The five lectures of the first week



Sven Bachmann

Aspects of Topological Quantum Transport


Since the foundational works of Thouless and Laughlin on the quantum Hall effect in the 80’s, topological properties of condensed matter have been extensively studied by both physicists and mathematicians. Most of the mathematical work focussed on the physics of non-interacting systems at zero temperature, namely on the understanding of families of Fermi projections. Although some fundamental progress towards a general theory for interacting physics was initiated in the 90’s, the topic has seen tremendous increase in activity in the last decade. I will review some recent results for interacting quantum lattice systems and the mathematical methods underlying them. I will also point out connections to the problem of the classification of quantum phases.

Here are the slides of lectures.


Jürg Fröhlich

Note: Because of covid, Jürg cannot come to Venice and his programme had to be changed into two 1-hour Zoom lectures. His original programme, and their slides, can be found at the end of the page.

Quantum Dynamics of Systems Under Repeated Observation

Slides.

The Classical Periphery of Quantum Physics - The Example of Particle Tracks in Detectors

Slides.


Vojkan Jaksic

Some Remarks on Mathematical Theory of Non-equilibrium Quantum Statistical Mechanics


The modern developments in non-equilbrium quantum statistical mechanics started with 2001 papers of Ruelle and Jaksic-Pillet on quantum entropy production. I will sketch the line of these developments focusing on loose ends of the emerging theory.

Here are the slides of lectures.


Bruno Nachtergaele

Locality, Quantum Many-Body Dynamics, and Gapped Phases


I start with a brief introduction to quantum lattice systems with an emphasis on a quantitative description of the quasi-local structure of observables. Next, I discuss the Heisenberg dynamics of infinite systems, its locality properties and Hastings’ quasi-adiabatic evolution. With this we introduce the notion of gapped ground state phases and review a number of recent results on the description and classification of such phases.

Here are the slides of Lectures 1,2 and the slides of Lectures 3,4,5.

Here is An Introduction to Quantum Spin Systems, lecture notes written with Bob Sims. And here is a video lecture of Hal Tasaki titled Variations on a Theme by Lieb, Schultz, and Mattis


Marcello Porta

The Correlation Energy of Mean-Field Fermi Gases


I will discuss the ground state properties of three-dimensional, homogeneous Fermi gases, in the mean-field regime. For large values of the number of particles, Hartree-Fock theory provides an effective description of the large scale behavior of such systems, based on the restriction of the space of fermionic wave functions to the set of Slater determinants. The main limitation of the Hartree-Fock approximation is that it neglects correlations among the particles besides those introduced by Pauli principle. In this course I will discuss a description of fermionic correlations based on a rigorous bosonization method. This approach allows to study the low-energy excitations of the Fermi gas in terms of emergent, quasi-free bosonic particles. I will use the method to compute the correlation energy, defined as the difference between many-body and Hartree-Fock ground state energies. As the number of particles goes to infinity, the correlation energy converges to the ground state energy of a quasi-free Bose gas, as predicted by the random-phase approximation. Based on joint works with N. Benedikter, P. T. Nam, B. Schlein and R. Seiringer.

Here are the slides of lectures.


Jürg Fröhlich's original programme

The Classical Periphery of Quantum Physics - an Attempt to Complete the Present Quantum Mechanics

1. The Passage from Classical Physics to Quantum Physics - a Leisurely Introduction

It seems clear that the present quantum mechanics is not in its final form. (P.A.M. Dirac)

I present a short account of the passage from classical physics to quantum physics – from the Platonic Realm, where strict causality, determinism and reversibility prevail, to the Aristotelian Realm, where chance occupies center stage, the future is uncertain and the flow of time is irreversible. This passage represents a revolution not only in our conception of the paradigms underlying natural science and our description and manipulation of Nature, but also in the area of new technologies born from quantum science, such as lasers, semi-conductors, transistors (and their many applications, e.g. in computers), superconductors, nuclear magnetic resonance imaging, nuclear power plants, atomic weapons, ...

Here are the slides for Lectures 1 & 2.

2. The Classical No-Go Theorems: Kochen-Specker and Bell

As my main task in this talk, I will attempt to explain to you the Kochen-Specker Theorem, which says that there does not exist a hidden variables theory reproducing the contents of Quantum Mechanics (QM), and Bell's Inequalities. The mathematics underlying the Kochen-Specker theorem is related to Gleason’s theorem. I will mention an extension of Gleason’s theorem to general von Neumann algebras.

Here are the slides for Lectures 1 & 2.

3. The Inadequacy of the Schrödinger Equation - Wigner's Friend and How to Get Beyond it in the "ETH-Approach to Quantum Mechanics."

The purpose of this lecture is to extend the standard formalism of QM and complete it (Dirac!) in such a way that the resulting theory makes sense. The extension, yielding a new Law of Nature, is called "ETH - Approach to QM."
The ETH - Approach to QM supplies the fourth one of four pillars QM rests upon:
(i) Physical quantities characteristic of a physical system are represented by s.-a. linear operators.
(ii) The time evolution of operators representing physical quantities is given by the Heisenberg equations;
(iii) Introduction of meaningful notions of Potential and Actual Events and of states.
(iv) Proposal of a general statistical Law for the Time Evolution of states.
Core of lecture: Besides sketching the ETH-Approach to QM, I will discuss simple models of a very heavy atom coupled to the radiation field in a limit where the speed of light tends to ∞, illustrating the ETH-Approach.
General goal: I am determined to remove some of the enormous confusion befuddling many colleagues who claim to work on the foundations of QM... Of course, hardly anybody expects that I will succeed – but I do!

Here are the slides for Lectures 3 & 4.

4. ETH-Approach to Quantum Mechanics - Non-Relativistic and Relativistic

I will explain why neither (classical) Relativistic Theories, nor Quantum Theory enable one to predict the future with certainty. I will then sketch why “Einstein causality”, or locality, is an essential property of relativistic Quantum Theories. I will then continue to present the “ETH approach” to Quantum Theory – for non-relativistic Quantum Mechanics and, to conclude, for relativistic Quantum Theory.

Here are the slides for Lectures 3 & 4.

5. Indirect Measurements in Quantum Mechanics - From Haroche-Raymond to Darwin & Mott

I will study the effective quantum dynamics of systems under repeated observation, more specifically ones interacting with a chain of independent probes (such as photons, neutrons, atoms, ...) which, afterwards, are subject to a projective measurement and are then lost. This leads to a theory of indirect measurements of time-independent quantities (non-demolition measurements). Subsequently, a theory of indirect weak measurements of time-dependent quantities is outlined, and a new family of diffusion processes, dubbed quantum jump processes, is described.

Here are the slides for Lecture 5.