Abstract
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.
Highlights
In [1] it was proposed to classify the states of a relativistic particle by means of the invariant operators (p = momentum, p0 = m2c4 + c2p2, 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 + α2)
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
Summary
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 = αh /mc). In this approach the variable ρ was interpreted as the relativistic generalization of a relative coordinate. [3,4] it has been shown that the ρn-representation may 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. This will allow us to obtain an exact expression for the energy levels of the particle and an expression for the spectrum of the Ad S3 radius
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