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q-Deformed Harmonic Oscillator in Phase Space

A. K. Aringazin·K. M. Aringazin·S. Baskoutas·G. Brodimas·A. Jannussis·E. Vlachos
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Abstract

Relation between Bopp-Kubo formulation and Weyl-Wigner-Moyal symbol calculus, and non-commutative geometry interpretation of the phase space representation of quantum mechanics are studied. Harmonic oscillator in phase space via creation and annihilation operators, both the usual and $q$-deformed, is investigated. We found that the Bopp-Kubo formulation is just non-commuting coordinates representation of the symbol calculus. The Wigner operator for the $q$-deformed harmonic oscillator is shown to be proportional to the 3-axis spherical angular momentum operator of the algebra $su_{q}(2)$. The relation of the Fock space for the harmonic oscillator and double Hilbert space of the Gelfand-Naimark-Segal construction is established. The quantum extension of the classical ergodiicity condition is proposed.

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A. K. Aringazin, K. M. Aringazin, S. Baskoutas, G. Brodimas, A. Jannussis, E. Vlachos. 1998-11-02. q-Deformed Harmonic Oscillator in Phase Space. https://arxiv.org/abs/hep-th/9811027

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