Program

In this mini-course we study the so-called non relativistic quantum electrodynamics (NR QED). NR QED describes non relativistic particles coupled to the quantized electromagnetic field and it is a low energy approximation of quantum electrodynamics. The course is intended to be accessible to students that have no background on the mathematical formalism of quantum field theory. For this reason, basic mathematical tools will be presented. We will cover, for example, operators defined on Fock spaces (including creation and annihilation operators) and we will state and prove some basic properties and results. In the last part of the course, we will present more advanced topics such as resonances.

I will consider the time evolution of N interacting Bosons (two body interaction) via a potential of the form N^{3β}V(N^β|x − y|) where 0 < β ≤ 1. I will derive what is called the Hartree-Fock-Bogoliubov approximation which results in a coupled system of Schrödinger type equations. These equations describe the coupling between the mean field (Nonlinear cubic Schrödinger equation) with the wave function of a pair of particles. Next I will explain some of the analysis needed to establish global in time existence of the limit equation (as N → ∞) of the coupled system. This part involves establishing Strichartz type estimates independent of N where N is the number of particles, since we are interested in the limit N → ∞ of the equations. This work is in collaboration with Matei Machedon, Zehua Zhang, Jacky Chong as well as some recent seminal work of X. Huang.

While the theory of quantum mechanics describes interactions between constituents of matter at the microscopic scale, the effects of these interactions may lead to fascinating quantum mechanics phenomena at a macroscopic scale. The discovery of such phenomena, including superfluidity and superconductivity as two pioneering examples, has led to extraordinary developments in realizing new materials that exploit the principles of quantum mechanics to exhibit innovative behaviours. In these lectures, we will discuss the challenge of developing mathematical models that rigorously describe how these macroscopic effects emerge from the microscopic scale, focusing on the paradigmatic example of the interacting Bose gas.