Acta Polytechnica https://doi.org/10.14311/AP.2021.61.0689 Acta Polytechnica 61(6):689–702, 2021 © 2021 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague DIRAC OSCILLATOR IN DYNAMICAL NONCOMMUTATIVE SPACE Ilyas Haouam Université Frères Mentouri, Laboratoire de Physique Mathématique et de Physique Subatomique (LPMPS), Constantine 25000, Algeria correspondence: ilyashaouam@live.fr Abstract. In this paper, we address the energy eigenvalues of two-dimensional Dirac oscillator perturbed by a dynamical noncommutative space. We derived the relativistic Hamiltonian of Dirac oscillator in the dynamical noncommutative space, in which the space-space Heisenberg-like commutation relations and noncommutative parameter are position-dependent. Then, we used this Hamiltonian to calculate the first-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 τ. Knowing that, with a set of two-dimensional Bopp-shift transformation, we mapped the noncommutative problem to the standard commutative one. Keywords: Dynamical noncommutative space, τ-space, noncommutative Dirac oscillator, perturbation theory. 1. Introduction In the last few decades, physicists and mathematicians have developed a mathematical theory called noncommu- tative geometry, which has quickly become a topic of great interest and has been finding applications in many areas of modern physics, such as high energy [1], cosmology [2, 3], gravity [4], quantum physics [5–7] and field theory [8, 9]. Substantially, the study on noncommutative (NC) spaces is very important for understanding phenomena at a tiny scale of physical theories. The idea behind the extension of noncommutativity to the coordinates was first suggested by Heisenberg in 1930 as a solution to remove the infinite quantities of field theories. The NC space-time structures were first mentioned by Snyder in 1947 [10, 11], in which he introduced noncommutativity in the hope of regularizing the divergencies that plagued quantum field theory. Motivated by the attempts to understand the 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 a two-dimensional dynamical noncommutative (DNC) space (also known as a position-dependent NC space). Unlike the simplest possible type of NC spaces, in which the NC parameter is constant, here we talk about a different 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 first paper by Snyder himself [10], the noncommutativity parameter was taken to depend on the coordinates and the momenta. Considerable different possibilities have been explored since then, especially in the Lie-algebraic approaches [12], κ-Poincaré noncommutativity [13], other fuzzy spaces [14]. Besides, more recently in position-dependent approach [15–17], the authors considered Θµν to be a function of the position coordinates, i.e., Θ→Θ(X,Y ). The relativistic DO has a great potential both for the theoretical and practical applications. The potential term is introduced linearly, by substitution −→p → −→p − imβω−→r in free Dirac Hamiltonian, this was considered for the first time by Ito et al. [18], with −→r being the position vector and m, β, ω > 0 being the rest mass of the particle, Dirac matrix and constant oscillator frequency, respectively. It was named Dirac oscillator by Moshinsky and Szczepaniak [19] because it is a relativistic generalization of the non-relativistic harmonic oscillator and, exactly in a 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 an interaction of the anomalous magnetic moment with a linear electric field [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 field has been widely studied. However, we mention, for instance, the following: In ref. [19], the spectra of (3+1)-dimensional DO are solved and the 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 offer a new approach to study quantum optics, where it was found 689 https://doi.org/10.14311/AP.2021.61.0689 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Ilyas Haouam Acta Polytechnica that there is an exact map from (2+1)-dimensional DO to Jaynes-Cummings (JC) model [24], which describes the atomic transitions in a two level system. Subsequently, it was found that this model can be mapped either to JC or anti-JC models, depending on the magnitude of the magnetic field [25]. Basically, DO became more and more important since the experimental observations. For instance, we mention that Franco-Villafañe et al. [26] came with the proposal of a first-experimental microwave realisation of the one-dimensional DO. The experiment depends on a relation of the DO to a corresponding tight-binding system. The experimental results obtained show that 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. More precisely, that 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 2, the DNC geometry is briefly reviewed. In section 3, the two-dimensional DNC DO is investigated, where in sub-section 3.2, the energy spectrum in NC space is obtained. In sub-section 3.3, based on the perturbation theory and Fock basis, the energy spectrum including the dynamical noncommutativity effect is obtained, therefore, we summarize the results and discussions. Section 4, is then devoted to the conclusions. 2. Review of dynamical noncommutativity Let us present the essential formulas of the DNC space algebra we need in this study. As is 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 [ xncµ ,x nc ν ] = iΘµν, (1) with Θµν being an anti-symmetric tensor. In simplest way, the deformation parameter is considered a real constant. But, in general, Θµν can be a function of coordinates. Fring et al. [16] made a generalization of an NC space to a position-dependent space by introducing a set of new variables X, Y , Px, Py and converting the constant Θ into a function of coordinates θ(X,Y ) = Θ ( 1 + τY 2 ) . As another example of θ(X,Y ), we also mention that Gomes et al. [17] chose in their study θ(X,Y ) = Θ/[1 + Θα ( 1 + Y 2 ) ]. However, as a deformation of this NC parameter form will almost inevitably lead to non-Hermitian coordinates, it was pointed out [30] that these types of structures are related directly to non-Hermitian Hamiltonian systems. Later, it is explained how this problem was solved. In the new type of the two-dimensional NC space, which is known as the DNC space or τ−space, the commutation relations are [16] [X,Y ] = iΘ ( 1 + τY 2 ) , [Y,Py ] = iℏ ( 1 + τY 2 ) , [X,Px] = iℏ ( 1 + τY 2 ) , [Y,Px] = 0, [X,Py ] = 2iτY (ΘPy + ℏX) , [Px,Py ] = 0. (2) It is interesting to note that √ Θ and √ τ have dimensions of L and L−1, respectively. In the limit τ → 0, we obtain the following non-dynamical NC commutation relations [xnc,ync] = iΘ, [ ync,pncy ] = iℏ, [xnc,pncx ] = iℏ, [ync,pncx ] = 0,[ xnc,pncy ] = 0, [ pncx ,p nc y ] = 0. (3) The coordinate X and the momentum Py are not Hermitian, which makes the Hamiltonian that includes these variables non-Hermitian. We represent algebra (2) in terms of the standard Hermitian NC variable operators xnc,ync,pncx ,pncy as X = ( 1 + τ (ync)2 ) xnc, Y = ync, Py = ( 1 + τ (ync)2 ) pncy , Px = pncx . (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 ηOη−1 = o = O† with η = (1 +τY 2)− 12 , as stated in [16]. Therefore, we express the new Hermitian variables x, y, px and py in terms of NC variables as follows 690 vol. 61 no. 6/2021 Dirac oscillator in dynamical noncommutative space x = ηXη−1 = (1 + τY 2)− 12 X(1 + τY 2) 12 = (1 + τ (ync)2) 12 xnc(1 + τ (ync)2) 12 , y = ηY η−1 = (1 + τ (ync)2)− 12 ync(1 + τ (ync)2) 12 = ync, px = ηPxη−1 = (1 + τ (ync) 2)− 12 pncx (1 + τ (ync) 2) 12 = pncx , py = ηPyη−1 = (1 + τ (ync) 2)− 12 Py (1 + τ (ync) 2) 12 = (1 + τ (ync)2) 12 pncy (1 + τ (ync) 2) 12 . (5) These new Hermitian DNC variables satisfy the following commutation relations [x,y] = iΘ ( 1 + τy2 ) , [y,py ] = iℏ ( 1 + τy2 ) , [x,px] = iℏ ( 1 + τy2 ) , [y,px] = 0, [x,py ] = 2iτy (Θpy + ℏx) , [px,py ] = 0. (6) Now, using Bopp-shift transformation [31], one can express the NC variables in terms of the standard commutative variables [5] xnc = xs − Θ2ℏp s y, p nc x = psx, ync = ys + Θ2ℏp s y, p nc y = psy, (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]: △Xmin = Θ √ τ √ 1 + τ ⟨Y ⟩2ρ, (8) as well, in a simultaneous Y , Py measurement, we find a minimal momentum as △ (Py )min = ℏ √ τ √ 1 + τ ⟨Y ⟩2ρ. (9) The motivation and interesting physical consequence for position-dependent noncommutativity is that objects in two-dimensional spaces are string-like [16]. However, investigating the DO in DNC geometry gives rise to some phenomenological consequences that may be very important and useful. 3. Two-dimensional Dirac oscillator in dynamical noncommutative space 3.1. Extension to dynamical noncommutative space The dynamics of the DO in the presence of a uniform external magnetic field is governed by the following Hamiltonian HD = c−→α. ( −→p s − e c −→ As − imcωβ−→r s ) + βmc2, (10) where −→ As ( Asx,A s y,A s z ) is the vector potential produced by the external magnetic field and e is the charge of the DO (the electron charge). The α⃗ matrices, in two dimensions, are represented by the following Pauli matrices α1 = σx = ( 0 1 1 0 ) , α2 = σy = ( 0 −i i 0 ) , β = σz = ( 1 0 0 −1 ) , (11) which satisfy the following relations α2i = β2 = 1, αiαj + αjαi = 0, αiβ + βαi = 0. i = 1, 2, 3 (12) In two dimensions, equation (10) becomes 691 Ilyas Haouam Acta Polytechnica HD = c ( α1p s x + α2p s y ) − e ( α1A s x + α2A s y ) − imcω (α1βxs + α2βys) + βmc2. (13) Let us consider −→ B to be along the z axis, thus the vector potential A⃗s is given in the Landau gauge by −→ A = B 2 (−ys,xs, 0) , (14) therefore, we have HD (xsi ,p s i ) = c ( α1p s x + α2p s y ) + e B 2 (α1ys − α2xs) − imcω (α1βxs + α2βys) + βmc2. (15) The above Hamiltonian in DNC space turns to HD (xi,pi) = c (α1px + α2py ) + e B 2 (α1y − α2x) − imcω (α1βx + α2βy) + βmc2. (16) Now, using equation (5), we express the Hamiltonian above in terms of NC variables HD (xnci ,p nc i ) = βmc 2 + c[α1pncx + α2 ( 1 + τ (ync)2 )1 2 pncy ( 1 + τ (ync)2 )1 2 ] + e B 2 [α1ync − α2 ( 1 + τ (ync)2 )1 2 xnc ( 1 + τ (ync)2 )1 2 ] − imcω[α1β ( 1 + τ (ync)2 )1 2 xnc ( 1 + τ (ync)2 )1 2 + α2βync]. (17) Since τ is very small, the parentheses can be expanded to the first-order through ( 1 + τ (ync)2 )1 2 = 1 + 1 2 τ (ync)2 , (18) so that equation (17) turns to HD (xnci ,p nc i ) = c [ α1p nc x + α2 { pncy + 1 2 τ (ync)2 pncy + 1 2 τpncy (y nc)2 }] + βmc2 + e B 2 [ α1y nc − α2 { xnc + 1 2 τ (ync)2 xnc + 1 2 τxnc (ync)2 }] − imcω [ α2βy nc + α1β { xnc + 1 2 τ (ync)2 xnc + 1 2 τxnc (ync)2 }] . (19) Using the Bopp-shift transformation (7), Hamiltonian (19) can be expressed in terms of the standard commutative variables HD (xsi , p s i ) = cα1p s x + βmc 2 + cα2psy + cα2 { 1 2 τ ( y s + Θ 2ℏ p s x )2 p s y + 1 2 τ p s y ( y s + Θ 2ℏ p s x )2} + eB 2 [ α1 ( y s + Θ 2ℏ p s x ) − α2 { τ 2 ( y s + Θ 2ℏ p s x )2 ( x s − Θ 2ℏ p s y ) +xs − Θ 2ℏ p s y + 1 2 τ ( x s − Θ 2ℏ p s y )( y s + Θ 2ℏ p s x )2}] − imcω [ α2β ( y s + Θ 2ℏ p s x ) + α1β { x s − Θ 2ℏ p s y + τ 2 ( y s + Θ 2ℏ p s x )2 ( x s − Θ 2ℏ p s y ) + τ 2 ( x s − Θ 2ℏ p s y )( y s + Θ 2ℏ p s x )2}] . (20) Therefore, to the first-order in Θ and τ, we have (noting that terms containing Θτ are neglected too) 692 vol. 61 no. 6/2021 Dirac oscillator in dynamical noncommutative space HD (xsi ,p s i ) = c [ α1p s x + α2 { psy + τ 2 (ys)2 psy + 1 2 τpsy (y s)2 }] + βmc2 + e B 2 [ α1 ( ys + Θ 2ℏ psx ) − α2 { xs − Θ 2ℏ psy + τx s (ys)2 }] − imcω [ α2β ( ys + Θ 2ℏ psx ) + α1β { xs − Θ 2ℏ psy + τx s (ys)2 }] , (21) which can be written as HD = H0 + HΘ + Hτ , (22) with H0 = cα1psx + cα2p s y + eB 2 (α1ys − α2xs) − imcω (α1βxs + α2βys) + βmc2, (23) HΘ = Θ 2ℏ [ eB 2 ( α1p s x + α2p s y ) − imcω ( α2βp s x − α1βp s y )] , (24) Hτ = 1 2 τ [ cα2 (ys) 2 psy + cα2p s y (y s)2 − (eBα2xs (ys) 2 − i2mcωα1βxs (ys) 2 ] = τ 2 [α2V1 + α2V2 − (eBα2 + i2mcωα1β) V3] , (25) where V1 = (ys) 2 psy, V2 = p s y (y s)2 , and V3 = xs (ys) 2 . (26) Knowing that Hτ is the perturbation Hamiltonian in which it reflects the effects of dynamical noncommuta- tivity of space on the DO Hamiltonian. We can also treat the term proportional to Θ, given in equation (24) as a perturbation term. But here, and in a different way, we will accurately calculate the energy of the deformed system H0 + HΘ and employ it to test the effect of the DNC space on the DO. Thus, we consider the following unperturbed system HU N P = H0 + HΘ. (27) While the noncommutativity parameter τ is non-zero and very small, one can use the perturbation theory to find the spectrum of the systems in question. The two-dimensional DO equation in the DNC space is written as follows HD |ψD ⟩ = (HU N P + Hτ ) |ψD ⟩ = EΘ,τ |ψD ⟩ , (28) with |ψD ⟩ = (|ψ1⟩ , |ψ2⟩) T , (29) being the wave function of the system in question. 693 Ilyas Haouam Acta Polytechnica 3.2. Unperturbed eigenvalues and eigenvectors We introduce the following complex coordinates zs = xs + iys, zs = xs − iys, (30) psz = −iℏ d dzs = 12 ( psx − ipsy ) , psz = −iℏ d dzs = 12 ( psx + ipsy ) , (31) where [zs,psz ] = [z̄ s,psz ] = iℏ, [z s,psz ] = [z̄ s,psz ] = 0. (32) Using equation (11), our unperturbed system (24), in the complex formalism, merely becomes HU N P = [ mc2 2cΩpsz + imczsω̃ 2cΩpsz − imcωz sω̃ −mc2 ] , (33) where Ω = 1 + m Θω̃ 2ℏ with ω̃ = ω − ωc 2 , (34) knowing that ωc = |e|Bmc is the cyclotron frequency. Now, let us introduce the following creation and annihilation operators a = i ( Ω √ mωℏ psz − i 2Ω √ mω ℏ zs ) , (35) a† = −i ( Ω √ mωℏ psz + i 2Ω √ mω ℏ zs ) , (36) that satisfy the following commutations relations [ a,a† ] = 1, [a,a] = [ a†,a† ] = 0. (37) Thus, in terms of the creation and annihilation operators, the Hamiltonian (33) takes the following form HΘ = [ mc2 i2c √ mωℏa† −i2c √ mωℏa −mc2 ] = [ mc2 iga† −iga −mc2 ] , (38) with g = 2c √ mℏω being a parameter that describes the coupling between different states in NC space, and ω = Ωω̃ being the correction of the frequency ω̃ of the commutative space. In addition, the parameter g = 2c √ mω̃ℏ describes the coupling between different states in the commutative space. Now, we solve the following equation HU N P ∣∣ψD〉 = EΘ ∣∣ψD〉 , (39) where EΘ, ∣∣ψD〉 are the eigenenergy and wave function of the Dirac equation above, respectively. By inserting equation (29) in (39), we obtain the following system of equations ( mc2 iga† −iga −mc2 )( | ψ1 > | ψ2 > ) = EΘ ( | ψ1 > | ψ2 > ) , (40) 694 vol. 61 no. 6/2021 Dirac oscillator in dynamical noncommutative space where ( mc2 − EΘ ) | ψ1 > +iga † | ψ2 >= 0, (41) −iga | ψ1 > − ( mc2 + EΘ ) | ψ2 >= 0. (42) From the equations (41) and (42), we have | ψ2 >= −iga EΘ + mc2 | ψ1 >, (43) subsequently [ g2a†a + m2c4 − (EΘ) 2 ] | ψ1 >= 0. (44) On the basis of the second quantization, of which | ψ1 >≡| n >, we have [ g2n + m2c4 − (EΘ) 2 ] | n >= 0, a†a | n >= n | n > . (45) Thus, the energy spectrum is given by E±Θ,n = ± √ m2c4 + g2n, (46) which can be rewritten as E±Θ,n = ±mc 2 √ 1 + 4ℏω̃ mc2 ( 1 + m Θω̃ 2ℏ ) n, n = 0, 1, 2, ... (47) Furthermore, we have the reduced energy spectrum E±n E0 = ± √ 1 + 4w ( 1 + 1 2 qw ) n, (48) where the non-relativistic limit feature is reduced in w = ℏω̃ mc2 , which is a parameter that controls the non- relativistic limit within the NC space (as well as in commutative case, if Θ = 0), and E0 = mc2 is a background energy, which corresponds to n = 0. And q = ΘΘ0 with Θ0 = ( ℏ mc )2of the dimension [ ℏ mc ]2 = L2 ≡ m2 . The corresponding wave function is written as a function of the basis | n >= (a †)n √ n! | 0 >, and it is given by the following formula | ψ ± n >= c ± n | n; 1 2 > +id±n | n − 1; − 1 2 >, (49) where the coefficients c±n and d±n are determined from the normalization condition. We thus obtain [24] c±n = √ E+n ± mc2 2E+n , d±n = ∓ √ E+n ∓ mc2 2E+n . (50) In the limit Θ → 0, the NC energy spectrum becomes commutative one, i.e., equation (47) turns to equation (10) of ref. [32], which confirms that we are in good agreement. As well, in ref. [33] Boumali et al. made a study of a DO in an NC phase-space, where, if Θ → 0, the energy eigenvalues (eq:50) will be similar as ours in equation (47). 695 Ilyas Haouam Acta Polytechnica We plot the reduced energy spectrum in terms of quantum number n, for the cases w = 1, q = 1; w = 1, q = 2 and the commutative case with w = 1. The E ± n E0 , as a function of quantum number n of equation (48) in both commutative (Θ = q = 0) and NC (q = 1; q = 2) spaces, is illustrated in Fig. 1. Knowing that Fig. 1 discloses that the influence of the NC parameter on the energy spectrum is considerable and significant. Figure 1. A reduced energy versus quantum number in both cases of NC and commutative spaces. The following figure shows the coupling parameters g and ḡ, between different levels for the two cases in the NC space. Figure 2. The coupling parameter between different levels: (a) in the case of commutative space, (b) in the case of 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 energy. Knowing that, it will be an infinite degeneracy when there is a contribution of an element related to the third dimension, such as kz or pz . In more detail and indirectly (in other sense), the energy spectrum is degenerated. As is known, this is related to the Landau problem, and it is known that there is an infinite degeneracy. Nevertheless, we consider the energy spectrum non-degenerated, because we do not rely on the states with different angular momentum, which is not useful here. The reason is that when we use chiral creation and annihilation operators (al, a†l and ar , a†r ), we see that the number of particles nr created by right operators does not appear in the form of energy, we see only number of particles nl generated by left operators. However, right operators create excitations with a definite 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, differently, 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 depends on p2z (check eq. 13 in this cited reference), which is the underlying reason for an infinite degeneracy of all levels. 3.3. Perturbed system In this sub-section, we aim to determine the correction of first-order energy by using first-order energy shift formulas. To explain the structure of our spectrum, we will use time-independent perturbation theory for 696 vol. 61 no. 6/2021 Dirac oscillator in dynamical noncommutative space small values of the parameter τ. In view that energies are non-degenerated, we use the non-degenerated time-independent perturbation theory ∣∣ψn〉 = ∣∣∣ψ(0)n 〉 + τ ∣∣∣ψ(1)n 〉 + τ2 ∣∣∣ψ(2)n 〉 + ... (51) En = E(0)n + τE (1) n + τ 2E(2)n + ... (52) Here, the (0) superscript denotes the quantities that are associated with the unperturbed system. The first-order correction to the eigenvalues and eigenvectors in perturbation theory are simply given by E(1)n = △En =< ψ (0) n | 1 τ Hτ | ψ (0) n >, (53) ∣∣∣ψ(1)n 〉 = ∑ k ̸=n < ψ (0) k | 1 τ Hτ | ψ (0) n > E (0) n − E (0) k ∣∣∣ψ(0)k 〉 . (54) Inserting equation (25) into the equation above, we find E(1)n =< ψ (0) n | 1 2 {α2(V1 + V2) − (eBα2 + i2mcωα1β) V3} | ψ (0) n >, (55) 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 | ψ (0) n >=| nx,ny > . (56) The perturbation matrix is given by M =< nx,ny | i 2 ( 0 −V1 − V2 + ΥV3 V1 + V2 − ΥV3 0 ) | n ′ x,n ′ y >, (57) with Υ = eB + 2mcω. To calculate the influence of Vi (i = 1, .., 3) on the element of the Fock basis, we conveniently use the following bj , b†j (j = x,y) operators bj = √ mω 2ℏ ( xsj + i psj mω ) and b†j = √ mω 2ℏ ( xsj − i psj mω ) , (58) where [ bj,b † j ] = 1, with b†jbj = Nj. (59) The above creation and annihilation operators are, in fact, extracted from the one in 3.2, when (a,a†) ={ i (ax + iay ) , −i ( a†x − ia†y )} |Θ→0→ (b,b†) = { i (bx + iby ) , −i ( b†x − ib†y )} (because in 3.3, we deal only with τ). In fact, we have only one integer, which is n, but with the feature n = nx + ny . We deliberately use nx, ny instead of n because in the perturbed Hamiltonian, we cannot use a complex formalism, thus we divide n into nx and ny . With the help of the following definitions of eigenkets and central properties of creation and annihilation operators [35] bj | nj > = √ nj | nj − 1 >, b † j | nj > = √ nj + 1 | nj + 1 >, b2j | nj > = √ nj (nj − 1) | nj − 2 >, b †2 j | nj > = √ (nj + 1) (nj + 2) | nj + 2 >, b3j | nj > = √ nj (nj − 1) (nj − 2) | nj − 3 >, b †3 j | nj > = √ (nj + 1) (nj + 2) (nj + 3) | nj + 3 >, ... (60) 697 Ilyas Haouam Acta Polytechnica with < n ′ j | nj >= δn′ j ,nj and [ b†b,b† ] = b†bb† − b†2b = b† and [ b†b,b ] = b†b2 − bb†b = −b, (61) b†bb† | n >= Nb† | n >= (n + 1) b† | n >, (62) b†bb | n >= Nb | n >= (n − 1) b | n >, (63) [N,b] = −b and bN = Nb + b, (64) [ N,b† ] = b† and b†N = Nb† − b†. (65) Knowing that xsj = √ ℏ 2mω ( bj + b†j ) and psj = i √ ℏmω 2 ( b † j − bj ) . (66) The contributions of the different parts of the perturbed Hamiltonian are as follows < nx,ny | V1 | n ′ x,n ′ y >=< nx,ny | (y s)2 psy | n ′ x,n ′ y > = −iℏ 2 √ ℏ 2mω δnx,n′x {√ n ′ y ( n ′ y − 1 )( n ′ y − 2 ) δny ,n′y −3 − √( n ′ y + 1 )( n ′ y + 2 )( n ′ y + 3 ) δny ,n′y +3 + ( n ′ y − 2 )√ n ′ yδny ,n′y −1 − ( n ′ y + 3 )√ n ′ y + 1δny ,n′y +1 } . (67) < nx,ny | V2 | n ′ x,n ′ y >=< nx,ny | p s y (y s)2 | n ′ x,n ′ y > = −iℏ 2 √ ℏ 2mω δnx,n′x {√ n ′ y ( n ′ y − 1 )( n ′ y − 2 ) δny ,n′y −3 − √( n ′ y + 1 )( n ′ y + 2 )( n ′ y + 3 ) δny ,n′y +3 + ( n ′ y + 2 )√ n ′ yδny ,n′y −1 + ( 3 − n ′ y )√ n ′ y + 1δny ,n′y +1 } . (68) < nx,ny | V3 | n ′ x,n ′ y >=< nx,ny | x s (ys)2 | n ′ x,n ′ y > = ( ℏ 2mω )3 2 (√ n ′ xδnx,n′x−1 + √ n ′ x + 1δnx,n′x+1 ) × (√ n ′ y ( n ′ y − 1 ) δny ,n′y −2 + √( n ′ y + 1 )( n ′ y + 2 ) δny ,n′y +2 + ( 1 + 2n ′ y ) δny ,n′y ) . (69) The relevant perturbation matrix is given by M = i ( 0 W12 W21 0 ) , (70) with W12 = −W21 =< nx,ny | 1 2 {ΥV3 − (V1 + V2)} | n ′ x,n ′ y > . (71) 698 vol. 61 no. 6/2021 Dirac oscillator in dynamical noncommutative space The DO Hamiltonian HD = H0 + HΘ + Hτ may be represented by a square matrix as follows (we have used the basis given by unperturbed energy eigenkets) HD ≡ ( E (0) n,+ (Θ) iτW12 iτW21 E (0) n,− (Θ) ) . (72) The eigenvalues of the problem above are ( E1 E2 ) = E (0) − + E (0) + 2 ± √√√√(E(0)− − E(0)+ )2 4 + λ2 |W12| 2 . (73) Here, we set λ = iτ. Supposing that λ |W12| is small as compared to relevant energy scale, so that the difference of the energy eigenvalues of the unperturbed system equals λ |W12| < ∣∣∣E(0)− − E(0)+ ∣∣∣ . (74) To obtain the expansion of the energy eigenvalues in the presence of a perturbation, namely (a perturbation expansion always exists for a sufficiently weak perturbation) E1 = E (0) − + λ2|W12|2 E (0) − −E (0) + + . . . E2 = E (0) + + λ2|W12|2 E (0) + −E (0) − + . . . (75) We terminate the calculation by the radius of convergence of series expansion (75), so while λ is a complex variable, λ is increased from zero, branch points are encoutered at [34] λ |W12| = ±i ( E (0) − − E (0) + ) 2 , (76) the condition for the convergence of (75) for the λ = 1 full strength case is |W12| = ∣∣∣E(0)− − E(0)+ ∣∣∣ 2 . (77) If this condition is not met, the expansion (75) 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 τ → 0, for instance, equations (51) and (52) give the same values as the eigenvalues and eigenvectors in the NC space, i.e., equations (47) and (49), respectively. It may be also useful to mention that the DO in deformed spaces (including NC spaces) has been investigated in [36–43]. We can regard equation (75) as the eigenvalues of our system, where we restrict ourselves to the first-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 effect on DO energy levels. This effect is reduced in the energy shifts obtained, hence we do the following sample, with a1 = Υ 2 ( ℏ 2mω )3 2 ,a2 = iℏ 2ω ( ℏ 2mω )1 2 ,a3 = iℏ ( ℏ mω )1 2 ,a4 = iℏ 2 ( ℏ mω )1 2 . (78) In Table 1, all numerical values of the energy are in units of eV. It may be worth to underline that thanks to the Kronecker’s delta, the elements of the perturbed Hamiltonian will seldom take many values. Now, the DNC and non-DNC effects on the energy levels of the DO are illustrated in Fig. 3. The upper bound on the value of the NC parameter Θ is √ Θ ≤ 2 × 10−20 m [44], as well for τ is √ τ ≤ 10−17 eV [45]. The bound on √ τ is consistent with the accuracy in the energy measurement 10−12 eV. 699 Ilyas Haouam Acta Polytechnica ( nx,ny,n ′ x,n ′ y ) W12 E1 E2 △E (1, 1, 1, 1) 0 −5, 11.105 5, 11.105 0 (1, 0, 0, 0) a1 −5, 11.105 + a21τ 2 10,22.105 5, 11.10 5 − a 2 1τ 2 10,22.105 a21τ 2 10,22.105 (0, 0, 0, 1) a2 −5, 11.105 + a22τ 2 10,22.105 5, 11.10 5 − a 2 2τ 2 10,22.105 a22τ 2 10,22.105 (0, 1, 0, 2) a3 −5, 11.105 + a23τ 2 10,22.105 5, 11.10 5 − a 2 3τ 2 10,22.105 a23τ 2 10,22.105 (0, 2, 0, 1) -a4 −5, 11.105 + a24τ 2 10,22.105 5, 11.10 5 − a 2 4τ 2 10,22.105 a24τ 2 10,22.105 Table 1. Energy levels due to DNC space, where we suffice with eigenvalues corrections to the ground state, i.e. E1,2 = ±mc2. Figure 3. Diagram of splittings for energy levels due to DNC and non-DNC spaces. It is important to clarify that the presence of τ2 in the eigenvalues is not due to the action of the second-order correction, but rather due to the Dirac matrices in the perturbed Hamiltonian term. In Fig. 3, we see the energy levels are splitting as the Θ and τ parameters turn on. Values of the eigenvalues E1 and E2 are shown in Table 1. First, in Fig. 3, we show the effect of the NC space when the perturbation parameter is off (τ = 0), where the effect of noncommutativity is very significative as explained in Fig. 1. Thereafter, the effect of noncommutativity is more significative when the DNC perturbation term is present (τ ̸= 0), the presence of this term exhibits the energy shifts. The last part of Fig. 3 shows the combined effect of both the DNC and NC parameters, where the effect is very evident. 4. Conclusion In conclusion, we have investigated the effects of a DNC space on the 2D DO in the presence of an external mag- netic field in terms of creation and annihilation operator languages 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 a two-dimensional NC space to find the exact energy spectrum and wave functions. Therefore, we have employed these obtained results to find the first-order correction to the eigenvalues and eigenvectors. It is worth noting that we addressed the system in the NC space as an unperturbed system instead of considering the fundamental system in a commutative space 700 vol. 61 no. 6/2021 Dirac oscillator in dynamical noncommutative space and noncommutativity as a perturbation. The first-order correction for the ground state of the DO due to the noncommutativity of space is zero for the non-DNC case while it has a nonvanishing value in the DNC case. Knowing that, the result reduces to that of a usual DO in the limits of τ→0, Θ→0. As mentioned in section 2, some operators in the DNC space are non-Hermitian. This mixture of DNCS, the non-Hermiticity theory and the string theory can lead to fundamental new insights in these three fields. Distinctly, there are plenty of interesting problems arising from our investigation, such as the investigation of further possibilities of consistent deformations, and studies of additional models in terms of the DNC variables. References [1] N. Seiberg, E. Witten. String theory and noncommutative geometry. Journal of High Energy Physics (9):032, 1999. https://doi.org/10.1088/1126-6708/1999/09/032. [2] D. M. Gingrich. Noncommutative geometry inspired blackholes in higher dimensions at the LHC. Journal of High Energy Physics 2010:22, 2010. https://doi.org/10.1007/JHEP05(2010)022. [3] P. Nicolini. Noncommutative black holes, the final appeal to quantum gravity: a review. International Journal of Modern Physics A 24(7):1229–1308, 2009. https://doi.org/10.1142/S0217751X09043353. [4] J. M. Gracia-Bondia. New Paths Towards Quantum Gravity, chap. Notes on quantum gravity and noncommutative geometry, pp. 3–58. Springer, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-11897-5_1. [5] M. Chaichian, M. M. Sheikh-Jabbari, A. Tureanu. Hydrogen atom spectrum and the lamb shift in noncommutative QED. Physical Review Letters 86(13):2716–2719, 2001. https://doi.org/10.1103/PhysRevLett.86.2716. [6] I. Haouam. Analytical solution of (2+1) dimensional Dirac equation in time-dependent noncommutative phase-space. Acta Polytechnica 60(2):111–121, 2020. https://doi.org/10.14311/AP.2020.60.0111. [7] I. Haouam. On the noncommutative geometry in quantum mechanics. Journal of Physical Studies 24(2):1–10, 2002. https://doi.org/10.30970/jps.24.2002. [8] M. R. Douglas, N. A. Nekrasov. Noncommutative field theory. Reviews of Modern Physics 73(4):977–1029, 2001. https://doi.org/10.1103/RevModPhys.73.977. [9] I. Haouam. On the Fisk-Tait equation for spin-3/2 fermions interacting with an external magnetic field in noncommutative space-time. Journal of Physical Studies 24(1):1801, 2020. https://doi.org/10.30970/jps.24.1801. [10] H. S. Snyder. Quantized space-time. Physical Review 71(1):38–41, 1947. https://doi.org/10.1103/PhysRev.71.38. [11] H. S. Snyder. The electromagnetic field in quantized space-time. Physical Review 72(1):68–71, 1947. https://doi.org/10.1103/PhysRev.72.68. [12] N. Sasakura. Space-time uncertainty relation and Lorentz invariance. Journal of High Energy Physics (05):015, 2000. https://doi.org/10.1088/1126-6708/2000/05/015. [13] J. Lukierski, H. Ruegg, A. Nowicki, V. N. Tolstoi. q-deformation of Poincaré algebra. Physics Letters B 264(3-4):331–338, 1991. https://doi.org/10.1016/0370-2693(91)90358-W. [14] A. P. Balachandran, S. Vaidya. Lectures on fuzzy and fuzzy susy physics, 2007. World Scientific. IISc/CHEP/11/05. [15] S. A. Alavi, N. Rezaei. Dirac equation, hydrogen atom spectrum and the Lamb shift in dynamical non-commutative spaces. Pramana 88:77, 2017. https://doi.org/10.1007/s12043-017-1381-4. [16] A. Fring, L. Gouba, F. G. Scholtz. Strings from position-dependent noncommutativity. Journal of Physics A: Mathematical and Theoretical 43:345401, 2010. https://doi.org/10.1088/1751-8113/43/34/345401. [17] M. Gomes, V. G. Kupriyanov. Position-dependent noncommutativity in quantum mechanics. Physical Review D 79:125011, 2009. https://doi.org/10.1103/PhysRevD.79.125011. [18] D. Itô, K. Mori, E. Carriere. An example of dynamical systems with linear trajectory. Nuovo Cimento A (1965-1970) 51:1119–1121, 1967. https://doi.org/10.1007/BF02721775. [19] M. Moshinsky, A. Szczepaniak. The Dirac oscillator. Journal of Physics A: Mathematical and General 22(17):L817, 1989. https://doi.org/10.1088/0305-4470/22/17/002. [20] R. P. Martínez-y-Romero, A. L. Salas-Brito. Conformal invariance in a Dirac oscillator. Journal of Mathematical Physics 33(5):1831, 1992. https://doi.org/10.1063/1.529660. [21] J. Benitez, P. R. Martnez y Romero, H. N. Núnez-Yépez, A. L. Salas-Brito. Solution and hidden supersymmetry of a Dirac oscillator. Physical Review Letters 64:1643, 1990. https://doi.org/10.1103/PhysRevLett.64.1643. [22] C. Quesne, M. Moshinsky. Symmetry Lie algebra of the Dirac oscillator. Journal of Physics A: Mathematical and General 23(12):2263, 1990. https://doi.org/10.1088/0305-4470/23/12/011. [23] O. L. de Lange. Shift operators for a Dirac oscillator. Journal of Mathematical Physics 32(5):1296, 1991. https://doi.org/10.1063/1.529328. 701 https://doi.org/10.1088/1126-6708/1999/09/032 https://doi.org/10.1007/JHEP05(2010)022 https://doi.org/10.1142/S0217751X09043353 https://doi.org/10.1007/978-3-642-11897-5_1 https://doi.org/10.1103/PhysRevLett.86.2716 https://doi.org/10.14311/AP.2020.60.0111 https://doi.org/10.30970/jps.24.2002 https://doi.org/10.1103/RevModPhys.73.977 https://doi.org/10.30970/jps.24.1801 https://doi.org/10.1103/PhysRev.71.38 https://doi.org/10.1103/PhysRev.72.68 https://doi.org/10.1088/1126-6708/2000/05/015 https://doi.org/10.1016/0370-2693(91)90358-W https://doi.org/10.1007/s12043-017-1381-4 https://doi.org/10.1088/1751-8113/43/34/345401 https://doi.org/10.1103/PhysRevD.79.125011 https://doi.org/10.1007/BF02721775 https://doi.org/10.1088/0305-4470/22/17/002 https://doi.org/10.1063/1.529660 https://doi.org/10.1103/PhysRevLett.64.1643 https://doi.org/10.1088/0305-4470/23/12/011 https://doi.org/10.1063/1.529328 Ilyas Haouam Acta Polytechnica [24] A. Bermudez, M. Martin-Delgado, E. Solano. Exact mapping of the 2+1 Dirac oscillator onto the Jaynes-Cummings model: Ion-trap experimental proposal. Physical Review A 76:041801(R), 2007. https://doi.org/10.1103/PhysRevA.76.041801. [25] A. Bermudez, M. A. Martin-Delgado, A. Luis. Chirality quantum phase transition in the Dirac oscillator. Physical Review A 77(6):063815, 2008. https://doi.org/10.1103/PhysRevA.77.063815. [26] J. A. Franco-Villafañe, E. Sadurní, S. Barkhofen, et al. First experimental realization of the Dirac oscillator. Physical Review Letters 111:170405, 2013. https://doi.org/10.1103/PhysRevLett.111.170405. [27] C. Quimbay, P. Strange. arXiv:1311.2021. [28] C. Quimbay, P. Strange. arXiv:1312.5251. [29] A. Boumali. Thermodynamic properties of the graphene in a magnetic field via the two-dimensional Dirac oscillator. Physica Scripta 90(4):045702, 2015. https://doi.org/10.1088/0031-8949/90/4/045702. [30] B. Bagchi, A. Fring. Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems. Physics Letters A 373(47):4307–4310, 2009. https://doi.org/10.1016/j.physleta.2009.09.054. [31] I. Haouam. The non-relativistic limit of the DKP equation in non-commutative phase-space. Symmetry 11(2):223, 2019. https://doi.org/10.3390/sym11020223. [32] B. Mandal, S. Verma. Dirac oscillator in an external magnetic field. Physics Letters A 374(8):1021–1023, 2010. https://doi.org/10.1016/j.physleta.2009.12.048. [33] A. Boumali, H. Hassanabadi. The thermal properties of a two-dimensional Dirac oscillator under an external magnetic field. The European Physical Journal Plus 128:124, 2013. https://doi.org/10.1140/epjp/i2013-13124-y. [34] A. Bermudez, M. A. Martin-Delgado, E. Solano. Mesoscopic superposition states in relativistic landau levels. Physical Review Letters 99:123602, 2007. https://doi.org/10.1103/PhysRevLett.99.123602. [35] J. J. Sakurai. Modern Quantum Mechanics. (revised edition). Addison-Wesley, 1994. [36] S. Cai, T. Jing, G. Guo, et al. Dirac oscillator in noncommutative phase space. International Journal of Theoretical Physics 49:1699–1705, 2010. https://doi.org/10.1007/s10773-010-0349-7. [37] B. P. Mandal, S. K. Rai. Noncommutative Dirac oscillator in an external magnetic field. Physics Letters A 376(36):2467–2470, 2012. https://doi.org/10.1016/j.physleta.2012.07.001. [38] M. Hosseinpour, H. Hassanabadi, M. de Montigny. The Dirac oscillator in a spinning cosmic string spacetime. The European Physical Journal C 79:311, 2019. https://doi.org/10.1140/epjc/s10052-019-6830-4. [39] M. de Montigny, S. Zare, H. Hassanabadi. Fermi field and Dirac oscillator in a Som-Raychaudhuri space-time. General Relativity and Gravitation volume 50:47, 2018. https://doi.org/10.1007/s10714-018-2370-8. [40] K. Bakke, H. Mota. Dirac oscillator in the cosmic string spacetime in the context of gravity’s rainbow. The European Physical Journal Plus 133:409, 2018. https://doi.org/10.1140/epjp/i2018-12268-6. [41] H. Chen, Z.-W. Long, Y. Yang, C.-Y. Long. Study of the Dirac oscillator in the presence of vector and scalar potentials in the cosmic string space-time. Modern Physics Letters A 35(21):2050179, 2020. https://doi.org/10.1142/S0217732320501795. [42] F. Ahmed. Interaction of the Dirac oscillator with the Aharonov-Bohm potential in (1+2)-dimensional Gürses space-time backgrounds. Annals of Physics 415:168113, 2020. https://doi.org/10.1016/j.aop.2020.168113. [43] D. F. Lima, F. M. Andrade, L. B. Castro, et al. On the 2D Dirac oscillator in the presence of vector and scalar potentials in the cosmic string spacetime in the context of spin and pseudospin symmetries. The European Physical Journal C 79:596, 2019. https://doi.org/10.1140/epjc/s10052-019-7115-7. [44] O. Bertolami, J. G. Rosa, C. M. L. de Aragão, et al. Noncommutative gravitational quantum well. Physical Review D 72:025010, 2005. https://doi.org/10.1103/PhysRevD.72.025010. [45] S. A. Alavi, M. A. Nasab. Gravitational radiation in dynamical noncommutative spaces. General Relativity and Gravitation 49:5, 2017. https://doi.org/10.1007/s10714-016-2167-6. 702 https://doi.org/10.1103/PhysRevA.76.041801 https://doi.org/10.1103/PhysRevA.77.063815 https://doi.org/10.1103/PhysRevLett.111.170405 http://arxiv.org/abs/1311.2021 http://arxiv.org/abs/1312.5251 https://doi.org/10.1088/0031-8949/90/4/045702 https://doi.org/10.1016/j.physleta.2009.09.054 https://doi.org/10.3390/sym11020223 https://doi.org/10.1016/j.physleta.2009.12.048 https://doi.org/10.1140/epjp/i2013-13124-y https://doi.org/10.1103/PhysRevLett.99.123602 https://doi.org/10.1007/s10773-010-0349-7 https://doi.org/10.1016/j.physleta.2012.07.001 https://doi.org/10.1140/epjc/s10052-019-6830-4 https://doi.org/10.1007/s10714-018-2370-8 https://doi.org/10.1140/epjp/i2018-12268-6 https://doi.org/10.1142/S0217732320501795 https://doi.org/10.1016/j.aop.2020.168113 https://doi.org/10.1140/epjc/s10052-019-7115-7 https://doi.org/10.1103/PhysRevD.72.025010 https://doi.org/10.1007/s10714-016-2167-6 Acta Polytechnica 61(6):689–702, 2021 1 Introduction 2 Review of dynamical noncommutativity 3 Two-dimensional Dirac oscillator in dynamical noncommutative space 3.1 Extension to dynamical noncommutative space 3.2 Unperturbed eigenvalues and eigenvectors 3.3 Perturbed system 4 Conclusion References