### Multi Parametric Deformed Heisenberg Algebras: A Route to ComplexityMulti Parametric Deformed Heisenberg Algebras: A Route to Complexity

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 Author Curado, E. M. F. ♦ Rego-Monteiro, M. A. Source arXiv.org Content type Text File Format PDF Date of Submission 2000-11-14 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Natural sciences & mathematics ♦ Mathematics ♦ Physics Subject Keyword High Energy Physics - Theory ♦ Condensed Matter - Statistical Mechanics ♦ Mathematical Physics ♦ Nonlinear Sciences - Exactly Solvable and Integrable Systems ♦ Quantum Physics ♦ math ♦ nlin ♦ physics:cond-mat ♦ physics:hep-th ♦ physics:math-ph ♦ physics:quant-ph 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. Description Reference: J.Phys.A34:3253-3264,2001 Educational Use Research Learning Resource Type Article Page Count 17