The four lectures of the first week
Jürg Fröhlich
A tentative completion of Quantum Mechanics - Introduction to the subject and applications
I describe ideas about how to complete Quantum Mechanics (QM) to a theory
that actually makes sense. My proposal is called "ETH - Approach/Completion
of QM" - "E" standing for Events, "T" for Trees, and "H" for Histories.
This approach supplies the last one of three pillars QM can be constructed
upon, which are:
(i) Physical quantities characteristic of a system are represented by
self-adjoint operators. Their time evolution is given by the
Heisenberg equations.
(ii) Appropriate notions of states and of potential and actual events.
(iii) A general quantum-mechanical Law for the Time Evolution of States
of a system.
Besides providing some details on pillars (ii) and (iii), I will sketch how the
ETH-Approach to QM can be reconciled with Relativity Theory. Furthermore, I
report on applications of this approach to the phenomenon of fluorescence of atoms.
Time permitting, I will also report some recent results on the theory of indirect
measurements and on the emergence of particle tracks in detectors.
My general goal is to attempt to remove some of the enormous confusion befuddling
many students of QM and many physicists who claim to work on the foundations of QM.
Slides:
1. From Classical Physics to the Miracles of the Quantum World
2. A Tentative Completion of Quantum Mechanics
3. Quantum Poisson Processes
4. The Quantum Mechanics of Particle Tracks in Detectors
5. Quantum Dynamics of Systems Under Repeated Observation
6. The Problem of the Unsolved Problems
Antti Knowles
Random graphs as models of quantum disorder
A disordered quantum system is mathematically described by a large Hermitian random matrix.
One of the most remarkable phenomena expected to occur in such systems is a localization-delocalization transition for the eigenvectors.
Originally proposed in the 1950s to model conduction in semiconductors with random impurities, this phenomenon is now recognized as a general feature of wave transport in disordered media, and is one of the most influential ideas in modern condensed matter physics.
A simple and natural model of a disordered quantum system is given by the adjacency matrix of a random graph.
This simple model turns out to be remarkably amenable to rigorous analysis.
In the first part of these lectures, I will explain the emergence of fully localized and fully delocalized phases, which are separated by a mobility edge.
I will also describe how the solution of the time-dependent Schrödinger equation splits into localized modes and a delocalized wave.
In the second part, I will give an overview of the main tools and ideas behind some of the proofs.
Support material. Part of the course will be based on the articles
Delocalization transition for critical Erdös–Rényi graphs and
Localized phase for the Erdös-Rényi graph.
Mathieu Lewin
Theory of inhomogeneous classical Coulomb systems
I will discuss classical systems of charged particles, in the particular situation that the energy is minimized while fixing the shape of the one-particle density.
After explaining the link with (quantum) density functional theory, I will describe results that can be proved using optimal transport methods.
Then I will turn to the Lieb-Oxford inequality and mention two conjectures about its best constant.
Finally I will discuss the link with the infinite homogeneous electron gas.
Slides:
Theory of Inhomogeneous Classical Coulomb Systems
Armen Shirikyan
The Ruelle-Lanford approach in the theory of large deviations
The theory of large deviations underwent spectacular development starting from the sixties of the last century thanks to the contributions of numerous researchers.
The aim of this course is to introduce some basic concepts and fundamental results of the theory and to describe the Ruelle--Lanford approach, which can be applied in a variety of situations.
We shall present some general theorems, discuss the historical context in which the approach was introduced, and study concrete examples.
Slides:
Handwritten slides of lectures 1 and 2
Handwritten slides of lectures 3 and 4
Handwritten slides of lecture 5
Here are
exercises.