Pauli equation, where the effect of the NCPS appeared widely in the obtained equation, and it modified most of the equation terms, and affected especially the spin and the orbital angular momentum terms that characterize the Pauli equation. Taking into account the fact that the non-commutativity influence was injected using both the Bopp-shift transformation through the Heisenberg-like commutation relations and the Gronewold-Moyal product. The use of the FW transformation always enables bringing the system of relativistic quantum mechanics to a non-relativistic regime, and it is confirmed by our present work that the FW transformation is applicable even when the non-commutativity is considered.
In the topic of the DKP theory, historically, the first authors who have studied the non-relativistic limit of the DKP equation were Nikitin and Fushchych in their paper, in which they used a different technique for diagonalizing the Hamiltonian as they pointed out in their paper [38], and others also have investigated the non-relativistic Kemmer equation through a Galilean covariance approach [14], in which they used the Galilean covariance to diagonalize the Kemmer Hamiltonian, without forgetting the authors Moshin and Tomazelli who have investigated the non-relativistic of the DKP equation in a commutative space [39].
We may compare our results with that of the other authors as follows: Firstly, we compared our results with that of the authors Moshin and Tomazelli [39]. Under the condition (η = Θ = 0), and by taking into account only terms until the order of ( 1 m ) 2 (terms up to or equal to ( 1 m ) 3 can be neglected), we found almost the same results. Secondly, we made a comparison with the work of the author Silenko [40]. We found that the author has based research on the equation of the particle spin motion described by the Bargmann-Michel-Telegdi equation. Then, in order to check the wave equations for the spin-1 particles, the author took the Lagrangian that describes the spin effects for the particles of an arbitrary spin which interacted with an EMF. Note that, in the general form of his Hamiltonian, he considered an additional term with the odd and even terms in the Hamiltonian to make the application of the FW transformation easier, so that Equation (19) is similar to that of ours with some exceptions, as in our transformed Hamiltonian in the case of the commutativity (η = Θ = 0) (but with a second FW transformation to eliminate the first term of our transformed Hamiltonian). Our Hamiltonian is more detailed than that of Silenko, and it contains corrections that are related to the order of ( 1 m ) 3 . The author has done two of the FW transformations. On the other hand, we made only one single FW transformation (it was enough for us to use a single transformation to find what was interesting).
Funding: This research received no external funding.

Conflicts of Interest:
The author declares no conflict of interest.

Appendix A. The Simplification of Vector Squared of Non-Commutative Momentum
Starting with considering only the 1st order of the phase non-commutativity 0(η 2 ).