2,990 research outputs found
Alternative Structures and Bihamiltonian Systems
In the study of bi-Hamiltonian systems (both classical and quantum) one
starts with a given dynamics and looks for all alternative Hamiltonian
descriptions it admits.In this paper we start with two compatible Hermitian
structures (the quantum analog of two compatible classical Poisson brackets)
and look for all the dynamical systems which turn out to be bi-Hamiltonian with
respect to them.Comment: 18 page
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 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
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