This is an introductory course on Quantum Field Theory, a theoretical framework combining quantum mechanics, special relativity and field theory to describe particles in high energy physics and condensed matter systems. This course focus on four dimensional space-time and applications in high energy physics, such as Quantum Electrodynamics (QED). These lectures are adapted from Luc Frappat’s lecture notes Théorie Quantique des Champs (in French).

The study of integrable quantum many body systems is a cornerstone of mathematical-physics, and led to spectacular advances in many fields of research, including statistical physics,quantum physics, condensed matter, cold atoms, quantum field theory, and more recently string theory. These lectures serve as an introduction to quantum integrable systems and the advanced mathematical methods that have been developed to study them, with a strong emphasis on algebraic methods based on the representation theory of quantum groups.

These lectures are aimed at Master and PhD student who want to do research in mathematical physics, in particular the study of quantum systems or statistical physics. They also serve as a good introduction to quantum groups and their representation theory for students in pure mathematics.

The goal of this series of lectures is to present a set of algebraic techniques inspired by Quantum Field Theory and based on the notions of free fields and vertex operators. These techniques can be applied to a wide range of mathematical problems in representation theory (free field representations), quantum systems and lattice models (quantum groups, Jimbo-Miwa vertex operators), symmetric functions (Hall-Littlewood and Macdonald polynomials) and differential equations (integrable hierarchies). Besides, these lecture will give an overview of different important research topics in mathematical-physics, and can serve as an introduction to essentials tools of quantum field theory and string theory.