Photonic realization of the kappa-deformed Dirac equation
Parisa Majari, Emerson Sadurni, Mohammad Reza Setare, John Alexander Franco-Villafane, Thomas H. Seligman
PPhotonic realization of the κ -deformed Dirac equation P. Majari ∗ Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, Cuernavaca 62210, M´exico
E. Sadurn´ı † Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apartado Postal J-48, 72570 Puebla, M´exico
M. R. Setare ‡ Department of Science, Campus of Bijar, University of Kurdistan, Bijar, Iran
J. A. Franco-Villafa˜ne § CONACYT-Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı, 78290 San Luis Potos´ı, SLP., M´exico
T. H. Seligman ¶ Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, Cuernavaca 62210, M´exico,Centro Internacional de Ciencias, Cuernavaca-62210, M´exico (Dated: December 14, 2020)We show an implementation of a κ -deformed Dirac equation in tight-binding arrays of photonicwaveguides. This is done with a special configuration of couplings extending to second nearestneighbors. Geometric manipulations can control these evanescent couplings. A careful study ofwave packet propagation is presented, including the effects of deformation parameters on Zitter-bewegung or trembling motion. In this way, we demonstrate how to recreate the effects of a flatnoncommutative spacetime –i.e., κ Minkowski spacetime– in simple experimental setups. We touchupon elastic realizations in the section of Conclusions.
I. INTRODUCTION
The field of quantum simulations has been of pivotal importance in the study of physical systems that are experi-mentally out of reach or beyond natural observation. Notable examples in cold matter can be found in [1, 2] and [3–5]using trapped ions. In this paper, we address the possibility of emulating the effects of Lorentz algebraic deformationson the motion of relativistic electrons [6, 7]. In theory, such deformations are associated with a noncommutative ge-ometry of spacetime and a generalized uncertainty principle [8–11]. On physical grounds, the hypothetical correctionsstem from a fundamental length scale, which is of a quantum mechanical nature.Let us describe the general features of our emulations to put our work into context. Within the class of complexquantum simulations, there are simpler systems whose properties can be studied with single body dynamics. Thesesystems, in turn, contain all mesoscopic realizations of quantum mechanical wave equations in microwaves [12–17] andphotonics [18–25]. For optical fiber couplings see [26–29]. Such table-top experiments provide flexible configurationsand easy tuning of parameters, including recent constructions of effective relativistic systems. Examples of condensedmatter realizations and their emulations also abound [30–36]. The success of dynamical analogies between quantummechanical equations and electromagnetic waves in various important subjects –quantum graphs, chaotic scatteringand billiards, tight-binding arrays and crystals, including graphene and phosphorene– has led us to consider newexperimental challenges towards high-energy extensions of the Dirac equation, ruling the motion of ultra relativisticfermions. Indeed, Dirac Hamiltonians in 1+1 and 2+1 dimensions have been produced effectively in a variety oftight-binding systems supported by honeycomb lattices, and in more general settings, by any bipartite (spinorial)lattice entailing conical energy relations, also dubbed Dirac points.These precedents become important when we look at the so-called κ -deformed algebras at hand [6, 7, 37, 38] andeven in the field of q deformations [39, 40]. Their effects on quantum field theory have been carefully studied [41–48] ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: jofravil@ifisica.uaslp.mx ¶ Electronic address: [email protected] a r X i v : . [ h e p - l a t ] D ec including particle statistics [49]. In this regard, it has been proved that quadratic corrections in the momentum of aparticle will modify the usual Dirac Hamiltonian defined in empty space, even when it represents a physical situationfree of interactions [50–52]. As previously mentioned, this is known to take place in the presence of a postulatedminimal length, presumably in the order of G (cid:126) /c ∼ . × − cm, i.e., the Planck scale. The correspondingcorrections would imply a modified energy-momentum relation and extraordinary dispersion relations of matter wavespropagating under the effects of new physics. The idea is to emulate them.In connection with fundamental aspects of physics, some words are in order. Presumably, the existence of afundamental length is the outcome of a foamy space consistent with theories that deal with the very nature ofspacetime or its emergent properties starting from string theory. However, it is still unknown whether such structuresactually underlie our physical world, and it is even more uncertain whether we shall be able to observe the actualconsequences of their existence in high energy experiments or cosmological observations. Important efforts in thephenomenology of deformations and minimal lengths can be found in [53–55], including a plausible stringy origin [56]and Planck scale phenomenology in [57, 58].For this reason, we recreate here the conditions in which the aforementioned effects can be observed. Our aim isto engineer the corresponding dispersion relations with a tight-binding scheme consistent with previous successfulemulations of relativistic wave equations. In what touches wave propagation, our emulations shall be able to producetwo important outcomes effectively: i) A modified energy spectrum in accordance with predictions from potentiallynew physics and ii) a corrected evolution of wave packets with modified group velocities in empty space. The firstresult can be easily achieved by introducing second-neighbor interactions –or hopping amplitudes– in a crystal whereDirac points are initially ensured; this shall be done by simple geometric manipulations of resonators in variousrealizations, such as optical fibers and ceramic disks. The second result will be tested by a close inspection of aphenomenon known as Zitterbewegung [59], already produced artificially in Dirac lattices [1, 2, 4, 60] and calculatedin previous treatments [61, 62], where some of them cover carefully the full energy band [63]. This reaches wellbeyond the conical region of the emulated spectrum. With our treatment, we shall be able to compare the oscillationfrequency of a wavepacket’s width – as well as its decay in amplitude – with the expected theoretical predictions,finding thereby new significant effects coming from a hypothetical minimal length, together with corrected trajectoriesof electrons obtained as average positions in the κ -deformed Heisenberg picture.Structure of the paper: In section II we revisit the emergence of the κ deformed Dirac equation and obtainthereby the first corrections in the Dirac Hamiltonian due to a fundamental length a . In section III we presenta careful construction of arrays made of coupled optical fibers disposed in a strip resembling a triangular lattice,fulfilling thus a Dirac-like dynamical equation with conical points. In section IV we study the effects of thedeformation on the trembling motion of wave packets, including the corrections in the width coming from a . Adetailed full-band computation of Zitterbewegung for tight-binding arrays with second neighbors is offered in IV A.We conclude in V. II. THE κ -DEFORMED DIRAC EQUATION Noncommutative (NC) geometry was envisaged by Gelfand when he showed that a space is determined by thealgebra of the functions acting on it. The notion of space is then tied to the nature of its algebra; therefore anoncommutative spacetime (NCST) follows from a noncommutative algebra. This has been extensively studied inthe framework of Hopf algebras and the so-called quantum groups [38, 41, 42]. In our case, the coordinate functions x µ , satisfy the commutation relations of the form [ x µ , x ν ] = i (Θ µν + Θ λµν x λ + ... ) = i Θ µν ( x ), a relation that has beeninstrumental in the construction of a deformed quantum field theory [43]. Among all the possible ways of representingNCST, we are particularly interested in the κ -Minkowski spacetime employed in [43–46]. In other words, the κ -Minkowski spacetime is a Lie algebraic deformation of the usual Minkowski (flat) spacetime where the deformationparameter can be related to a length scale in which quantum gravity might take place. The corresponding κ -Poincar´e-Hopf algebraic relations can be written in terms of a deformation parameter κ = 1 / | a | , as [ˆ x µ , ˆ x ν ] = i ( a µ ˆ x ν − a ν ˆ x µ ).Moreover, the existence of such a fundamental scale can be encoded in Dirac operators acting on spinor fields. Thedeformed Dirac equation obtained in the framework of the κ -Poincar´e-Hopf algebra and its equivalent in periodicarrays of coupled waveguides will be studied in the following.Let us start with the algebra related to NC spaces satisfying the following relations [50, 51]:[ M i , ˆ x ] = − ˆ x i + iaM i , (1)[ M i , ˆ x j ] = − δ ij ˆ x + iaM ij , (2)where M µν contains the rotation and boost generators of the κ -Poincar´e algebra and ˆ x µ denotes the NC coordinates.This algebraic structure also entails the following relations:[ˆ x µ , ˆ x ν ] = iC µνλ ˆ x λ = i ( a µ ˆ x ν − a ν ˆ x µ ) , (3)where the structure constants are written in terms of a Minkowski vector a µ and the flat metric η µν : C µνλ = a µ η νλ − a ν η µλ . In some frame of reference a i = 0, a = a and ˆ x i = x i φ where φ is so far free. In the κ -Poincar´ealgebra, the modified derivative operators D µ , the so-called Dirac derivatives, are given by [42, 47]: D = ∂ (cid:18) sinh( A ) A (cid:19) + ia ∇ e − A φ , (4)and D i = ∂ i (cid:18) e − A φ (cid:19) , (5)with A = − ia∂ and a = κ − . This leads to the following relations [49]:[ M µν , D λ ] = η νλ D µ − η µλ D ν , (6)[ D µ , D ν ] = 0 , (7)where the metric’s signature convention is fixed as η µν = diag( − , , , D µ as [ iγ µ D µ + m ] ψ = 0 . (8)With the special choice φ = e − A , we eliminate the deformation in the spatial derivatives, leaving us only with anew (corrected) time component D . We note here that this choice, together with the definition of A , turns φ into a nonlocal operator in time. By substituting (4), (5) in the above equation and after a few straightforwardmanipulations, the following κ -Dirac equation is deduced: (cid:20) iγ (cid:18) ia sinh( A ) + ia ∇ (cid:19) + iγ i ∂ i + m (cid:21) ψ = 0 (9)and this equation is, in fact, nonlocal due to the obvious relation φψ ( t ) = ψ ( t + ia ) for any wave function ψ . Onphysical grounds, we may take only the first corrections in a with the aim of describing a slightly perturbed Diracoperator (note however, that this concession is not made on mathematical grounds, because infinite order differentialequations cannot be truncated without dire consequences on the oscillatory behavior of their solutions) leading to (cid:20) iγ (cid:18) ∂ + ia ∇ (cid:19) + iγ i ∂ i + m (cid:21) ψ = 0 . (10)It follows from the above equation that the corresponding Hamiltonian is H = cα · p + a ∇ + mβ. (11)Here, the explicit representation of Dirac matrices in 1 + 1 dimensions given by α = σ and β = − σ is possible andit is consistent with our choice of metric signature. It is also obvious that the undeformed Dirac equation is obtainedin the limit a →
0. However, we underscore the fact that, experimentally, the deformation parameter a is greatlylimited by an upper bound of the order of a < − m [31].Several effects can be investigated using this new Hamiltonian. It might well be that the presence of a modifies thespectrum of a relativistic particle, but it is not easy to gain access to such energy scales in accelerators. There areother effects that could be amplified in other settings, e.g. wave packet evolution. We shall explore this possibility infurther sections.Figure 1: Schematic view of a binary array made of two types of waveguides, A and B, arranged in a triangularlattice along a strip. III. AN ARRAY OF PHOTONIC WAVEGUIDES
The tight-binding model of the κ -deformed Weyl equation can be implemented on a macroscopic experiment usingan array of microwave resonators, as shown in figure 2. The resonators can be built as cylinders of the same size,but with two different dielectric constants, for example, Exxilia Temex Ceramics E2000 and E3000 with (cid:15) = 36 and34, respectively. An induced mass parameter α around 0 .
28 GHz is expected for cylinders of 8 mm diameter, andtheir length much larger than their diameter. The coupling parameter ξ between the cylinders can be set between0 . ω (cid:48) and ω . The corresponding Zitterbewegungoscillation lengths that we named here λ (cid:48) and λ respectively will be of order λ (cid:48) ∼ cm and λ ∼ m. Those scalesmake the Zitterbewegung effect observable on a macroscopic scale, thanks to our tight-binding representation of thecorresponding wave operator. In order to excite the resonators, we propose to use an array of antennas. Each antennawill be placed near the end of each fiber or cylinder. The antennas should all be parallel and oriented in such a waythat they are capable of exciting an electric field perpendicular to the optical axis and to the horizontal axis shown infig. 2. The array of antennas has to be excited by the same microwave frequency but, at the same time, allowing thecontrol of the input power independently in each antenna. One possible and inexpensive way to feed the antennas isto use a direct digital synthesizer (DDS) that provides independent frequency, phase, and amplitude control on eachchannel. The propagation of an optical field of disordered waveguide arrays by using tight-binding approximation isgiven by: i dE n dz + ( − n αE n + C n − E n − + C n E n +2 + C (cid:48) n E n +1 + C (cid:48) n − E n − = 0 , (12)where E n ≡ E ( n, z ) is the electric field amplitude at the nth waveguide. Next, we let upper waveguide array haveodd numbers. Then i dE n dz + αE n + C n − E n − + C n E n +2 + C (cid:48) n E n +1 + C (cid:48) n − E n − = 0 , (13)and let the lower array label by even ones: i dE n − dz − αE n − + C n − E n − + C n − E n +1 + C (cid:48) n − E n + C (cid:48) n − E n − = 0 . (14)Now by setting E n = ( − n ψ ( n, z ) ≡ ( − n ψ ( n ) and E n − = − i ( − n ψ ( n, z ) ≡ − i ( − n ψ ( n ), Eqs (13) and(14) can be written as: i dψ ( n ) dz + αψ ( n ) − C n − ψ ( n − − C n ψ ( n + 1) + iC (cid:48) n ψ ( n + 1) − iC (cid:48) n − ψ ( n ) = 0 , (15)and i dψ ( n ) dz − αψ ( n ) − C n − ψ ( n − − C n − ψ ( n + 1) + iC (cid:48) n − ψ ( n ) − iC (cid:48) n − ψ ( n −
1) = 0 , (16)It is straightforward to show that by considering C n ± i = η and C (cid:48) n − i = ξ with i = 0 , , ... these equations reduce to i ddz (cid:20) ψ ( n ) ψ ( n ) (cid:21) = (cid:20) − αψ ( n ) + ηψ ( n −
1) + ηψ ( n + 1) − iξψ ( n + 1) + iξψ ( n )+ αψ ( n ) + ηψ ( n −
1) + ηψ ( n + 1) − iξψ ( n ) + iξψ ( n − (cid:21) = H (cid:20) ψ ( n ) ψ ( n ) (cid:21) (17)with a Hamiltonian operator defined as H = (cid:20) − α + ηT − + ηT − iξT + iξ − iξ + iξT − α + ηT − + ηT (cid:21) , (18)where T is the translation operator in one unit of n . In order to reduce the system of equations to a 2 × ψ ( n, z ) , ψ ( n, z )) † = ( A ( z ) e ikn , B ( z ) ikn ) † which satisfies Bloch’s theorem T e ikn = e ik ( n +1) . Now, the Hamiltonian18 is reduced to H = (cid:20) − α + 2 η cos( k ) − iξe ik + iξ − iξ + iξe − ik α + 2 η cos( k ) (cid:21) (cid:39) (cid:20) − α + 2 η − ηk ξkξk α + 2 η − ηk (cid:21) (19)Therefore, the energy eigenvalues are given by the formula E = 2 η cos( k ) + s (cid:115) ξ sin (cid:18) k (cid:19) + α (20)where s = ±
1. This holds even beyond the Dirac (conical) points. The eigenfunctions are two-component spinors ofthe form (cid:18) u u (cid:19) = 1 (cid:112) E − η cos( k )) (cid:18) (cid:112) E − α − η cos( k ) e − ik/ (cid:112) E + α − η cos( k ) (cid:19) . (21)By setting k → p x , i.e. in units where (cid:126) = 1, we obtain: H = − ηp x I + ξp x σ x − ασ z + 2 ηI = H + V (22)where V = 2 η is a constant potential, therefore irrelevant in the dynamics. Finally, we note that after the formalchange a → η , m → α and 1 → ξ , the expression for the Hamiltonian in (11) can be mapped to H previously writtenin (22). IV. EVOLUTION OF POSITION IN κ -DEFORMED DIRAC THEORY Now we would like to investigate one of the special features of the Dirac equation, the trembling motion known asZitterbewegung. To clarify this effect in the κ -Dirac equation, we must calculate the time evolution of the positionoperator under the strict conditions η (cid:54) = 0, α (cid:54) = 0. In the absence of rest mass (Weyl equation) we know that there isno visible effect, for the evolution of x would be trivial. The calculation of x ( t ) for the more general case α (cid:54) = 0 is,however, straightforward and we shall proceed in this direction. In the Heisenberg picture, we have [31]: x ( t ) = x (0) − ηp x t + ξ p x ( H (cid:48) ) − t + iξ H (cid:48)− (cid:2) σ x − ξp x ( H (cid:48) ) − (cid:3) ( e − iH (cid:48) t − , (23) (a) (b) Figure 2: Time development of (a) (cid:104) x (cid:105) ψ and (b) variance with ξ = α = 4 and a few values of η .where H (cid:48) = − ασ z + ξσ x p x . To see the dependence on the deformation parameter more clearly, we focus now on thewidth of wave packets (cid:104) (∆ x ) (cid:105) = (cid:104) x (cid:105) − (cid:104) x (cid:105) . Due to Ehrenfest’s theorem, the time average (cid:104) x (cid:105) suffers the samemodifications as the classical trajectory of a particle governed by a κ -deformed energy momentum relation. On theother hand, the second term (cid:104) x (cid:105) ψ provides an important modification to the wave-like behavior of the particle. Itsexplicit form is given by (cid:104) x (cid:105) ψ = F t − (cid:114) πtτ (cid:18) ηξ απ sin( ω (cid:48) t ) ω (cid:48) (cid:19) Γ+ 1 √ t (cid:34)(cid:114) πτ (cid:18) ξ α sin( ω (cid:48) t ) ω (cid:48) (cid:19) Λ + (cid:114) παξ (cid:18) ξ α sin( ωt ) ω (cid:19) Λ (cid:35) (24)as shall be derived later on.The time development of (cid:104) x (cid:105) ψ , with x (0) = 1 . F . We also see in the results shown in fig. (2) (a) that the damping of the tremblingpart disappears with the envelope 1 / √ t , but some of its oscillations are ’prolonged’ by an increase of η , from 0 to0.3. Indeed, the envelope √ t is affected by η in the second term of (29), while the rest of the contributions remainunperturbed due to a phase cancellation of the η component of the energy. This direct proportionality in Γ amplifiesthe phenomenon in time, but in case of realistic values of η within experimental bounds, it would be too challengingto detect them in experiments with electrons. In optical realizations, such a parameter is at our disposal, withrecommended values shown in the inset of fig. 3 and 2 (b), for a better appreciation. A. Zitterbewegung in the photonic lattice: computations
We derive the time evolution of position operator for photonic waveguide arrays. It is important to do so withoutapproximations in the tight-binding dispersion relations for an honest comparison with deformed theories. Since thestationary phase approximation will be required in the derivation of averages, it is important to analyze the energylandscape in (Bloch) momentum space or Brillouin zone in search for vanishing group velocities. A comparison ofenergy curves for some values of η is given in fig. 3.The time average of the position using the Dirac theory (23) can be written as (cid:104) x ZB (cid:105) ψ = (cid:28)(cid:26) iξ H (cid:48)− [ σ x − ξp x ( H (cid:48) ) − ]( e − iH (cid:48) t − (cid:27)(cid:29) ψ , (25)but using the wave packet decomposition with Fourier coefficients ψ k,s , one also hasFigure 3: A comparison of dispersion relations for waves governed by our tight-binding array (solid lines) and the κ -deformed Dirac equation (dashed lines). Mass α = 0 .
5, coupling ξ = 1. There is good agreement near the Diracpoint at k = 0. Deformation parameters: η = 0 (black) 0.2 (blue) 0.4 (red). The tight-binding relation displays newpoints of vanishing group velocity in the upper band, marked by light blue vertical lines. The phase factors,however, cancel out in the computation of averages. (cid:104) x ZB (cid:105) ψ = α ξ (cid:88) s (cid:90) π dk k sin(2 E (cid:48) t )2 E (cid:48) | ψ k,s | + ξ (cid:88) s,s (cid:48) (cid:90) π dk (cid:20) iξαE (cid:48) sin ( E (cid:48) t )( u ∗ u − u ∗ u ) − (26) − ξα E (cid:48) sin(2 E (cid:48) t )( u ∗ u + u ∗ u ) (cid:21) ( ψ k,s ψ k,s (cid:48) ∗ ) (27)where E (cid:48) = s (cid:113) ξ sin ( k ) + α . Now by using the stationary phase approximation for k = 0 , π we obtain (cid:104) x ZB (cid:105) ψ (cid:39) (cid:32) πt | ξ ω (cid:48) | (cid:33) e − itω (cid:48) F ( α, ξ ) + (cid:32) πt | ξ ω | (cid:33) e − itω G ( α, ξ ) + (cid:32) πt | ξ ω (cid:48) | (cid:33) e − itω (cid:48) L ( α, ξ ) + c . c . (28)where ω (cid:48) = s (cid:112) ξ + α and ω = sα . It is clear that the trembling motion vanishes with an envelope curve 1 / √ t [63].We note that the amplitudes and frequencies of oscillation are independent of the deformation parameter. This resultis valid only for the Dirac approximation and, in general, the corrections due to η have an impact on (cid:104) x ZB (cid:105) for thetight-binding system with second neighbors. On the other hand, the width of the wave packet needs the average ofthe squared position operator (cid:104) x (cid:105) ψ which is given by (cid:104) x (cid:105) ψ = F t − (cid:114) πtτ (cid:18) ηξ απ sin( ω (cid:48) t ) ω (cid:48) (cid:19) Γ+ 1 √ t (cid:34)(cid:114) πτ (cid:18) ξ α sin( ω (cid:48) t ) ω (cid:48) (cid:19) Λ + (cid:114) παξ (cid:18) ξ α sin( ωt ) ω (cid:19) Λ (cid:35) (29)where the following shorthands are used:We introduce the variable τ = ξ √ α + ξ π − ξ π ( α + ξ π ) / . The amplitude proportional to t is given by F = (cid:88) s (cid:90) π dk (cid:18) k α ξ E (cid:48) + k ξ E (cid:48) + k η (cid:19) | ψ k,s | + (cid:88) s (cid:90) π dk (cid:18) k ηαξ E (cid:48) (cid:19) ( u ∗ u − u ∗ u ) | ψ k,s | − (30) − (cid:88) s,s (cid:48) (cid:90) π dk (cid:18) k ηξ E (cid:48) (cid:19) ( u ∗ u + u ∗ u ) ψ k,s ψ k,s (cid:48) , (31)while the coefficient Γ of the √ t envelope isΓ = e − π/ (cid:104) sin( ω (cid:48) t + πτ αω (cid:48) cos( ω (cid:48) t + πτ (cid:105) . (32)Finally, the coefficients Λ , in the brackets of (29) (ruled by the 1 / √ t envelope) are as follows:Λ = e − π/ (cid:20) sin( ω (cid:48) t ) sin( ω (cid:48) t + πτ ω (cid:48) t + πτ ) cos( ω (cid:48) t )( ω (cid:48) ) ( α + ξ π ) (cid:21) (33)and Λ = sin( ωt ) sin (cid:18) ωt + πξ α (cid:19) + α cos( ωt + πξ α ) cos( ωt ) ω . (34)These results support our previous discussion on wave packet expansion. V. CONCLUSIONS
We have succeeded in our emulation and theoretical study of deformed Dirac equations by means of photonicwaveguide arrays. Once more, our tight-binding approach to coupling engineering has led to satisfactory resultsregarding spectrum and wavefunction simulation. We have shown how to surround subtle obstacles regarding thecorrespondence of photonic Zitterbewegung and relativistic trembling motion, as they differ by small but visibleamounts when full frequency-band computations are employed. We were able to confirm the η corrections, due toalgebraic deformations, in the evolution of localized wave packets. Interestingly, at the end of the day, it was the √ t envelope what bore the κ deformation, leading to a persistent oscillatory effect in the width and prolonged by anincrease of η as a direct proportionality. All other trembling components of the width remained untouched, includingthe rarely seen –i.e. short-lived but always present– envelope 1 / √ t . We also found a strong dependence on η of theballistic part t , controlling the overall speed of expansion, but such an effect already appears in the evolution ofscalar particles, as it has little to do with spin.From a technological point of view, we have shown that photonic waveguides may enable experimentalists to studythe effects of noncommutative spacetime in the lab. We should also comment on a renewed interest in elastic systemsdue to the flexibility of their experimental setups. The construction of elastic waveguides using Aluminum plates makesacoustic transport an attractive area in which emulations may play an interesting role, given the rich phenomenologyof vibrational transport using various types of polarizations [64–67]. As we mention above, microwave experience mayalso be consider for the realization of κ -deformed Dirac equation. Acknowledgments
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