Dynamics of relativistic quantum systems
Welcome to the webpage of the ANR DYRAQ ANR-17-CE40-0016-01
This project is funded by the French National Research Agency (Agence Nationale de la Recherche).
The aim of this project is the study of relativistic quantum systems in interaction within the framework of time-dependent partial differential equations (PDEs). We will derive and analyze effective equations in various asymptotic regimes, obtaining a simpler description of complex physical phenomena.
Relativistic systems are described using the Dirac operator, a matrix-valued first order differential operator. Contrary to its nonrelativistic analogue, the Laplacian, the Dirac operator is associated with an indefinite energy functional. In the systems we consider, the interactions are modeled by coupling the Dirac equation to other evolution equations or via a nonlinearity in the Dirac equation. Both the indefiniteness of the energy and the presence of interactions seriously complicate the analysis regarding issues of well-posedness, long time behavior of solutions and accurate estimates in the parameters of the problem.
On the mathematically rigorous basis of the analysis of nonlinear PDEs and spectral theory, we aim to develop original methods to increase the theoretical knowledge of relativistic quantum systems and their asymptotic analysis.
The scientific coordinator of this project, Simona Rota Nodari, is Maître de conférences (assistant professor) at the Institut de Mathématiques de Bourgogne, part of the Université de Bourgogne Franche-Comté, since September 2015. She works on nonlinear PDEs arising in Mathematical Physics. Using a large variety of tools ranging from variational methods to numerical analysis, she already obtained outstanding results on nonlinear effective models from relativistic quantum physics involving the Dirac operator. Recent works include results on orbital stability for infinite-dimensional Hamiltonian systems.
The team working on the project consists of the following members:
Julien Sabin is Maître de conférences at the Laboratoire de Mathématiques d’Orsay (Université Paris-Sud) since September 2014. He studied the dispersive effects in nonlinear PDEs describing infinite quantum systems, as well as the electron-positron pair creation phenomenon in relativistic quantum mechanics. He has been extending results in harmonic analysis, like Strichartz inequalities, to the context of infinite quantum systems.
Jonas Lampart is chargé de recherche CNRS at the Laboratoire Interdisciplinaire Carnot de Bourgogne (Université de Bourgogne Franche-Comté) since October 2016. His research is focused on problems in mathematical physics related to partial differential equations, functional analysis and differential geometry. He has worked successfully on the derivation of effective equations in adiabatic problems and the semiclassical limit of Dirac equations. He has also obtained results on the long-time dynamics of many-body quantum systems.
Loïc Le Treust is Maître de conférences at the Institut de Mathématiques de Marseille (Université Aix-Marseille) since September 2016. He has used variational methods for relativistic equations to prove the existence of solitary-wave solutions. He is an expert on modeling and numerical analysis for evolutionary problems.
Institut de Mathématiques de Bourgogne, Université de Bourgogne
UFR Sciences et Techniques
Faculté des Sciences Mirande, Aile A
9 avenue Alain Savary
21078 Dijon Cedex