In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.

La quantification par déformation et la correspondance de McKay forment les grands thèmes de l'étude qui porte sur des variétés algébriques singulières, des quotients d'algèbres de polynômes et des algèbres de polynômes invariants sous l'action d'un groupe fini. Nos principaux outils sont les cohomologies de Poisson et de Hochschild et la théorie des représentations. Certains calculs formels sont effectués avec Maple et GAP. Nous calculons les espaces d'homologie et de cohomologie de Hochschild des surfaces de Klein, en développant une généralisation du Théorème de HKR au cas de variétés non lisses et utilisons la division multivariée et les bases de Gröbner. La clôture de l'orbite nilpotente minimale d'une algèbre de Lie simple est une variété algébrique singulière sur laquelle nous construisons des star-produits invariants, grâce à la décomposition BGS de l'homologie et de la cohomologie de Hochschild, et à des résultats sur les invariants des groupes classiques. Nous explicitons les générateurs de l'idéal de Joseph associé à cette orbite et calculons les caractères infinitésimaux. Pour les algèbres de Lie simples B, C, D, nous établissons des résultats généraux sur l'espace d'homologie de Poisson en degré 0 de l'algèbre des invariants, qui vont dans le sens de la conjecture d'Alev et traitons les rangs 2 et 3. Nous calculons des séries de Poincaré à 2 variables pour des sous-groupes finis du groupe spécial linéaire en dimension 3, montrons que ce sont des fractions rationnelles, et associons aux sous-groupes une matrice de Cartan généralisée pour obtenir une correspondance de McKay algébrique en dimension 3. Toute l'étude a donné lieu à 4 articles / Deformation quantization and McKay correspondence form the main themes of the study which deals with singular algebraic varieties, quotients of polynomial algebras, and polynomial algebras invariant under the action of a finite group. Our main tools are Poisson and Hochschild cohomologies and representation theory. Certain calculations are made with Maple and GAP. We calculate Hochschild homology and cohomology spaces of Klein surfaces by developing a generalization of HKR theorem in the case of non-smooth varieties and use the multivariate division and the Groebner bases. The closure of the minimal nilpotent orbit of a simple Lie algebra is a singular algebraic variety : on this one we construct invariant star-products, with the help of the BGS decomposition of Hochschild homology and cohomology, and of results on the invariants of the classical groups. We give the generators of the Joseph ideal associated to this orbit and calculate the infinitesimal characters. For simple Lie algebras of type B, C, D, we establish general results on the Poisson homology space in degree 0 of the invariant algebra, which support Alev's conjecture, then we are interested in the ranks 2 and 3. We compute Poincaré series of 2 variables for the finite subgroups of the special linear group in dimension 3, show that they are rational fractions, and associate to the subgroups a generalized Cartan matrix in order to obtain a McKay correspondence in dimension 3. All the study comes from 4 papers

This thesis provides the explicit expression of the star-exponential for the action of normal j-groups on their coadjoint orbits, and of the so-called modified star-exponential defined by Gayral et al. Using this modified star-exponential as the kernel of a functional transform between the group and its coadjoint orbits yields an adapted Fourier transform which is also detailed here. The normal j-groups arise in the work of Pytatetskii-Shapiro, who established the one-to-one correspondence with homogeneous bounded domains of the complex space; these groups are also the central element of the deformation formula recently developed by Bieliavsky & Gayral (a non abelian analog of the strict deformation quantization theory of Rieffel). Since these groups are exponential, the results given in this text illustrate the general work of Arnal & Cortet on the star-representations of exponential groups.<p> As this work is meant to be as self-contained as possible, the first chapter reproduces many definitions introduced by Bieliavsky & Gayral, in order to obtain the expression of the symplectic symmetric space structure on normal j-groups, and of their unitary irreducible representations. The Weyl-type quantizer associated to this symmetric structure is then computed, thus yielding the Weyl quantization map for which the composition of symbols is precisely the deformed product defined by Bieliavsky-Gayral on normal j-groups. A detailed proof of the structure theorem of normal j-groups is also provided.<p> The second chapter focuses on the expression and properties of the star-exponential itself, and exhibits a useful tool for the computation, namely the resolution of the identity associated to square integrable unitary irreducible representations of the groups. The result thus obtained satisfies the usual integro-differential equation defining the star-exponential. A criterion for the existence of a tempered pair underlying a given tempered structure on Lie groups is proven; the star-exponential functions are also shown to belong to the multiplier algebra of the Schwartz space associated to the tempered structure. Before that, it is shown that all Schwartz spaces that appear in this work are isomorphic as topological vector spaces.<p> The modified version of this star-exponential is computed in chapter three, first for elementary normal j-groups and then for normal j-groups. It is then used to define an adapted Fourier transform between the group and the dual of its Lie algebra. This transform generalizes (to all normal j-groups) a Fourier transform that was already studied in the “ax+b” case by Gayral et al. (2008), as well as by Ali et al. (2003) in the context of wavelet transforms. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold (\\\"phase space\\\"). His algorithm gives a non-commutative, but associative, product (a so-called \\\"star-product\\\") between smooth phase space functions parameterized by Planck\\\''s constant ℏ, which is treated as a deformation parameter. In the limit as ℏ goes to zero, the star product commutator goes to ℏ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical observables. In this work, we develop a generalization of Fedosov\\\''s method which applies to the infinite-dimensional symplectic \\\"manifolds\\\" that occur in Lagrangian field theories. We show that the procedure remains mathematically well-defined, and we explain the relationship of this method to more standard perturbative quantization schemes in quantum field theory.