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.
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.