Harmonic oscillator chains as Wigner Quantum Systems: periodic and fixed wall boundary conditions in gl(1|n) solutions

We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the $n$th oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner Quantum Systems (WQS), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anti-commutators. These triple relations have a solution in terms of the Lie superalgebra gl(1|n). We study a particular class of gl(1|n) representations V(p), the so-called ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the n oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position and momentum of each oscillator) have a finite spectrum

On the eigenvalue problem for arbitrary odd elements of the Lie superalgebra gl(1|n) and applications

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible representation W. We show that the eigenvalue problem can be solved by the decomposition of W with respect to the branching gl(1|n) --> gl(1|1) + gl(n-1). The eigenvector problem is much harder, since the Gel'fand-Zetlin basis of W is involved, and the explicit actions of gl(1|n) generators on this basis are fairly complicated. Using properties of the Gel'fand-Zetlin basis, we manage to present a solution for this problem as well. Our solution is illustrated for two special classes of unitary gl(1|n) representations: the so-called Fock representations and the ladder representations

The paraboson Fock space and unitary irreducible representations of the Lie superalgebra osp(1|2n)

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators $b^\pm_i$. In particular, with the usual star conditions, this implies that the ``parabosons of order p'' correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Apart from the simple cases p=1 or n=1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching $osp(1|2n) \supset sp(2n) \supset u(n)$, we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters.Comment: typos correcte

U.S. Department of Energy Synfuels Program Overview

The Administration has proposed shifting the focus of the Government\u27s synfuels program to the Synthetic Fuels Corporation (SFC) and would assign to the SRC and responsibility to assist major synfuel plant construction projects. The U.S. Department of Energy (DOE), currently involved in the management and would discontinue its activities in this area

Dupin v. France: the ECtHR going old school in its appraisal of inclusive education?

In Dupin v. France the European Court of Human Rights saw itself confronted with one of the key conflicts in education law: when parents and state officials disagree on which educational trajectory is best for a child with a disability, who gets the final say? This case concerned a mother fighting the decision of the French authorities to refuse her child, who has Autism Spectrum Disorder, access to a general school (through a form of inclusive education). Instead, the child was referred to an ‘Institut medico-éducatif’, an institution established to provide care and a specialized type of education to children with an intellectual impairment. Seemingly going back on its prior case law, the Court did not consider the right to education of the child to be violated

The Wigner function of a q-deformed harmonic oscillator model

The phase space representation for a q-deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the $q$-oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case $h \to 0$ ($q \to 1$), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q-deformed model shows that, when $q \ll 1$, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on $q$ and $n$. The ground-state like behaviour of the distribution functions for excited states in the q-deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.Comment: 16 pages, 24 EPS figures, uses IOP style LaTeX, some misprints are correctd and journal-reference is adde

Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency $\omega$, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner Quantum System, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton's equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1|M). Then we study the properties and spectra of the physical operators in a class of unitary representations of gl(1|M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain

The thanatophoric dysplasia type II mutation hampers complete maturation of fibroblast growth factor receptor 3 (FGFR3), which activates signal transducer and activator of transcription 1 (STAT1) from the endoplasmic reticulum

The K650E substitution in the fibroblast growth factor receptor 3 (FGFR3) causes constitutive tyrosine kinase activity of the receptor and is associated to the lethal skeletal disorder, thanatophoric dysplasia type II (TDII). The underlying mechanisms of how the activated FGFR3 causes TDII remains to be elucidated. FGFR3 is a transmembrane glycoprotein, which is synthesized through three isoforms, with various degrees of N-glycosylation. We have studied whether immature FGFR3 isoforms mediate the abnormal signaling in TDII. We show that synthesis of TDII-FGFR3 presents two phosphorylated forms: the immature non-glycosylated 98-kDa peptides and the intermediate 120-kDa glycomers. The mature, fully glycosylated 130-kDa forms, detected in wild type FGFR3, are not present in TDII. Endoglycosidase H cleaves the sugars on TDII intermediates thus indicating their intracellular localization in the endoplasmic reticulum. Accordingly, TDII-FGFR3-GFP co-localizes with calreticulin in the endoplasmic reticulum. Furthermore, following TDII transfection, signal transducer and activator of transcription 1 (STAT1) is phosphorylated in the absence of FGFR3 ligand and brefeldin A does not inhibit its activation. On the contrary, the cell membrane-anchored FRS2alpha protein is not activated in TDII cells. The opposite situation is observed in stable TDII cell clones where, despite the presence of phosphorylated mature receptor, STAT1 is not activated whereas FRS2alpha is phosphorylated. We speculate that the selection process favors cells defective in STAT1 activation through the 120-kDa TDII-FGFR3, thus allowing growth of the TDII cell clones. Accordingly, apoptosis is observed following TDII-FGFR3 transfection. These observations highlight the importance of the immature TDII-FGFR3 proteins as mediators of an abnormal signaling in TDII