Acta Polytechnica DOI:10.14311/AP.2020.60.0111 Acta Polytechnica 60(2):111–121, 2020 © Czech Technical University in Prague, 2020 available online at https://ojs.cvut.cz/ojs/index.php/ap ANALYTICAL SOLUTION OF (2+1) DIMENSIONAL DIRAC EQUATION IN TIME-DEPENDENT NONCOMMUTATIVE PHASE-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 article, we studied the system of a (2+1) dimensional Dirac equation in a time-dependent noncommutative phase-space. More specifically, we investigated the analytical solution of the corresponding system by the Lewis-Riesenfeld invariant method based on the construction of the Lewis-Riesenfeld invariant. Knowing that we obtained the time-dependent Dirac Hamiltonian of the problem in question from a time-dependent Bopp-Shift translation, then used it to set the Lewis- Riesenfeld invariant operators. Thereafter, the obtained results were used to express the eigenfunctions that lead to determining the general solution of the system. Keywords: Lewis-Riesenfeld invariant method, time-dependent Bopp-Shift translation, Bopp’s Shift; time-dependent Dirac equation, time-dependent noncommutative phase-space. 1. Introduction It is known that Heisenberg suggested the option of noncommutative (NC) space-time in 1930, and in 1947, Snyder presented it [1, 2] to the necessity of regularizing the divergence of the quantum field theory. Then, in recent years, noncommutative geometry (NCG) became very interesting for studying several physical problems, and it became clear that there is a strong connection between the NCG and string theories. Studies of this geometric type and its involvement have been incorporated with important physical concepts and tools, and have been useful for highlighting in various fields of physics, particularly in matrix theory (matrix model BFSS (1997)) [3]. The NCG was involved also in the description of quantum gravity theories [4], Aharonov-Bohm effect [5], Aharonov-Casher effect [6], etc [7]. Knowing that the origins of the NCG related to the investigations for topological spaces of C∗-algebras of functions. Later this type of geometry was theorized by A. Connes and others in 1985 [8–12] by studying and defining a cyclical cohomology. It has been shown that the differential calculus on manifolds had an NC equivalent. Next, the NCG found great encouragement through several mathematical results, such as the K-theory of C∗-algebras, Gelfand-Naïmark theorem on the C∗-algebras, characterizations of commutative von Neumann algebras, cyclic cohomology of the C∞(M) algebra, relations between Dirac operators and Riemannian metrics, Serre-Swan theorem, etc. The idea of the phase-space noncommutativity is largely motivated by the foundations of quantum mechanics through the canonical quantization. It is easy to apply the phase-space noncommutativity using the ordinary product with Weyl operators (Weyl-Wigner maps)[12], or by replacing the ordinary product with the Moyal-Weyl product (?-product) in the functions and actions of our systems[13, 14], also the Bopp-shift linear transformations [15, 16], and the Seiberg-Witten maps [8, 10, 17]. Studying physics within the NCG has attracted a lot of interest in recent years, because noncommutativity is necessary when considering the low-energy efficiency of D-brane with a background magnetic field, and also in a tiny scale of strings or in conditions of a very high energy, the effects of noncommutativity may appear. Besides, one of the strong motivations of the NCG, is to obtain a coherent mathematical framework in which it would be possible to describe quantum gravitation. For all these reasons and advantages, we carry out this work in the NC formalism. In addition, it is interesting to find other models in which noncommutativity emerges. Several scientific works have focused on the time-independent noncommutativity. Experimental research has considered the parameters of the noncommutativity of fixed values in the context of cosmic microwave background radiation, perhaps considered approximately fixed to the celestial sphere, for example, as proposed in ref [18]. However, differently, in our work, our obvious intention is to involve the time-dependency in the NC parameters because of the possibility that NC parameters may show time-dependency. For instance, physical measurements must take into account the effect of the Earth’s rotation around its axis, which causes a time-dependency in the NC parameters. However, the motivations for choosing to study the (2+1) dimensional Dirac equation are due to several important works in this context, such as the investigations of Landau levels [19], the oscillation of magnetization 111 https://doi.org/10.14311/AP.2020.60.0111 https://ojs.cvut.cz/ojs/index.php/ap Ilyas Haouam Acta Polytechnica [20], Weiss oscillation [21], de Haas-van Alphen effect [22], analysis through coherent states [23], the movement of electrons transporters in graphene and other materials [24], etc. Particularly, the 2 dimensional Dirac equation in interaction with a homogeneous magnetic field has various applications in graphene, such as in refs [25, 26], and in studying quantum Hall effect and fractional Hall effects [27, 28], Berry phase [29], etc. In graphene and other materials such as in Weyl semimetals, an important phenomenon takes place if the magnetic field and the uniform electric field are introduced. Specifically, the spacing between different Landau levels decreases if the electric field strength reaches a critical value [30, 31]. In our study about a (2+1) dimensional Dirac system, the noncommutativity will be considered time- dependent through a time-dependent Darboux transformation “Bopp’s shift”. This, in turn, makes the studied system a time-dependent one, H(xnci ,p nc i ) −→H(t). Solving the system of equations in interaction with time-dependent potentials has attracted many physicists over the past years. In addition to the essential mathematical benefit, this topic is related to a lot of physical problems and applications for its applicability. For instance, in quantum transport [32–34], quantum optics [35–37], quantum information [38], the degenerate parametric amplifier [39], also spintronics [40, 41], and in the description of the two trapped cold ions dynamics in the Paul trap [42]. To study systems of time-dependent equations, there are many methods like the evolution operator, the change of representation, the unitary transformations, path integral, second quantization, Lewis-Riesenfeld (LR) invariant approach. Also, there are other used techniques, as in refs [43, 44]. The LR method [45, 46] is a technique that allows obtaining a set of solutions of time-dependent equation systems, through the eigenstates of LR invariants. These invariants are built to find the solutions of such systems of equations, where Lewis and Riesenfeld in their original paper presented a technique to obtaining a group of exact wave-functions for the time-dependent harmonic oscillator in Hilbert space. The LR approach has been applied in many applications such as in mesoscopic R(t)L(t)C(t) electric circuits where the quantum evolution is described[47]. As well as in engineering, in shortcuts and adiabaticity [48]... A large variety of scientific papers concerning time-dependent systems were interested in the time-dependent harmonic oscillator, or in time-dependent linear potentials, but in our current work, to be more specific, we report the time-dependent Background of the NC phase-space. We consider a time-dependent Bopp-shift translation to transform the system to a time-dependent NC one, then, due to the LR invariant method, we obtain the LR invariant and its eigenstates to solve our system equations. 2. time-dependent noncommutativity In theory, the NCG space may not commute anymore (i.e. AB6=BA). In a d dimensional time-dependent NC phase-space, let us consider the operators of coordinates and momentum xncj and p nc k , respectively. These NC operators satisfy the deformed commutation relations [49][ xncj ,x nc k ] = iΘjk(t)[ pncj ,p nc k ] = iηjk(t), (j,k = 1, ..,d)[ xncj ,p nc k ] = i~effδjk , (1) the effective Planck constant being ~eff = ~ ( 1 + Θη 4~2 ) , (2) where Θη4~2 � 1 is the consistency condition in the usual quantum mechanics. δjk is the identity matrix, and Θjk, ηjk are real constant antisymmetric d×d matrices. In some studies concerning the NC parameters, as in the experiment by “Nesvizhevsky et al” [50, 51], we note that Θ ≈ 10−30m2 and η ≈ 1, 76.10−61Kg2m2s−2. Other bounds exist. For example, Θ ≈ 4.10−40m2 when assuming the natural units, ~ = c = 1 [52]. As well as when taking into account that the experimental energy resolution is related to the uncertainty principle because of the finite lifetime of the neutron, this leads to obtaining η ≈ 10−67Kg2m2s−2 (a kind of correction). These obtained results including the experiment by “Nesvizhevsky et al”, allow us to evaluate the consistency condition of the NC model ∣∣∣ Θη4~2 ∣∣∣ � 10−24. But if we consider the modifications introduced by noncommutativity over ~ value (the precision is about 10−9), which are at least about 24 orders of magnitude smaller than its value, with considering the corrected bounds of η, we have∣∣∣ Θη4~2 ∣∣∣ � 10−29 [53]. These values agree with the higher limits on the basic scales of coordinate and momentum. These limits will be suppressed if the used magnetic field in the experiment is weak, about B ≈ 5mG. As long as the system in which we investigate the effects of noncommutativity is 2 dimensional, we restrict 112 vol. 60 no. 2/2020 Analytical solution of (2+1) dimensional dirac equation. . . ourselves to the following NC algebra[ xncj ,x nc k ] = iΘeγt�jk[ pncj ,p nc k ] = iηe−γt�jk, (j,k = 1, 2)[ xncj ,p nc k ] = i~effδjk , (3) we have �12 = −�21 = 1, �11 = �22 = 0, and Θ, η are real-valued with the dimension of length2 and momentum2, respectively. While the space coordinates and momentum are fuzzy and fluid [54], they cannot be localized, unless for minus infinite times. The parameters Θ, η represent the fuzziness and γ represents the fluidity of the space. The above equation is the relation of the ordinary NCG except that the NC structure constants are considered as exponentially increasing functions with the evolution of time. Certainly, there is a multitude of other possibilities, such as Θ(t) = Θcos(γt), η(t) = ηsin(γt). The new deformed geometry can be described by the operators xnc1 = xnc = x− 1 2~ Θe γtpy, p nc 1 = pncx = px + 1 2~ηe −γty xnc2 = ync = y + 1 2~ Θe γtpx, p nc 2 = pncy = py − 1 2~ηe −γtx . (4) When γ = 0, the time-dependency in the structure of NC parameters vanishes. In addition, for Θ = η = 0, the NCG reduces to a commutative one and the coordinates xj and the momentum pk satisfy the ordinary canonical commutation relations [xj,xk] = 0 [pj,pk] = 0, (j,k = 1, 2) [xj,pk] = i~δjk . (5) 3. (2+1) D Explicitly time-dependent Dirac equation and its invariant operator 3.1. (2+1) D Dirac equation in time-dependent noncommutative phase-space In presence of an electromagnetic four-potential Aµ = (A0,Ai), the Dirac equation in (2+1) d is given by( cαi(pi − ecAi(x)) + eA0(x) + βmc 2 ) |ψ〉 = i~ ∂ ∂t |ψ〉 , (6) with |ψ〉 being the Dirac wave function, and pj =   pxpy pz   is the momentum. The Dirac matrices αj , β αj = ( 0 σj σj 0 ) ,α1 = σ1 = ( 0 1 1 0 ) ,α2 = σ2 = ( 0 −i i 0 ) ,β = σ3 = ( 1 0 0 −1 ) , I = ( 1 0 0 1 ) , (7) satisfy the following anticommutation relations {αi,αj} = 2δij , {αi,β} = 0 with α2i = β 2 = 1. (8) We consider the magnetic field −→ B along z-direction, and it is defined in terms of the symmetric potential Ai = B 2 (−y,x, 0) , with A0 = 0, (9) most research about time-dependent systems concerns the presence of an electric field. However, in our current work, we do not rely on the electric field. Using Eq.(9), the Hamiltonian of the system becomes H(x,y,pxpy) = cα1px + cα2py + eα1 B 2 y −eα2 B 2 x + βmc2. (10) Achieving the NCG in the Dirac Hamiltonian (10) as follows H ( xnc,ync,pncx ,p nc y ) = cα1pncx + cα2p nc y −eα2 B 2 xnc + eα1 B 2 ync + βmc2, (11) by applying Eq.(4), we necessarily express the new NC Hamiltonian using the commutative variables {x,y,px,py}, and by assuming that ~ = c = 1 (natural units) to simplify the calculations, then we obtain Hnc(x,y,pxpy, t) = α1(1 + eB 4 Θeγt)px−α2( eB 2 + η 2 e−γt)x+α2(1 + eB 4 Θeγt)py +α1( eB 2 + η 2 e−γt)y +βm. (12) 113 Ilyas Haouam Acta Polytechnica The time-dependent Dirac equation in NC phase-space is given by i ∂ ∂t ∣∣ψ̄ (t)〉 = Hnc (t) ∣∣ψ̄ (t)〉 , (13) where ∣∣ψ̄ (t)〉 is the Dirac NC wave function. 3.2. The construction of the Lewis-Riesenfeld invariants To solve Eq.(13), we use the LR invariant method, which assumes the existence of a quantum-mechanical invariant I(t) which satisfies dI(t) dt = −i [I(t),Hnc (t)] + ∂I(t) ∂t = 0, (14) with i ∂ ∂t ( I(t) ∣∣ψ̄ (t)〉) = Hnc (t) I(t) ∣∣ψ̄ (t)〉 . (15) The Eq.(14) is called the invariance condition for the dynamical invariant operator I(t), which is a Hermitian operator I(t) = I+(t). (16) Assuming that I(t) = A1(t)px + B1(t)x + A2(t)py + B2(t)y + C(t), (17) with A1(t), B1(t), A2(t), B2(t), C(t) are time-dependent matrices. The substitution of Eqs.(17, 11) into Eq.(14) and using the properties of the commutation relations lead to [I,Hnc] + i ∂I ∂t = [A1px,Hnc] + [B1x,Hnc] + [A2py,Hnc] + [B2y,Hnc] + [C,Hnc] + i ∂I ∂t = 0, (18) for simplicity, we take fΘ(t) = 1 + eB4 Θe γt and fη(t) = eB2 + η 2 e −γt, which are not matrices, then we have [A1,α1fΘ] p2x + [A2,α2fΘ] p2y − [B1,α2fη] x2 + [B2,α1fη] y2 + {[A1,α2fΘ] + [A2,α1fΘ]}pxpy +{[A1,α1fη] + [B2,α1fΘ]}ypx + { [A1,βm] + [C,α1fΘ] + i∂A1∂t } px + { [A2,βm] + [C,α2fΘ] + ∂A2∂t } py + { [B1,βm] − [C,α2fη] + i∂B1∂t } x + { [B2,βm] + [C,α1fη] + i∂B2∂t } y + {− [A1,α2fη] + [B1,α1fΘ]}xpx +{[B1,α2fΘ] − [A2,α2fη]}xpy + {[B1,α1fη] − [B2,α2fη]}xy + {[A2,α1fη] + [B2,α2fΘ]}ypy +iA1α2fη + iB1α1fΘ − iA2α1fη + iB2α2fΘ − i [B1,α1fΘ] − i [B2,α2fΘ] + [C,βm] + i∂C∂t = 0 . (19) Then, to satisfy Eq.(14), and always taking advantage of the properties of commutation relations, with pipj = pjpi, xipj = pjxi if i 6=j ∈{1,2}, else pxx = xpx − i, pyy = ypy − i. We demand [A1,α1fΘ] = 0, (20) [A2,α2fΘ] = 0, (21) [B1,α2fη] = 0, (22) [B2,α1fη] = 0, (23) [A1,β] m + [C,α1fΘ] + i ∂A1 ∂t = 0, (24) [A2,β] m + [C,α2fΘ] + i ∂A2 ∂t = 0, (25) [B1,β] m− [C,α2fη] + i ∂B1 ∂t = 0, (26) [B2,β] m + [C,α1fη] + i ∂B2 ∂t = 0, (27) [A1,α2fΘ] + [A2,α1fΘ] = 0, (28) [B1,α1fΘ] − [A1,α2fη] = 0, (29) [B1,α2fΘ] − [A2,α2fη] = 0, (30) [B1,α1fη] − [B2,α2fη] = 0, (31) [B2,α1fΘ] + [A1,α1fη] = 0, (32) 114 vol. 60 no. 2/2020 Analytical solution of (2+1) dimensional dirac equation. . . [A2,α1fη] + [B2,α2fΘ] = 0, (33) iA1α2fη + iB1α1fΘ − iA2α1fη + iB2α2fΘ − i{[B1,α1fΘ] + [B2,α2fΘ]} + [C,βm] + i ∂C ∂t = 0. (34) From the relations (20 - 23), and as long as from Eq.(20), we have A1 = a0(t) + a1(t)α1 + a2(t)α21 + a3(t)α 3 1 + a4(t)α 4 1 + ... = a ′ 0(t) + a ′ 1(t)α1, with a ′ i(t) = ai(t), (35) therefore, we obtain A1 = a1 + a2α1, (36) A2 = a3 + a4α2, (37) B1 = b1 + b2α2, (38) B2 = b3 + b4α1. (39) From Eqs.(24 - 27) and with the same manner, supposing that C is written in terms of α1, α2 and β as follows C = c1 + c2α1 + c3α2 + c4β, (40) where aj, bj and cj (with j = 1, .., 4) are supposed to be time-dependent arbitrary functions. Substituting Eqs.(40, 36) into Eq.(24) and Eqs.(40, 37) into Eq.(25), and taking into consideration Eq.(8), yields ∂a1 ∂t = 0, ∂a3 ∂t = 0, a2 = a4 = c2 = c3 = c4 = 0 , (41) thereafter, substituting Eqs.(40, 38) into Eq.(26) and Eqs.(40, 39) into Eq.(27), and taking into consideration Eq.(8), yields ∂b1 ∂t = 0, ∂b3 ∂t = 0, b2 = b4 = c2 = c4 = 0 . (42) From the Eqs.(41, 42) we note that a1, a3, b1, b3 are time-independent constants. We have A1 = a1, A2 = a3, B1 = b1, B2 = b3, C = c1 . (43) In addition, from Eqs.(30, 32), and assuming that there exist χ(t) and ϕ(t), which are time-dependent matrices, with [χ(t),α2] = [ϕ(t),α1] = 0. The time-dependency may appear as follows b1fΘ −a3fη = χ(t) b3fΘ + a1fη = ϕ(t) . (44) Now, substituting Eq.(43) into Eq.(34) and using Eq.(44) gives us ∂c1 ∂t = −{a1fη + b3fΘ}α2 −{b1fΘ −a3fη}α1, (45) using system of relations (44), we find ∂c1 ∂t = 0 and χ = ϕ = 0. (46) Last but not least, the dynamical invariant (17) of the time-dependent NC Dirac equation can be written as follows I = a1px + b1x + a3py + b3y + c1, (47) we inferred that Eq.(14) is verified and c1 should be a constant. We may also note that all the spin-dependent parts, which are proportional to αj, β disappear. Which means that I has no spin-dependency, but it is proportional to the matrix of identity in the spinor of space. 115 Ilyas Haouam Acta Polytechnica 3.3. Eigenvalues and eigenstates of I and H(t) Supposing that the invariant in general I(t) is a complete set of eigenfunctions |φ(λ,k)〉 (in this subsection, the analysis is not concerning only on time-independent invariants), with λ being the corresponding eigenvalue (spectrum of the operator), and k represents all other necessary quantum numbers to specify the eigenstates, the eigenvalues equation is written as I(t) |φ(λ,k)〉 = λ |φ(λ,k)〉 , (48) where |φ(λ,k)〉 are an orthogonal eigenfunctions〈 φ(λ,k) | φ(λ ′ ,k ′ ) 〉 = δλλ′ δkk′ . (49) According to Eq.(16), the eigenvalues are real and not time-dependent. Deriving Eq.(48) in time, we find ∂I ∂t |φ(λ,k)〉 + I ∂ ∂t |φ(λ,k)〉 = ∂λ ∂t |φ(λ,k)〉 + λ ∂ ∂t |φ(λ,k)〉 , (50) we apply Eq.(14) over the eigenfunctions |φ(λ,k)〉, we have i ∂I ∂t |φ(λ,k)〉 + IHnc |φ(λ,k)〉−Hncλ |φ(λ,k)〉 = 0, (51) the scalar product of Eq.(51) by 〈 φ(λ ′ ,k ′ ) ∣∣∣ is i 〈 φ(λ ′ ,k ′ ) ∣∣∣∣∂I∂t ∣∣∣∣φ(λ,k) 〉 + ( λ ′ −λ )〈 φ(λ ′ ,k ′ ) |Hnc|φ(λ,k) 〉 = 0, (52) which implies 〈 φ(λ ′ ,k ′ ) ∣∣∣∣∂I∂t ∣∣∣∣φ(λ,k) 〉 = 0, (53) the scalar product of Eq.(50) by 〈 φ(λ ′ ,k ′ ) ∣∣∣is〈 φ(λ ′ ,k ′ ) ∣∣∣∣∂I∂t ∣∣∣∣φ(λ,k) 〉 = ∂λ ∂t , (54) from Eq.(53), the Eq.(54) shows that 〈 φ(λ ′ ,k ′ ) ∣∣∣∣∂I∂t ∣∣∣∣φ(λ,k) 〉 = ∂λ ∂t = 0. (55) While the eigenvalues are time-independent, the eigenstates should be time-dependent. In order to find the link between the eigenstates of the invariant I(t) and the solutions of the relativistic Dirac equation, we first start with writing the motion equation of |φ(λ,k)〉, so that, using Eq.(50) and Eq.(55), we obtain ∂I ∂t |φ(λ,k)〉 = (λ− I) ∂ ∂t |φ(λ,k)〉 , (56) by using the scalar product with 〈 φ(λ ′ ,k ′ ) ∣∣∣, and taking Eq.(52) to eliminate 〈φ(λ′,k′ ) ∣∣∂I∂t ∣∣φ(λ,k)〉, then we obtain i 〈 φ(λ ′ ,k ′ ) ∣∣∣∣(λ−λ′) ∂∂t ∣∣∣∣φ(λ,k) 〉 = ( λ−λ ′ )〈 φ(λ ′ ,k ′ ) |Hnc|φ(λ,k) 〉 , (57) for λ ′ 6= λ, we deduce i 〈 φ(λ ′ ,k ′ ) ∣∣∣∣ ∂∂t ∣∣∣∣φ(λ,k) 〉 = 〈 φ(λ ′ ,k ′ ) |Hnc|φ(λ,k) 〉 , (58) then we deduce immediately that |φ(λ,k)〉 satisfy the Dirac equation, that is to say |φ(λ,k)〉 are particular solutions of the Dirac equation. It is assumed that a phase has been taken, but it is still always possible to multiply it by an arbitrary time-dependent phase factor, which means that we can define a new set of I(t) eigenstates linked to our overall by a time-dependent gauge transformation, and |φ(λ,k)〉α = e iαλ(t) |φ(λ,k)〉 , (59) 116 vol. 60 no. 2/2020 Analytical solution of (2+1) dimensional dirac equation. . . where αλ(t) is a real time-dependent function arbitrarily chosen called LR phase, |φλ(x,y,t)〉αare eigenstates of I(t) which are orthonormal and associated with λ. By putting Eq.(59) in Eq.(58) and using Eq.(49), we find ∂αλ,k ∂t δλλ′ δkk′ = 〈 φ(λ ′ ,k ′ ) ∣∣∣i ∂ ∂t −Hnc |φ(λ,k)〉 . (60) All the eigenstates of the invariant are also solutions of the time-dependent Dirac equation, it was shown in [46] that its general solution is done by ∣∣ψ̄(t)〉 = ∑ λ,k Cλ,ke iαλ,k(t) |φ(λ,k,t)〉 , (61) we remark that Eq.(61) is also spin-independent in its state. But maybe the spin-dependent part is entangled in the coefficient C. |φ(λ,k,t)〉 are the orthonormal eigenstates of I(t), with Cλ,k being time-independent coefficients, which correspond to |ψ(0)〉 Cλ,k = 〈λ,k | ψ(0)〉 . (62) For a discrete spectrum of I(t), with λ = λ ′ , k = k ′ , and from Eq.(60) the LR phase is defined as α(t) = � t 0 〈 φ(λ,k,t ′ ) ∣∣∣i ∂ ∂t ′ −Hnc(t ′ ) ∣∣∣φ(λ,k,t′ )〉dt′. (63) But in the continuous spectrum case, the general expression of the phase is ∂αλ,k ∂t 〈 φ(λ ′ ,k ′ , t ′ | φ(λ,k,t) 〉 = 〈 φ(λ ′ ,k ′ , t ′ ) ∣∣∣i ∂ ∂t −Hnc |φ(λ,k,t)〉 , (64) where k is an index that varies continuously in the real values, thus〈 φ(λ ′ ,k ′ , t ′ | φ(λ,k,t) 〉 = δλλ′ δ(k −k ′ ), (65) substituting Eq.(65) in Eq.(64) yields α(t) = � � t 0 〈 φ(λ,k ′ , t ′ ) ∣∣∣i ∂ ∂t ′ −Hnc ∣∣∣φ(λ,k,t′ )〉dt′dk′. (66) Once the expression of the phase α(t) is found, we can write the particular solution of our NC time-dependent Dirac equation (61). For simplicity, we use the notation of the discrete spectrum of I(t). We see that the eigenfunction of I(t) has the form of [55, 56] |φλ,k(x,y,t)〉∝ |λ,k〉exp [ i ( ξ1(t)x + ξ2(t)y + ξ3(t)x2 + ξ4(t)y2 )] , (67) where ξ1(t), ξ2(t), ξ3(t), ξ4(t) are arbitrary time-dependent functions. By substituting Eq.(67) into Eq.(63) yields α(t) = ϑ− � t 0 Encdt ′ , (68) with ϑ(x,y,t) = (ξ1(0) − ξ1(t)) x + (ξ2(0) − ξ2(t)) y + (ξ3(0) − ξ3(t)) x2 + (ξ4(0) − ξ4(t)) y2, (69) and Enc is the eigenvalue of the Hamiltonian (12). Finally, the solution of the NC Dirac equation (13) is [46] ∣∣ψ̄(t)〉 = ∑ λ,k Cλ,ke i[ϑ− � t 0 E ncdt ′ ] |φ(λ,k,t)〉 , (70) 117 Ilyas Haouam Acta Polytechnica 3.4. The exact form of the solutions of the problem As agreed [55–57], the wave function of the NC Dirac equation is given by the following trial function∣∣ψ̄(x,y,t)〉 = F(t) |φ(x,y,t〉 , (71) where F is a time-dependent vector of 2 components (2 × 1) F(t) = ( F1(t) F2(t) ) , (72) as long as I(t) is independent in time, Eq.(15) goes to Eq.(13). Then the substitution of Eq.(71) into Eq.(13), and using Eqs.(67, 7) give { i∂F1 ∂t −F1 ∂ξ1∂t x−F1 ∂ξ2 ∂t y −F1 ∂ξ3∂t x 2 −F1 ∂ξ4∂t y 2 i∂F2 ∂t −F2 ∂ξ1∂t x−F2 ∂ξ2 ∂t y −F2 ∂ξ3∂t x 2 −F2 ∂ξ4∂t y 2 } ={ m α1fΘpx −α2fηx + α2fΘpy + α1fηy α1fΘpx −α2fηx + α2fΘpy + α1fηy −m } × ( F1 F2 ) , (73) then, we obtain i∂F1 ∂t −F1 ∂ξ1∂t x−F1 ∂ξ2 ∂t y −F1 ∂ξ3∂t x 2 −F1 ∂ξ4∂t y 2 = fΘF2px + ifηF2x− ifΘF2py + fηF2y + mF1 i∂F2 ∂t −F2 ∂ξ1∂t x−F2 ∂ξ2 ∂t y −F2 ∂ξ3∂t x 2 −F2 ∂ξ4∂t y 2 = fΘF1px − ifηF1x + ifΘF1py + fηF1y −mF2 , (74) by solving the above system of equations, we find ∂F1 ∂t = −imF1, ∂F2 ∂t = imF2, (75) F1 ∂ξ1 ∂t = −i{ eB 2 + η 2 e−γt}F2, (76) F1 ∂ξ2 ∂t = −{ eB 2 + η 2 e−γt}F2, (77) ∂ξ3 ∂t = ∂ξ4 ∂t = 0, (78) which leads to obtaining F1 = e−imt+q1, F2 = eimt+q2, (79) ∂ξ1 ∂t = i ∂ξ2 ∂t = −i{ eB 2 + η 2 e−γt}ei2mt+q2−q1, (80) ξ1 = iξ2 = −i{ κ 4l2Bim ei2mt + ηκ 4im− 2γ e(−γ+i2m)t}, (81) with q1, q2 and κ = eq2−q1 being real constants, l−1B = √ eB is the magnetic length [58]. In commutative case (Θ = η = γ = 0), then the above relations (79, 81) return to that of general quantum mechanics ξ1(t) = iξ2(t) = − κ 4l2Bm ei2mt, |φ(x,y,t〉 |η=γ=0 ∼ e − iκ 4l2 B m ei2mt(x+iy)+o1ix2+o2iy2 , (82) with o1, o2 are real constants, and in t = 0 ξ1(t = 0) = iξ2(t = 0) = − κ 4l2Bm , and F1 = κF2 = eq1. (83) 4. Conclusion In conclusion, the dynamics of the system of a time-dependent NC Dirac equation has been analysed and formulated using the LR invariant method. We introduced the time-dependent noncommutativity using a time- dependent Bopp-shift translation. Knowing that the NC structure constants postulated expanding exponentially with the evolution of time, and the time-dependency have a multitude of other possibilities. We benefit from the dynamical invariant following the standard procedure allowed to construct and to obtain an analytical solution of the system. 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Physical Review Letters 98:197403, 2007. doi:10.1103/PhysRevLett.98.197403. 121 http://dx.doi.org/10.1103/PhysRevLett.87.141601 http://dx.doi.org/10.1103/PhysRevD.72.025010 http://dx.doi.org/10.1088/1742-6596/103/1/012020 http://dx.doi.org/10.4236/jmp.2013.47126 http://dx.doi.org/10.1088/0253-6102/64/3/263 http://dx.doi.org/10.1063/1.2747609 http://dx.doi.org/10.1103/PhysRevLett.98.197403 Acta Polytechnica 60(2):1–11, 2020 1 Introduction 2 time-dependent noncommutativity 3 (2+1) D Explicitly time-dependent Dirac equation and its invariant operator 3.1 (2+1) D Dirac equation in time-dependent noncommutative phase-space 3.2 The construction of the Lewis-Riesenfeld invariants 3.3 Eigenvalues and eigenstates of I and H(t) 3.4 The exact form of the solutions of the problem 4 Conclusion Acknowledgements References