The Master Thesis of Mosh\'e Flato

The genesis and impact of the M.Sc. thesis by Mosh\'e Flato is analysed. In this connection, the fruitful passage of Mosh\'e in Lyon, capitale mondiale de la gastronomie, is evoked. Finally, some basic elements for a model in crystal- field theory are given as a important step on the way opened by Mosh\'e.Comment: 10 pages, Tex file. Contributed paper submitted for the proceedings of the `Conf\'erence Mosh\'e Flato 1999' held in Dijon (France), 5 - 8 September 1999. The proceedings will be published by Kluwer Academic Publishers in the series `Mathematical Physics Studies', eds. G. Dito and D. Sternheimer, founded by the late Professor Mosh\'e Flat

Miscellaneous Physical Applications of Quantum Algebras

Some ideas about phenomenological applications of quantum algebras to physics are reviewed. We examine in particular some applications of the algebras $U_ q (su_2)$ and $U_{qp}({\rm u}_2)$ to various dynamical systems and to atomic and nuclear spectroscopy. The lack of a true (unique) $q$- or $qp$-quantization process is emphasized.Comment: 16 pages, Te

Miscellaneous Applications of Quons

This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We show the interest of such algebras for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU_2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).Comment: This is a contribution to the Proc. of the 3-rd Microconference "Analytic and Algebraic Methods III"(June 19, 2007, Prague, Czech Republic), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On Two Ways to Look for Mutually Unbiased Bases

Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in C(d) to the search for d(d+1) vectors in C(d*d) satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.Comment: three pages to be published in Acta Polytechnica (Czech Technical University in Prague

The Impact of Giulio Racah on Crystal- and Ligand-field Theories

This paper focuses on the impact of Racah on crystal- and ligand-field theories, two branches of molecular physics and condensed matter physics (dealing with ions embedded in aggregates of finite symmetry). The role of Racah and some of his students in developing a symmetry-adapted weak-field model for crystal-field theory is examined. Then, we discuss the extension of this model to a generalized symmetry-adapted weak-field model for ligand-field theory. Symmetry considerations via the use of the Wigner-Racah algebra for chains of type SU(2) > G is essential for these weak-field models. Therefore, the basic ingredients for the Wigner-Racah algebra of a finite or compact group are reviewed with a special attention paid to the SU(2) group in a SU(2) > G basis. Finally, as an unexpected application of nonstandard SU(2) bases, it is shown how SU(2) bases adapted to the cyclic group allow to build bases of relevance in quantum information

Phase states and coherent states for generalized Weyl-Heisenberg algebras

This paper is concerned with the construction of phase operators, phase states, vector phase states, and coherent states for a generalized Weyl-Heisenberg algebra. This polynomial algebra (that depends on real parameters) is briefly described. The various states are defined on a finite- or infinite-dimensional space depending on the parameters. This report constitutes an introduction to three papers published by the authors in J. Phys. A [43 (2010) 115303 and 45 (2012) 244036] and J. Math. Phys. [52 (2011) 082101]. See these three papers for the relevant references.Comment: 6 pages, The XXIXth International Colloquium on Group-Theoretical Methods in Physics, Tianjin : China (2012