DIRAC OSCILLATOR IN DYNAMICAL NONCOMMUTATIVE SPACE

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 ( -space), in which the space-space Heisenberg like commutation relations and noncommutative parameter are position-dependent. Then used this Hamiltonian to calculate the rst-order correction to the eigenvalues and eigenvectors, based on the language of creation and annihilation operators and using the perturbation theory. It is shown that the energy shift depends on the dynamical noncommutative parameter . Knowing that with a set of twodimensional Bopp-shift transformation, we mapped the noncommutative problem to the standard commutative one.


INTRODUCTION
In the last few decades, physicists and mathematicians have developed a mathematical theory called noncommutative geometry, then quickly became a topic of great interest and has been nding applications in many areas of modern physics such as high energy [1], cosmology [2,3], gravity [4], quantum physics [57] and eld theory [8,9]. Substantially, the study on noncommutative (NC) spaces is very important for understanding phenomena at tiny scale of physical theories. Knowing that the idea behind extension of noncommutativity to the coordinates was rst suggested by Heisenberg in 1930 as a solution to remove the innite quantities of eld theories. Besides, NC space-time structures are initiated by Snyder in 1947 [10,11], in which he introduced noncommutativity in the hope of regularizing the divergencies that plagued quantum eld theory.
Motivated by the attempts to understand string theory, the quantum gravitation and black holes through NC spaces and by seeking to highlight more phenomenological implications, we consider the Dirac oscillator (DO) within two-dimensional dynamical noncommutative (DNC) space (known also as position-dependent NC space).
Unlike the simplest possible type of NC spaces, in which NC parameter is constant, here we talk about a dierent type of NC spaces, where the deformation parameter will no longer be constant. However, there are many other possibilities that cannot be excluded. In fact, in the rst paper by Snyder himself [10], the noncommutativity parameter was taken to depend on the coordinates and the momenta. Considerable dierent possibilities have been explored since then especially in the Lie-algebraic approaches [12], Poincaré noncommutativity [13], other fuzzy spaces [14]. Besides, more recently £ ilyashaouam@live.fr ; ilyashaouam@ymail.com in position-dependent approach [1517], the authors considered ¢ to be a function of the position coordinates, i.e. ¢¢@X; Y A.
The relativistic DO is very important potential for both theory and application. The potential term is introduced linearly by substitution 3 p 3 3 p im! 3 r in free Dirac Hamiltonian, this was considered for the rst time by Ito et al [18], with 3 r being the position vector and m, , ! > H are the rest mass of the particle, Dirac matrix and constant oscillator frequency respectively. It is known as Dirac oscillator by Moshinsky and Szczepaniak [19] because it is a relativistic generalization of the nonrelativistic harmonic oscillator, exactly in non-relativistic limit it reduces to a standard harmonic oscillator with a strong spin-orbit coupling term.
Physically, DO has attracted a lot of attention because of its considerable physical applications, it is widely studied and illustrated. It can be shown that it is a physical system, which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric eld [20]. In addition, it can be associated with the electromagnetic potential [21]. As an exactly solvable model, DO in the background of a perpendicular uniform magnetic eld have been wildly studied. However, we mention, for instance, the following: In ref. [19], the spectra of (3+1)-dimensional DO are solved and non-relativistic limit is discussed, as well, in ref [22], the symmetrical properties of the DO are studied. The operators of shift for symmetries are constructed explicitly [23]. Interestingly, the DO may aord a new approach to study quantum optics, where it was found that there is an exact map from (2+1)-dimensional DO to JaynesCummings (JC) model [24], which describes the atomic transitions in a two level system. Subsequently, it found be that this model can be mapped either to JC or anti-JC models, depending on the magnitude of the magnetic eld [25].
Basically, DO became more and more important since the experimental observations. For instance, we mention that Franco-Villafañe et al [26] exposed the proposal of the rst-experimental microwave realization of the onedimensional DO. The experiment depends on a relation of the DO to a corresponding tight-binding system. The experimental results obtained, where the spectrum of the one-dimensional DO is in good agreement with that of the theory. Quimbay et al [27,28] show that the DO may describe a naturally occurring physical system. Precisely, the case of a two-dimensional DO can be used to describe the dynamics of the charge carriers in graphene, and hence its electronic properties [29].
This paper is organized as follows. In section II, the DNC geometry is briey reviewed. In section III, the two-dimensional DNC DO is investigated, where in subsection III B, the energy spectrum in noncommutative space is obtained. In sub-section III C, based on the perturbation theory and Fock basis, the energy spectrum including dynamical noncommutativity eect is obtained, therefore, we summarize the results and discussions. Section IV, is devoted to the conclusions.

II.
REVIEW OF DYNAMICAL NONCOMMUTATIVITY Let us present the essential formulas of the DNC space algebra we need in this study. As known at the tiny scale (string scale), the position coordinates do not commute with each other, thus the canonical variables satisfy the following deformed Heisenberg commutation relation ¢ x nc ; x nc £ a i¢ ; (1) with ¢ is an anti-symmetric tensor. The deformation parameter is a real constant and has the dimension of length P . Dierently, here in the new version of NC spaces, ¢ is taken to be a function of coordinates. However, as a deformation of this NC parameter form will almost inevitably lead to non-Hermitian coordinates, it was pointed out recently [30] that these types of structures are related directly to non-Hermitian Hamiltonian systems. Thus, it is another subject to deal with later. Recently, Fring et al [16] made a generalization of NC space to a position-dependent space by introducing a set of new variables X, Y , P x ,P y and convert the constant ¢ into a function ¢ 3 ¢@X; Y A, by choosing as one possibility @X; Y A a ¢ I C Y P ¡ . In addition, Gomes M et all chose in their study [17] @X; Y A a ¢=I C ¢ I C Y P ¡ .
It is interesting to note that p ¢ has the dimension of length (L), while p has the dimension of energy (or L =I m =I , see eq.(2)).
The new version of noncommutativity known as the DNC space or space. We restrict ourselves here to the two-dimensional space, the commutation relations (Lie brackets) are [16] X; Y a i¢ I C Y P ¡ ; Y; P y a i~ I C Y P ¡ ; X; P x a i~ I C Y P ¡ ; Y; P x a H; X; P y a PiY @¢P y C~XA ; P x ; P y a H: (2) In the limit 3 H, it should be noted that we recover the non-DNC variables, therefore the NC variables satisfy the following Lie brackets x nc ; y nc a i¢; ¢ y nc ; p nc y £ a i~; x nc ; p nc x a i~; y nc ; p nc x a H; ¢ x nc ; p nc y £ a H; ¢ p nc x ; p nc y £ a H: (3) The coordinate X and the momentum P y are not Hermitian, which make the Hamiltonian that includes these variables non-Hermitian. We may represent algebra (2) in terms of the standard Hermitian NC variables operators x nc ; y nc ; p nc x ; p nc y as X a I C @y nc A P x nc ; Y a y nc ; P y a I C @y nc A P p nc y ; P x a p nc x : (4) From this representation, we can see that some of the operators involved above are no longer Hermitian. However, to convert the non-Hermitian variables into a Hermitian one, we use a similarity transformation as a Dyson map O I a o a O y (with a @I C Y P A 1 2 ), as stated in [16]. Therefore, we express the new Hermitian variables x, y, p x and p y in terms of NC variables as follows x a X I a @I C Y P A 1 2 X@I C Y P A 1 2 a @I C @y nc A P A 1 2 x nc @I C @y nc A P A 1 2 y a Y I a @I C @y nc A P A 1 2 y nc @I C @y nc A P A 1 2 a y nc p x a P x I a @I C @y nc A P A 1 2 p nc x @I C @y nc A P A 1 2 a p nc x p y a P y I a @I C @y nc A P A 1 2 P y @I C @y nc A P A 1 2 a @I C @y nc A P A 1 2 p nc y @I C @y nc A P A 1 2 : (5) These new Hermitian DNC variables satisfy the following commutation relations x; y a i¢ I C y P ¡ ; y; p y a i~ I C y P ¡ ; x; p x a i~ I C y P ¡ ; y; p x a H; x; p y a Piy @¢p y C~xA ; p x ; p y a H: (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]: RX min a ¢ p q I C hY i P ; (8) as well, in a simultaneous Y , P y measurement we nd a minimal momentum as R @P y A min a~p q I C hY i P : (9) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 June 2021 The motivation and the interesting physical consequence for position-dependent noncommutativity is that objects in two-dimensional space (geometry) are stringlike [16]. However, investigating DO in DNC geometry gives rise to some phenomenological consequences, which can aid understanding and enhancing string theory.  (11) which, satisfy the commutation relation P i a P a I i j C j i a H i C i a H ; i a I; P; Q: (12) In two dimensions, equation (10) becomes H D a c I p s x C P p s y ¡ e I A s x C P A s y ¡ imc! @ I x s C P y s A C mc P : (13) Let us choose the direction of the eld 3 B according to (Oz), then the vector potentialÃ s is given, in the Landau gauge, by 3 A a B P @ y s ; x s ; HA ; (14) therefore, we have H D @x s i ; p s i A a c I p s x C P p s y ¡ C e B P @ I y s P x s A imc! @ I x s C P y s A C mc P : (15) The Hamiltonian of a two-dimensional DO in DN space is given by H D @x i ; p i A a c @ I p x C P p y A C e B P @ I y P xA imc! @ I x C P yA C mc P : (16) Now, using equation (5), we express the Hamiltonian above in terms of NC variables HD @x nc i ; p nc i A a mc P CcIp nc x C P I C @y nc A P ¡ 1 2 p nc y I C @y nc A P ¡ 1 2 Ce B P Iy nc P I C @y nc A P ¡ 1 2 x nc I C @y nc A P ¡ 1 2 imc!I I C @y nc A P ¡ 1 2 x nc I C @y nc A P ¡ 1 2 C Py nc : (17) Since is very small, the parentheses can be expanded to the rst order using I C @y nc A P 1 2 a I C I P @y nc A P ; (18) so that, equation (17)  Ip nc x C P¨p nc y C I P @y nc A P p nc y C I P p nc y @y nc A P ©£ Cmc P C e B P ¢ Iy nc P¨x nc C I P @y nc A P x nc C I P x nc @y nc A P ©£ imc! ¢ Py nc C I¨x nc C I P @y nc A P x nc C I P x nc @y nc A P ©£ : (19) Using the Bopp-shift transformation (7) (20) Therefore, to the rst order in ¢ and , we have (noting that terms containing ¢ are also neglected) H D @x s i ; p s i A a c h I p s x C P n p s y C P @y s A P p s y C I P p s y @y s A P oi Cmc P C e B P h I y s C ¢  (22) with H H a c I p s x C c P p s y C eB P @ I y s P x s A imc! @ I x s C P y s A C mc P ;  (24) H a I P h c P @y s A P p s y C c P p s y @y s A P @eB P x s @y s A P iPmc! I x s @y s A P i a P P I C P P @eB P C iPmc! I A Q ; (25) where I a @y s A P p s y ; P a p s y @y s A P ; and Q a x s @y s A P : (26) Knowing that H is the perturbation Hamiltonian in which it reects the eects of dynamical noncommutativity of space on the DO Hamiltonian. We also can treat the term proportional to ¢ given in equation (24) as a perturbation term. But here and in a dierent way, we will accurately calculate the energy of the deformed system H H C H ¢ and employed it to test the eect of the DNC space on DO. Thus, we consider the following unperturbed system H UNP a H H C H ¢ : (27) While the noncommutativity parameter is non-zero and very small, we use the perturbation theory to nd the spectrum of the systems in question.
The two-dimensional DO equation in DNC space is written as follow H D j D i a @H UNP C H A j D i a E ¢; j D i ; (28) with j D i a @j I i ; j P iA T ; (29) is the wave function of the system in question.

B. Unperturbed Eigenvalues and Eigenvectors
We introduce the following complex coordinates z s a x s C iy s ; z s a x s iy s ; (30) p s z a i~d dz s a I P p s x ip s y ¡ ; p s z a i~d dz s a I P p s x C ip s y ¡ ; where z s ; p s z a " z s ; p s z a i~; z s ; p s z a " z s ; p s z a H: (32) Using equation (11) with! a ! ! c P ; (34) where ! c a jejB mc is the cyclotron frequency. By putting a I C m ¢! P~t hen our Hamiltonian dened by equation The operators above satisfy the following commutations relations ¢ a; a y £ a I; a; a a ¢ a y ; a y £ a H: (38) Thus, our Hamiltonian (35) subsequently h g P a y a C m P c R @E ¢ A P i j I >a H: In the basis of the second quantization, of which j I >j n >, we have h g P n C m P c R @E ¢ A P i j n >a H; a y a j n >a n j n > : where w a~~! mc 2 , is a parameter that controls the nonrelativistic limit within noncommutative space, and E H a mc P is a background energy, which corresponds to n a H. And q a ¢ ¢0 with ¢ H a m c ¡ P of the dimension ¢m c £ P a L P m P .
The corresponding wave function is written as a function of the basis j n >a @a y A n p n3 j H >, and it is given by the following formula j ¦ n >a c ¦ n j nY I P > Cid ¦ n j n IY I P >; where the coecients c ¦ n and d ¦ n are determined from the normalization condition. We thus obtain [24] c ¦ We plot the reduced energy spectrum in terms of quantum number n, for the cases w a I, q a I ; w a I, q a P and the commutative case with w a I.
The E ¦ n E0 as a function of the quantum number n of equation (49) in both commutative (¢ a q a H) and NC (q a I; q a P) spaces are illustrated in g. 1. Knowing that g. 1 discloses that the inuence of the NC parameter on the energy spectrum is considerable and signicant.
The following gure shows the coupling parameters g and " g , between dierent levels for the two cases in NC space.  While n are non-negative integers, we explicitly observe that our eigenvalues are non-degenerated (the spectrum has no degeneracy), this case can be explained by the fact that the particle is restricted to moving in two dimensions, and the third dimension does not contribute in the form of the energy. Knowing that, it will be an innite degeneracy when there is a contribution of an element related to the third dimension such as k z or p z .
In more detail, however, indirectly (in other sense), the energy spectrum is degenerated. As we all know this is related to the Landau problem, and it is known that there is an innite degeneracy. Nevertheless, considering the energy spectrum non-degenerated, because we do not rely on the states with dierent angular momentum, which is not useful here. The reason is that when we use chiral creation and annihilation operators (a l , a y l and a r , a y r ) we see that the number of particles n r created by right operators does not appear in the form of the energy, we see only number of particles n l generated by left operators. However, right operators create excitations with denite angular momentum in one or the other direction; thus, in this sense we have the degeneracy.
This point is very important to clarify because our calculations in the perturbation theory depend on this point. As many researchers have dealt with this sensitive point and considered that the spectrum has no degeneracy such as [33]. Besides, dierently, for instance, energy levels can appear explicitly degenerated, as in a study [34] about the mesoscopic states in a relativistic Landau levels, the authors found that the energy spectrum is dependent on p P z (check eq. 13 in this cited reference), which is the underlying reasons for innite degeneracy of all levels.

C. Perturbed System
We aim in this sub-section to determine the correction of rst-order energy by using rst-order energy shift formulas. To explain the structure of our spectrum, we will use time-independent perturbation theory for small values of the NC parameter . In view that energies are non-degenerated, we use the non-degenerated timeindependent perturbation theory n a @HA n i C @IA n i C P @PA n i C ::: E n a E @HA n C E @IA n C P E @PA n C ::: Here the (0) superscript denote the quantities that are associated with the unperturbed system.
The rst-order correction to the eigenvalues and eigenvectors in perturbation theory are simply given by E @IA n a RE n a< @HA n j I H j @HA n >; Inserting equation (25) into equation above, we nd E @IA n a< @HA n j I P fP@ICPA @eBP C iPmc!IA Qg j @HA n >; the operator method can also be used to obtain the energy shift in Fock space. In our scenario, we require adopting the notation of the state as follows j @HA n >aj n x ; n y > : The perturbation matrix is given by w a< nx; ny j i P H I P C §Q I C P §Q H The above creation and annihilation operators in fact are extracted from the used one in III B when @a; a y A a¨i @a x C ia y A ; i a y x ia y y ¡© j ¢H 3 @b; b y A ä i @b x C ib y A ; i b y x ib y y ¡© (because in III C we deal only with ). Besides in fact, we have only one integer, which is n, but with the feature n a n x C n y . We deliberately use n x ; n y instead of n because in the perturbaed Hamiltonian we can not use a complex formalism thus we spread n into n x and n y .
With the help of the following denitions of eigenkets and central properties of creation and annihilation operators [35] b j j n j > a p n j j n j I >; b y j j n j > a p n j C I j n j C I >; b P j j n j > a p n j @n j IA j n j P >; b yP j j n j > a p @n j C IA @n j C PA j n j C P >; b Q j j n j > a p n j @n j IA @n j PA j n j Q >; b yQ j j n j > a p @n j C IA @n j C PA @n j C QA j n j C Q >; . . . The contributions of the dierent parts of the perturbed Hamiltonian are as follows < n x ; n y j I j n H x ; n H y >a< n x ; n y j @y s A P p s y j n H < n x ; n y j Q j n H x ; n H y >a< n x ; n y j x s @y s A P j n H Here we set a i. Supposing that j IP j is small compared to relevant energy scale, so that the dierence of the energy eigenvalues of the umperturbed system j IP j < E @HA E @HA C : To obtain the expansion of the energy eigenvalues in the presence of perturbation, namely (a perturbation expansion always exists for a suciently weak perturbation) E I a E @HA C 2 j12j 2 E (0) E (0) + C : : : E P a E @HA C C 2 j12j 2 E (0) + E (0) C : : : We terminate the calculation by the radius of convergence of series expansion (76), so while is a complex variable, is increased from zero, branch points are encoutered at [34] j IP j a ¦i E @HA E @HA C P ; the condtion for the convergence of (76) while the a I full strength case is j IP j a E @HA E @HA C P : If this condition is not met, expansion (76) is meaningless.
It can be checked that all the results of the NC case can be obtained from the DNC case directly by taking the limit of 3 H, for instance equations (52) and (53) give the same values as the eigenvalues and eigenvectors in NC space, i.e. equations (48) and (50), respectively.
It may be also useful to mention that the NC DO (in non-DNC space) has been investigated in [3638].
We can regard equation (76) as the eigenvalues of our system, where we restrict ourselves to the rst-order correction to the eigenvalues and eigenvectors, which leads the energy shift for the ground state. Besides, it is easy to obtain the eigensolutions for excited states.
It is interesting to illustrate the DNC eect on DO energy levels. This eect is reduced in the energy shifts obtained, hence we do the following sample, with a I a § P P The upper bound on the value of the NC parameter ¢ is p ¢ P ¢ IH =PH m [39], as well for is p IH IU eV [40]. The bound on p is consistent with the accuracy in the energy measurement IH IP eV.
It is important to clarify that the presence of P in the eigenvalues is not due to the act of the second-order correction, but rather to the Dirac matrices in the perturbed Hamiltonian term.
In g. 3

IV. CONCLUSION
In conclusion, the DO has been investigated in twodimensions in presence of an external magnetic eld in DNC space in terms of creation and annihilation operators language and through properly chosen canonical pairs of coordinates and its corresponding momenta in a complex NC space. However, the dynamical noncommutativity was treated as a perturbation. More precisely, we have solved the DO problem in two-dimensional NC space to nd the exact energy spectrum and wave functions. Therefore, we have employed these obtained results to nd the rst-order correction to the eigenvalues and eigenvectors. So that the correction due to dynamical noncommutativity on the energy of the quantum system can be stated in terms of . It is worth noting that we addressed the system in NC space as a fundamental system instead of considering the fundamental system in a commutative space and noncommutativity is a perturbation. The rst-order correction for the ground state of the DO due to noncommutativity of space is zero for a non-DNC case while it has a nonvanishing value in DNC case. Knowing that the result reduces to that of usual DO in commutative space in the limits of H, ¢H.
As mentioned in section II, some operators in DNC space are non-Hermitian. This mixture of DNCS and the non-Hermiticity theory together with the string theory can lead to fundamental new insights in these three elds. Distinctly, there are plenty of interesting problems arising from our investigation, such as the investigation of further possibilities of consistent deformations, the construction of the solution for the DKP oscillator, as well Klein-Gordon oscillator. The study of additional models in terms this newly used DNC variables.  Table I. Energy levels due to DNC space, where we suce with eigenvalues corrections to the ground state, i.e. EI;P a ¦mc P . Figure 3. Diagram of splittings for energy levels due to DNC and non-DNC spaces.