421 research outputs found
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
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
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
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
Quantum Tomography twenty years later
A sample of some relevant developments that have taken place during the last
twenty years in classical and quantum tomography are displayed. We will present
a general conceptual framework that provides a simple unifying mathematical
picture for all of them and, as an effective use of it, three subjects have
been chosen that offer a wide panorama of the scope of classical and quantum
tomography: tomography along lines and submanifolds, coherent state tomography
and tomography in the abstract algebraic setting of quantum systems
Stochastic evolution of finite level systems: classical vs. quantum
Quantum dynamics of the density operator in the framework of a single
probability vector is analyzed. In this framework quantum states define a
proper convex quantum subset in an appropriate simplex. It is showed that the
corresponding dynamical map preserving a quantum subset needs not be stochastic
contrary to the classical evolution which preserves the entire simplex.
Therefore, violation of stochasticity witnesses quantumness of evolution.Comment: 8 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
On pseudo-stochastic matrices and pseudo-positive maps
Stochastic matrices and positive maps in matrix algebras proved to be very
important tools for analysing classical and quantum systems. In particular they
represent a natural set of transformations for classical and quantum states,
respectively. Here we introduce the notion of pseudo-stochastic matrices and
consider their semigroup property. Unlike stochastic matrices,
pseudo-stochastic matrices are permitted to have matrix elements which are
negative while respecting the requirement that the sum of the elements of each
column is one. They also allow for convex combinations, and carry a Lie group
structure which permits the introduction of Lie algebra generators. The quantum
analog of a pseudo-stochastic matrix exists and is called a pseudo-positive
map. They have the property of transforming a subset of quantum states
(characterized by maximal purity or minimal von Neumann entropy requirements)
into quantum states. Examples of qubit dynamics connected with "diamond" sets
of stochastic matrices and pseudo-positive maps are dealt with.Comment: 15 pages; revised versio
On Reduced Time Evolution for Initially Correlated Pure States
A new method to deal with reduced dynamics of open systems by means of the
Schr\"odinger equation is presented. It allows one to consider the reduced time
evolution for correlated and uncorrelated initial conditions.Comment: accepted in Open Sys. Information Dy
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