This book collects lecture courses and seminars given at the Les Houches Summer School 2010 on "Quantum Theory: From Small to Large Scales." Fundamental quantum phenomena appear on all scales, from microscopic to macroscopic. Some of the pertinent questions include the onset of decoherence, the dynamics of collective modes, the influence of external randomness and the emergence of dissipative behaviour. Our understanding of such phenomena has been advanced by the study of model systems and by the derivation and analysis of effective dynamics for large systems and over long times. In this field, research in mathematical physics has regularly contributed results that were recognized as essential in the physics community. During the last few years, the key questions have been sharpened and progress on answering them has been particularly strong. This book reviews the state-of-the-art developments in this field and provides the necessary background for future studies. All chapters are written from a pedagogical perspective, making the book accessible to master and PhD students and researchers willing to enter this field.

The book is based on the lectures given at the CIME school "Quantum many body systems" held in the summer of 2010. It provides a tutorial introduction to recent advances in the mathematics of interacting systems, written by four leading experts in the field: V. Rivasseau illustrates the applications of constructive Quantum Field Theory to 2D interacting electrons and their relation to quantum gravity; R. Seiringer describes a proof of Bose-Einstein condensation in the Gross-Pitaevski limit and explains the effects of rotating traps and the emergence of lattices of quantized vortices; J.-P. Solovej gives an introduction to the theory of quantum Coulomb systems and to the functional analytic methods used to prove their thermodynamic stability; finally, T. Spencer explains the supersymmetric approach to Anderson localization and its relation to the theory of random matrices. All the lectures are characterized by their mathematical rigor combined with physical insights.

The Luttinger Model is the only model of many-fermion physics with legitimate claims to be both exactly and completely solvable. In several respects it plays the same role in many-body theory as does the 2D Ising model in statistical physics. Interest in the Luttinger model has increased steadily ever since its introduction half a century ago. The present volume starts with reprints of the seminal papers in which it was originally introduced and solved, and continues with several contributions setting out the landscape of the principal advances of the last fifty years and of prominent new directions.

The notion of renormalization is at the core of several spectacular achievements of contemporary physics, and in the last years powerful techniques have been developed allowing to put renormalization on a firm mathematical basis. This book provides a self-consistent and accessible introduction to the sophisticated tools used in the modern theory of non-perturbative renormalization, allowing an unified and rigorous treatment of Quantum Field Theory, Statistical Physics and Condensed Matter models. In particular the first part of this book is devoted to Constructive Quantum Field Theory, providing a mathematical construction of models at low dimensions and discussing the removal of the ultraviolet and infrared cut-off, the verification of the axioms and the validity of Ward Identities with the relative anomalies. The second part is devoted to lattice 2D Statistical Physics, analyzing in particular the theory of universality in perturbed Ising models and the computation of the non-universal critical indices in Vertex or Ashkin-Teller models. Finally the third part is devoted to the analysis of complex quantum fluids showing Luttinger of Fermi liquid behavior, like the 1D or 2D Hubbard model.