Linear confinement of generalized KG-oscillator with a uniform magnetic field in Kaluza-Klein theory and Aharonov-Bohm effect
aa r X i v : . [ g r- q c ] J a n Linear confinement of generalized KG-oscillator with auniform magnetic field in Kaluza-Klein theory andAharonov-Bohm effect
Faizuddin AhmedNational Academy, Gauripur-783331, Assam, India
E-mail: [email protected]
Abstract
In this paper, we solve generalized KG-oscillator interacts with auniform magnetic field in five-dimensional space-time background pro-duced by topological defects under a linear confining potential usingthe Kaluza-Klein theory. We solve this equation and analyze an ana-logue of the Aharonov-Bohm effect for bound states. We observe thatthe energy level for each radial mode depend on the global param-eters characterizing the space-time, the confining potential, and themagnetic field which shows a quantum effect. keywords:
Kaluza-Klein theory, Relativistic wave equation, electromag-netic interactions, Aharonov-Bohm effect, special functions.
PACS Number:
The first proposal of a unified theory of fundamental interactions was elab-orated by Kaluza [1] and Klein [2] (see also, [3]). This new proposal es-tablished that the electromagnetism can be introduced through an extra(compactified) dimension in the space-time, where the spatial dimensionbecomes five-dimensional. This geometrical unification of gravitation andelectromagnetism in five-dimensional version of general relativity gave some1nteresting 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 andEinstein–Gauss–Bonnet equations has studied in [5]. Few examples of thesesolutions 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 ofphysics. For example, in Khaler fields [9], in the presence of torsion [10, 11],in the Grassmannian context [12, 13, 14], in the description of geometricphases 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 ofthe Lorentz symmetry violation [20, 21, 22]. In addition, the Kaluza-Kleintheory has studied in the relativistic quantum mechanics, for example, theKG-oscillator on curved background in [23], the KG-oscillator field interactswith 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 stringwith a linear confining potential in [26], quantum dynamics of a scalar par-ticle in the background of magnetic cosmic string and chiral cosmic stringin [27, 28], bound states solution for a relativistic scalar particle subject toCoulomb-type potential in the Minkowski space-time in five dimensions in[29], investigation of a scalar particle with position-dependent mass subjectto a uniform magnetic field and quantum flux in the Minkowski space-timein five dimensions in [30], and quantum dynamics of KG-scalar particle sub-ject to linear and Coulomb-type central potentials in the five-dimensionalMinkowski space-time [31]. Furthermore, effects of rotation on KG-scalarfield 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 largedistances in a uniformly rotating frame on the clocks and on a rotating body,2andau et al. [34] made a transformation such that it introduces a uniformrotation 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-timewith torsion [40]. Study of non-inertial effects on KG-oscillator within theKaluza-Klein theory will be our next work.The Klein-Gordon oscillator (KGO) [41, 42] was inspired by the Diracoscillator (DO) [43] applied to spin- particle. Several authors have studiedKGO on background space-times, for example, in cosmic string, G¨odel-typespace-times etc. ( e. g. [44, 45, 46]). In the context of KKT, the KG-oscillatorin 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 backgroundspace-time induced by a spinning cosmic string coupled to a magnetic fieldincluding quantum flux [47], in magnetic cosmic string background under theeffects of Cornell-type potential [25] and a linear confining potential [26] usingKKT, in the presence of Coulomb-type potential in (1+2)-dimensions G¨ursesspace–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 theeffects of a linear confining potential, and analyze a relativistic analogueof the Aharonov-Bohm effect for bound states. The Aharonov-Bohm effect[51, 52, 53, 54] is a quantum mechanical phenomena that describe phaseshifts of the wave-function of a quantum particle due to the presence of aquantum flux produced by topological defects space-times. This effect hasinvestigated by several authors in different branches of physics, such as, inNewtonian theory [55], in bound states of massive fermions [56], in scatter-ing of dislocated wave-fronts [57], on position-dependent mass system under3orsion 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 stringspace-time background [23, 25, 26, 27, 28, 29, 30, 31, 32, 33]. The basic idea of the Kaluza-Klein theory [1, 2, 3, 27] was to postulateone extra compactified space dimension and introducing pure gravity in new(1 + 4)-dimensional space-time. It turns out that the five-dimensional gravitymanifests in our observable (1 + 3)-dimensional space-time as gravitational,electromagnetic and scalar filed. In this way, we can work with generalrelativity in five-dimensions. The information about the electromagnetism isgiven by introducing a gauge potential A µ in the space-time [27, 28, 24] as ds = − dt + dr + α r dφ + dz + [ dy + κ A µ ( x µ ) dx µ ] , (1)where µ = 0 , , , x = t is the time-coordinate, x = y is the coordinateassociated with fifth additional dimension having ranges 0 < y < π a where, a is the radius of the compact dimension of y , ( x = r, x = φ, x = z ) are thecylindrical coordinates with the usual ranges, and κ is the gauge couplingor Kaluza constant [27]. The parameter α = (1 − µ ) [59] characterizingthe wedge parameter where, µ is the linear mass density of the string. Weassume the values of the parameter α lies in the range 0 < α < B and quantum flux Φ through the line-element of the cosmic string space-time(1) in the following form ds = − dt + dr + α r dφ + dz + (cid:20) dy + (cid:18) − α B r + Φ2 π (cid:19) dφ (cid:21) , (2)4here the gauge field given by A φ = κ − (cid:18) − α B r + Φ2 π (cid:19) (3)gives rise to a uniform magnetic field ~B = ~ ∇ × ~A = − κ − B ˆ z [60], ˆ z is theunitary 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 scalarpotential S ( r ) by modifying the mass term in the form m → m + S ( r ) asdone in [25, 26, 46] in five-dimensional case is described by [24, 29, 30, 31]: (cid:20) √− g ∂ M ( √− g g MN ∂ N ) − ( m + S ) (cid:21) Ψ = 0 , (4)where M, N = 0 , , , ,
4, with g = det g = − α r is the determinant ofmetric tensor g MN with g MN its inverse for the line element (2) and m is restmass of the particle.To couple generalized Klein-Gordon oscillator with field, following changein the radial momentum operator is considered [47, 25, 26, 48, 49, 50, 62] ~p → ~p − i m Ω f ( r )ˆ r or ∂ r → ∂ r + m Ω f ( r ) , (5)where Ω is the oscillator frequency and we can write ~p → ( ~p + i m Ω f ( r )ˆ r )( ~p − i m Ω f ( r )ˆ r ). Therefore, the KG-equation becomes (cid:20) √− g ( ∂ M + m Ω X M ) √− g g MN ( ∂ N − m Ω X N ) − ( m + S ) (cid:21) Ψ = 0 , (6)where X M = (0 , f ( r ) , , , g MN = − α r − K A φ α r − K A φ α r K A φ α r . (7)5y considering the line-element (2) into the Eq. (6), we obtain the fol-lowing differential equation :[ − ∂ ∂t + ∂ ∂r + 1 r ∂∂r + 1 α r (cid:18) ∂∂φ − κ A φ ∂∂y (cid:19) + ∂ ∂z + ∂ ∂y − m Ω (cid:18) f ′ + fr (cid:19) − m Ω f ( r ) − ( m + S ) ] Ψ( t, r, φ, z, y ) = 0 . (8)Since the line-element (2) is independent of t, φ, z, x . One can choose thefollowing ansatz for the function Ψ as:Ψ( t, r, φ, z, y ) = e i ( − E t + l φ + k z + q y ) ψ ( r ) , (9)where E is the total energy of the particle, l = 0 , ± , ± , .. ∈ Z , and k, q are constants.Substituting the ansatz (9) into the Eq. (8), we obtain the followingequation: (cid:20) d dr + 1 r ddr + E − k − q − m Ω (cid:18) f ′ + fr (cid:19) − m Ω f ( r ) − ( l − K q A φ ) α r (cid:21) ψ ( r )= ( m + S ) ψ ( r ) . (10) 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 potentialis given by S ( r ) = η L r (11)where η L is a constant that characterizes the linear confining potential.Below, we choose two types of function f ( r ) for the studies of generalizedKG-oscillator in the considered relativistic system subject to linear confiningpotential. 6 ase A : Cornell-type function f ( r ) = b r + b r Substituting eqs. (3) and (11) into the Eq. (10) and using the abovefunction, we obtain the following equation: (cid:20) d dr + 1 r ddr + λ − j r − ω r − b r (cid:21) ψ ( r ) = 0 , (12)where λ = E − k − q − m − m ω c ( l − q Φ2 π ) α − m Ω b − m Ω b b ,ω = q m ω c + η L + m Ω b ,j = s ( l − q Φ2 π ) α + m Ω b ,ω c = q B m ,b = 2 m η L . (13)Introducing a new variable ρ = √ ω r , Eq. (12) becomes (cid:20) d dρ + 1 ρ ddρ + ζ − j ρ − ρ − θ ρ (cid:21) ψ ( ρ ) = 0 , (14)where ζ = λω , θ = bω . (15)Let us impose the requirement that the wave-function ψ ( ρ ) → ρ → ρ → ∞ . Suppose the possible solution to Eq. (14) is ψ ( ρ ) = ρ j e − ( ρ + θ ) ρ H ( ρ ) . (16)Substituting the solution Eq. (16) into the Eq. (14), we obtain H ′′ ( ρ ) + (cid:20) γρ − θ − ρ (cid:21) H ′ ( ρ ) + (cid:20) − βρ + Θ (cid:21) H ( ρ ) = 0 , (17)7here γ = 1 + 2 j, Θ = ζ + θ − j ) ,β = θ j ) . (18)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. Weconsider the power series solution [73] H ( ρ ) = ∞ X i =0 c i ρ i (19)Substituting the above power series solution into the Eq. (17), we obtain thefollowing recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) [ { β + θ ( n + 1) } c n +1 − (Θ − n ) c n ] . (20)And the various coefficients are c = θ c ,c = 14 (1 + j ) [( β + θ ) c − Θ c ] . (21)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 , , ... ) c n +1 = 0 . (22)By analyzing the condition Θ = 2 n , we get the following second degree8xpression of the energy eigenvalues E n,l : λω + θ − j ) = 2 n ⇒ E n,l = ± { k + q + m + 2 ω n + 1 + s ( l − q Φ2 π ) α + m Ω b +2 m ω c ( l − q Φ2 π ) α − m η L ω + 2 m Ω b (1 + m Ω b ) } . (23)Note that the Eq. (23) does not represent the general expression foreigenvalues. One can obtain the individual energy eigenvalues one by one,that is, E , E , E ,.. by imposing the additional recurrence condition C n +1 =0 on the eigenvalue as done in [58, 46, 26, 25]. For n = 1, we have Θ = 2and c = 0 which implies from Eq. (21) c = 2 β + θ c ⇒ θ β + θω ,l = (cid:20) b j ) (cid:21) = (cid:20) m η L j ) (cid:21) (24)a constraint on the parameter ω ,l . The relation given in Eq. (24) givesthe value of the parameter ω ,l that permit us to construct a first degreepolynomial solution of H ( ρ ) for the radial mode n = 1. Note that theparameter ω ,l depends on the linear confining potential η L and its valuechanges for each quantum number { n, l } , so we have labeled ω → ω n,l and η L → η L . Besides, we have adjusted the magnetic field B ,l and the linearconfining potential η L such that Eq. (24) can be satisfied and we havesimplified by labelling: ω ,lc = 1 m q ω ,l − η L − m Ω b ↔ B ,l = 2 q q ω ,l − η L − m Ω b . (25)It is noteworthy that the allowed value of the magnetic field B ,l for loweststate of the system given by (25) is defined for the radial mode n = 1. We9an note from Eq. (25) that the magnetic field B depends on the quantumnumbers { n, l } of the relativistic system which shows a quantum effect.Therefore, the ground state energy level for n = 1 is given by E ,l = ± { k + q + m + 2 ω ,l s ( l − q Φ2 π ) α + m Ω b +2 m ω ,lc ( l − q Φ2 π ) α − m η L ω ,l + 2 m Ω b (1 + m Ω b ) } . (26)And the radial wave-functions is ψ ,l = ρ r ( l − q Φ2 π )2 α + m Ω b e − m ηLω ,l + ρ ρ ( c + c ρ ) , (27)where c = 1 r + q ( l − q Φ2 π ) α + m Ω b c . (28)Then, by substituting the magnetic field (25) into the Eq. (26), one canobtain 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 theranges 0 < α <
1, thus, the degeneracy of the energy is broken and shiftedthe energy level in comparison to the case of five-dimensional Minkowskispace-time.
Case B : Coulomb-type function f ( r ) = b r In that case, the radial wave-equation (12) becomes (cid:20) d dr + 1 r ddr + ˜ λ − j r − ˜ ω r − b r (cid:21) ψ ( r ) = 0 , (29)where ˜ λ = E − k − q − m − m ω c ( l − q Φ2 π ) α , ˜ ω = q m ω c + η L . (30)10ntroducing a new variable ρ = √ ˜ ω r , Eq. (12) becomes (cid:20) d dρ + 1 ρ ddρ + ˜ ζ − j ρ − ρ − ˜ θ ρ (cid:21) ψ ( ρ ) = 0 , (31)where ˜ ζ = ˜ λ ˜ ω , ˜ θ = b ˜ ω . (32)Let the possible solution to Eq. (31) is ψ ( ρ ) = ρ j e − ( ρ +˜ θ ) ρ H ( ρ ) . (33)Substituting solution Eq. (33) into the Eq. (31), we obtain H ′′ ( ρ ) + (cid:20) γρ − ˜ θ − ρ (cid:21) H ′ ( ρ ) + " − ˜ βρ + ˜Θ H ( ρ ) = 0 , (34)where ˜Θ = ˜ ζ + ˜ θ − j ) , ˜ β = ˜ θ j ) . (35)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), weobtain the following recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) hn ˜ β + ˜ θ ( n + 1) o c n +1 − ( ˜Θ − n ) c n i . (36)And the various coefficients are c = ˜ θ c ,c = 14 (1 + j ) [ (cid:16) ˜ β + ˜ θ (cid:17) c − ˜Θ c ] . (37)11e must truncate the power series by imposing the following two condi-tions [58, 46, 29, 30, 31, 26, 25]:˜Θ = 2 n, ( n = 1 , , ... ) c n +1 = 0 . (38)By analyzing the condition ˜Θ = 2 n , we get the following second degreeexpression of the energy eigenvalues E n,l :˜ λ ˜ ω + ˜ θ − j ) = 2 n ⇒ E n,l = ± { k + q + m + 2 ˜ ω n + 1 + s ( l − q Φ2 π ) α + m Ω b +2 m ω c ( l − q Φ2 π ) α − m η L ˜ ω } . (39)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 = 2and c = 0 which implies from Eq. (21) c = 2˜ β + ˜ θ c ⇒ ˜ θ β + ˜ θ ˜ ω ,l = (cid:20) b j ) (cid:21) (40)a constraint on the parameter ˜ ω ,l . The magnetic field B ,l is so adjustedthat Eq. (24) can be satisfied and we have simplified by labelling: ω ,lc = 1 m q ˜ ω ,l − η L ↔ B ,l = 2 q q ˜ ω ,l − η L . (41)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 theconfining potential parameter. 12herefore, the ground state energy level for n = 1 is given by E ,l = ± { k + q + m + 2 ˜ ω ,l s ( l − q Φ2 π ) α + m Ω b +2 m ω ,lc ( l − q Φ2 π ) α − m η L ˜ ω ,l } . (42)And the radial wave-functions is ψ ,l = ρ r ( l − q Φ2 π )2 α + m Ω b e − m ηL ˜ ω ,l + ρ ρ ( c + c ρ ) , (43)where c = 1 r + q ( l − q Φ2 π ) α + m Ω b c . (44)Then, by substituting the real solution from Eq. (40) into the Eq. (41), itis possible to obtain the allowed values of the relativistic energy levels forthe radial mode n = 1 of the system. We can see that the lowest energystate 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 tothe presence of linear confining potential in the relativistic system. Since thewedge parameter α are in the ranges 0 < α <
1, thus, the degeneracy of therelativistic energy eigenvalue here also is broken and shifted the energy levelin comparison to the case of five-dimensional Minkowski space-time.
In this work, we have investigated generalized Klein-Gordon oscillator witha 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 of13uarks in particle physics and other branches of physics including relativis-tic quantum mechanics. For suitable total wave-function, we have derivedthe radial wave-equation for a Cornell-type function in sub-section 2.1 andfinally reached a biconfluent Heun’s differential equation form. By power se-ries method we have solved this equation and by imposing condition we haveobtained the non-compact expression of the energy eigenvalues (23). By im-posing the recurrence condition c n +1 = 0 for each radial mode, for example n = 11, we have obtained the lowest state energy level and wave-function byEqs. (26)–(28), and others are in the same way.In sub-section 2.2 , we have considered a Coulomb-type function on thesame relativistic system with a linear confining potential. Here also we havereached a biconfluent Heun’s equation form and following the similar tech-nique as done earlier, we have obtained the non-compact expression of theenergy eigenvalues (39). By imposing the additional recurrence condition c n +1 = 0 on the eigenvalue, one can obtained the individual energy level andthe corresponding wave-function, as for example, for the radial mode n = 1by 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 arein the ranges 0 < α <
1, and thus, the degeneracy of each energy level isbroken and shifted the relativistic energy level in comparison to the case offive-dimensional Minkowski space-time. When we tries to analyze c n +1 = 0for the radial mode n = 1, an observation is noted, where certain parameteris constraint, for example, ω ,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 onlinear confining potential η L . Another interesting observation that we madein this work is the quantum effect which arises due to the dependence of themagnetic field B n,l on the quantum number { n, l } of the relativistic system.14e have observed in this work that the angular momentum number l ofthe system is shifted, l → l = α ( l − q Φ2 π ), an effective angular quantum num-ber. Therefore, the relativistic energy eigenvalues depends on the geometricquantum phase [51, 60]. Thus, we have that, E n,l (Φ+Φ ) = E n,l ∓ τ (Φ), whereΦ = ± πq τ with τ = 0 , , , ... . This dependence of the relativistic energylevel on the geometric quantum phase gives us a relativistic analogue of theAharonov-Bohm effect for bound states.The Kaluza-Klein theory lead to many new unified field theories, forexample, connecting this theory with supergravity resulted in improved su-persymmetric Kaluza-Klein theory, multi-dimensional unified theories usingthe idea of Kaluza-Klein theory to postulate extra (compactified) space di-mensions, superstring theory as well as the M-theory [74] model using theKaluza-Klein theory in higher dimensions. The five-dimensional theory yieldsa geometrical interpretation of the electromagnetic field and electric charge.The relation of the five-dimensional KKT with the structure of fiber bundleswas first made in [75] and the relationship between principal fiber bundlesand higher dimensional theories is in [76]. Acknowledgement
Author sincerely acknowledge the anonymous kind referee(s) for their valuecomments 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
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