Model of a relativistic oscillator in a generalized Schrödinger picture

Abstract

In a generalized Schrödinger picture, we consider the motion of a relativistic particle in an external field (like in the case of a harmonic oscillator). In this picture the analogs of the Schrödinger operators of the particle are independent of both the time and the space coordinates. These operators induce operators which are related to Killing vectors of the Anti de Sitter (AdS) space. We also consider the nonrelativistic limit.

On the One‐Dimensional Transition State Theory and the Relation between Statistical and Deterministic Oscillation Frequencies of Anharmonic Energy Wells

“…The results are also useful in the context of relativistic quantum mechanics. [69,70] We are able to show that this relationship is exactly as in the classical case, thus proving its thermodynamic ground, independent of the physical theory considered.…”

One particle quantum equation in a de Sitter spacetime

“…Here, in analogy to (11) for the external field which corresponds to a repulsive force we use the following operators…”

“…In the paper [5], it was shown that equations like (5) may be used to describe a particle in an Anti-de Sitter spacetime. It was found that operators with an external field in the ρn-representation which corresponds to an attractive force induce operators K a (t, x i ) which are related to Killing vectors of the AdS spacetime (a = 1, 2, ...10; {x i }, i = 1, 2, 3.…”

A model of the two-dimensional quantum harmonic oscillator in an $$AdS_3$$ A d S 3 background

“…[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.…”

“…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 S 4 spacetime (d = 4; a = 1, 2, .…”