5,963 research outputs found
From the Jordan product to Riemannian geometries on classical and quantum states
The Jordan product on the self-adjoint part of a finite-dimensional
-algebra is shown to give rise to Riemannian metric
tensors on suitable manifolds of states on , and the covariant
derivative, the geodesics, the Riemann tensor, and the sectional curvature of
all these metric tensors are explicitly computed. In particular, it is proved
that the Fisher--Rao metric tensor is recovered in the Abelian case, that the
Fubini--Study metric tensor is recovered when we consider pure states on the
algebra of linear operators on a finite-dimensional
Hilbert space , and that the Bures--Helstrom metric tensors is
recovered when we consider faithful states on .
Moreover, an alternative derivation of these Riemannian metric tensors in terms
of the GNS construction associated to a state is presented. In the case of pure
and faithful states on , this alternative geometrical
description clarifies the analogy between the Fubini--Study and the
Bures--Helstrom metric tensor.Comment: 32 pages. Minor improvements. References added. Comments are welcome
Schwinger's Picture of Quantum Mechanics I: Groupoids
A new picture of Quantum Mechanics based on the theory of groupoids is
presented. This picture provides the mathematical background for Schwinger's
algebra of selective measurements and helps to understand its scope and
eventual applications. In this first paper, the kinematical background is
described using elementary notions from category theory, in particular the
notion of 2-groupoids as well as their representations. Some basic results are
presented, and the relation with the standard Dirac-Schr\"odinger and
Born-Jordan-Heisenberg pictures are succinctly discussed.Comment: 32 pages. Comments are welcome
Geometrical Structures for Classical and Quantum Probability Spaces
On the affine space containing the space of quantum states of
finite-dimensional systems there are contravariant tensor fields by means of
which it is possible to define Hamiltonian and gradient vector fields encoding
relevant geometrical properties of . Guided by Dirac's analogy
principle, we will use them as inspiration to define contravariant tensor
fields, Hamiltonian and gradient vector fields on the affine space containing
the space of fair probability distributions on a finite sample space and
analyse their geometrical properties.
Most of our considerations will be dealt with for the simple example of a
three-level system.Comment: 16 page
Manifolds of classical probability distributions and quantum density operators in infinite dimensions
The manifold structure of subsets of classical probability distributions and
quantum density operators in infinite dimensions is investigated in the context
of -algebras and actions of Banach-Lie groups. Specificaly, classical
probability distributions and quantum density operators may be both described
as states (in the functional analytic sense) on a given -algebra
which is Abelian for Classical states, and non-Abelian for
Quantum states. In this contribution, the space of states of a
possibly infinite-dimensional, unital -algebra is
partitioned into the disjoint union of the orbits of an action of the group
of invertible elements of . Then, we prove that the
orbits through density operators on an infinite-dimensional, separable Hilbert
space are smooth, homogeneous Banach manifolds of
, and, when admits a
faithful tracial state like it happens in the Classical case when we
consider probability distributions with full support, we prove that the orbit
through is a smooth, homogeneous Banach manifold for .Comment: 35 pages. Revised version in which some imprecise statements have
been amended. Comments are welcome
Dynamical aspects in the Quantizer-Dequantizer formalism
The use of the quantizer-dequantizer formalism to describe the evolution of a
quantum system is reconsidered. We show that it is possible to embed a manifold
in the space of quantum states of a given auxiliary system by means of an
appropriate quantizer-dequantizer system. If this manifold of states is
invariant with respect to some unitary evolution, the quantizer-dequantizer
system provides a classical-like realization of such dynamics, which in general
is non linear. Integrability properties are also discussed. Weyl systems and
generalized coherente states are used as a simple illustration of these ideas.Comment: 15 page
Schwinger's Picture of Quantum Mechanics IV: Composition and independence
The groupoids description of Schwinger's picture of quantum mechanics is
continued by discussing the closely related notions of composition of systems,
subsystems, and their independence. Physical subsystems have a neat algebraic
description as subgroupoids of the Schwinger's groupoid of the system. The
groupoids picture offers two natural notions of composition of systems: Direct
and free products of groupoids, that will be analyzed in depth as well as their
universal character. Finally, the notion of independence of subsystems will be
reviewed, finding that the usual notion of independence, as well as the notion
of free independence, find a natural realm in the groupoids formalism. The
ideas described in this paper will be illustrated by using the EPRB experiment.
It will be observed that, in addition to the notion of the non-separability
provided by the entangled state of the system, there is an intrinsic
`non-separability' associated to the impossibility of identifying the entangled
particles as subsystems of the total system.Comment: 32 pages. Comments are welcome
Consumersâ Attitudes on Services of General Interest in the EU: Accessibility, Price and Quality 2000-2004
The research question addressed by this paper is a simple one: are European consumers happy with the services provided by the utilities after two decades of reforms? We focus on electricity, gas, water, telephone in the EU 15 Member States. The variables we analyse are consumersâ satisfaction with accessibility, price and quality, as reported in three waves of Eurobarometer survey, 2000-2002-2004, comprising around 47,000 observations. We use ordered logit models to analyze the impact of privatization and regulatory reforms, as represented by an OECD dataset, controlling for individual and country characteristics. Our results do not support a clear association between consumersâ satisfaction and a standard reform package of privatization, vertical disintegration, liberalization.Consumersâ Satisfaction, Gas, Electricity, Telephone, Water, Eurobarometer
Parkinsonâs disease motor disorganization and temporal processing
Motor control is essential for everyday life and highly contributes to the development and organisation of higher cognitive functions. Embodied cognition endemically approaches cognitive activities, grounding on sensory-motor processes and the ability to switch from each other in response to specific context and situations. In this view, it is possible to deliberate higher functions such as âexpertiseâ and âdecision makingâ as the ability to reactivate, deconstruct and reconstruct different motor plans in their subroutines to plastically react to external or internal environmental requirements.peer-reviewe
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