Properties of the interacting Fermi Gas in the Random Phase Approximation
Niels Benedikter
Università degli Studi di Milano
Abstract: I am going to introduce approximate bosonization as a rigorous approach to the random phase approximation for interacting fermions, which allows us to prove results beyond the precision of mean-field (i.e., Hartree-Fock) theory. In particular I am going to discuss predictions for the momentum distribution and the renormalization of the Fermi surface.
The half-wave maps equation in one space dimension
Patrick Gérard
Université Paris-Saclay
Abstract: The half wave maps equation in one space dimension is a continuum limit of the classical Haldane—Shastry spin chain. We prove global wellposedness for rational data, and long time soliton resolution for a dense subclass. The main tools are a Lax pair structure and an explicit formula derived from it.
This is a joint work with Enno Lenzmann (Basel).
Von Neumann algebras and a Bekenstein-type bound in AQFT
Roberto Longo
Università di Roma Tor Vergata
Abstract: After a brief introduction to operator algebras, I will present my recent entropy/energy ratio local bound in Quantum Field Theory. This bound is model-independent and rigorous; it follows solely from first principles in the framework of translation covariant, local Quantum Field Theory on the Minkowski spacetime.
A brief overview of Efimov physics and related mathematical problems
Pascal Naidon
RIKEN - Nishina Centre
Abstract: I will present an overview of the so-called Efimov physics, which designates the physics of few- and many-body systems of particles in which the Efimov effect occurs. The Efimov effect is a quantum effect for particles with short-range interactions, according to which an infinite number of three-body bound states exist close to the dissociation of a two-body bound state.
I will first explain the effect, and then present various theoretical extensions and experimental verifications. If time allows, I will also discuss about some considerations and problems related to the mathematical description of these physical systems.
RAGE and the machine
Constanza Rojas-Molina
CY Cergy Paris Université
Abstract: In the study of electron wave propagation in materials, the presence of impurities induces, under certain conditions, an insulator behavior known as Anderson localization. Mathematically, the system is modeled by random Schrödinger operator and showing localization amounts to showing that wave packets evolving according to Schrödinger's equation, under the action of this random operator, remain localized in space for all times. In the search for a mathematically rigorous description of this phenomenon, several approaches have been developed, including the Multiscale Analysis (MSA, the machine). The MSA is a technique that studies the decay of resolvent of the operator at different scales in space, and yields bounds that allow to show the absence of wave propagation and, in particular, pure point spectrum, thanks to results developed by Ruelle, Amrein, Georgescu and Enss (RAGE).
In the last 40 years the MSA method has been reviewed and refined constantly until its most performing version, but we will see in this talk that its story is far from over and, when encountering variants of the usual Anderson model, new techniques are needed...and new machines.
How to express the Uncertainty Principle of Quantum Mechanics?
Jos Uffink
University of Minnesota
Abstract: This talk will start with a brief review of the early history of the expression of Werner Heisenberg’s famous (1927) Uncertainty Principle for position (Q) and momentum (P) in Quantum Mechanics. In his 1927 paper, Heisenberg only gave a qualitative formulation of the Uncertainty Principle, but within a few months, the physical community settled on a mathematical expression of this principle in the inequality ∆Q∆P ≥ ħ/2, first presented by E.H. Kennard (1927) where the ∆’s are defined as standard deviations. This is still the most common and well-known version of an uncertainty relation today.
Nevertheless, it will be shown in this talk that Kennard’s inequality, is not strong enough to provide a foundation for most , if not all, of the common examples used in textbooks as illustrations of the Uncertainty Principle. I will argue, therefore, that conceptually stronger inequalities are needed for the purpose. I will review several approaches to obtain such inequalities: the Landau & Pollak inequalities (1961); the entropic uncertainty inequality (Bialynicki-Birula & Mycielski,1975) and a statistico-geometric approach to the uncertainty relation by (Hilgevoord & Uffink,1991), and discuss their strengths and weaknesses vis-a-vis the common textbook examples.
