Multi Parametric Deformed Heisenberg Algebras: A Route to Complexity

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.

latex, 17 pages, 5 PS figures; to be published in J. Phys. A: Math and Gen (2001); a few sentences were added in order to clarify some points

References in corpus (3)

• Thermostatistics of q-deformed boson gas
• Thermodynamic Properties of a Solid Exhibiting the Energy Spectrum given by the Logistic Map
• q-deformed dynamics of q-deformed oscillators

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