arXiv:2101.09168v1 [gr-qc] 21 Jan 2021 lcrmgeimi v-iesoa eso fgnrlrltvt g relativity general gravitatio of dimen of version five-dimensional unification spatial in es geometrical the electromagnetism proposal This where an new space-time, five-dimensional. through This the becomes introduced [3]). in be also, dimension can (see (compactified) electromagnetism [2] the Klein that and interaction tablished fundamental [1] of Kaluza theory by unified orated a of proposal first The Introduction 1 ei neatos hrnvBh ffc,seilfunctions. special effect, Aharonov-Bohm interactions, netic iercnnmn fgnrlzdK-siltrwt a with KG-oscillator generalized of confinement Linear nfr antcfil nKlz-li hoyand theory Kaluza-Klein in field magnetic uniform ASNumber: PACS keywords: h nrylvlfrec ailmd eedo h lblpar effect. global quantum the a potenti on shows confining which depend the field mode magnetic space-time, radial the each characterizing observe We for eters level states. bound energy for analyze the effect and Aharonov-Bohm equation the potent this of confining solve logue We linear theory. a Kaluza-Klein under the defects topological by backg space-time duced five-dimensional in field magnetic uniform ainlAaey arpr733,Asm India Assam, Gauripur-783331, Academy, National nti ae,w ov eeaie Goclao interact KG-oscillator generalized solve we paper, this In auaKenter,Rltvsi aeeuto,electromag- equation, wave Relativistic theory, Kaluza-Klein -al [email protected] E-mail: 36.w 36.m 36.e 12.d 04.50.Cd, 11.27.+d, 03.65.Ge, 03.65.Pm, 03.65.-w, hrnvBh effect Aharonov-Bohm azdi Ahmed Faizuddin Abstract 1 l n the and al, on pro- round a using ial iha with s nana- an a elab- was s that am- v some ave and n extra sion - interesting results. The idea behind introducing additional space-time di- mensions has found wide applications in quantum field theory [4]. Station- ary cylindrically symmetric solutions to the five-dimensional Einstein and Einstein–Gauss–Bonnet equations has studied in [5]. Few examples of these solutions are the five-dimensional generalizations of cosmic string, chiral cos- mic string [6, 7], and magnetic flux string [8] space-times. The Kaluza-Klein theory (KKT) has investigated in several branches of physics. For example, in Khaler fields [9], in the presence of torsion [10, 11], in the Grassmannian context [12, 13, 14], in the description of geometric phases in graphene [15], in Kaluza-Klein reduction of a quadratic curva- ture model [16], in the presence of fermions [17, 18, 19], and in studies of the Lorentz symmetry violation [20, 21, 22]. In addition, the Kaluza-Klein theory has studied in the relativistic quantum mechanics, for example, the KG-oscillator on curved background in [23], the KG-oscillator field interacts with Cornell-type potential in [24], generalized KG-oscillator in the back- ground of magnetic cosmic string with scalar potential of Cornell-type in [25], generalized KG-oscillator in the background of magnetic cosmic string with a linear confining potential in [26], quantum dynamics of a scalar par- ticle in the background of magnetic cosmic string and chiral cosmic string in [27, 28], bound states solution for a relativistic scalar particle subject to Coulomb-type potential in the Minkowski space-time in five dimensions in [29], investigation of a scalar particle with position-dependent mass subject to a uniform magnetic field and quantum flux in the Minkowski space-time in five dimensions in [30], and quantum dynamics of KG-scalar particle sub- ject to linear and Coulomb-type central potentials in the five-dimensional Minkowski space-time [31]. Furthermore, effects of rotation on KG-scalar field subject Coulomb-type interaction and on KG-oscillator using Kaluza- Klein theory in the Minkowski space-time in five dimensions has also inves- tigated [32]. In order to describe singular behavior for a system at large distances in a uniformly rotating frame on the clocks and on a rotating body,

2 Landau et al. [34] made a transformation such that it introduces a uniform rotation in the Minkowski space-time in cylindrical system. Non-inertial ef- fects related to rotation have been investigated in several quantum systems, such as, in Dirac particle [35], on a neutral particle [36], on the Dirac oscil- lator [37], in cosmic string space-time [38, 39], in cosmic string space-time with torsion [40]. Study of non-inertial effects on KG-oscillator within the Kaluza-Klein theory will be our next work. The Klein-Gordon oscillator (KGO) [41, 42] was inspired by the Dirac 1 oscillator (DO) [43] applied to spin- 2 particle. Several authors have studied KGO on background space-times, for example, in cosmic string, G¨odel-type space-times etc. (e. g. [44, 45, 46]). In the context of KKT, the KG-oscillator in five-dimensional cosmic string and magnetic cosmic string background [23], under a Cornell-type potential in five-dimensional Minkowski space-time [24] have investigated. In addition, generalized KGO on curved background space-time induced by a spinning cosmic string coupled to a magnetic field including quantum flux [47], in magnetic cosmic string background under the effects of Cornell-type potential [25] and a linear confining potential [26] using KKT, in the presence of Coulomb-type potential in (1+2)-dimensions G¨urses space–time [48], in cosmic string space-time with a spacelike dislocation [49], and in cosmic string space-time [50] have investigated. In this work, we study a generalized KGO by introducing a uniform mag- netic field in the cosmic string line element using KKT [1, 2, 3] under the effects of a linear confining potential, and analyze a relativistic analogue of the Aharonov-Bohm effect for bound states. The Aharonov-Bohm effect [51, 52, 53, 54] is a quantum mechanical phenomena that describe phase shifts of the wave-function of a quantum particle due to the presence of a quantum flux produced by topological defects space-times. This effect has investigated by several authors in different branches of physics, such as, in Newtonian theory [55], in bound states of massive fermions [56], in scatter- ing of dislocated wave-fronts [57], on position-dependent mass system under

3 torsion effects [40, 46, 58], in bound states solution of spin-0 scalar particles [47]. In addition, this effect has investigated using KKT with or without in- teractions of various kind in five-dimensional the Minkowski or cosmic string space-time background [23, 25, 26, 27, 28, 29, 30, 31, 32, 33].

2 Interactions of generalized KGO with scalar potential using the KKT

The basic idea of the Kaluza-Klein theory [1, 2, 3, 27] was to postulate one extra compactified space dimension and introducing pure gravity in new (1+4)-dimensional space-time. It turns out that the five-dimensional gravity manifests in our observable (1 + 3)-dimensional space-time as gravitational, electromagnetic and scalar filed. In this way, we can work with general relativity in five-dimensions. The information about the electromagnetism is given by introducing a gauge potential Aµ in the space-time [27, 28, 24] as

ds2 = dt2 + dr2 + α2 r2 dφ2 + dz2 + [dy + κ A (xµ) dxµ]2 , (1) − µ where µ = 0, 1, 2, 3, x0 = t is the time-coordinate, x4 = y is the coordinate associated with fifth additional dimension having ranges 0

Based on [23, 24, 25, 27, 29, 30], we introduce a uniform magnetic field B0 and quantum flux Φ through the line-element of the cosmic string space-time (1) in the following form

1 Φ 2 ds2 = dt2 + dr2 + α2 r2 dφ2 + dz2 + dy + α B r2 + dφ , (2) − −2 0 2π    

4 where the gauge field given by 1 Φ A = κ−1 α B r2 + (3) φ −2 0 2π   gives rise to a uniform magnetic field B~ = ~ A~ = κ−1 B zˆ [60],z ˆ is the ∇ × − 0 unitary vector in the z-direction. Here Φ = const is quantum flux [51, 60] through the core of the topological defects [61]. The relativistic quantum dynamics of spin-0 scalar particle with a scalar potential S(r) by modifying the mass term in the form m m + S(r) as → done in [25, 26, 46] in five-dimensional case is described by [24, 29, 30, 31]: 1 ∂ (√ ggMN ∂ ) (m + S)2 Ψ=0, (4) √ g M − N −  −  where M, N = 0, 1, 2, 3, 4, with g = det g = α2 r2 is the determinant of MN − metric tensor gMN with g its inverse for the line element (2) and m is rest mass of the particle. To couple generalized Klein-Gordon oscillator with field, following change in the radial momentum operator is considered [47, 25, 26, 48, 49, 50, 62]

~p ~p i m Ω f(r)ˆr or ∂ ∂ + m Ω f(r), (5) → − r → r where Ω is the oscillator frequency and we can write ~p 2 (~p+i m Ω f(r)ˆr)(~p → − i m Ω f(r)ˆr). Therefore, the KG-equation becomes 1 (∂ + m Ω X ) √ ggMN (∂ m Ω X ) (m + S)2 Ψ=0, (6) √ g M M − N − N −  −  where XM = (0, f(r), 0, 0, 0). For the metric (2)

10 0 0 0 − 01 0 0 0 MN  1 KAφ  g = 0 0 2 2 0 2 2 . (7) α r − α r  00 0 1 0   2 2   KAφ K Aφ   0 0 2 2 0 1+ 2 2   − α r α r    5 By considering the line-element (2) into the Eq. (6), we obtain the fol- lowing differential equation :

∂2 ∂2 1 ∂ 1 ∂ ∂ 2 ∂2 ∂2 [ + + + κ A + + −∂t2 ∂r2 r ∂r α2 r2 ∂φ − φ ∂y ∂z2 ∂y2   f m Ω f ′ + m2 Ω2 f 2(r) (m + S)2]Ψ(t,r,φ,z,y)=0. (8) − r − −   Since the line-element (2) is independent of t,φ,z,x. One can choose the following ansatz for the function Ψ as:

Ψ(t,r,φ,z,y)= ei (−E t+lφ+kz+qy) ψ(r), (9) where E is the total energy of the particle, l = 0, 1, 2, .. Z, and k, q ± ± ∈ are constants. Substituting the ansatz (9) into the Eq. (8), we obtain the following equation: d2 1 d f (l K q A )2 + + E2 k2 q2 m Ω f ′ + m2 Ω2 f 2(r) − φ ψ(r) dr2 r dr − − − r − − α2 r2     =(m + S)2 ψ(r). (10)

2.1 Linear confining potential

In this work, we consider linear confining potential that studies in the con- finement of quarks [63], in the relativistic quantum mechanics [45, 26, 46, 64, 65, 66, 67, 68, 69], and in atomic and molecular physics [70]. This potential is given by

S(r)= ηL r (11) where ηL is a constant that characterizes the linear confining potential. Below, we choose two types of function f(r) for the studies of generalized KG-oscillator in the considered relativistic system subject to linear confining potential.

6 b2 Case A : Cornell-type function f(r)= b1 r + r Substituting eqs. (3) and (11) into the Eq. (10) and using the above function, we obtain the following equation:

d2 1 d j2 + + λ ω2 r2 br ψ(r)=0, (12) dr2 r dr − r2 − −   where (l q Φ ) λ = E2 k2 q2 m2 2 mω − 2π 2 m Ω b 2 m2 Ω2 b b , − − − − c α − 1 − 1 2 2 2 2 2 2 2 ω = m ωc + ηL + m Ω b1, q (l q Φ )2 j = − 2π + m2 Ω2 b2, s α2 2 q B ω = 0 , c 2 m b = 2 m ηL. (13)

Introducing a new variable ρ = √ωr, Eq. (12) becomes

d2 1 d j2 + + ζ ρ2 θ ρ ψ(ρ)=0, (14) dρ2 ρ dρ − ρ2 − −   where λ b ζ = , θ = 3 . (15) ω ω 2 Let us impose the requirement that the wave-function ψ(ρ) 0 for both → ρ 0 and ρ . Suppose the possible solution to Eq. (14) is → →∞ j − 1 (ρ+θ) ρ ψ(ρ)= ρ e 2 H(ρ). (16)

Substituting the solution Eq. (16) into the Eq. (14), we obtain

γ β H′′(ρ)+ θ 2 ρ H′(ρ)+ +Θ H(ρ)=0, (17) ρ − − − ρ    

7 where

γ =1+2 j, θ2 Θ= ζ + 2(1+ j), 4 − θ β = (1+2 j). (18) 2 Equation (17) is the biconfluent Heun’s differential equation [58, 46, 26, 25, 71, 72] and H(ρ) is the Heun polynomials. The above equation (17) can be solved by the Frobenius method. We consider the power series solution [73]

∞ i H(ρ)= ci ρ (19) i=0 X Substituting the above power series solution into the Eq. (17), we obtain the following recurrence relation for the coefficients: 1 c = [ β + θ (n + 1) c (Θ 2 n) c ] . (20) n+2 (n + 2)(n +2+2 j) { } n+1 − − n

And the various coefficients are θ c = c , 1 2 0 1 c = [(β + θ) c Θ c ]. (21) 2 4(1+ j) 1 − 0

We must truncate the power series by imposing the following two condi- tions [58, 46, 29, 30, 31, 26, 25]:

Θ = 2 n, (n =1, 2, ...)

cn+1 = 0. (22)

By analyzing the condition Θ = 2 n, we get the following second degree

8 expression of the energy eigenvalues En,l:

λ θ2 + 2(1+ j)=2 n ω 4 − (l q Φ )2 E = k2 + q2 + m2 +2 ω n +1+ − 2π + m2 Ω2 b2 ⇒ n,l ±{  s α2 2

q Φ 2 2   (l 2π ) m ηL 1 +2 mω − +2 m Ω b (1 + m Ω b ) 2 . (23) c α − ω2 1 2 } Note that the Eq. (23) does not represent the general expression for eigenvalues. One can obtain the individual energy eigenvalues one by one, that is, E1, E2, E3,.. by imposing the additional recurrence condition Cn+1 = 0 on the eigenvalue as done in [58, 46, 26, 25]. For n = 1, we have Θ = 2 and c2 = 0 which implies from Eq. (21) 2 θ 2 c = c = 1 β + θ 0 ⇒ 2 β + θ 1 1 b2 3 m2 η2 3 ω = (3+2 j) = 1 L (3+2 j) (24) 1,l 8 2     a constraint on the parameter ω1,l. The relation given in Eq. (24) gives the value of the parameter ω1,l that permit us to construct a first degree polynomial solution of H(ρ) for the radial mode n = 1. Note that the parameter ω1,l depends on the linear confining potential ηL and its value changes for each quantum number n, l , so we have labeled ω ω and { } → n,l η η . Besides, we have adjusted the magnetic field B1,l and the linear L → 1 L 0 confining potential η1 L such that Eq. (24) can be satisfied and we have simplified by labelling: 1 2 ω1,l = ω2 η2 m2 Ω2 b2 B1,l = ω2 η2 m2 Ω2 b2. (25) c m 1,l − 1 L − 1 ↔ 0 q 1,l − 1 L − 1 q q 1,l It is noteworthy that the allowed value of the magnetic field B0 for lowest state of the system given by (25) is defined for the radial mode n = 1. We

9 can note from Eq. (25) that the magnetic field B0 depends on the quantum numbers n, l of the relativistic system which shows a quantum effect. { } Therefore, the ground state energy level for n = 1 is given by

(l q Φ )2 E = k2 + q2 + m2 +2 ω 2+ − 2π + m2 Ω2 b2 1,l ±{ 1,l  s α2 2

q Φ 2 2   (l ) m ηL 1 1,l 2π 2 +2 mωc − 2 +2 m Ω b1 (1 + m Ω b2) . (26) α − ω1,l } And the radial wave-functions is

1 2 m ηL q Φ −  +ρ ρ (l− )2 2 3 2π m2 2 b2 ω 2 r 2 + Ω 2  1,l  ψ1,l = ρ α e   (c0 + c1 ρ) , (27) where 1 c1 = c0. (28) (l− q Φ )2 3 2π 2 2 2 2 + α2 + m Ω b2 r q Then, by substituting the magnetic field (25) into the Eq. (26), one can obtain the allowed values of the relativistic energy level for the radial mode n = 1 of the system. As the values of the wedge parameter α are in the ranges 0 <α< 1, thus, the degeneracy of the energy is broken and shifted the energy level in comparison to the case of five-dimensional Minkowski space-time.

b2 Case B : Coulomb-type function f(r)= r

In that case, the radial wave-equation (12) becomes d2 1 d j2 + + λ˜ ω˜ 2 r2 br ψ(r)=0, (29) dr2 r dr − r2 − −   where (l q Φ ) λ˜ = E2 k2 q2 m2 2 mω − 2π , − − − − c α 2 2 2 ω˜ = m ωc + ηL. (30) q 10 Introducing a new variable ρ = √ωr˜ , Eq. (12) becomes

d2 1 d j2 + + ζ˜ ρ2 θ˜ ρ ψ(ρ)=0, (31) dρ2 ρ dρ − ρ2 − −   where ˜ ˜ λ ˜ b ζ = , θ = 3 . (32) ω˜ ω˜ 2 Let the possible solution to Eq. (31) is

j − 1 (ρ+θ˜) ρ ψ(ρ)= ρ e 2 H(ρ). (33)

Substituting solution Eq. (33) into the Eq. (31), we obtain

γ β˜ H′′(ρ)+ θ˜ 2 ρ H′(ρ)+ + Θ˜ H(ρ)=0, (34) ρ − − − ρ   " # where θ˜2 Θ=˜ ζ˜ + 2(1+ j), 4 − θ˜ β˜ = (1+2 j). (35) 2 Equation (34) is the biconfluent Heun’s differential equation [58, 46, 26, 25, 71, 72] and H(ρ) is the Heun polynomials. Substituting the above power series solution (19) into the Eq. (34), we obtain the following recurrence relation for the coefficients: 1 c = β˜ + θ˜(n + 1) c (Θ˜ 2 n) c . (36) n+2 (n + 2)(n +2+2 j) n+1 − − n hn o i And the various coefficients are θ˜ c = c , 1 2 0 1 c = [ β˜ + θ˜ c Θ˜ c ]. (37) 2 4(1+ j) 1 − 0  

11 We must truncate the power series by imposing the following two condi- tions [58, 46, 29, 30, 31, 26, 25]:

Θ˜ = 2 n, (n =1, 2, ...)

cn+1 = 0. (38)

By analyzing the condition Θ=2˜ n, we get the following second degree expression of the energy eigenvalues En,l:

λ˜ θ˜2 + 2(1+ j)=2 n ω˜ 4 − (l q Φ )2 E = k2 + q2 + m2 +2˜ω n +1+ − 2π + m2 Ω2 b2 ⇒ n,l ±{  s α2 2

q Φ 2 2   (l 2π ) m ηL 1 +2 mω − 2 . (39) c α − ω˜2 } Following the similar technique as done earlier, we want to find the indi- vidual energy level and wave-function. For example n = 1 we have Θ = 2 and c2 = 0 which implies from Eq. (21)

2 θ˜ 2 c1 = c0 = β˜ + θ˜ ⇒ 2 β˜ + θ˜ 1 b2 3 ω˜ = (3+2 j) (40) 1,l 8   1,l a constraint on the parameterω ˜1,l. The magnetic field B0 is so adjusted that Eq. (24) can be satisfied and we have simplified by labelling: 1 2 ω1,l = ω˜2 η2 B1,l = ω˜2 η2 . (41) c m 1,l − L ↔ 0 q 1,l − L q q We can see from Eq. (41) that the possible values of the magnetic field B depend on the quantum numbers n, l of the system as well as on the 0 { } confining potential parameter.

12 Therefore, the ground state energy level for n = 1 is given by

(l q Φ )2 E = k2 + q2 + m2 +2˜ω 2+ − 2π + m2 Ω2 b2 1,l ±{ 1,l  s α2 2

q Φ 2 2   (l ) m ηL 1 1,l 2π 2 +2 mωc − 2 . (42) α − ω˜1,l } And the radial wave-functions is

1 2 m ηL q Φ −  +ρ ρ (l− )2 2 3 2π m2 2 b2 ω˜ 2 r 2 + Ω 2  1,l  ψ1,l = ρ α e   (c0 + c1 ρ) , (43) where 1 c1 = c0. (44) (l− q Φ )2 3 2π 2 2 2 2 + α2 + m Ω b2 r q Then, by substituting the real solution from Eq. (40) into the Eq. (41), it is possible to obtain the allowed values of the relativistic energy levels for the radial mode n = 1 of the system. We can see that the lowest energy state defined by Eqs. (40)–(41) plus the expression given in Eqs. (42)–(44) is for the radial mode n = 1, instead of n = 0. This effect arises due to the presence of linear confining potential in the relativistic system. Since the wedge parameter α are in the ranges 0 <α< 1, thus, the degeneracy of the relativistic energy eigenvalue here also is broken and shifted the energy level in comparison to the case of five-dimensional Minkowski space-time.

3 Conclusions

In this work, we have investigated generalized Klein-Gordon oscillator with a uniform magnetic field subject to a linear confining potential in a topolog- ical defect five-dimensional space-time in the context of Kaluza-Klein the- ory. Linear confining potential has many applications such as confinement of

13 quarks in particle physics and other branches of physics including relativis- tic quantum mechanics. For suitable total wave-function, we have derived the radial wave-equation for a Cornell-type function in sub-section 2.1 and finally reached a biconfluent Heun’s differential equation form. By power se- ries method we have solved this equation and by imposing condition we have obtained the non-compact expression of the energy eigenvalues (23). By im- posing the recurrence condition cn+1 = 0 for each radial mode, for example n = 11, we have obtained the lowest state energy level and wave-function by Eqs. (26)–(28), and others are in the same way. In sub-section 2.2, we have considered a Coulomb-type function on the same relativistic system with a linear confining potential. Here also we have reached a biconfluent Heun’s equation form and following the similar tech- nique as done earlier, we have obtained the non-compact expression of the energy eigenvalues (39). By imposing the additional recurrence condition cn+1 = 0 on the eigenvalue, one can obtained the individual energy level and the corresponding wave-function, as for example, for the radial mode n = 1 by Eqs. (42)–(44), and others are in the same way. In sub-section 2.1 –2.2, we have seen that the presence of linear confining potential allow the forma- tion of bound states solution of the considered relativistic system and hence, the lowest energy state is defined by the radial mode n = 1, instead of n = 0. Also in gravitation and cosmology, the values of the wedge parameter α are in the ranges 0 <α< 1, and thus, the degeneracy of each energy level is broken and shifted the relativistic energy level in comparison to the case of

five-dimensional Minkowski space-time. When we tries to analyze cn+1 = 0 for the radial mode n = 1, an observation is noted, where certain parameter is constraint, for example, ω1,l that appears in Eq. (24) in sub-section 2.1 (Case A) depend on the quantum number n, l of the system as well as on { } linear confining potential ηL. Another interesting observation that we made in this work is the quantum effect which arises due to the dependence of the magnetic field Bn,l on the quantum number n, l of the relativistic system. 0 { }

14 We have observed in this work that the angular momentum number l of the system is shifted, l l = 1 (l q Φ ), an effective angular quantum num- → 0 α − 2 π ber. Therefore, the relativistic energy eigenvalues depends on the geometric quantum phase [51, 60]. Thus, we have that, En,l(Φ+Φ0)= En,l∓τ (Φ), where Φ = 2π τ with τ = 0, 1, 2, .... This dependence of the relativistic energy 0 ± q level on the geometric quantum phase gives us a relativistic analogue of the Aharonov-Bohm effect for bound states. The Kaluza-Klein theory lead to many new unified field theories, for example, connecting this theory with supergravity resulted in improved su- persymmetric Kaluza-Klein theory, multi-dimensional unified theories using the idea of Kaluza-Klein theory to postulate extra (compactified) space di- mensions, superstring theory as well as the M-theory [74] model using the Kaluza-Klein theory in higher dimensions. The five-dimensional theory yields a geometrical interpretation of the electromagnetic field and electric charge. The relation of the five-dimensional KKT with the structure of fiber bundles was first made in [75] and the relationship between principal fiber bundles and higher dimensional theories is in [76].

Acknowledgement

Author sincerely acknowledge the anonymous kind referee(s) for their value comments and suggestions which have greatly improved this present manuscript.

Conflict of Interest

There is no potential conflict of interest regarding publication this manuscript.

Contribution Statement

Faizuddin Ahmed has done the whole work.

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