The DKP oscillator with a linear interaction in the cosmic string space-time

We study the relativistic quantum dynamics of a DKP oscillator field subject to a linear interaction in cosmic string space-time in order to better understand the effects of gravitational fields produced by topological defects on the scalar field. We obtain the solution of DKP oscillator in the cosmic string background. Also, we solve it with an ansatz in presence of linear interaction. We obtain the eigenfunctions and the energy levels of the relativistic field in that background.


Introduction
The Dirac equation including the linear harmonic potential was initially studied by Ito et al. [1], Cook [2] and Ui et al. [3]. This system was latterly called by Moshinsky and Szczepaniak as Dirac oscillator [4], because it behaves as an harmonic oscillator with a strong spin-orbit coupling in the non-relativistic limit. This model is based on the dynamics of a harmonic oscillator for spin-1/2 particles by introducing a nonminimal prescription into free Dirac equation [4]. Physically, it can be shown that the Dirac oscillator interaction is a physical system, which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field [5,6].
As a relativistic quantum mechanical system, the Dirac oscillator has been widely studied. Because it is an exactly solvable model, several investigations have been developed in the context of this theoretical framework in the last years. Although the Dirac oscillator is normally introduced within the context of many body theory, relativistic quantum mechanics and quantum chromodynamics (in particular as an interquark potential and also as the confining part of the phenomenological Cornell potential). The interest in this issue appears in different contexts such as quantum optics [7][8][9], supersymmetry [5,10,11], nuclear reactions [12], the hadronic spectrum using the two-body Dirac oscillator [13,14] a new representation for its solutions using the Clifford algebra [15,16], noncommutative space [17,18], thermodynamic properties [19] Lie Algebra symmetries [20], supersymmetric (non-relativistic) quantum mechanics [21] the super symmetric path integral formalism [22] the chiral phase transition in presence of a constant magnetic field [8],the relativistic Landau in presence of external magnetic field [23] the Aharonov-Bohm effect [24,25] condensed matter physical phenomena and graphene [26].
The DKP oscillator embedded in a magnetic cosmic string background has inspired a great deal of research in last years. A cosmic string is a linear defect that changes the topology of the medium when viewed globally. The spacetime around a cosmic string is locally flat but not globally. The theory of general relativity predicts that gravitation is manifested as curvature of spacetime. This curvature is characterized by the Riemann tensor. There are connections between topological properties of the space and local physical laws. The nontrivial topology of spacetime, as well as its curvature, leads to a number of interesting gravitational effects. For example, it has been known that the energy levels of an atom placed in a gravitational field will be shifted as a result of the interaction of the atom with spacetime curvature. Therefore, we have to consider the topology of the spacetime in order to describe completely the physics of system.
In this work, we examine the relativistic quantum dynamics of DKP oscillator in the presence of the linear interaction, on the curved spacetime of a cosmic string. From the corresponding DKP equation, we analyze the influence of the topological defect on the equation of motion, the energy spectrum and the wave-function. In Sec. 2, we introduce the covariant DKP equation. In Sec. 3 we present the covariant DKP oscillator in cosmic string background and obtain the soloution of DKP oscillator In Sec. 4, we present solution of DKP oscillator presence linear interaction. In the next section we present our conclusions.

Covariant form of the DKP equation in the cosmic string background
The cosmic string spacetime with an internal magnetic field in cylindrical coordinates is described by the line element (units such that with −∞ < z < ∞, ρ ≥ 0 and 0 ≤ ϕ ≤ 2π. The angular parameter 0 < α < 1 is related to the linear mass density µ of the string as α = 1 − 4µ. From the geometrical point of view, the metric (1) describes a Minkowski space-time with a conical singularity.
The DKP equation in the cosmic string spacetime (1) reads [47][48][49] The covariant derivative in (2) is given by [50] where Γ µ are the spinorial affine connections given by The matrices β a (x) are the standard Dirac matrices in Minkowski spacetime and ω a µb represents the spin connection given by The nonnull components of the spin connection are We can build the local reference frame through a non-coordinate basis with eā µ where eā µ and e The vierbeins form our local reference frame that satisfy the orthonormality conditions And satisfy The Kemmer matrices in curved spacetime are related to their Minkowski counterparts via In terms of the Minkowski flat spacetime coordinates, these matrices can be cast into the form The matrices β µ (x) in (6) where β • , β r , β ϕ and β z are the general form of the Kemmer matrices in this spacetime.

Solution of DKP oscillator in cosmic string background
In this section, we concentrate our efforts in the interaction called DKP oscillator. For this external interaction we use the non-minimal substitution where ω is the oscillator frequency and η 0 = 2β 0 2 − 1. Considering only the radial component, with the non-minimal substitution one gets As the interaction is time-independent one can write the spinor as where E is the energy of the scalar boson, and the five-component spinor can be written as Ψ(r ) = ψ 1 (r ), . . . , ψ 5 (r ) T and the DKP equation for scalar bosons becomes (for compactness of next equations, we momentarily drop the r dependence in the spinor components) By solving the above system of equations in favour of ψ 1 (r ) we get Combining these results we obtain an equation of motion for the first component of the DKP spinor In order to solve the above equation, we employ the change of variable: s = r 2 , thus we rewrite the radial equation (21) in the form where which gives the energy levels of the relativistic Klein-Gordon equation from [51] α 2 n−(2n+1)α 5 +(2n+1)( α 9 +α 3 α 8 )+n(n−1)α 3 +α 7 +2α 3 α 8 +2 α 8 α 9 = 0, where As the final step, it should be mentioned that the corresponding wave function is ψ 1 (r ) = N r 2α 12 e α 13 r 2 L α 10 −1 n (α 11 r 2 ). (27) where N is the normalization constant.

Solution of the DKP oscillator in presence linear interaction
Let us now to analyse the situation when a DKP field interacts with a scalar potential U (r ), which is introduced via the substitution M → M +U (r ). Thus, (16) becomes Here we are interested in studying the linear scalar potential: In order to solve Eq. (28) we make the following change of variables: This leads us to an equation without first-order derivative term The next step, is to write R n,m (r ) in the following form R n,m (r ) = R n (r )e g m (r ) .
Thus, the r -dependent terms in Eq. (32) suggest that we take g m (r ) as where the five constants b 1 , . . . , b 4 are to be expressed in terms of the physical constants α, a, ω, M, M m, k z and E . For nodeless states, with n = 0, we have R n (r ) = 1, consequently [52][53][54] R 0,m (r ) = e g m (r ) , In this manner, from Eq. (26) we have Thus substituting Eq. (33) into Eq. (35), the latter equation becomes If we compare Eq. (36) with Eq. (31), we acquire the following six equations (displayed here along with their respective factors in (r ): These equations can be solved for b 1 , b 2 , b 3 and b 4 they also provide constraints on the physical parameters, in particular, the energy E . Equation (37) admit the following solutions: Finally, the solution for DKP oscillator interacting with a linear scalar potential can be written as Ψ n,m (t , r, ϕ, z) = N n,m e i (k z +mϕ−E t) e b 1 r +b 2 r 2 r b 3 + 1−α 2α (M + ar ) b 4 + 1 2 , where N n,m is normalization constant.

Conclusion
In this contribution, we have investigated the relativistic quantum dynamics of DKP oscillator in the presence of the linear interaction, on the curved spacetime of a cosmic string. From the corresponding DKP equation, we analyze the influence of the topological defect on the equation of motion, the energy spectrum and the wave function. We established and found solutions of covariant DKP oscillator in cosmic string background. We present solution of DKP oscillator presence linear interaction. We solved the DKP oscillator analytically by using a proper ansatz solution to the equation.