Quantum Bi-Hamiltonian systems, alternative Hermitian structures and Bi-Unitary transformations

We discuss the dynamical quantum systems which turn out to be bi-unitary with respect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbert space. We give a necessary and sufficient condition so that the Hermitian structures are in generic position. Finally the transformations of the bi-unitary group are explicitly obtained.Comment: Note di Matematica vol 23, 173 (2004

Alternative Algebraic Structures from Bi-Hamiltonian Quantum Systems

We discuss the alternative algebraic structures on the manifold of quantum states arising from alternative Hermitian structures associated with quantum bi-Hamiltonian systems. We also consider the consequences at the level of the Heisenberg picture in terms of deformations of the associative product on the space of observables.Comment: Accepted for publication in Int. J. Geom. Meth. Mod. Phy

Poisson Lie Group Symmetries for the Isotropic Rotator

We find a new Hamiltonian formulation of the classical isotropic rotator where left and right $SU(2)$ transformations are not canonical symmetries but rather Poisson Lie group symmetries. The system corresponds to the classical analog of a quantum mechanical rotator which possesses quantum group symmetries. We also examine systems of two classical interacting rotators having Poisson Lie group symmetries.Comment: 22pp , Latex fil

The Quantum-Classical Transition: The Fate of the Complex Structure

According to Dirac, fundamental laws of Classical Mechanics should be recovered by means of an "appropriate limit" of Quantum Mechanics. In the same spirit it is reasonable to enquire about the fundamental geometric structures of Classical Mechanics which will survive the appropriate limit of Quantum Mechanics. This is the case for the symplectic structure. On the contrary, such geometric structures as the metric tensor and the complex structure, which are necessary for the formulation of the Quantum theory, may not survive the Classical limit, being not relevant in the Classical theory. Here we discuss the Classical limit of those geometric structures mainly in the Ehrenfest and Heisenberg pictures, i.e. at the level of observables rather than at the level of states. A brief discussion of the fate of the complex structure in the Quantum-Classical transition in the Schroedinger picture is also mentioned.Comment: 19 page

Quantum Systems and Alternative Unitary Descriptions

Motivated by the existence of bi-Hamiltonian classical systems and the correspondence principle, in this paper we analyze the problem of finding Hermitian scalar products which turn a given flow on a Hilbert space into a unitary one. We show how different invariant Hermitian scalar products give rise to different descriptions of a quantum system in the Ehrenfest and Heisenberg picture.Comment: 18 page

A tomographic setting for quasi-distribution functions

The method of constructing the tomographic probability distributions describing quantum states in parallel with density operators is presented. Known examples of Husimi-Kano quasi-distribution and photon number tomography are reconsidered in the new setting. New tomographic schemes based on coherent states and nonlinear coherent states of deformed oscillators, including q-oscillators, are suggested. The associated identity decompositions providing Gram-Schmidt operators are explicitly given, and contact with the Agarwal-Wolf $\Omega$-operator ordering theory is made.Comment: A slightly enlarged version in which contact with the Agarwal-Wolf $\Omega$-operator ordering theory is mad