Syllabus

Starting from a precise mathematical formulation of quantum mechanics (no previous physics background is assumed), including in particular the famous uncertainty and exclusion principles, we will aim to rigorously prove the stability of ordinary matter. This is something which most of us take for granted and physics textbooks rarely discuss but which is quite subtle and requires proper analysis. Various generally useful functional inequalities such as the Hardy, Sobolev and Lieb-Thirring inequalities will be introduced and proved during the course.

Here is a preliminary plan for the lectures based on the table of contents of the lecture notes. The lecture number 1-17 is indicated in square brackets. Not all topics will be covered.

1. Introduction [1]
2. Some preliminaries and notation [1,2]
2.1. Hilbert spaces
2.2. Lebesgue spaces
2.3. Fourier transform
2.4. Sobolev spaces
2.5. Forms and operators
3. A very brief mathematical formulation of classical and quantum mechanics [3,4]
3.1. Some classical mechanics
3.2. The instability of classical matter
3.3. Some quantum mechanics
3.4. The one-body problem
3.5. The two-body problem and the hydrogenic atom
3.6. The N-body problem
4. Uncertainty principles [5,6,7,8]
4.1. Heisenberg
4.2. Hardy
4.3. Sobolev
4.4. Application to the stability of the hydrogen atom
4.5. Poincaré
4.6. Poincaré-Sobolev?
4.7. IMS localization?
4.8. Local uncertainty
5. Exclusion principles [9,10]
5.1. Fermions
5.2. Repulsive bosons
5.3. Local exclusion
5.4. Anyons
6. The Lieb–Thirring inequality [11,12,13]
6.1. Some history
6.2. Covering lemma
6.3. Local proof of LT for fermions
6.4. Local proof of LT for inverse-square repulsive bosons
6.5. One-body formulations
6.6. Some applications of LT
6.7. Connections between Hardy–Sobolev–LT?
7. The stability of matter [14,15,16]
7.1. Some history
7.2. Stability of the first kind
7.3. Some electrostatics
7.4. Proof of stability of the second kind

Examination: There will be homework assignments and, if aiming for a higher grade, an individual project. More information will become available when the course starts.