1. Introduction

The studies of the relativistic generalization of the harmonic oscillator has drawn much attention in recent years. The wellknown relativistic model of the harmonic oscillator was revived by Moshinsky and Szczepaniak [¹], who gave it the name of Dirac oscillator (DO) because, in the non-relativistic limit, it becomes a harmonic oscillator with a very strong spin-orbit coupling term. The Dirac relativistic oscillator is an important potential both the theoretical and application implications. It was for the first time studied by Ito et al. [²]. They considered a Dirac equation in which the momentum p→ is replaced by p→-imβωr→, with r→ being the position vector, m the mass of particle, and ω the frequency of the oscillator. Physically, it can be shown that the DO interaction represents a physical system, which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field [³,⁴]. The electromagnetic potential associated with the DO has been found by Benitez et al. [⁵]. The Dirac oscillator has attracted a lot of interest both because it provides one of the examples of the Dirac’s equation exact solvability and because of its numerous physical applications (see [⁶] and references therein). Recently, Franco-Villafane et al. [⁷] exposed the proposal of the first experimental microwave realization of the one-dimensional DO. The experiment relies on a relation of the DO to a corresponding tight-binding system. The experimental results obtained, concerning the spectrum of the one-dimensional DO with and without the mass term, are in good agreement with those obtained in the theory. In addition, Yang and Piekarewicz [⁸] illustrate the power and flexibility of the Dirac oscillator and they suggest extensions to the study of systems without spherical symmetry as required in constrained calculations of nuclear excitations. Self-consistent calculations of binding energies and ground-state densities for a selected set of doubly-magic magic are performed using the Dirac-oscillator basis and are then compared against results obtained with the often-used Rung-Kutta method. The results obtained using the Dirac oscillator basis reproduced with high accuracy those derived using the Rung-Kutta method.

The unification between the general theory of relativity and the quantum mechanics is one of the most important problems in theoretical physics. This unification predicts the existence of a minimal measurable length on the order of the Planck length. All approaches of quantum gravity show the idea that near the Planck scale, the standard Heisenberg uncertainty principle should be reformulated. The minimal length uncertainty relation has appeared in the context of the string theory, where it is a consequence of the fact that the string cannot probe distances smaller than the string scale ℏβ, where β is a small positive parameter called the deformation parameter. This minimal length can be introduced as an additional uncertainty in position measurement, so that the usual canonical commutation relation between position and momentum operators becomes

This commutation relation leads to the standard Heisenberg uncertainty relation

which clearly implies the existence of a non-zero minimal △xmin=ℏβ∼lp where l _p is the Planck length. This modification of the uncertainty relation is usually termed the generalized uncertainty principle (GUP) or the minimal length uncertainty principle [⁹-¹²]. Note here that Saavedra and Utreras [¹³] were the first to propose a generalization of the canonical commutation relations of quantum mechanics which should be important at high energies. So a new (high energy) uncertainty principle was obtained.

In this direction, some remarks can be made about Eqs. (1) and (2): (i) according to the woks of [¹⁴-¹⁸], one important observation was that various observable effects of the minimal length uncertainty relation are non-perturbative in the deformation parameter β, (i.e., contain all orders in β) even though β appears only to linear order in Eqs. (1) and (2) [¹⁶-¹⁸] . (ii) In Ref. [¹⁵], the authors study the effect of the minimal length on the thermal properties of a Dirac oscillator when the position and momentum obey Eq. (1), and studied the possible constraint that can be placed on β. By using the properties of the Epstein zeta function, this study leads to a minimal length in the interval of 10⁻¹⁶ < ∆x < 10⁻¹⁴ m with the following physically acceptable condition β > β ₀ = (1/m ² c ²). (iii) Kempf [¹⁰,¹¹,¹⁹] has shown that Eq. (1) follows naturally when the canonical commutation relation between position and momentum operators is modified according to the Eq. (1). So, we may select the position x^ and momentum p^ operators in terms of the undeformed observables x and p, i.e.[x,p]=iℏ, as

One should note that the construction of models in these spaces would not be an easy task as the operators x^ and p^ in the deformed commutation relation (1) are in general not Hermitian x^†=x^+2iβℏp^ and p^†=p^, albeit the simplified version (3) still allows one operator to remain Hermitian [²⁰]. According to the works of Bagchia et al. [²¹] the positivity of β becomes important, as it ensures the absence of singularities in the metric.

According to the Kempf’s prescription [¹⁰-¹², ¹⁹], the position and momentum operators satisfying Eq. (1) can be also represented by

where the operators x and p satisfy the canonical commutation relation [x,p] =iℏ. In this case, the internal product in Fourier space becomes

The parameter γ appears in both Eqs. (4) and (5) is an arbitrary constant which does not affect the observable quantities; its choice determines only the weight function in the definition of the scalar product [²²]. In this work, we have opted with the Kempf method, and so we chose γ = 0.

Nowadays, the reconsideration of the relativistic quantum mechanics in the presence of a minimal measurable length have been studied extensively. In this context, many papers were published where a different quantum system in space with Heisenberg algebra was studied. They are: the Abelian Higgs model [¹⁹], the thermostatics with minimal length [²³], the one-dimensional Hydrogen atom [²⁴], the Casimir effect in minimal length theories [²⁵], the effect of minimal lengths on electron magnetism [²⁶-²⁸], the solutions of a two-dimensional Dirac equation in presence of an external magnetic field [²⁹], the non-commutative phase space Schrödinger equation [³⁰], and the Schrödinger equation with Harmonic potential in the presence of a magnetic field [³¹].

The study of NC spaces and their implications in physics is an extremely active area of research. It has been argued in various instances that non-commutativity should be considered as a fundamental feature of space-time at the Planck scale. On the other side, the study of quantum systems in an NC space has been the subject of much interest in past years, assuming that non-commutativity may be, in fact, a result of quantum gravity effects. In these studies, some attention has been paid to the models of NC quantum mechanics (NCQM). The interest in this approach lies on the fact that NCQM is a fruitful theoretical laboratory where we can get some insight on the consequences of non-commutativity in field theory by using standard calculation techniques of quantum mechanics. Various NC field theory models have been discussed as well as many extensions of quantum mechanics. Of particular interest is the so-called phase space noncommutativity, which has been investigated in the context of quantum cosmology, black holes physics, and the singularity problem. This specific formulation is necessary to implement the Bose-Einstein statistics in the context of NCQM (see [³²-³⁸]).

In addition, investigating the influence of the minimal length assumption on the energy spectrum of quantum systems has become an interesting issue primarily for two reasons. First, this may help to set some upper bounds on the value of the minimal length. In this context, we can cite some studies of the hydrogen atom and a two dimensional Dirac equation in an external magnetic field. Moreover, the classical limit has also provided some interesting insights into some cosmological problems. Second, it has been argued that quantum mechanics with a minimal length may also be useful to describe non-point-like particles, such as quasi- particles and various collective excitations in solids, or composite particles (see Ref [²⁵] and references therein).

The purpose of this work is to investigate the formulation of a two-dimensional Klein Gordon oscillator (KGO) in the presence of a magnetic field by solving fundamental equations in the framework of relativistic quantum mechanics with minimal length in the NC space. To do this we first mapped the problem in question into a commutative space by using an appropriate transformations. Then, we solved it in the presence of a minimal length.

The paper is organized as follows. In Sec. 2, we solve the KGO in the presence of magnetic field in noncommutative phase space. Then, in Sec. 3, we study this problem in the framework of relativistic quantum mechanics with minimal length. Finally, Sec. 4 will be a conclusion.

2. The solutions within habitual quantum mechanics in NC space

To begin with, we note that the non-commutative phase space is characterized by the fact that their coordinate operators satisfy the equation [³³-³⁸]

where Θ _µν is an antisymmetric tensor of space dimension. In order to obtain a theory which includes the aspects of being unitary and causal, we choose Θ _0ν = 0 , which implies that the time remains as a parameter and the non-commutativity affects only the physical space. The noncommutative models specified by Eq. (1) can be realized in terms of a ⋆-product: the commutative algebra of functions with the usual product f(x)g(x) is replaced by the ⋆-product Moyal algebra [³⁷]:

Since the system in which we study is two-dimensional, we limit our analysis to the xy plane, where the noncommutative algebra is written by

where ϵ _ij is the two-dimensional Levi-civita tensor. Instead of solving the NC Klein-Gordon equation by using the star product procedure, we use Bopp’s shift method, that is, we replace the star product by the usual product by making a Bopp’s shift

Hence, in the two dimensional non-commutative phasespace, Eq. (9) becomes

In this case, the two-dimensional Klein-Gordon oscillator , in commutative space, which is written by

is modified and transformed into

with

Here we have used that

In the literature (see Ref [⁶] and references therein), the upper bound on the value of the coordinate commutator found is Θ ≤ 4×10⁻⁴⁰ m². So, as an approximation, all terms with the square of Θ have been neglected.

Now, by putting Eqs. (13) and (14) in (12), we obtain

In order to solve the last equation, and for the sake of simplicity, we bring the problem into the momentum space. Recalling that

and when passing onto polar coordinates [⁹]

Eqs. (17) and (18) become

Putting Eqs. (20) and (21) in Eq. (16), the last equation can be written explicitly in polar coordinates by

where

With the help of the following relation [³⁹]

Eq. (22) is transformed into

with

Now, noting that [³⁹]

then the differential equation

is transformed into the Kummer equation

which by using, instead of p, the variable xt = kp ². The solution of this equation is the confluent series ₁ F ₁ (α;|l| + 1;t), with

The confluent series becomes a polynomial if and only if α = −n, (n = 0,1,2,).

Thus, following this, we have the solutions

with N = 2n + |l| is the principal quantum number, and r=(ℏω/m0c2) the parameter that controls the non-relativistic limit.

We can see that the presence of the parameter Θ, in the spectrum of energy, breaks the degeneracy of the energy levels. Furthermore, by taking that Θ = 0 , we reach the exact result of the two-dimensional Klein-Gordon oscillator in Minkowski space-time

This form of energy is in a good agreement with that obtained in the literature (see Ref. [³⁷]).

3. The solutions in the presence of a minimal length

In the minimal length formalism, the Heisenberg algebra is given by [¹¹-²²]

where 0 < β ≤ 1 is minimal length and p is magnitude of the momentum. When the energy is much smaller than the Planck mass, β goes to zero and we recover Heisenberg uncertainty principle.

A representation of x^i and p^i which satisfies Eq. (35), may be taken as

By using the Eqs. (36) and (37), Eq. (16) becomes

Now, when we put that

with j = 0,±1,±2,..., the Eq. (38) is transformed into

with

In order to solve Eq. (40), we adopt the following substitution as used by Jana et al., [³⁸]

After these susbsitutions, Eq. (40) becomes

with

with ρ(p) = p ^−1/2 .

In order to simplify the function V (p), we use the change of variable

In this case, the form V (p) with the new variable q becomes

Consequently, the final form of our differential equation is

where

Thus, Eq. (48) is brought to

with

and where

Finally, we have

where U ₀ = α ² with α=λβ

In this stage, we can see that Eq. (54) is the well-known Schrödinger equation in a Pöschl-Teller potential with the following potential [³⁹]

and where we have the conditions that (ζ ₁ ,ζ ₂) > 1.

By comparing Eq. (48) with Eq. (54), we obtain

Now, in order to solve Eq. (48), we introduce the new variable

In this case, Eq. (48) can be rewritten as

With the new wave function φ, defined by

we arrive at

The general solution of this equation is

with

With the condition a' = −n, we obtain

In order to obtain the energy spectrum, it should be noted that in the limit β → 0, the energy spectrum should regenerate to the no-GUP result.

Thus, the exact form of ζ ₁ and ζ ₂ are

where j ≠ 0.

With the aid of Eqs. (56), (57) and (64), we obtain the final form of the spectrum of energy: it is expressed explicitly by

From Eq. (67), the presence of both β and Θ breaks the degeneracy of our spectrum of energy. In addition, the presence of the spectrum of energy with the parameter β, contrarily to the case of Θ, leads to the dependence of the energy levels on n ²: as mentioned by Nouicer [²⁷], this dependence on n ² is a feature of hard confinement. Finally, in the limit where β → 0, we reach the following equation

which has the same form as the Eq. (33). The corresponding wave function is

with N_norm is the constant of normalization.

4. The solutions in the presence of a magnetic field

In a magnetic field, when we choose the potential vectors as

the two-dimensional Klein-Gordon oscillator in non-commutative space is

or in the explicit form as

By using Bopp’s shift method, Eq. (72) is transformed into

Putting Eqs. (36), (37) and (39) in (73), we obtain

with ς ² = (E ² - (m ₀ c ²)² /c ²). Following the same procedure as in the above section, we have

or with the new variable q,

In this case, the form of V is

or with the variable q

Where ζ1' and ζ2' are defined by

Finally, our differential equation can be written as

where U ₀ = α ² and ξ¯'2=ς'2+(1+[eBΘ4ℏc])2(1/β) with α=β(λ2+ℏ2[eB/2c]2).

As in the above case, Eq. (78) is the well-known Schrödinger equation in a Pöschl-Teller potential [³⁹]. with the constraint that (ζ1',ζ2')>1, the exact form of ζ ₁ and ζ ₂ are

So, the general solution of Eq. (78) is

with z = sin² (αq) and where

With the condition a' = −n, and by using Eq. (64), we obtain.

Now, from Eq. (83) the final form of the spectrum of energy is

with

Also, we can see that the presence of both parameters β and Θ with a magnetic field breaks the degeneracy of the spectrum of energy. In addition, the presence of a term on n ² shows the existence of hard confinement. This dependence is only clear when we introduce minimal length in the problem in question.

Finally, the corresponding wave function is

with N_norm the constant of normalization.

Now, to achieve a deeper understanding of our spectrum of energy presented by Eq. (84), we have plotted the variation of the positive energy E with quantum number n and magnetic field B for different various situations. As the spectrum of energy is symmetric, any comments on the case of the positive energy can be extended to the case of the negative energy.

Figure 1 shows the behavior of the spectrum of energy versus n by changing the three parameters appearing in the expression for the energy spectrum, i.e, the magnetic field B, the NC parameter θ and the minimal length β. This figure has been depicted for two cases: j = 0 and j = 1. Here, three remarks can be made

In Fig. 2, we represent the variation of the positive energy levels with magnetic field for four levels n = 0,1,2,3. For each level of energy, we have chosen j = 0,1,2,3. With the exception of the case j = 0, every curve show approximately a linear form with a magnetic field B.

Finally, in order to show that the problem in question has relevance from a physical point of view, we make the following remarks. Firstly, we can mention some studies that showed the existence of a close relation between graphene and the Dirac oscillator model. [⁶,⁴⁰-⁴²].

Bastos et al., [³³] consider a non-commutative description of graphene. This description consists of a Dirac equation for massless Dirac fermions plus non-commutative corrections, which are treated in the presence of an external magnetic field. They argue that, being a two-dimensional Dirac system, graphene is particularly interesting to test nonommutativity. They find that momentum non-commutativity affects the energy levels of graphene and they obtain a bound for the momentum non-commutative parameter. Their results show that momentum non-commutativity yields interesting results also at low-energies and that its implications are not restricted to quantum cosmology and black holes physics.

In the same context, Falomir et al. [⁴³] employ a simple nonrelativistic model to describe the low-energy excitation of graphene. The model is based on a deformation of the Heisenberg algebra which makes the commutator of momenta proportional to the pseudo-spin. As a result, the resulting Hall conductivity is consistent with the anomalous integer quantum Hall effect found in graphene.

Jallel [⁴⁴] describes the lattice deformation in graphene under strain effect by considering the spacial-moment coordinates do not commute. Within such framework, he (i) build a new model describing Dirac fermions interacting with an external source that is non commutative parameter dependent, and (ii) the eigenvalues are showing Landau levels in similar way to the case of a real magnetic field applied to graphene.

Iorio and Pais [⁴⁵], in their paper “Generalized uncertainty principle in graphene” showed that, by going beyond the low-energy approximation for which the dispersion relations of graphene are linear, the corresponding emergent field theory is a specific generalization a Dirac field theory. The generalized Dirac Hamiltonians one obtains are those compatible with specific generalizations of the uncertainty principle. They also comment on the compatibility of the latter with non-commuting positions, and on their possible physical realization.

Thus, following these works and maybe others which are not cited here, we can expect that our study can be regarded as a contribution to the case of graphene by introducing the minimal length in the commutations rules of Heisenberg.