Project description
Probing the macroscopic behaviour of many-body quantum systems
Understanding how macroscopic properties of quantum systems relate to microscopic parameters is one of the most complex questions in mathematical physics. The EU-funded CLaQS project will focus on the study of important physical properties of many-body quantum systems. Its primary aim is to develop new mathematical tools to describe correlations between interacting particles and understand their effects on large quantum systems. The project will examine ground-state properties and low-energy excitations as well as non-equilibrium dynamics. Long-term goals include a mathematical proof of the emergence of Bose-Einstein condensation in the thermodynamic limit and a rigorous justification of the Boltzmann equation for fermions in the weak coupling limit.
Objective
This project is devoted to the mathematical analysis of important physical properties of many-body quantum systems. We will be interested in properties of the ground state and low-energy excitations but also of non-equilibrium dynamics. We are going to consider systems with different statistics and in different regimes. The questions we are going to address have a common aspect: correlations among particles play a crucial role. Our main goal consists in developing new tools that allow us to correctly describe many-body correlations and to understand their effects. The starting point of our proposal are ideas and techniques that have been introduced in a series of papers establishing the validity of Bogoliubov theory for Bose gases in the Gross-Pitaevskii regime, and in a recent preprint showing how (bosonic) Bogoliubov theory can also be used to study the correlation energy of Fermi gases. In this project, we plan to develop these techniques further and to apply them to new contexts. We believe they have the potential to approach some fundamental open problem in mathematical physics. Among our most ambitious objectives, we include the proof of the Lee-Huang-Yang formula for the energy of dilute Bose gases and of the corresponding Huang-Yang formula for dilute Fermi gases, as well as the derivation of the Gell-Mann--Brueckner expression for the correlation energy of a high density Fermi system. Furthermore, we propose to work on long-term projects (going beyond the duration of the grant) aiming at a rigorous justification of the quantum Boltzmann equation for fermions in the weak coupling limit and at a proof of Bose-Einstein condensation in the thermodynamic limit, two very challenging and important questions in the field.
Fields of science
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Funding Scheme
ERC-ADG - Advanced GrantHost institution
8006 Zurich
Switzerland