Analytical Solution of (2+1) Dimensional Dirac Equation in Time-Dependent Noncommutative Phase-Space

In this article, we studied the system of (2+1) dimensional Dirac equation in time-dependent noncommutative phase-space. Exactly, 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 it used to set the Lewis-Riesenfeld invariant operators. Thereafter, the obtained results used to express the eigenfunctions that lead to determining the general solution of the system.


I. INTRODUCTION
It is known that Heisenberg suggested the opinion of noncommutative (NC) space-time in 1930, and in 1947, Snyder presented it [1,2] to the necessity to 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 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 in highlighting various fields of physics, particularly in matrix theory (matrix model BFSS (1997)) [3]. NCG 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 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][9][10][11][12] by studying and defining a cyclical cohomology. It has been shown that the differential calculus on manifolds had a NC equivalent. Next, the NCG found great encouragement through several mathematical results such as of 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 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 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 the very high energy, the effects of noncommutativity may appear. Besides, one of the strong motivations of 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 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 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.
On the other hand, 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 [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. Exactly, the spacing between different Landau levels decreases if the electric field strength reaches a critical value [30,31].
In our study about (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 timedependent one, H(x nc i , p nc i ) −→ H(t). Solving the system of equations in interaction with time-dependent potentials has attracted many physicists for a long time. 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][33][34], quantum optics [35][36][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 to 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 in 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.

II. TIME-DEPENDENT NONCOMMUTATIVITY
In the theory of NCG space may not commute anymore (i.e. AB =BA). In a d dimensional time-dependent NC phase-space let us consider the operators of coordinates and momentum x nc j and p nc k respectively. These NC operators satisfy the deformed commutation relations [49] x nc j , x nc the effective Planck constant being where Θη 4 2 ≪ 1 is the consistency condition in the usual quantum mechanics. δ jk is the identity matrix, and Θ jk , η jk 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 −30 m 2 and η ≈ 1, 76.10 −61 Kg 2 m 2 s −2 . Other bounds exist. For example Θ ≈ 4.10 −40 m 2 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 −67 Kg 2 m 2 s −2 (kind of a correction). These obtained results including of the experiment by "Nesvizhevsky et al", allow us to evaluate the consistency condition of the NC model Θη  [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 ourselves to the following NC algebra we have ǫ 12 = −ǫ 21 = 1, ǫ 11 = ǫ 22 = 0, and Θ, η are real-valued with the dimension of length 2 and momentum 2 , respectively. While the space coordinates and momentum are fuzzy and fluid [54], they can not 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 NC structure constants are considered as exponentially increasing functions with the evolution of time. Certainly, there are a multitude of other possibilities, such as The new deformed geometry can be described by the operators When γ = 0, the time-dependency in the structure of NC parameters vanishes. In Addition, for Θ = η = 0, the NCG reduces to commutative one, yonder the coordinates x j and the momentum p k satisfy the ordinary canonical commutation relations (II.5) In presence of an electromagnetic four-potential A µ = (A 0 , A i ), the Dirac equation in (2+1) d is given by with |ψ is the Dirac wave function, and p j = satisfy the following anticommutation relations We consider the magnetic field − → B along z-direction, and it is defined in terms of the symmetric potential The time-dependent Dirac equation in NC phase-space is giving by where ψ (t) is the Dirac NC wave function.

B. The construction of the Lewis-Riesenfeld invariants
To solve Eq.(III.8), we use the LR invariant method, which assumes the existence of a quantum-mechanical invariant I(t) which satisfies (III.10) The Eq.(III.9) is called the invariance condition for the dynamical invariant operator I(t), which is a Hermitian operator (III.11) Assuming that for simplicity we take f Θ (t) = 1 + eB 4 Θe γt and f η (t) = eB 2 + η 2 e −γt , which are not matrices, then we have . (III.14) Then, to satisfy Eq.(III.9), and always taking advantage of the properties of commutation relations, with From the relations (III.15 -III.18), and as long as from Eq.(III.15), we have therfore, we obtain (III.32) From Eqs.(III. ) and with the same manner, supposing that C is written in terms of α 1 , α 2 and β as follows where a j , b j and c j (with j = 1, ..   we inferred that Eq.(III.9) is verified and c 1 should be a constant. We may note also that all the spin-dependent parts which are proportional to α j , β disappear. Which means that I has no spin-dependency, but it is proportional to the matrix of identity in the spinor of space.

C. 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 where |φ(λ, k) are an orthogonal eigenfunctions According to Eq.(III.11), the eigenvalues are real and not time-dependent. Deriving Eq.(III.43) in time, we find we apply Eq.(III.9) over the eigenfunctions |φ(λ, k) , we have 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, firstly, we start with writing the motion equation of |φ(λ, k) , so that using Eq.(III.45) and Eq.(III.50), we obtain by using the scalar product with φ(λ ′ , k ′ ) , and taking Eq.(III.47) to eliminate φ(λ ′ , k ′ ) ∂I ∂t φ(λ, k) , then we obtain then we deduce immediately that |φ(λ, k) satisfy the Dirac equation, that is to say |φ(λ, k) are particular solutions of Dirac equation. It is assumed that, a phase has been taken, but it 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 iα λ (t) |φ(λ, k) , (III. 54) where α λ (t) is a real time-dependent function arbitrarily chosen called LR phase, |φ λ (x, y, t) α are eigenstates of I(t) which are orthonormal and associated with λ. By putting Eq.(III.54) in Eq.(III.53) and using Eq.(III.44), we find 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 (III. 56) we remark that Eq.(III.56) 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 C λ,k are time-independent coefficients, which correspond to |ψ(0) For a discrete spectrum of I(t), with λ = λ ′ , k = k ′ , and from Eq.(III.55) the LR phase is defined as But in the continuous spectrum case, the general expression of the phase is where k is an index which varies continuously in the real values, thus substituting Eq.(III.60) in Eq.(III.59) yields Once found the expression of the phase α(t), we can write the particular solution of our NC time-dependent Dirac equation (III.56).
We use for simplicity the notation of the discrete spectrum of I(t). We see that the eigenfunction of I(t) has the form of [55,57] |φ λ,k (x, y, t) ∝ |λ, k exp i ξ 1 (t)x + ξ 2 (t)y + ξ 3 (t)x 2 + ξ 4 (t)y 2 , (III.62) where ξ 1 (t), ξ 2 (t), ξ 3 (t), ξ 4 (t) are arbitrary time-dependent functions. By substituting Eq.(III.62) into Eq.(III.58) yields As agreed [55][56][57], the wave function of the NC Dirac equation is given by the following trial function ψ (x, y, t) = F (t) |φ(x, y, t , (III.66) where F is a time-dependent vector of 2 components (2 × 1) then, we obtain (III.69) by solving the above system of equations, we find with q 1 , q 2 and κ = e q2−q1 are real constants, l −1 B = √ eB is the magnetic length [58]. In commutative case (Θ = η = γ = 0), then the above relations (III.74, III.76) return to that of general quantum mechanics In conclusion, the dynamics of the system of time-dependent NC Dirac equation has been analysed and formulated using 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. Having obtained the explicit solutions could help also to investigate and reformulate the modified version of Heisenberg's uncertainty relations emerging from non-vanishing commutation relations (II.1). The uncertainty for the observables A, B has to satisfy the inequality △A△B | ψ ≥ 1 2 | ψ| [A, B] |ψ | with △A | 2 ψ = ψ| A 2 |ψ − ψ| A |ψ 2 and the same for B for any state. Depending on these results, we are planning to study the pair creation process, and investigate its implications in quantum optics.

ACKNOWLEDGMENTS
The author wishes to express thanks to Pr Lyazid Chetouani for his interesting comments and suggestions, also would like to appreciate anonymous reviewers for their careful reading of the manuscript and their insightful comments.