Dynamics of relativistic quantum systems – DYRAQ
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.
The two main themes of the project are the analysis of coupled/nonlinear time-dependent Dirac equations and the dynamical study of the Dirac sea. In the first theme, several asymptotic regimes will be considered in order to derive simpler, effective equations. Various dynamical properties (well-posedness, stationary solutions and their stability) of these effective equations will be studied and the results will be used to enhance our understanding of to the original equations. In the second theme, an additional step in the modeling of relativistic quantum systems is made by taking into account the so-called Dirac sea. The analysis of the dynamical properties of the Dirac sea amounts to considering a quantum system with infinitely many particles (and infinite energy). This provides a description closer to relativistic quantum field theory, where the vacuum is a polarizable medium (the Dirac sea). From a mathematical point of view, this amounts to studying PDEs where the unknown is an operator rather than a wavefunction. In both themes, an emphasis is made on the derivation of effective, simpler equations which account for the complicated physical behavior of these systems. While the first theme can be connected to many existing works in related topics in PDEs, the second theme has not been investigated as much and thus is an exciting new direction of research.
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. Through a precise partition of each theme into different tasks, we will answer questions about the behavior of relativistic systems relevant both from the mathematical and physical point of view.
The investigator will be supported by a team of young collaborators with complementary viewpoints and a common background in relativistic quantum systems.
Madame Simona Rota Nodari (Institut de Mathématiques de Bourgogne)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
IMB Institut de Mathématiques de Bourgogne
Help of the ANR 62,316 euros
Beginning and duration of the scientific project: January 2018 - 36 Months