Multi Parametric Deformed Heisenberg Algebras: A Route to Complexity

TL;DR

This paper generalizes the Heisenberg algebra using parametric functions, exploring linear, polynomial, and quadratic cases, revealing connections to q-oscillators, multi-parameter deformations, and chaotic dynamics with implications for complex quantum systems.

Contribution

It introduces a broad class of deformed Heisenberg algebras based on functional parameters, analyzing their representations and revealing links to chaos and non-continuous deformations.

Findings

Linear case corresponds to q-oscillators with q related to the slope.

Polynomial cases lead to multi-parameter deformations.

Quadratic case exhibits chaotic behavior and non-deformable representations.

Abstract

We introduce a generalization of the Heisenberg algebra which is written in terms of a functional of one generator of the algebra, $f(J_0)$, that can be any analytical function. When $f$ is linear with slope $\theta$, we show that the algebra in this case corresponds to $q$-oscillators for $q^2 = \tan \theta$. The case where $f$ is a polynomial of order $n$ in $J_0$ corresponds to a $n$-parameter deformed Heisenberg algebra. The representations of the algebra, when $f$ is any analytical function, are shown to be obtained through the study of the stability of the fixed points of $f$ and their composed functions. The case when $f$ is a quadratic polynomial in $J_0$, the simplest non-linear scheme which is able to create chaotic behavior, is analyzed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of Heisenberg algebra.