A model of the two-dimensional quantum harmonic oscillator in an $AdS_3$ background

In this paper we study a model of the two-dimensional quantum harmonic oscillator in a 3-dimensional anti-de Sitter background. We use a generalized Schr\"odinger picture in which the analogs of the Schr\"odinger 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 $AdS_3$ spacetime. In this picture, we have a metamorphosis of the Heisenberg's uncertainty relations.


Introduction
In [1] it was proposed to classify the states of a relativistic particle by means of the invariant operators (p = momentum, p 0 = √ m 2 c 4 + c 2 p 2 , m=mass,) 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 < ∞, n 2 (θ, ϕ) = 1) The functions ξ (0) (p, α, n) are the eigenfunctions of the operator C 1 (p), (C 1 (p)=⇒1 + α 2 ). The boost and rotation generators of the Lorentz group have the form (spin = 0) The operator C 2 (p) vanishes for a spinless particle. 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 ρn-representation, ρ = αh/mc) . In this approach the variable ρ was interpeted 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 our previous papers [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 representation of the Poincaré algebra; the p and the ρn-representation. In the GS-picture the ρn-representation may be used to describe extendent objects like strings.
In the case of the one-dimensional momentum space representation (p = momentum, m = mass, p 2 0 − c 2 p 2 = m 2 c 4 ) the eigenfunctions of the boost generator N(p) = ip 0 ∂ cp , (N=⇒ mc h ρ) may be written in the form The following expansion leads to the functions ψ(ρ) in the ρ-representation. In the ρ-representation the Hamilton operator H and the momentum operator P of the particle have the form (λ =h mc ) and satisfy the commutation relations of the Poincaré algebra 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 We obtain where ψ(p, t, In the case of a point particle (ρ = 0) we have the Fourier transform in relativistic quantum mechanics In (9), the spacetime coordinates appear in the states in the ρ and in the p-representation. We have a metamorphosis of the Heisenberg's uncertainty relation ∆x·∆p ≥h/2. From [ρ, P ] = ihH/mc 2 in (7) follows that instead of ∆x·∆p ≥h/2, we have ∆ρ · ∆p ≥h/2.
The GS-picture may be used in a quantum theory of gravity in which objects need a sharply defined frame. In the paper [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 AdS 4 spacetime (d = 4; a = 1, 2, ...10; Here Φ denotes the wave function of the particle. The operators of Killing vector field K a (t, x i ) satisfy the same commutation rules as the spacetime independent operators B a (ρ, n), except for the minus signs on the right-hand sides. The equations (12) are valid for any d. In the present paper we use these equations to describe the motion of a particle in AdS 3 spacetime. In the case of d = 3 we need six spacetime independent operators of the particle. In Sect. 2 we will now show that the operators of a relativistic model of the two-dimensional quantum harmonic oscillator in the ρn-representation can be used in Eqs. (12). This will allow us to obtain an exact expression for the energy levels of the particle and an expression for the spectrum of the AdS 3 radius.

One particle quantum equation in AdS 3 spacetime
In the two-dimensional momentum space representation, the first Casimir operator of the Lorentz group C 1 (p) has the eigenfuctions ( 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] H(ρ, n) = mc 2 cosh iλ∂ ρ + ihc 2ρ The operators H, P, and the three operators of the Lorentz algebra in the ρn-representation satisfy the commutation relations of the Poincare algebra. For the particle in an external field like the two-dimensional harmonic oscillator potential we use the following operatorŝ where (ω = f requency, i = 1, 2) In the nonrelativistic limit the operatorP 0 (ρ) − mc 2 assume the form The operatorsP and L, N i (ρ, n) satisfy the commutations rules of the Lie algebra so(2, 2) For the Casimir operator we have The explicit forms of the six operators K a (t, x i ) depend on the realisation in terms of the spacetime coordinates. We have the problem of determining observables in the GS-picture. In order to interpret the operatorP 0 as Hamilton operator, we chose the following realisation (t, x 1 , x 2 (x 1 = r cos ϕ, x 2 = r sin ϕ, i = 1, 2)), The set of the operators {K 03 , K i3 , K i0 , K ij } determines the same Lie algebra as the operators {P 0 ,P i , N i , L} except for the minus signs on the right-hand sides The operators {K 03 , K i3 , K i0 , K ij } form a basis for the SO(2, 2) group generators and related to Killing vectors of the AdS 3 spacetime with metric Here, the constant ω/c is related to the radius κ of the AdS 3 spacetime (κ = c/ω). We can introduce the equation which define the operatorP 0 (ρ, n) as Hamilton operator of the particle. A general solution of Φ(ρ, n; t, x 1 , x 2 ) can be written as a sum of separated solutions or the eigenfunctions of the operatorsP 0 (ρ, n) and the Casimir operator (τ = ωt, tan σ = ωr/c) The eigenfunctions of C(τ, σ, ϕ) are (n = 0, 1, 2.., | m| = 0, 1, 2.., M = 2, 3, 4, .., λ = 2n + | m| + M) where N M n| m| m are the normalization constants and are the hypergeometric functions. Thus we find For the spectrum of ih ∂ ∂t we have The eigensolutions of the Hamilton operatorP 0 (ρ, n) are ( ρ = mcρ h , M = 1 + 1/4 + ( mc 2 hω ) 2 , l = |m| = 0, 1, 2.., ) where c M nl are normalization constants and S n ( ρ) are the Hahn polynomials S n ( ρ 2 ; l + 1/2, M − 1/2), 1/2)) = Γ(l + M + n)Γ(l + M) Γ(l + 1 + n)Γ(l + 1) × 3 F 2 (−n, l + 1/2 + i ρ, l + 1/2 − i ρ; l + M, l + 1; 1).
For the function Φ(ρ, n; t, x 1 , x 2 ), we have (m = m, l = | m|) From follows that the oscillator frequency is discrete and for higher M decreases accoding to The energy spectrum of the particle can be written as For the AdS 3 radius κ = c/ω, we have κ M = (M 2 − 2M + 3/4)h/mc.

Conclusion
In this paper we have shown that a generalized Schrödinger picture may be used to describe a relativistic particle in a 3-dimensional anti-de Sitter spacetime. A specific feature of this picture is that the frame itself becomes dynamical. It was found that in this picture we have a metamorphosis of the Heisenberg's uncertainty relations. We have shown that, the energy of the particle and the anti-de Sitter radius are discrete.