A model of the two-dimensional quantum harmonic oscillator in an \(AdS_3\) background

Eur. Phys. J. C (2016) 76:551 DOI 10.1140/epjc/s10052-016-4381-5 Regular Article - Theoretical Physics A model of the two-dimensional quantum harmonic oscillator in an Ad S3 background R. Fricka Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77, 50937 Cologne, Germany Received: 24 March 2016 / Accepted: 15 September 2016 / Published online: 8 October 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In this paper we study a model of the twodimensional quantum harmonic oscillator in a threedimensional anti-de Sitter background. We use a generalized Schrödinger picture in which the analogs of the Schrödinger operators of the particle are independent of both the time and the space coordinates in different representations. The spacetime independent operators of the particle induce the Lie algebra of Killing vector fields of the Ad S3 spacetime. In this picture, we have a metamorphosis of the Heisenberg uncertainty relations. 1 Introduction In [1] it was proposed to classify the states of a relativistic particle by means of the invariant operators (p = momentum, p0 = m 2 c4 + c2 p2 , m = mass,) C1 (p) = N2 − L2 , C2 (p) = N · L (1) characterizing the infinite-dimensional unitary representations of the Lorentz group, and to carry out the expansion of the wave function in the momentum space representation over the functions (0 ≤ α < ∞, n2 (θ, ϕ) = 1) ξ (0) (p, α, n) := [( p0 − cp · n)/mc2 ]−1+iα . (2) The functions ξ (0) (p, α, n) are the eigenfunctions of the operator C1 (p), (C1 (p)⇒1 + α 2 ). The boost and rotation generators of the Lorentz group have the form (spin = 0) N i = i p0 ∂ , ∂cpi L i = ii jk pk ∂ . ∂ pj The operator C2 (p) vanishes for a spinless particle. a e-mail: The expansion proposed in [1] does not include any dependence on the time t and space coordinates x, i.e. it is ”spacetime independent”. In [2], in the framework of a two-particle equation of the quasipotential type, the expansion over the functions ξ ∗ (p, α, n) was used to introduce the “relativistic configurational” representation (in following the ρn-representation, ρ = α h̄/mc). In this approach the variable ρ was interpreted as the relativistic generalization of a relative coordinate. It was shown that the corresponding operators of the Hamiltonian H (ρ, n) and the 3-momentum P(ρ, n), defined on the functions ξ ∗ (p, ρ, n), has a form of the differential-difference operators. In Refs. [3,4] it has been shown that the ρn-representation may also be used in a so-called generalized Schrödinger picture in which the analogs of the Schrödinger operators of a particle are independent of both the time and the space coordinates in different representations. It was found that the operators H (ρ, n), P(ρ, n), L(n), and N(ρ, n) = ρn + (n × L − L × n)/2mc satisfy the commutations relations of the Poincaré algebra in the ρn-representation. We have two spacetime independent representations of the Poincaré algebra; the p and the ρn-representation. In the GS-picture the ρn-representation may be used to describe extended objects like strings. In the case of the one-dimensional momentum space representation ( p = momentum, m = mass, p02 − c2 p 2 = m 2 c4 ) the eigenfunctions of the boost generator N ( p) = i p0 ∂cp , (N ⇒ mc h̄ ρ) may be written in the form mc ξ1 ( p, ρ) = [( p0 − cp)/mc2 ]i h̄ ρ . (4) The expansion (3) 1 ψ(ρ) = (2π )1/2  dp ψ( p) ξ ∗ 1 ( p, ρ) p0 (5) leads to the functions ψ(ρ) in the ρ-representation. In the ρrepresentation the Hamilton operator H and the momentum 123 551 Page 2 of 4 Eur. Phys. J. C (2016) 76:551 h̄ operator P of the particle have the form (λ̃ = mc ) H (ρ) = mc2 cosh(−i λ̃∂ρ ), P(ρ) = mc sinh(−i λ̃∂ρ ), (6) and they satisfy the commutation relations of the Poincaré algebra [ρ, P] = i h̄ h̄ H, [P, H ] = 0, [H, ρ] = −i P. 2 mc m (7) 2 One particle quantum equation in Ad S3 spacetime For a free particle in the Minkowski spacetime of two dimensions (d = 2), the coordinates t, x may be introduced in the states with the help of the transformation S(t, x) = exp[−i(t H − x P)/h̄]. 1  ixp dp ψ( p, t)e h̄ . p0 (9) (10) In (9), the spacetime coordinates appear in the states in the ρ- and in the p-representation. We have a metamorphosis of the Heisenberg uncertainty relation, x· p ≥ h̄/2. From [ρ, P] = i h̄ H/mc2 in (7) it follows that instead of x· p ≥ h̄/2, we have ρ· p ≥ h̄/2. (11) The GS-picture may be used in a quantum theory of gravity in which objects need a sharply defined frame. In Ref. [4], this picture was used to describe the motion of a relativistic particle in anti-de Sitter spacetime (d = 2, d = 4). It was found that the spacetime independent operators of the particle in an external field (like in the case of a harmonic oscillator) induce the Lie algebra of Killing vector fields of the Ad S4 spacetime (d = 4; a = 1, 2, . . . , 10; {x i }, i = 1, 2, 3.) K a (t, x i )(ρ, n, t, x i ) = Ba (ρ, n)(ρ, n, t, x i ). (12) (13) where (n 1 = cos ϕ, n 2 = sin ϕ). The Hamilton operator and the momentum operators of the particle defined on the functions ξ ∗ 2 (p, ρ, n) have the form [5]  i h̄c    H (ρ, n) = mc2 cosh i λ̃∂ρ + sinh i λ̃∂ρ 2ρ 2 (−i h̄∂ϕ ) (14) ei λ̃∂ρ , − mρ(2ρ + i λ̃) P1 (ρ, n) = mcn 1 (H/mc2 − ei λ̃∂ρ ) + i h̄n 2 · ∂ϕ i λ̃∂ρ , (15) e ρ + 2i λ̃ P2 (ρ, n) = mcn 2 (H/mc2 − ei λ̃∂ρ ) − i h̄n 1 · ∂ϕ i λ̃∂ρ . (16) e ρ + 2i λ̃ The operators H , P, and the three operators of the Lorentz algebra in the ρn-representation   i N1 (ρ, n) = n 1 ρ − λ̃ − i λ̃n 2 ∂ϕ , (17) 2   i N2 (ρ, n) = n 2 ρ − λ̃ + i λ̃n 1 ∂ϕ , L = −i h̄∂ϕ , (18) 2 satisfy the commutation relations of the Poincaré algebra. For the particle in an external field like the two-dimensional harmonic oscillator potential we use the following operators:  P̂0 (ρ, n) = H (ρ, n) + H0 (ρ), (19) P̂i (ρ, n) = Pi (ρ, n) + Pi (ρ, n), (20) where (ω = frequency, i = 1, 2)    mω2 i  H0 = ρ − λ̃ ρ − i λ̃ e−i λ̃∂ρ , 2 2    2 i mω  ρ − λ̃ ρ − i λ̃ e−i λ̃∂ρ . Pi = n i 2c 2 (21)  Here  denotes the wave function of the particle. The operators of the Killing vector field K a (t, x i ) satisfy the same commutation rules as the spacetime independent operators Ba (ρ, n), except for the minus signs on the right-hand sides. Equations (12) are valid for any d. In the present paper we 123 mc ξ2 (p, ρ, n) := [( p0 − cp · n)/mc2 ]− 2 +i h̄ ρ , where ψ( p, t, x) = ψ( p)exp[−i(t p0 − x p)/h̄]. In the case of a point particle (ρ = 0) we have the Fourier transform in relativistic quantum mechanics, 1 ψ(t, x) = (2π )1/2 In the two-dimensional momentum space representation, the first Casimir operator of the Lorentz group C1 (p) has the eigenfuctions ( p02 − c2 p12 − c2 p22 = m 2 c4 ) (8) We obtain S(t, x)ψ(ρ) = ψ(ρ, t, x)  1 dp = ψ( p, t, x) ξ1∗ ( p, ρ), (2π )1/2 p0 use these equations to describe the motion of a particle in Ad S3 spacetime. In the case of d = 3 we need six spacetime independe (...truncated)