Star Products on Coadjoint Orbits

We study properties of a family of algebraic star products defined on coadjoint orbits of semisimple Lie groups. We connect this description with the point of view of differentiable deformations and geometric quantization.Comment: Talk given at the XXIII ICGTMP, Dubna (Russia) August 200

Notions of Infinity in Quantum Physics

In this article we will review some notions of infiniteness that appear in Hilbert space operators and operator algebras. These include proper infiniteness, Murray von Neumann's classification into type I and type III factors and the class of F{/o} lner C*-algebras that capture some aspects of amenability. We will also mention how these notions reappear in the description of certain mathematical aspects of quantum mechanics, quantum field theory and the theory of superselection sectors. We also show that the algebra of the canonical anti-commutation relations (CAR-algebra) is in the class of F{/o} lner C*-algebras.Comment: 11 page

Quantum twistors

We compute explicitly a star product on the Minkowski space whose Poisson bracket is quadratic. This star product corresponds to a deformation of the conformal spacetime, whose big cell is the Minkowski spacetime. The description of Minkowski space is made in the twistor formalism and the quantization follows by substituting the classical conformal group by a quantum group.Comment: 47 pages. references added, some parts rewritten. To appear in 'p-adic Numbers, Ultrametric Analysis and Applicarions

Torsion formulation of gravity

We make it precise what it means to have a connection with torsion as solution of the Einstein equations. While locally the theory remains the same, the new formulation allows for topologies that would have been excluded in the standard formulation of gravity. In this formulation it is possible to couple arbitrary torsion to gauge fields without breaking the gauge invariance.Comment: AMS-LaTeX, 25 pages. Appendices have been eliminated and the necessary concepts have been inroduced in the text. We have added some reference

Considerations on Super Poincare Algebras and their Extensions to Simple Superalgebras

We consider simple superalgebras which are a supersymmetric extension of \fspin(s,t) in the cases where the number of odd generators does not exceed 64. All of them contain a super Poincar\'e algebra as a contraction and another as a subalgebra. Because of the contraction property, some of these algebras can be interpreted as de Sitter or anti de Sitter superalgebras. However, the number of odd generators present in the contraction is not always minimal due to the different splitting properties of the spinor representations under a subalgebra. We consider the general case, with arbitrary dimension and signature, and examine in detail particular examples with physical implications in dimensions $d=10$ and $d=4$.Comment: 16 pages, AMS-LaTeX. Version to appear in the Reviews in Mathematical Physic

Twisted duality of the CAR-Algebra

We give a complete proof of the twisted duality property M(q)'= Z M(q^\perp) Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.Comment: 32 pages, Latex2e, to appear in Journal of Mathematical Physic

A Comparative Analysis of 2D and 3D Tasks for Virtual Reality Therapies Based on Robotic-Assisted Neurorehabilitation for Post-stroke Patients

Post-stroke neurorehabilitation based on virtual therapies are performed completing repetitive exercises shown in visual electronic devices, whose content represents imaginary or daily life tasks. Currently, there are two ways of visualization of these task. 3D virtual environments are used to get a three dimensional space that represents the real world with a high level of detail, whose realism is determinated by the resolucion and fidelity of the objects of the task. Furthermore, 2D virtual environments are used to represent the tasks with a low degree of realism using techniques of bidimensional graphics. However, the type of visualization can influence the quality of perception of the task, affecting the patient's sensorimotor performance. The purpose of this paper was to evaluate if there were differences in patterns of kinematic movements when post-stroke patients performed a reach task viewing a virtual therapeutic game with two different type of visualization of virtual environment: 2D and 3D. Nine post-stroke patients have participated in the study receiving a virtual therapy assisted by PUPArm rehabilitation robot. Horizontal movements of the upper limb were performed to complete the aim of the tasks, which consist in reaching peripheral or perspective targets depending on the virtual environment shown. Various parameter types such as the maximum speed, reaction time, path length, or initial movement are analyzed from the data acquired objectively by the robotic device to evaluate the influence of the task visualization. At the end of the study, a usability survey was provided to each patient to analysis his/her satisfaction level. For all patients, the movement trajectories were enhanced when they completed the therapy. This fact suggests that patient's motor recovery was increased. Despite of the similarity in majority of the kinematic parameters, differences in reaction time and path length were higher using the 3D task. Regarding the success rates were very similar. In conclusion, the using of 2D environments in virtual therapy may be a more appropriate and comfortable way to perform tasks for upper limb rehabilitation of post-stroke patients, in terms of accuracy in order to effectuate optimal kinematic trajectories

Scherk-Schwarz Reduction of D=5 Special and Quaternionic Geometry

We give the N=2 gauged supergravity interpretation of a generic D=4, N=2 theory as it comes from generalized Scherk-Schwarz reduction of D=5, N=2 (ungauged) supergravity. We focus on the geometric aspects of the D=4 data such as the general form of the scalar potential and masses in terms of the gauging of a ``flat group''. Higgs and super-Higgs mechanism are discussed in some detail.Comment: final version to be published on Class.Quant.Gra

On Central Charges and Hamiltonians for 0-brane dynamics

We consider general properties of central charges of zero branes and associated duality invariants, in view of their double role, on the bulk and on the world volume (quantum-mechanical) theory. A detailed study of the BPS condition for the mass spectrum arising from toroidal compactifications is given for 1/2, 1/4 and 1/8 BPS states in any dimensions. As a byproduct, we retreive the U-duality invariant conditions on the charge (zero mode) spectrum and the orbit classification of BPS states preserving different fractions of supersymmetry. The BPS condition for 0-branes in theories with 16 supersymmetries in any dimension is also discussed.Comment: 23 pages, latex fil

Integration of massive states as contractions of non linear σ\sigma-models

We consider the contraction of some non linear sigma models which appear in effective supergravity theories. In particular we consider the contractions of maximally symmetric spaces corresponding to N=1 and N=2 theories, as they appear in certain low energy effective supergravity actions with mass deformations. The contraction procedure is shown to describe the integrating out of massive modes in the presence of interactions, as it happens in many supergravity models after spontaneous supersymmetry breaking.Comment: AMS-LaTeX, 33 page