aa r X i v : . [ qu a n t - ph ] M a y Torsion and noninertial effects on a nonrelativistic Dirac particle
K. Bakke ∗ Departamento de F´ısica, Universidade Federal da Para´ıba,Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, PB, Brazil.
Abstract
We investigate torsion and noninertial effects on a spin-1 / PACS numbers: 03.65.Ge, 04.62.+v, 61.72.LkKeywords: Torsion, Screw Dislocation, Noninertial effects, cosmic dislocation, Fermi-Walker reference frame, Hard-Wall confining potential ∗ Electronic address: kbakke@fisica.ufpb.br . INTRODUCTION In recent decades, a great deal of works has studied the influence of torsion on several physicalsystems [1–11]. The interaction between fermions and torsion and possible physical effects havebeen discussed in Refs. [4–6]. In crystalline solids, torsion has been studied in the continuumpicture of defects by using the differential geometry in order to describe the strain and the stressinduced by the defect in an elastic medium [12–17]. The study of the influence of torsion onquantum systems has been extended to the electronic properties of graphene sheets [18], Berry’sphase [19], quantum scattering [20], Landau levels for a nonrelativistic scalar particle [21] andholonomic quantum computation [22]. The influence of torsion on a two-dimensional quantumring has been discussed in Ref. [23] and on a quantum dot in Ref. [24].In this paper, we discuss torsion effects on the spectrum of energy of a nonrelativistic spin-1 / / / II. TORSION EFFECTS ON THE NONRELATIVISTIC QUANTUM DYNAMICS OF ADIRAC PARTICLE IN A NONINERTIAL FRAME
In this section, we study the confinement of a nonrelativistic Dirac particle to a hard-wall con-fining potential under the influence of noninertial effects and torsion. We present the mathematicaltools to describe spinors under the influence of torsion and noninertial effects, when we considerthe local reference frame of the observers is a Fermi-Walker reference frame. In this paper, we workwith the units ~ = c = 1. We start by writing the line element of the cosmic dislocation spacetime[9, 10, 44] (in the rest frame of the observers): ds = − d T + d R + R d Φ + ( d Z + ζ d Φ) . (1)The parameter ζ is a constant, and it is related to the torsion of the defect. From the crystallographylanguage, the parameter ζ is related to the Burgers vector ~b = b ˆ z , where ζ = b π [7, 9, 44].In the following, we consider a coordinate transformation given by T = t , R = ρ , Φ = ϕ + ω t ,and Z = z , where ω is the constant angular velocity of the rotating frame. Thus, the line element(1) becomes ds = − (cid:0) − ω ρ − ζ ω (cid:1) dt + (cid:0) ωρ + 2 ζ ω (cid:1) dϕ dt + dρ + ρ dϕ + 2 ζω dz dt + ( dz + ζ dϕ ) . (2)We can note that the line element (2) is defined for values of the radial coordinate inside the range:0 < ρ < p − ζ ω ω . (3)Therefore, for all values of the radial coordinate given by ρ ≥ √ − ζ ω ω , the line element (2)is not well-defined. Observe that for values of the radial coordinate given by ρ ≥ √ − ζ ω ω , wehave that the component g tt of the metric tensor becomes positive. The meaning of this restrictionof the radial coordinate is that, in the rotating system, the coordinate system becomes singularat ρ → √ − ζ ω ω , which is associated with the velocity of the particle would be greater than the3elocity of the light as discussed in Ref. [40]. Moreover, the range (3) shows that the restrictionof the radial coordinate in the rotating frame depends on the angular velocity and the parameterassociated with the torsion of the defect. Therefore, based on the coordinate singularity givenin Eq. (3), we analyse the behaviour of the quantum particle under the influence of torsion andnoninertial effects inside the region 0 < ρ < p − ζ ω /ω by imposing that the wave functionvanishes at ρ → √ − ζ ω ω , which yields bound states analogous to the confinement of a spin-1 / ψ ′ ( x ) = D (Λ ( x )) ψ ( x ),where D (Λ ( x )) corresponds to the spinor representation of the infinitesimal Lorentz group andΛ ( x ) corresponds to the local Lorentz transformations [49]. Locally, the reference frame of theobservers can be build via a noncoordinate basis ˆ θ a = e aµ ( x ) dx µ , where the components e aµ ( x ) arecalled tetrads and satisfy the relation: g µν ( x ) = e aµ ( x ) e bν ( x ) η ab [49–51], where η ab = diag( − +++) is the Minkowski tensor. The inverse of the tetrads are defined as dx µ = e µa ( x ) ˆ θ a , and therelations e aµ ( x ) e µb ( x ) = δ ab and e µa ( x ) e aν ( x ) = δ µν are satisfied. In the Fermi-Walker referenceframe, noninertial effects can be observed from the action of external forces without any influenceof arbitrary rotations of the spatial axis of the local frame [52]. This reference frame can be builtby taking ˆ θ = e t ( x ) dt , which means that the components of the noncoordinate basis form a restframe for the observers at each instant, and the spatial components of the noncoordinate basis ˆ θ i must be chosen in such a way that they do not rotate [52]. With these conditions, we can writeˆ θ = dt ; ˆ θ = dρ ; ˆ θ = ρ ω dt + ρ dϕ ; ˆ θ = ζ ω dt + ζ dϕ + dz. (4)Recently, the Fermi-Walker reference frame has been used in studies of the analogue effect of theAharonov-Casher effect [53], and the Dirac oscillator [54]. Hence, by solving the Maurer-Cartanstructure equations in the presence of torsion T a = d ˆ θ a + ω ab ∧ ˆ θ b [51], where T a = T aµν dx µ ∧ dx ν is the torsion 2-form, ω ab = ω aµ b ( x ) dx µ is the connection 1-form, the operator d correspondsto the exterior derivative and the symbol ∧ means the wedge product, we obtain the followingnon-null components of the connection 1-form: T = 2 πζ δ ( ρ ) δ ( ϕ ) dρ ∧ dϕ ; ω ϕ ( x ) = − ω ϕ ( x ) = − ω t ( x ) = − ω t ( x ) = − ω.
4n order to write the Dirac equation in a curved spacetime background and in the presence oftorsion, we need to take into account that the partial derivative becomes the covariant derivative,where the covariant derivative is given by ∂ µ → ∇ µ = ∂ µ + Γ µ ( x ) + K µ ( x ), with Γ µ ( x ) = i ω µab ( x ) Σ ab being the spinorial connection [50, 51], and K µ ( x ) = i K µab ( x ) Σ ab , with Σ ab = i (cid:2) γ a , γ b (cid:3) . The indices ( a, b, c = 0 , , ,
3) indicate the local reference frame. The connection 1-form K µab ( x ) is related to the contortion tensor by [5]: K µab = K βνµ h e νa ( x ) e βb ( x ) − e νb ( x ) e βa ( x ) i .Following the definitions of Ref. [5], the contortion tensor is related to the torsion tensor via K βνµ = (cid:16) T βνµ − T βν µ − T βµ ν (cid:17) , where we have that the torsion tensor is antisymmetric in the last twoindices, while the contortion tensor is antisymmetric in the first two indices. Moreover, it is usuallyconvenient to write the torsion tensor into three irreducible components: the trace vector ¯ T µ = T βµβ ,the axial vector S α = ǫ αβνµ T βνµ and in the tensor q βνµ , which satisfies the conditions q βµβ = 0 and ǫ αβνµ q βνµ = 0. Thus, the torsion tensor becomes: T βνµ = (cid:0) ¯ T ν g βµ − ¯ T µ g βν (cid:1) − ǫ βνµγ S γ + q βνµ .The γ a matrices are defined in the local reference frame and correspond to the Dirac matrices inthe Minkowski spacetime [50, 55], i.e. , ~ Σ = ~σ ~σ ; γ = ˆ β = I − I ; γ i = ˆ β ˆ α i = σ i − σ i , (6)with ~ Σ and I being the spin vector and the 2 × ~σ arethe Pauli matrices and satisfy the relation (cid:0) σ i σ j + σ j σ i (cid:1) = 2 η ij . The γ µ matrices are related tothe γ a matrices via γ µ = e µa ( x ) γ a [50]. In this way, the general expression for the Dirac equationdescribing torsion effects on a quantum particle in the Fermi-Walker reference frame (4) is mψ = iγ ∂ψ∂t − iω γ ∂ψ∂ϕ + iγ ∂ψ∂ρ + i γ ρ (cid:20) ∂∂ϕ − ζ ∂∂z (cid:21) ψ + iγ ∂ψ∂z + iγ µ Γ µ ( x ) ψ (7)+ 18 S γ γ ψ − ~ Σ · ~S ψ. From the results obtained in Eq. (5), we can calculate the non-null components of the spinorialconnection Γ µ ( x ), and obtain iγ µ Γ µ = i γ ρ [42]. Moreover, from the definition of the axial vector S α given above and the results given in Eq. (5), we have that the only non-null component of theaxial vector is S = − πζρ δ ( ρ ) δ ( ϕ ) [56]. Hence, for ρ = 0, the Dirac equation (7) becomes i ∂ψ∂t = m ˆ βψ + iω ∂ψ∂ϕ − i ˆ α (cid:18) ∂∂ρ + 12 ρ (cid:19) ψ − i ˆ α ρ (cid:18) ∂∂ϕ − ζ ∂∂z (cid:19) ψ − i ˆ α ∂ψ∂z . (8)Now, let us discuss the nonrelativistic behaviour of the spin-1 / / ψ = e − imt φχ , (9)where φ and χ are two-spinors, and we consider φ being the “large” component and χ beingthe “small” component [55]. Substituting (9) into the Dirac equation (8), we obtain two coupledequations of φ and χ . The first coupled equation is i ∂φ∂t − iω ∂φ∂ϕ = (cid:20) − i σ ∂∂ρ − i σ ρ − i σ ρ (cid:18) ∂∂ϕ − ζ ∂∂z (cid:19) − i σ ∂∂z (cid:21) χ, (10)while the second coupled equation is i ∂χ∂t + 2 mχ − iω ∂χ∂ϕ = (cid:20) − i σ ∂∂ρ − i σ ρ − i σ ρ (cid:18) ∂∂ϕ − ζ ∂∂z (cid:19) − i σ ∂∂z (cid:21) φ. (11)With χ being the “small” component of the wave function, we can consider | mχ | ≫ (cid:12)(cid:12)(cid:12) i ∂χ∂t (cid:12)(cid:12)(cid:12) , and | mχ | ≫ (cid:12)(cid:12)(cid:12) iω ∂χ∂ϕ (cid:12)(cid:12)(cid:12) , thus, we can write χ ≈ m (cid:20) − i σ ∂∂ρ − i σ ρ − i σ ρ (cid:18) ∂∂ϕ − ζ ∂∂z (cid:19) − i σ ∂∂z (cid:21) φ. (12)Substituting χ of the expression (12) into (10), we obtain a second order differential equationgiven by i ∂φ∂t = − m " ∂ ∂ρ + 1 ρ ∂∂ρ + 1 ρ (cid:18) ∂∂ϕ − ζ ∂∂z (cid:19) + ∂ ∂z φ + i m σ ρ (cid:18) ∂∂ϕ − ζ ∂∂z (cid:19) φ (13)+ 18 mρ φ + iω ∂φ∂ϕ , which corresponds to the Schr¨odinger-Pauli equation for a spin-1 / φ is an eigenfunction of σ in Eq. (13), whose eigenvalues are s = ±
1. Thus, we canwrite σ φ s = ± φ s = sφ s . We can see that the operators ˆ p z = − i∂ z and ˆ J z = − i∂ ϕ [57] commutewith the Hamiltonian of the right-hand side of (13), thus, we write the solution of (13) in terms ofthe eigenvalues of the operator ˆ p z = − i∂ z , and the z -component of the total angular momentumˆ J z = − i∂ ϕ [68]: φ s = e − i E t e i ( l + ) ϕ e ikz R + ( ρ ) R − ( ρ ) , (14)where l = 0 , ± , ± , . . . and k is a constant. Substituting the solution (14) into the Schr¨odinger-Pauli equation (13), we obtain two noncoupled equations for R + and R − . After some calculations,6e can write the noncoupled equations for R + and R − in the following compact form: R ′′ s + 1 ρ R ′ s − ν s ρ R s + η R s = 0 , (15)where we have defined the following parameters: ν ± = ν s = l + 12 (1 − s ) − ζ k (16) η = 2 m (cid:20) E + ω (cid:18) l + 12 (cid:19) − k m (cid:21) . The second order differential equation (15) corresponds to the Bessel differential equation [58].The general solution of (15) is given by: R s ( ρ ) = A J ν s ( ηρ ) + B N ν s ( ηρ ), where J ν s ( ηρ ) and N ν s ( ηρ ) are the Bessel function of first and second kinds [58]. In order to have a regular solutionat the origin, we must take B = 0 in the general solution of Eq. (15), since the Neumann functiondiverges at the origin. Thus, the solution of (15) becomes: R s ( ρ ) = A J | ν s | ( ηρ ). Moreover, we wishto obtain a normalized wave function inside the region 0 < ρ < √ − ω ζ ω , therefore we considerthe spin-1 / ρ → ρ = √ − ω ζ ω , that is, R s ρ → ρ = p − ω ζ ω ! = 0 . (17)Then, by assuming that ηρ ≫ ρ = √ − ω ζ ω ), and we can take [43, 58] J | ν s | ( ηρ ) → r πηρ cos (cid:18) ηρ − | ν s | π − π (cid:19) . (18)Hence, substituting (18) into (17), we obtain E n, l ≈ m ω (1 − ω ζ ) (cid:20) n π + π (cid:12)(cid:12)(cid:12)(cid:12) l + 12 (1 − s ) − ζ k (cid:12)(cid:12)(cid:12)(cid:12) + 3 π (cid:21) + k m − ω [ l + 1 / . (19)Equation (19) is the spectrum of energy of a nonrelativistic Dirac particle confined to a hard-wall confining potential under the influence of torsion and noninertial effects. We can observethe influence of torsion on the energy levels (19) in the effective angular momentum given by ν s = l + (1 − s ) − ζ k , and by the presence of the fixed radius ρ = √ − ω ζ ω . Note that thespectrum of energy obtained in Eq. (19) is proportional to n (parabolic energy spectrum) incontrast to recent studies of the influence of noninertial effects on the Landau quantization forneutral particles [41], whose spectrum of energy is proportional to the quantum number n . Thisresults from the imposition of having the wave function being normalized in the space restricted7y the range (3), where we have considered the wave function vanishing at ρ → ρ = √ − ω ζ ω and assumed ηρ ≫
1. It is worth mentioning an analogy between the present study and studiesof confinement of quantum particle to a quantum dot in condensed matter systems. The resultobtained in Eq. (19) agrees with the studies of the confinement of quantum particles to a quantumdot made in Refs. [45–48] where the quantum dot models provide a parabolic spectrum of energy(in relation to the quantum number n ). The main difference between the present work and thequantum dot models of Refs. [45–48] is that the hard-wall confining potential is determined bythe geometry of the manifold (described by the line element (2)). Hence, the geometrical approachyielded in the present work can be useful in studies of noninertial effects in condensed mattersystems possessing the presence of screw dislocations [9, 43].We also obtain in Eq. (19) the Page-Werner et al. term [28–30], which corresponds to thecoupling between the angular velocity ω and the quantum number l . Returning to the analogywith condensed matter systems, we have that the presence of the Page-Werner et al. term in Eq.(19) agrees with the studies of the confinement of a neutral particle to a quantum dot analogousto the Tan-Inkson model [59] made in Ref. [42]. Finally, we can see in the last term of Eq. (19)that there is no influence of the torsion on the Page-Werner et al. term [28–30]. III. CONCLUSIONS
In this brief report, we have discussed torsion and noninertial effects on the confinement of anonrelativistic Dirac particle to a hard-wall confining potential. We have seen, in the rotatingsystem, that the coordinate system becomes singular at ρ → √ − ω ζ ω , where the restriction ofthe values of the radial coordinate depends on the parameter related to the torsion of the defectand the angular velocity. We have also shown that by considering the wave function vanishing at ρ → ρ = √ − ω ζ ω and ηρ ≫
1, we can obtain bound states whose spectrum of energy is parabolicin relation to the quantum number n . We also have shown that the influence of torsion on theenergy levels is given by the presence of an effective angular momentum ν s = l + (1 − s ) − ζ k andby a fixed radius ρ = √ − ω ζ ω . We have also obtained the coupling between the angular velocity ω and the quantum number l , which is known as the Page-Werner et al. term [28–30]. Moreover,this study has shown that there exists no influence of the torsion on the Page-Werner et al. term[28–30].We would like to add a comment on the influence of torsion and noninertial effects in quantumsystems. It has been shown in Ref. [11] that the presence of torsion modifies the electromagnetic8eld in the rest frame of the observers. Following the study made in Ref. [11], it should beinteresting to consider either a charged particle or a neutral particle interacting with externalfields in the presence of torsion in a noninertial reference frame. Both torsion and noninertialeffects can yield new field configurations and, consequently, new contributions to the energy levelscan be obtained. Furthermore, the geometrical approach used in this work can be useful in studiesof quantum dots in condensed matter systems described by the Dirac equation such as graphene[18, 60], topological insulators [61] and cold atoms [62]. A different context of using the presentgeometrical approach can be on the behaviour of the Dirac oscillator [54] in the cosmic dislocationspacetime and the Casimir effect [63, 64]. Another interesting quantum effect arises from thepresence of a quantum flux in the energy levels of bound states called persistent currents. Forinstance, in Ref. [65], persistent currents arise from the dependence of the energy levels on theBerry phase [66] and the Aharonov-Anandan quantum phase [67]. In Ref. [35], persistent currentshave been investigated in a quantum ring from rotating effects. Hence, the study of persistentcurrents should be interesting in a scenario where there exists the presence of torsion in an elasticmedium in a noninertial frame. Acknowledgments
The author would like to thank CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico eTecnol´ogico - Brazil) for financial support. [1] F. W. Hehl, Gen. Relat. Grav.
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