The Dirac oscillator in a spinning cosmic string spacetime

Eur. Phys. J. C (2019) 79:311 https://doi.org/10.1140/epjc/s10052-019-6830-4 Regular Article - Theoretical Physics The Dirac oscillator in a spinning cosmic string spacetime Mansoureh Hosseinpour1,a , Hassan Hassanabadi1,b , Marc de Montigny2,c 1 Faculty of Physics, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran 2 Faculté Saint-Jean, University of Alberta, Edmonton, AB T6C 4G9, Canada Received: 1 December 2018 / Accepted: 1 April 2019 © The Author(s) 2019 Abstract We examine the effects of gravitational fields produced by topological defects on a Dirac field and a Dirac oscillator in a spinning cosmic string spacetime. We obtain the eigenfunctions and the energy levels of the relativistic field in that background and consider the effect of various parameters, such as the frequency of the rotating frame, the oscillator’s frequency, the string density and other quantum numbers. 1 Introduction The Dirac equation with interactions linear in the coordinates was initially studied in Refs. [1–3]. Such a system was referred to as a ‘Dirac oscillator’ in Ref. [4], because, in the non-relativistic limit, it behaves as a harmonic oscillator with a strong spin-orbit coupling term. This model describes the dynamics of a harmonic oscillator for spin-half particles and is obtained by introducing a non-minimal prescription into the free Dirac Eq. [4]. It was observed that the Dirac oscillator interaction is a physical system which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field [5,6]. The authors of Ref. [5] also established the conformal invariance of the Dirac oscillator and the authors of Ref. [6] examined its covariance properties, applied the Foldy-Wouthuysen and the Cini-Toushek transformations. As a relativistic quantum mechanical system, the Dirac oscillator has been widely studied and, because it is an exactly solvable model, several investigations have been performed in its theoretical framework. Although the Dirac oscillator is normally utilized within the context of many-body theory, relativistic quantum mechanics and quantum chromodynamics (such as the interaction between quarks as well as the confining part of the a e-mail: b e-mail: c e-mail: 0123456789().: V,-vol phenomenological Cornell potential), the Dirac oscillator and related models have been applied in many other contexts as well, such as quantum optics [7–9], supersymmetry [5,10,11], nuclear reactions [12], the hadronic spectrum (with the two-body Dirac oscillator) [13,14], the Clifford algebra [15,16], non-commutative space [17,18], thermodynamic properties [19], Lie algebras [20], supersymmetric (non-relativistic) quantum mechanics [21], the supersymmetric path-integral formalism [22], chiral phase transitions in presence of a constant magnetic field [8], the relativistic Landau levels in presence of external magnetic field [23], the Aharonov-Bohm effect [24], and condensed matter physics phenomena and graphene [25]. Similar studies for the Duffin–Kemmer–Petiau (DKP) oscillator, which is analogous to the Dirac oscillator for spinless and spin-one particles, are in Refs. [26,27]. Finally, let us mention many studies of the Dirac oscillator with topological defects and cosmic string spacetimes in Refs. [28–34] and analogous investigations for scalar fields in Refs. [35–39]. Some studies of relativistic oscillators are in Refs. [40–47]. In this work, we examine the relativistic quantum dynamics of Dirac oscillator on the curved spacetime of a rotating cosmic string. From the corresponding Dirac equation, we analyze the influence of the topological defect on the equation of motion, the energy spectrum and the wave-functions. An analogous study for the Klein–Gordon equation is in Ref. [49]. In Sect. 2, we write down the covariant Dirac equation without oscillator in a spinning cosmic string spacetime, and find its wave-functions and energy eigenvalues. In Sect. 3, we present the covariant Dirac oscillator in the same spacetime and obtain the wave-functions and energy spectrum. We present concluding remarks in Sect. 4. 2 Dirac equation in the cosmic string spacetime The spacetime generated by a spinning cosmic string without internal structure, or ‘ideal’ spinning cosmic string, can be obtained from the line element 123 311 Page 2 of 7 Eur. Phys. J. C ds 2 = −dT 2 + d X 2 + dY 2 + d Z 2 (1) σ , can be obtained from symbols of the second kind, μν 1 μ i j = g μk by applying the coordinate transformation  2 T = t + aα −1 ϕ, X = r cosϕ, (2) Y = r sinϕ, ϕ = αϕ  , (2019) 79:311  ∂ g jk ∂ gi j ∂ gik , + − ∂q j ∂qi ∂qk (8) with coordinates (q1 , q2 , q3 ). With the metric in Eq. (3), the non-null components of Christoffel symbols are t rt ϕ = ϕr = −a , r r = −r α 2 , ϕϕ ϕ rϕϕ = ϕr = 1 . r (9) which leads to (see also Refs. [50–53]) ds 2 = −(dt + adϕ)2 + dr 2 + α 2 r 2 dϕ 2 + dz 2 = −dt 2 − 2adtdϕ + dr 2 +(α 2 r 2 − a 2 )dϕ 2 + dz 2 , (3) where −∞ < z < ∞ , r ≥ 0 and 0 ≤ ϕ ≤ 2π . We work with units such that c = 1. The angular parameter α runs in the interval (0, 1] and is related to the linear mass density μ of the string by α = 1 − 4μ, and to the deficit angle by γ = 2π(1 − α). We have also a = 4G j where G is the universal gravitation constant and j is the angular momentum of the spinning string; thus a is a length that represents the rotation of the cosmic string. Note that in this case, the source of the gravitational field of a spinning cosmic string possesses angular momentum and the metric (3) has an off-diagonal term involving time and space. The Dirac equation for a field  of mass M in the cosmic string spacetime described by Eq. (3) reads [54–57] μ (iγ (x) ∇μ − M) (x) = 0, (4) We can build the local reference frame through a nonā called tetrads or viercoordinate basis with components eμ beins which form our local reference frame. With the line μ ā (obtained in Ref. element (3), we can use tetrads eā and eμ [52]) as follows ⎛ 1 ⎜0 ā eμ = ⎜ ⎝0 0 ⎛ 1 ⎜ 0 μ eā = ⎜ ⎝0 0 0 cos ϕ sin ϕ 0 a −r αsin ϕ −r αcos ϕ 0 asin ϕ rα −acos ϕ rα −sin ϕ rα cos ϕ rα cos ϕ 0 sin ϕ 0 ⎞ 0 0⎟ ⎟, 0⎠ 1 ⎞ 0 0⎟ ⎟. 0⎠ 1 (10) The vierbeins satisfy the orthonormality conditions μ μ eā (x) eνā (x) = δν , ā (x) eμ (x) = δ ā , eμ b̄ b̄ (11) and with the covariant derivative ∇μ = ∂μ + μ (x) , ā gμν (x) = eμ (x) eνb̄ (x) ηāb̄ . (5) and the spinorial affine connections, μ =  1 ω γ ā , γ b̄ , 2 μā b̄ (6) where γ ā denotes the standard Dirac matrices in Minkowski spacetime with metric ηā b̄ = (−1, +1, +1, +1), and ωμā b̄ is the spin connection, given by ωμā b̄ = ηā c̄ ec̄ν eb̄σ σν μ − ηā c̄ eb̄ν eνc̄ . (7) We use greek indices μ, ν, etc. for the curved spacetime, and bar latin indices ā, b̄, etc. in Minkowski spacetime. As discussed in Refs. [58,59], the spin connection allows us to construct a local frame through the tetrad basis which gives the spinors in the curved spaceti (...truncated)