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 suq(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.
