36 research outputs found
Permutation Invariant Algebras, a Fock Space Realization and the Calogero Model
We study permutation invariant oscillator algebras and their Fock space
representations using three equivalent techniques, i.e. (i) a normally ordered
expansion in creation and annihilation operators, (ii) the action of
annihilation operators on monomial states in Fock space and (iii) Gram matrices
of inner products in Fock space. We separately discuss permutation invariant
algebras which possess hermitean number operators and permutation invariant
algebras which possess non-hermitean number operators. The results of a general
analysis are applied to the S_M extended Heisenberg algebra, underlying the
M-body Calogero model. Particular attention is devoted to the analysis of Gram
matrices for the Calogero model. We discuss their structure, eigenvalues and
eigenstates. We obtain a general condition for positivity of eigenvalues,
meaning that all norms of states in Fock space are positive if this condition
is satisfied. We find a universal critical point at which the reduction of the
physical degrees of freedom occurs. We construct dual operators, leading to the
ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point
of view, we briefly discuss the existence of mapping from the Calogero
oscillators to the free Bose oscillators and vice versa.Comment: 40 pages, Latex, no figures, accepted in EPJ
Interacting families of Calogero-type particles and SU(1,1) algebra
We study a one-dimensional model with F interacting families of Calogero-type
particles. The model includes harmonic, two-body and three-body interactions.
We emphasize the universal SU(1,1) structure of the model. We show how SU(1,1)
generators for the whole system are composed of SU(1,1) generators of arbitrary
subsystems. We find the exact eigenenergies corresponding to a class of the
exact eigenstates of the F-family model. By imposing the conditions for the
absence of the three-body interaction, we find certain relations between the
coupling constants. Finally, we establish some relations of equivalence between
two systems containing F families of Calogero-type particles.Comment: 16 pages, no figures, to be published in Mod.Phys.Lett.
On the Clebsch-Gordan coefficients for the two-parameter quantum algebra
We show that the Clebsch - Gordan coefficients for the -
algebra depend on a single parameter Q = ,contrary to the explicit
calculation of Smirnov and Wehrhahn [J.Phys.A 25 (1992),5563].Comment: 5 page
On Infinite Quon Statistics and "Ambiguous" Statistics
We critically examine a recent suggestion that "ambiguous" statistics is
equivalent to infinite quon statistics and that it describes a dilute,
nonrelativistics ideal gas of extremal black holes. We show that these two
types of statistics are different and that the description of extremal black
holes in terms of "ambiguous" statistics cannot be applied.Comment: Latex, 9 pages, no figures, to appear in Mod.Phys.Lett.
Calogero Model(s) and Deformed Oscillators
We briefly review some recent results concerning algebraical (oscillator) aspects of the N-body single-species and multispecies Calogero models in one dimension. We show how these models emerge from the matrix generalization of the harmonic oscillator Hamiltonian. We make some comments on the solvability of these models
Covariant - tensor method for quantum groups and applications I:
A covariant - tensor method for is described. This tensor method
is used to calculate q - deformed Clebsch - Gordan coefficients. The connection
with covariant oscillators and irreducible tensor operators is established.
This approach can be extended to other quantum groups.Comment: 18 page
Exclusion statistics,operator algebras and Fock space representations
We study exclusion statistics within the second quantized approach. We
consider operator algebras with positive definite Fock space and restrict them
in a such a way that certain state vectors in Fock space are forbidden ab
initio.We describe three characteristic examples of such exclusion, namely
exclusion on the base space which is characterized by states with specific
constraint on quantum numbers belonging to base space M (e.g.
Calogero-Sutherland type of exclusion statistics), exclusion in the
single-oscillator Fock space, where some states in single oscillator Fock space
are forbidden (e.g. the Gentile realization of exclusion statistics) and a
combination of these two exclusions (e.g. Green's realization of para-Fermi
statistics). For these types of exclusions we discuss extended Haldane
statistics parameters g, recently introduced by two of us in Mod.Phys.Lett.A
11, 3081 (1996), and associated counting rules. Within these three types of
exclusions in Fock space the original Haldane exclusion statistics cannot be
realized.Comment: Latex,31 pages,no figures,to appear in J.Phys.A : Math.Ge
Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
The development in the study of supersymmetric many-particle quantum systems
with inverse-square interactions is reviewed. The main emphasis is on quantum
systems with dynamical OSp(2|2) supersymmetry. Several results related to
exactly solved supersymmetric rational Calogero model, including shape
invariance, equivalence to a system of free superoscillators and non-uniqueness
in the construction of the Hamiltonian, are presented in some detail. This
review also includes a formulation of pseudo-hermitian supersymmetric quantum
systems with a special emphasis on rational Calogero model. There are quite a
few number of many-particle quantum systems with inverse-square interactions
which are not exactly solved for a complete set of states in spite of the
construction of infinitely many exact eigen functions and eigenvalues. The
Calogero-Marchioro model with dynamical SU(1,1|2) supersymmetry and a quantum
system related to short-range Dyson model belong to this class and certain
aspects of these models are reviewed. Several other related and important
developments are briefly summarized.Comment: LateX, 65 pages, Added Acknowledgment, Discussions and References,
Version to appear in Jouranl of Physics A: Mathematical and Theoretical
(Commissioned Topical Review Article
Biomimetic rehabilitation engineering: the importance of somatosensory feedback for brain-machine interfaces.
Brain-machine interfaces (BMIs) re-establish communication channels between the nervous system and an external device. The use of BMI technology has generated significant developments in rehabilitative medicine, promising new ways to restore lost sensory-motor functions. However and despite high-caliber basic research, only a few prototypes have successfully left the laboratory and are currently home-deployed.
The failure of this laboratory-to-user transfer likely relates to the absence of BMI solutions for providing naturalistic feedback about the consequences of the BMI's actions. To overcome this limitation, nowadays cutting-edge BMI advances are guided by the principle of biomimicry; i.e. the artificial reproduction of normal neural mechanisms.
Here, we focus on the importance of somatosensory feedback in BMIs devoted to reproducing movements with the goal of serving as a reference framework for future research on innovative rehabilitation procedures. First, we address the correspondence between users' needs and BMI solutions. Then, we describe the main features of invasive and non-invasive BMIs, including their degree of biomimicry and respective advantages and drawbacks. Furthermore, we explore the prevalent approaches for providing quasi-natural sensory feedback in BMI settings. Finally, we cover special situations that can promote biomimicry and we present the future directions in basic research and clinical applications.
The continued incorporation of biomimetic features into the design of BMIs will surely serve to further ameliorate the realism of BMIs, as well as tremendously improve their actuation, acceptance, and use