Symplectic Structures and Quantum Mechanics

Canonical coordinates for the Schr\"odinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schr\"odinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a {\sl recursion operator} which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone-von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel'fand-Levitan-Marchenko formula.Comment: 11 pages, LaTex fil

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

Alternative linear structures for classical and quantum systems

The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as "adapted" to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different non equivalent ways, "evading", so to speak, the von Neumann uniqueness theorem.Comment: 32 pages, two figures, to be published in IJMP

The Hamilton--Jacobi Theory and the Analogy between Classical and Quantum Mechanics

We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (HJ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJ theory on the tangent bundle of a configuration manifold, the quantum HJ theory, HJ problems for general differential operators and the HJ problem for Lie groups.Comment: 42 pages, LaTeX with AIMS clas

The space of density states in geometrical quantum mechanics

We present a geometrical description of the space of density states of a quantum system of finite dimension. After presenting a brief summary of the geometrical formulation of Quantum Mechanics, we proceed to describe the space of density states \D(\Hil) from a geometrical perspective identifying the stratification associated to the natural GL(\Hil)--action on \D(\Hil) and some of its properties. We apply this construction to the cases of quantum systems of two and three levels.Comment: Amslatex, 18 pages, 4 figure

Alternative structures and bi-Hamiltonian systems on a Hilbert space

We discuss transformations generated by dynamical quantum systems which are bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an infinite-dimensional complex Hilbert space. We introduce the notion of Hermitian structures in generic relative position. We provide few necessary and sufficient conditions for two Hermitian structures to be in generic relative position to better illustrate the relevance of this notion. The group of bi-unitary transformations is considered in both the generic and non-generic case. Finally, we generalize the analysis to real Hilbert spaces and extend to infinite dimensions results already available in the framework of finite-dimensional linear bi-Hamiltonian systems.Comment: 11 page

Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.Comment: AmsLaTeX, 37 pages, 8 figures. This paper is an expanded version of some lectures delivered by one of us (G. M.) at the ``Advanced Winter School on the Mathematical Foundation of Quantum Control and Quantum Information'' which took place at Castro Urdiales (Spain), February 11-15, 200

Tensorial description of quantum mechanics

Relevant algebraic structures for the description of Quantum Mechanics in the Heisenberg picture are replaced by tensorfields on the space of states. This replacement introduces a differential geometric point of view which allows for a covariant formulation of quantum mechanics under the full diffeomorphism group.Comment: 8 page