# Quantum dynamics of scalar bosons in a cosmic string background

## Abstract

The quantum dynamics of scalar bosons embedded in the background of a cosmic string is considered. In this work, scalar bosons are described by the Duffin–Kemmer–Petiau (DKP) formalism. In particular, the effects of this topological defect in the equation of motion, energy spectrum, and DKP spinor are analyzed and discussed in detail. The exact solutions for the DKP oscillator in this background are presented in closed form.

## Keywords

Cosmic String Quantum Dynamic Nonrelativistic Limit Scalar Boson Dirac Oscillator## 1 Introduction

The first-order Duffin–Kemmer–Petiau (DKP) formalism [1, 2, 3, 4] describes spin-zero and spin-one particles and has been used to analyze relativistic interactions of spin-zero and spin-one hadrons with nuclei as an alternative to their conventional second-order Klein–Gordon (KG) and Proca counterparts. Although the formalisms are equivalent in the case of minimally coupled vector interactions [5, 6, 7], the DKP formalism enjoys a richness of couplings not capable of being expressed in the KG and Proca theories [8, 9]. Recently, there has been increasing interest in the so-called DKP oscillator [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. The DKP oscillator considering minimal length [20, 21] and noncommutative phase space [22, 23, 24, 25] have also appeared in the literature. The DKP oscillator is a kind of tensor coupling with a linear potential which leads to the harmonic oscillator problem in the weak-coupling limit. Also, a sort of vector DKP oscillator (non-minimal vector coupling with a linear potential [26, 27, 28, 29, 30] has been a topic of recent investigation. Vector DKP oscillator is the name given to the system with a Lorentz vector coupling which exhibits an equally spaced energy spectrum in the weak-coupling limit. The name distinguishes it from the system called a DKP oscillator with Lorentz tensor couplings of Ref. [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

The DKP oscillator is an analogous to Dirac oscillator [31]. The Dirac oscillator is a natural model for studying properties of physical systems, it is an exactly solvable model, several research have been developed in the context of this theoretical framework in recent years. A detailed description for the Dirac oscillator is given in Ref. [32] and for other contributions see Refs. [33, 34, 35, 36, 37, 38, 39]. The Dirac oscillator embedded in a cosmic string background has inspired a great deal of research in last years [40, 41, 42, 43, 44, 45, 46]. A cosmic string is a linear defect that change the topology of the medium when viewed globally. The influence of this topological defect in the dynamics of spin-1 / 2 particles has been widely discussed in the literature. However, the same problem involves bosons via DKP formalism has not been established. Therefore, we believe that this problem deserves to be explored.

The main motivation of this work is inspired by the results obtained in Ref. [46]. As a natural extension, we address the quantum dynamics of scalar bosons (via DKP formalism) embedded in the background of a cosmic string. The influence of this topological defect in the equation of motion, energy spectrum and DKP spinor are analyzed and discussed in detail. The case of DKP oscillator in this background is also considered. In this case, the problem is mapped into a Schrödinger-like equation embedded in a three-dimensional harmonic oscillator for the first component of the DKP spinor and the remaining components are expressed in terms of the first one in a simple way. Our results are very similar to Dirac oscillator in a cosmic string background, except by the absence of terms that depend on the spin projection parameter.

This work is organized as follows. In Sect. 2, we consider the DKP equation in a curved space-time. We discuss conditions on the interactions which lead to a conserved current in a curved space-time (Sect. 2.1). In Sect. 3, we give a brief review on a cosmic string background and we also analyze the curved-space beta matrices and spin connection in this background. In Sect. 4, we concentrate our efforts in the interaction called DKP oscillator embedded in the background of a cosmic string. In particular, we focus the case of scalar bosons and obtain the equation of motion, energy spectrum and DKP spinor (Sect. 4.1). Finally, in Sect. 5 we present our conclusions.

## 2 Duffin–Kemmer–Petiau equation in a curved space-time

*tetrads*\(e_{\mu }\,^{\bar{a}}(x)\) satisfy the relations

### 2.1 Interaction in the Duffin–Kemmer–Petiau equation

*U*is written in terms of 25 (100) linearly independent matrices pertinent to the five- (ten)-dimensional irreducible representation associated to the scalar (vector) sector. In the presence of interaction, \(J^{\mu }\) satisfies the equation

*U*is Hermitian with respect to \(\eta ^{0}\) and the curved-space beta matrices are covariantly constant then the four-current will be conserved. The potential matrix

*U*can be written in terms of well-defined Lorentz structures. For the spin-zero sector there are two scalar, two vector, and two tensor terms [8], whereas for the spin-one sector there are two scalar, two vector, a pseudoscalar, two pseudovector, and eight tensor terms [9]. The condition (16) for the case of Minkowski space-time has been used to point out a misleading treatment in the recent literature regarding analytical solutions for non-minimal vector interactions [30, 50, 51, 52].

## 3 Cosmic string background

## 4 DKP oscillator in a cosmic string background

*E*is the energy of the scalar boson, in such a way that the time-independent DKP equation becomes

### 4.1 Scalar sector

*z*-direction and adopt the usual decomposition

*r*the square-integrable solution behaves as \(e^{-\lambda r^2/2}\), and thereby the solution for all

*r*can be expressed as

*A*and

*B*are arbitrary constants. The second term in (52) has a singular point at \(\rho =0\), so that we set \(B=0\). The asymptotic behavior of Kummer’s function is dictated by

*n*is a nonnegative integer and \(b\ne -\tilde{n}\), where \(\tilde{n}\) is also a nonnegative integer. In fact, \(M(-n,b,\rho )\) with \(b>0\) is proportional to the generalized Laguerre polynomial \(L_{n}^{(b-1)}(\rho )\), a polynomial of degree

*n*with

*n*distinct positive zeros in the range \([0,\infty )\). Therefore, the solution for all

*r*can be written as

*m*. This fact is associated to the fact that the DKP oscillator embedded in a cosmic string background does not distinguish particles from antiparticles. At this stage, due to invariance under rotation along the z-direction, without loss of generality we can fix \(k_{z}=0\). In general \(|E|>M\), except for \(\omega =0\); then the spectrum acquiesces to \(|E|=M\).

*n*and |

*m*|, the energy |

*E*| increases as \(\alpha \) decreases.

*n*and |

*m*|, the distribution has a maximum at \(r\approx 1.7\) for \(\alpha =1\), and this maximum decreases and moves to positive

*r*-direction as \(\alpha \) increases. In addition, comparison with \(|\phi _{1}|^{2}\) shows that \(\alpha =1\) tends to be better localized than \(\alpha <1\). From this, we can conclude that in the limit \(\alpha \rightarrow 0\) one has \(N_{n}\rightarrow 0\), so that the solutions \(\phi _{1}\) tends to disappear one after another as \(\alpha \rightarrow 0\). The comparison between the profiles of \(|\phi _{1}|^{2}\) for \(n=0\) and \(n=2\) are shown in Fig. 4 for \(|m|=1\) and different values of \(\alpha \). Figure 4 clearly shows the effects of \(\alpha \) on the excited modes, which are qualitatively similar to \(n=0\). Finally, Fig. 5 illustrates the behavior of \(|\phi _{1}|^{2}\) for \(n=2\), \(|m|=1\), and \(\alpha =0.5\) in polar coordinates. One can see that scalar bosons tend to be better localized at the blue region.

## 5 Conclusions

We studied the Duffin–Kemmer–Petiau (DKP) equation in a curved space-time and we found the general condition on the interactions which leads to a conserved current. This result is a generalization of [27] (Minkowski space-time). Furthermore, we showed that considering a cosmic string background and a DKP oscillator interaction, they furnish a conserved current.

Considering only scalar bosons, we showed that the motion equation which describes the quantum dynamics of a DKP oscillator in a cosmic string background was mapped into a Schrödinger-like equation embedded in a three-dimensional harmonic oscillator for the first component of the DKP spinor; and the remaining components were expressed in terms of the first one in a simple way. Our result is very similar to a Dirac oscillator in a cosmic string background, except for the absence of some terms that depend on the spin projection parameter [46].

We found the spectrum of energy for this background and we showed that the energy |*E*| increases as \(\alpha \) decreases. Both particle and antiparticle energy levels are members of the spectrum, and the particle and antiparticle spectra are symmetrical about \(E=0\). That fact implies that there is no channel for spontaneous boson–antiboson creation. We also found that both weak-coupling limit and nonrelativistic limit furnish an equally spaced energy spectrum, so that we concluded that this problem describes a genuine DKP oscillator.

The behavior of the solutions for this problem was discussed in detail. We showed that the cosmic string background influences the scalar bosons localization. As an important result, we showed that \(\alpha =1\) tends to be better localized than \(\alpha <1\) (see Fig. 3). Also, we showed that in the limit \(\alpha \rightarrow 0\) the solutions \(\phi _{1}\) tend to disappear.

Beyond investigating the quantum dynamics of scalar bosons in a cosmic string background, the results of this paper could be used, in principle, in condensed matter physics, owing to the analogy between cosmic strings and disclinations in solids [56]. Another physical application could be associated to Bose–Einstein condensates (BEC) [57, 58] and neutral atoms. It is well known that condensates can be exploited for building a coherent source of neutral atoms [59], which in turn can be used to study entanglement and quantum information processing [60].

Finally, it is worthwhile to mention that the natural extension of the present work is to consider more general backgrounds, as for instance a global monopole [61] and a spinning cosmic string [62, 63], among others.

## Notes

### Acknowledgments

The author would like to thank Edilberto O. Silva for fruitful discussions. Thanks also go to the referee for useful comments and suggestions. This work was supported in part by means of funds provided by CNPq, Brazil, Grants No. 455719/2014-4 (Universal) and No. 304105/2014-7 (PQ).

## References

- 1.G. Petiau, Acad. R. Belg. Cl. Sci. Mem. Collect.
**16**, 2 (1936)Google Scholar - 2.N. Kemmer, Proc. R. Soc. Lond. A
**166**, 127 (1938). doi: 10.1098/rspa.1938.0084. http://rspa.royalsocietypublishing.org/content/166/924/127.short - 3.R.J. Duffin, Phys. Rev.
**54**, 1114 (1938). doi: 10.1103/PhysRev.54.1114 - 4.N. Kemmer, Proc. R. Soc. Lond. A
**173**, 91 (1939). doi: 10.1098/rspa.1939.0131. http://rspa.royalsocietypublishing.org/content/173/952/91.short - 5.M. Nowakowski, Phys. Lett. A
**244**, 329 (1998). doi: 10.1016/S0375-9601(98)00365-X - 6.J.T. Lunardi, B.M. Pimentel, R.G. Teixeira, J.S. Valverde, Phys. Lett. A
**268**, 165 (2000). doi: 10.1016/S0375-9601(00)00163-8 - 7.L.B. Castro, A.S. de Castro, Phys. Rev. A
**90**, 022101 (2014). doi: 10.1103/PhysRevA.90.022101 - 8.R.F. Guertin, T.L. Wilson, Phys. Rev. D
**15**, 1518 (1977). doi: 10.1103/PhysRevD.15.1518 - 9.B. Vijayalakshmi, M. Seetharaman, P.M. Mathews, J. Phys. A Math. Gen.
**12**, 665 (1979). doi: 10.1088/0305-4470/12/5/015 - 10.N. Debergh, J. Ndimubandi, D. Strivay, Z. Phys. C
**56**, 421 (1992). doi: 10.1007/BF01565950 ADSCrossRefGoogle Scholar - 11.Y. Nedjadi, R.C. Barrett, J. Phys. A Math. Gen.
**27**, 4301 (1994). http://stacks.iop.org/0305-4470/27/i=12/a=033 - 12.Y. Nedjadi, S. Ait-Tahar, R.C. Barrett, J. Phys. A Math. Gen.
**31**, 3867 (1998). http://stacks.iop.org/0305-4470/31/i=16/a=014 - 13.Y. Nedjadi, R.C. Barrett, J. Phys. A Math. Gen.
**31**, 6717 (1998). http://stacks.iop.org/0305-4470/31/i=31/a=016 - 14.A. Boumali, L. Chetouani, Phys. Lett. A
**346**, 261 (2005). doi: 10.1016/j.physleta.2005.08.002 - 15.I. Boztosun, M. Karakoc, F. Yasuk, A. Durmus, J. Math. Phys.
**47**, 062301 (2006). doi: 10.1063/1.2203429 - 16.A. Boumali, Phys. Scr.
**76**, 669 (2007). http://stacks.iop.org/1402-4896/76/i=6/a=014 - 17.F. Yasuk, M. Karakoc, I. Boztosun, Phys. Scr.
**78**, 045010 (2008). http://stacks.iop.org/1402-4896/78/i=4/a=045010 - 18.A. Boumali, J. Math. Phys.
**49**, 022302 (2008). doi: 10.1063/1.2841324 - 19.Y. Kasri, L. Chetouani, Int. J. Theor. Phys.
**47**, 2249 (2008). doi: 10.1007/s10773-008-9657-6 MathSciNetCrossRefzbMATHGoogle Scholar - 20.M. Falek, M. Merad, J. Math. Phys.
**50**, 023508 (2009). doi: 10.1063/1.3076900 - 21.M. Falek, M. Merad, J. Math. Phys.
**51**, 033516 (2010). doi: 10.1063/1.3326236 - 22.M. Falek, M. Merad, Commun. Theor. Phys.
**50**, 587 (2008). http://stacks.iop.org/0253-6102/50/i=3/a=10 - 23.G. Guo, C. Long, Z. Yang, S. Qin, Can. J. Phys.
**87**, 989 (2009). doi: 10.1139/P09-060 ADSCrossRefGoogle Scholar - 24.Z.H. Yang, C.Y. Long, S.J. Qin, Z.W. Long, Int. J. Theor. Phys.
**49**, 644 (2010). doi: 10.1007/s10773-010-0244-2 MathSciNetCrossRefzbMATHGoogle Scholar - 25.H. Hassanabadi, Z. Molaee, S. Zarrinkamar, Eur. Phys. J. C
**72**, 2217 (2012). doi: 10.1140/epjc/s10052-012-2217-5 ADSCrossRefGoogle Scholar - 26.D.A. Kulikov, R.S. Tutik, A.P. Yaroshenko, Mod. Phys. Lett. A
**20**, 43 (2005). doi: 10.1142/S0217732305016324 - 27.T.R. Cardoso, L.B. Castro, A.S. de Castro, J. Phys. A Math. Theor.
**43**, 055306 (2010). doi: 10.1088/1751-8113/43/5/055306 - 28.T.R. Cardoso, L.B. Castro, A.S. de Castro, Nucl. Phys. B Proc. Suppl.
**199**(203), 2010 (2009). doi: 10.1016/j.nuclphysbps.2010.02.029. Proceedings of the International Workshop Light Cone (LC2009): Relativistic Hadronic and Particle Physics - 29.L.B. Castro, A.S. de Castro, Phys. Lett. A
**375**, 2596 (2011). doi: 10.1016/j.physleta.2011.05.067 - 30.L.B. Castro, L.P. de Oliveira, AdHEP 2014 (2014). doi: 10.1155/2014/784072
- 31.M. Moshinsky, A. Szczepaniak, J. Phys. A Math. Gen.
**22**, L817 (1989). http://stacks.iop.org/0305-4470/22/i=17/a=002 - 32.P. Strange, Relativistic Quantum Mechanics with Applications in Condensed Matter and Atomic Physics (Cambridge University Press, Cambridge, 1998). doi: 10.1017/CBO9780511622755
- 33.N. Ferkous, A. Bounames, Phys. Lett. A
**325**, 21 (2004). doi: 10.1016/j.physleta.2004.03.033 - 34.K. Nouicer, J. Phys. A Math. Gen.
**39**, 5125 (2006). http://stacks.iop.org/0305-4470/39/i=18/a=025 - 35.A.S. de Castro, P. Alberto, R. Lisboa, M. Malheiro, Phys. Rev. C
**73**, 054309 (2006). doi: 10.1103/PhysRevC.73.054309 - 36.J. Carvalho, C. Furtado, F. Moraes, Phys. Rev. A
**84**, 032109 (2011). doi: 10.1103/PhysRevA.84.032109 - 37.K. Bakke, C. Furtado, Ann. Phys. (N.Y)
**336**, 489 (2013). doi: 10.1016/j.aop.2013.06.007 - 38.F.M. Andrade, E.O. Silva, M.M. Ferreira Jr., E.C. Rodrigues, Phys. Lett. B
**731**, 327 (2014). doi: 10.1016/j.physletb.2014.02.054 - 39.L.B. Castro, A.S. de Castro, P. Alberto, Ann. Phys. (N.Y)
**356**, 83 (2015). doi: 10.1016/j.aop.2015.02.033 - 40.M. Alford, J. March-Russell, F. Wilczek, Nucl. Phys. B
**328**, 140 (1989). doi: 10.1016/0550-3213(89)90096-5 - 41.M.G. Alford, F. Wilczek, Phys. Rev. Lett.
**62**, 1071 (1989). doi: 10.1103/PhysRevLett.62.1071 - 42.C. Filgueiras, F. Moraes, Phys. Lett. A
**361**, 13 (2007). doi: 10.1016/j.physleta.2006.09.030 - 43.H. Belich, E.O. Silva, M.M. Ferreira, M.T.D. Orlando, Phys. Rev. D
**83**, 125025 (2011). doi: 10.1103/PhysRevD.83.125025 - 44.F.M. Andrade, E.O. Silva, M. Pereira, Phys. Rev. D
**85**, 041701 (2012). doi: 10.1103/PhysRevD.85.041701 - 45.F.M. Andrade, E.O. Silva, M. Pereira, Ann. Phys. (N.Y)
**339**, 510 (2013). doi: 10.1016/j.aop.2013.10.001 - 46.F.M. Andrade, E.O. Silva, Eur. Phys. J. C
**74**, 3187 (2014). doi: 10.1140/epjc/s10052-014-3187-6 ADSCrossRefGoogle Scholar - 47.J.T. Lunardi, B.M. Pimentel, R.G. Teixeira, Gen. Rel. Gravit.
**34**, 491 (2002). doi: 10.1023/A:1015540708007 MathSciNetCrossRefzbMATHGoogle Scholar - 48.R. Casana, J.T. Lunardi, B.M. Pimentel, R.G. Teixeira, Gen. Rel. Gravit.
**34**, 1941 (2002). doi: 10.1023/A:1020732611995 MathSciNetCrossRefzbMATHGoogle Scholar - 49.R.A. Krajcik, M.M. Nieto, Phys. Rev. D
**10**, 4049 (1974). doi: 10.1103/PhysRevD.10.4049 - 50.T.R. Cardoso, L.B. Castro, A.S. de Castro, Can. J. Phys.
**87**, 857 (2009). doi: 10.1139/P09-054 ADSCrossRefGoogle Scholar - 51.T.R. Cardoso, L.B. Castro, A.S. de Castro, Can. J. Phys.
**87**, 1185 (2009). doi: 10.1139/P09-082 ADSCrossRefGoogle Scholar - 52.T.R. Cardoso, L.B. Castro, A.S. de Castro, J. Phys. A Math. Theor.
**45**, 075302 (2012). doi: 10.1088/1751-8113/45/7/075302 - 53.E.R.B. de Mello, JHEP
**2004**(06), 016 (2004)Google Scholar - 54.Y. Nedjadi, R.C. Barrett, J. Phys. G Nucl. Part. Phys.
**19**, 87 (1993). doi: 10.1088/0954-3899/19/1/006 - 55.M. Abramowitz, I.A. Stegun,
*Handbook of Mathematical Functions*(Dover, Toronto, 1965)Google Scholar - 56.D.R. Nelson,
*Defects and Geometry in Condensed Matter Physics*(Cambridge University Press, Cambridge, 2002)Google Scholar - 57.R. Casana, V.Y. Fainberg, B.M. Pimentel, J.S. Valverde, Phys. Lett. A
**316**(1), 33 (2003). doi: 10.1016/S0375-9601(03)01018-1 - 58.L.M. Abreu, A.L. Gadelha, B.M. Pimentel, E.S. Santos, Phys. A Stat. Mech. Appl.
**419**(0), 612 (2015). doi: 10.1016/j.physa.2014.10.049 - 59.M.O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, W. Ketterle, Phys. Rev. Lett.
**78**, 582 (1997). doi: 10.1103/PhysRevLett.78.582 - 60.D. Jaksch, H.J. Briegel, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett.
**82**, 1975 (1999). doi: 10.1103/PhysRevLett.82.1975 - 61.A.L.C. de Oliveira, E.R.B. de Mello, Class. Quant. Gravit.
**23**(17), 5249 (2006). http://stacks.iop.org/0264-9381/23/i=17/a=009 - 62.G. ClTment, Ann. Phys. (N.Y)
**201**(2), 241 (1990). doi: 10.1016/0003-4916(90)90041-L - 63.C.R. Muniz, V.B. Bezerra, M.S. Cunha, Ann. Phys. (N.Y)
**350**(0), 105 (2014). doi: 10.1016/j.aop.2014.07.017

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