Sakarya University Journal of Science SAUJS e-ISSN 2147-835X | Period Bimonthly | Founded: 1997 | Publisher Sakarya University | http://www.saujs.sakarya.edu.tr/en/

Title: Effects of Rotating Frame on a Vector Boson Oscillator

Authors: Abdullah GUVENDİ

Recieved: 2021-04-07 17:07:23 Accepted: 2021-05-07 20:36:07 Article Type: Research Article Volume: 25 Issue: 3 Month: June Year: 2021 Pages: 834-840 How to cite Abdullah GUVENDİ; (2021), Effects of Rotating Frame on a Vector Boson Oscillator. Sakarya University Journal of Science, 25(3), 834-840, DOI: https://doi.org/10.16984/saufenbilder.911340 Access link http://www.saujs.sakarya.edu.tr/en/pub/issue/62736/911340

New submission to SAUJS http://dergipark.org.tr/en/journal/1115/submission/step/manuscript/new Sakarya University Journal of Science 25(3), 834-840, 2021

Effects of Rotating Frame on a Vector Boson Oscillator

Abdullah GUVENDI*1

Abstract

We analyze the effects of the and angular velocity of rotating frame on the dynamics of a relativistic vector boson oscillator ( ). To determine these effects on the energy of the we solve the corresponding vector boson equation in the rotating frame of 2+1 dimensional cosmic string-induced spacetime background. We obtain an exact energy spectrum, which depends on the angular velocity of the rotating frame and angular deficit parameter of the background. We show that the effects of angular deficit parameter on each energy level of the cannot be same and the angular velocity of the rotating frame couples with the spin of the . Furthermore, we have obtained that the angular velocity of rotating frame breaks the symmetry of the positive-negative energy states.

Keywords: Non-inertial effects, quantum fields in curved spaces, topological defects.

the symmetric spinor can be constructed by 1. INTRODUCTION kronocker product of two dirac spinors [1,4-6]. However, it is important to underline that this The relativistic wave equations for spin-1/2, spin- symmetric spinor does not include the spin-0 1 and spin-3/2 particles can be derived via sector in 2+1 [7]. The vector boson canonical quantization of the classical spinning equation ( ) corresponds to the spin-1 part of particle with zitterbewegung. These equations can the equation in 2+1 dimensions [1,4-7] and be obtained as possible quantum states from the its massless form can also be derived via the same mentioned classical quantization procedure [1]. quantization procedure [4]. It has been declared After demonstrating that there is a unifying that massless and Maxwell equations are principle for this class of relativistic wave equivalent [4], and moreover it has been clearly equations [1], it has been shown that the well- shown that the state functions for massless form known duffin-kemmer-proca equations and the of the have probability interpretation [4]. A many-body dirac equation (fully-covariant) [1-3] few applications of the can be found in the can be derived from similar action. Spin-1 sector refs. [5-8]. In this work, we introduce the vector of the duffin-kemmer-petiau ( ) equation has boson oscillator ( ) and investigate its been derived as excited state via the mentioned dynamics in the rotating frame of the 2+1 procedure provided that the spinor is a symmetric dimensional cosmic string-generated spacetime. spinor of rank-two [1,4,5]. Under this condition,

*Corresponding author: [email protected] 1Kutahya Health Sciences University, Faculty of Engineering and Natural Sciences, Department of Basic Sciences, 43100, Kutahya. E-Mail: [email protected] ORCID: https://orcid.org/ 0000-0003-0564-9899 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator

Cosmic strings are predicted to be linear where the particles can be placed [20-22]. It has topological defects that probably occurred in the been shown that the phase shift in neutron early stages of the universe [9]. After Cosmic interferometry is due to the coupling of angular strings were introduced in dimensional velocity of the earth with the angular momentum spacetime [10], the obtained solution was of the neutron [26]. The effects of rotating frame naturally expanded to dimensions where on the molecules were analized [27,28] and there is a dynamical symmetry [11]. The the obtained results have indicated that rapidly spacetime backgrounds of cosmic strings include rotating molecules can acquire a topological a conical singularity on the intersection point of phase [27,28]. the string with the space-like subspace [9-11]. Even though these spacetime backgrounds are In this contribution, we investigate the influences locally flat [9-11] out of the intersection point, of rotating frame on the relativistic dynamics of a they are not flat when viewed globally [9-11]. in the -dimensional cosmic string Because of this feature, the cosmic strings can be spacetime. In order to determine the effects of responsible for several physical events in the both uniformly rotating frame and spacetime universe [12-15]. The spacetime background of topology on the system in question we solve the the cosmic string is characterized by parameter, corresponding . We obtain an exact energy which is angular deficit parameter depending on spectrum depending on the parameters of the linear mass density ( ) of the cosmic string spacetime background. [12-15]. The gravitational [9-14] and electrostatic effects [12,15] of these topological defects on the 2. GENERALIZED FORM OF THE dynamics of physical systems depend on the parameter . The influences of topological defects on the dynamics of quantum mechanical In this part of the manuscript, we introduce the systems have been widely studied [15-17]. In one generalized form of the and arrive at a set of of such works, it has been shown that the spin- coupled equations for a in the rotating frame symmetric [18] quantum state of a positronium of 2+1 dimensional cosmic string-generated system can be used to detect the topological spacetime background.The generalized form of defects [15]. The effects of topological defects on the can be written as the following [1-8], the spectral lines of quantum systems are not same for each spectral line of the quantum systems [19]. Ψ(x)=0, This means that the effects of topological defects , are distinguishable from the other effects such as doppler effect and [15,19]. . (1)

On the other hand, the investigations about the In eq. (1), are the generalized dirac matrices, effects of rotating frames on the quantum systems are the spinorial affine connections for spin-1/2 have provided very interesting results and field, m is the rest mass of vector boson, c is the accordingly such studies have attracted great speed of , is usual planck constant, the attention for a long time [20-22] since these symbol indicates kronocker product [28], x is investigations have discovered very important the spacetime position vector, and stand for effects [23-29]. The sagnac effect [23], mashhoon the and dimensional identity matrices, effect [24], persistent currents in the quantum ring and the coupling between the angular momentum respectively, and is the symmetric spinor of of the quantum systems with the velocity of rank-two. The dimensional spacetime uniformly rotating frame [25] can be considered geometry of the uniformly rotating cosmic string- among these effects. One of the interesting results generated background is represented through the of these investigations is that the non-inertial following [20-22], effects stemming from the rotating frames restrict the physical region of the spacetime backgrounds

Sakarya University Journal of Science 25(3), 834-840, 2021 835 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator

- - . (2) eq. (2) the space-dependent dirac matrices are obtained, In eq. (2), is the angular deficit parameter, depending on the linear mass = , = , = + . (5) density of the cosmic string and newtonian gravitational constant ) [15]. The azimuthal The choice of space-independent dirac matrices is angle range is in this spacetime not unique and different choices can be preferred, background. Here, we deal with 0 case, of course. According to signature of eq. (2) the since the case describes an anti-conical space-independent dirac matrices can be chosen spacetime background with negative as = , = , = [29-31], in which [20-22]. Therefore, the case does not make , and are the pauli spin matrices. Also, sense in the context [20-22]. The we can obtain the spinorial connections for spin- eq. (2) describes a flat spacetime background, in 1/2 fields via the following expression [15], terms of polar coordinates, when and [15]. In eq. (2), we should notice that the radial coordinate must satisfy , since (휆,σ,μ,ν,a,b=0,1,2.), (6) this spacetime geometry can be defined for values of the inside the range: 0 . This where are the christoffel symbols obtained condition guarantees that the vector boson through =1/2 ( ) and considered is placed inside of the light-cone and are the spin operators determined by imposes a spatial constraint. That is, the wave [ ]. Now, one can obtain the function vanishes at = [20-22]. At this spinorial connections (non-vanishing) as the point, we should underline that the in eq. (2) is following, and is the angular velocity (not frequency) of the uniformly rotating frame. Therefore, the = , = . (7) rises to a hard-wall confining potential [20-22]. Now, one can obtain the tetrads [15], The coupling, describing the interaction of magnetic moment (anomalous) with a linearly varying electric field [3,16,17,21], can be introduced through the following non-minimal ( ). (3) substitution, In eq. (3), is the flat spacetime metric tensor + ( )r, and hence the indices indicate the local reference frame. The tetrads, which satisfy the ( ) = diag (1,-1,-1,1), (8) following orthonormality conditions, where is the oscillator frequency (coupling , , (4) strength) [3]. Eq. (2) allows to factorization of the spinor as the following, are obtained as the following,

Ψ(t,r,ϕ)= , (9) .

By using the tetrads one can obtain the where and are the total energy and spin of the generalized dirac matrices via the relation system under consideration. = where are the space-independent dirac matrices [15]. According to line element in

Sakarya University Journal of Science 25(3), 834-840, 2021 836 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator

3. ENERGY SPECTRUM obtained [7]. At any rate, the solution function of eq. (20) is found in terms of the whittaker function In this section, we obtain an equation system 푾퐴,퐺 as 휓(ξ) = 푁푾퐴,퐺 (ξ), where 푁 is an consisting of three equations for the components arbitrary constant [7]. The function 푾퐴,퐺 can only of the symmetric spinor and arrive at an exact be polynomial degree of n if the parameters 1 energy spectrum for the in question. To end satisfy the termination condition that reads as + 2 this, we substitute eq. (5), (7), (8) and eq. (9) into 퐺 − 퐴 = − n [7]. This termination condition the eq. (1) and then we can acquire a set of gives a condition for the quantization of the coupled equations, in the most symmetric form, energy of system in question and accordingly one as the following, can obtain the following non-perturbative energy spectrum, W –M 2 M – + , (13) W –B (10) which can be clarified as, here, , provided that m and the abbreviations are given as the following, (14)

W= j , , M= , κ= . if and The result in eq. (14) is exactly same with the previously announced By considering a new change of variable, ξ= κ result for one-dimensional kemmer oscillator [32] we can write the equation system in eq. (10) as the and form of the eq. (14) is same with the result of following, one-dimensional dirac oscillator [33]. In the latest comparison (see eq. (14) and [33]), one can realize that the difference is only mass factor of W –M the particles. Also, it is very interesting that the result in eq. (14) is found to be in a good M agreement with the recently announced results for a composite structure consisting of a fermion- – antifermion pair interacting via dirac oscillator coupling [3]. Therefore, eq. (13) gives an W –M (11) opportunity to analyze the effects of both uniformly rotating frame and spacetime topology on the energy of the . In eq. (13) we see that one of which is algebraic. By solving these the effects of and on each energy level of the equations in favor of we can arrive at a cannot be same. It is clear that the wave equation. Through an ansatz reads as, parameters and do not any impact on the , the wave equation becomes as the energy of when following, 4. RESULTS AND DISCUSSIONS , In this contribution, we have investigated the , . (12) influences of uniformly rotating frame and spacetime topology on a relativistic vector boson oscillator. In order to analyze the effects of angular velocity (Ω) of the rotating frame and Dividing each term in this equation by 휉2, the angular deficit of the spacetime background on well-known whittaker differential equation is

Sakarya University Journal of Science 25(3), 834-840, 2021 837 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator the energy of the vector boson oscillator between the particle and topological defect. we have solved the generalized vector boson Therefore, one can also conclude that the equation in the rotating frame of a 2+1 parameter restricts the relativistic dynamics of dimensional spacetime. The spacetime has been the quantum system under consideration whether considered as -dimensional static cosmic or . string-induced spacetime [15]. The rotating frame of this background is introduced in eq. (2), which Funding represents to the flat Minkowski spacetime in terms of polar coordinates when angular velocity The author has no received any financial support of the rotating frame and angular deficit of the for this research. background vanish. We have performed an exact solution for the system in question and The Declaration of Conflict of Interest/ accordingly we have arrived at a spectrum in Common Interest energy domain. This energy spectrum has given No conflict of interest or common interest has in eq. (13) and it can be reduced in the form of eq. been declared by the author. The author have fully (14) when and . We have verified disclosed conflict of interest situations to the that the eq. (14) is exactly same with the journal. previously announced results for one-dimensional kemmer oscillator [32] and form of the eq. (14) is Author' Contribution also same with the results obtained for one- dimensional dirac oscillator [33]. Furthermore, This article has been prepared by the author. eq. (14) has been found to be in a good agreement with the previously published results for a The Declaration of Ethics Committee Approval composite structure consisting of a fermion- antifermion pair that they interacts through dirac This study does not require ethics committee oscillator coupling [3]. The non-perturbative permission or any special permission. energy spectrum in eq. (13) includes the effects of spin coupling, uniformly rotating frame, angular The Declaration of Research and Publication deficit parameter of the spacetime background, Ethics frequency (coupling strength) of the oscillator and mass of the vector boson. This property of the The author declares that he complies with the energy spectrum provides a suitable basis to scientific, ethical and quotation rules of SAUJS in all processes of the present paper and that he does discuss the effects of both and on the system not make any falsification on the data collected. under consideration. The results have shown that Also, the author declares that Sakarya University the parameter of the rotating frame breaks the Journal of Science and its editorial board have no symmetry (positive-negative) of the energy levels responsibility for any ethical violations that may and it is also clear that the effects of and on be encountered, and that the present study has not each energy level of the system cannot be same. been evaluated in any academic publication As it can be seen in eq. (2) that the information environment other than Sakarya University about the spacetime topology is carried by the Journal of Science. angular coordinate and accordingly parameter of the background alters differently the energy REFERENCES levels corresponding to possible spin eigen-states, when . We can also see that [1] A. O. Barut, “Excited states of such a system does not sense the effects of the zitterbewegung,” Letters B, vol. parameter when and In this paper, 237, no. 3, pp. 436-439, 1990. all of the discussions have been made for the parameters defined in the range and [2] A. O. Barut and S. Komy, “Derivation of where is the distance nonperturbative Relativistic Two-Body

Sakarya University Journal of Science 25(3), 834-840, 2021 838 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator

Equations from the Action Principle in [12] B. Linet, “Force on a charge in the space- Quantumelectrodynamics,” Fortschritte der time of a cosmic string,” Physical Review Physik/Progress of Physics, vol. 33, no. 6. D, vol. 33, no. 6. pp 1833, 1986. pp 309-318, 1985. [13] D. D. Harari and V. D. Skarzhinsky, “Pair [3] A. Guvendi, “Relativistic Landau for a production in the gravitational field of a fermion-antifermion pair interacting cosmic string,” Physics Letters B, vol. 240, through Dirac oscillator interaction,” The no. 3-4. pp 322-326, 1990. European Physical Journal C, vol. 81, no. 2. pp 1-7, 2021. [14] L. Parker, “Gravitational particle production in the formation of cosmic [4] N. Ünal, “A simple model of the classical strings,” Physical Review Letters, vol. 59, zitterbewegung: photon wave no. 12. pp 1369, 1987. function,”Foundations of Physics, vol. 27, no. 5. pp 731-746, 1997. [15] A. Guvendi and Y. Sucu, “An interacting fermion-antifermion pair in the spacetime [5] Y. Sucu and N. Ünal, “Vector bosons in the background generated by static cosmic expanding universe,” The European string,” Physics Letters B, vol. 811, no. Physical Journal C, vol. 44, no. 2. pp 287- 135960. pp 135960, 2020. 291, 2005. [16] M. Hosseinpour, H. Hassanabadi and F. M. [6] Y. Sucu and C. Tekincay, “Photon in the Andrade,“The DKP oscillator with a linear Earth-ionosphere cavity: Schumann interaction in the cosmic string space-time,” resonances,” Astrophysics and Space The European Physical Journal C, vol. 78, Science, vol. 364, no. 4. pp 1-7, 2019. no. 2. pp 1-7, 2018.

[7] M. Dernek and S. G. Doğan and Y. Sucu [17] J. Carvalho, C. Furtado and F. Moreas, and N. Ünal, “Relativistic quantum “Dirac oscillator interacting with a mechanical spin-1 wave equation in 2+1 topological defect,” Physical Review A, dimensional spacetime,” Turkish Journal of vol. 84, no. 3. pp 032109, 2011. Physics, vol. 42, no. 5. pp 509-526, 2018. [18] A. Guvendi, “The lifetimes for each state of [8] G. Gecim and Y. Sucu, “The GUP effect on levels of the para-positronium,” Eur. tunneling of massive vector bosons from the Phys. J. Plus, vol. 136, no. 4. pp 1-10, 2021. 2+1 dimensional blackhole,” Advances in High Energy Physics, vol. 2018, no. 8. pp 1- [19] G. A. Marques and V. B. Bezerra, 8, 2018. “Hydrogen atom in the gravitational fields of topological defects,” Physical Review D, [9] A. Vilenkin, “Cosmic strings and domain vol. 66, no. 10. pp 105011, 2002. walls,” Physics Reports, vol. 121, no. 5. pp 263-315, 1985. [20] K. Bakke and C. Furtado, “Bound states for neutral particles in a rotating frame in the [10] S. Deser, R. Jackiw and G. Hooft, “Three- cosmic string spacetime,” Physical Review dimensional Einstein : dynamics of D, vol. 82, no. 8. pp 084025, 2010. flat space,” Annals of Physics, vol. 152, no. 1. pp 220-235, 1984. [21] S. Zare, H. Hassanabadi and M. de Montigny, “Non-inertial effects on a [11] J. R. Gott and M. Alpert, “General relativity generalized DKP oscillator in a cosmic in a (2+1)-dimensional space-time,” string space-time,” General Relativity and General Relativity and Gravitation, vol. 16, Gravitation, vol. 52, no. 3. pp 1-20, 2020. no. 3. pp 243-247, 1984.

Sakarya University Journal of Science 25(3), 834-840, 2021 839 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator

[22] L. C. N. Santos and C. C. Barros, “Scalar Uniform Magnetic Field,” Few-Body bosons under the influence of noninertial Systems, vol. 62, no. 1. pp 1-8, 2021. effects in the cosmic string spacetime,” The European Physical Journal C, vol. 77, no. 3. [32] A. Boumali, “One-dimensional thermal pp 1-7, 2017. properties of the Kemmer oscillator,” Physica Scripta, vol. 76, no. 6. pp 669, [23] E. J. Post, “Sagnac Effect,” Reviews of 2007. Modern Physics, vol. 39, no. 2. pp 475-493, 1967. [33] M. H. Pacheco, R. R. Landim and C. A. S. Almeida, “One-dimensional Dirac [24] B. Mashhoon, “Neutron interferometry in a oscillator in a thermal bath,” Physics Letters rotating frame of reference,” Physical A, vol. 311, no. 2-3. pp 93-96, 2003. Review Letters, vol. 61, no. 23. pp 2639- 2642, 1988.

[25] F. W. Hehl and W. T. Ni, “Inertial effects of a Dirac particle,” Physical Review D, vol. 42, no. 6. pp 2045-2048, 1990.

[26] S. A. Werner, J. L. Staudenmann and R. Corella, “Effects of Earth’s rotation on the Quantum Mechanical Phase of the Neutron,” Physical Review Letters, vol. 42, no. 17. pp 1103-1106, 1979.

[27] J. Q. Shen and S. L. He, “Geometric phases of electrons due to spin-rotation coupling in rotating molecules,” Physical Review B, vol. 68, no. 19. pp 195421, 2003.

[28] J. Q. Shen, S. L. He and F. Zhuang, “Aharonov-Carmi effect and energy shift of valence electrons in rotating molecules,” The European Physical Journal D, vol. 33, no. 1. pp 35-38, 2005.

[29] A. Guvendi, R. Sahin and Y. Sucu, “Exact solution of an exciton energy for a monolayer medium,” Scientific Reports, vol. 9, no. 1. pp 1-6, 2019.

[30] A. Guvendi, R. Sahin and Y. Sucu, “Binding energy and decaytime of exciton in dielectric medium,” The European Physical Journal B, vol. 94, no. 1. pp 1-7, 2021.

[31] A. Guvendi and S. G. Doğan, “Relativistic Dynamics of Oppositely charged Two Fermions Interacting with External

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