Institut Camille Jordan, UMR 5208

Séminaire de Physique Mathématique

Vendredi à 14h30

en salle 112, bâtiment 101 (Braconnier) la DOUA (Plan d'accès)

  Résumé : Un graphe quantique est constitué d'un ensemble de sommets reliés entre eux par des liens unidimensionnels. Ces modèles sont valables pour des systèmes dont épaisseur des guides d'onde est négligeable par rapport à leur longeur. Sur chaque lien on considère l'operateur de Schrödinger, et on impose des conditions aux bords à chaque sommet. Les graphes quantiques font l'objet des nombreux travaux en mécanque quantique, chimie quantique, nanotechnologie etc. Dans cet exposé on étudie les graphes quantiques dont tous les liens sont identiques. On démontre que la théorie spectrale de graphes quantiques de ce type est equivalent à la théorie spectrale des laplaciens magnetiques discrèts sur les graphes combinatoires.
  Résumé : This article is devoted to an extensive study of >the so-called Schrodinger- Virasoro algebra,introduced by M.Henkel >in \cite{Henk94}. We begin with its geometric construction, >integration into a group,and its relations with conformal and Poisson >geometry. >The third part is devoted to the construction of representations of this > algebra, coming from physical aspects, or from the coadjoint >orbit method. > The fourth part establishes some links with Cartan prolongation, it allows > constructions of coinduced modules.In the fifth part a detailed cohomological > study is given,and deformations and central charges are classified; >non local cocycles appear. In the sixth and last part ,we give > the construction of Verma modules and Kac formula, generalizing > the classical results for Virasoro algebra.
  Résumé : I plan to begin with the introduction into the first-quantized approach to gauge field theories (familiar e.g. in the string field theory context). This includes recalling basics of BRST quantization, understanding the quantum mechanics as a free classical field theory, and precise dictionary between the first-quantized and the field theory languages. Using the equivalence between first-quantized constrained systems and free local gauge field theories one can easily build apparently different formulations of a field theory in terms of the underlying constrained system. In particular, by using an appropriately generalized Fedosov quantization at the first-quantized level one arrives at the so-called ``parent form'' of the field theory. This can be reduced to a variety of apparently different formulations via equivalent reductions. In particular, it reproduces Vasiliev unfolded formulation well-known in the context of higher spin gauge theories. Another remarkable feature of the parent formulation is that it is well adapted to theories on generic space-time manifolds (backgrounds) in the same way as the Fedosov quantization is. I plan to illustrate these ideas with two examples: Klein-Gordon field and Fronsdal higher spin gauge fields. If I have time the plan is to describe possible extensions of these concepts to the nonlinear level.
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  Résumé : The general Hall resistance formula for Bloch electrons is derived by using the force-balance equation for statistical forces. It is shown that except of the free electron case it is not determined precisely by the carrier concentration. The deviations are ascibed to the effect of the local charge polarization. As an illustrative example the results obtained for two-dimensional electron gas with cosine energy dispersion is discussed in detail.