Department Mathematik



Interaction between Light and Matter


Project Overview

Our research is split into three pillars:

  1. Complex Systems: In many-body systems, such as large molecules and particle gases, direct numerical computations become impracticable due to their enormous amount of particles. We investigate special yet physically relevant situations, e.g., small excitations w.r.t. the ground state, in which it is possible to derive simpler, effective equations of motion. For most practical purposes these effective equations are all that is needed to arrive at desired predictions.
  2. Electron-Positron Pair-Creation: One of the most striking phenomenon that quantum electrodynamics predicts is the light-induced electron-positron pair-creation from the vacuum. To date our description of it does not go beyond mathematically poorly understood algorithms to extract numerical predictions for eigenstates and scattering events. We investigate mathematical rigorous models of quantum electrodynamics that allow for a dynamical description of pair-creation and non-perturbative mathematical tools in view of next generation experiments.
  3. Radiation Reaction: The motion of test charges in given fields and the generation of fields from prescribed trajectories of test charges is well-understood. However, the mathematical description of the fully coupled process in which a charge radiates a field that immediately reacts back on the charge turns out to be very problematic. The dynamics become either ill-defined or after renormalization unstable. Nevertheless, radiation reaction is needed to ensure that such a charge loses the same amount of kinetic energy that it radiates into its field. We investigate stable solution theories describing classical and quantum mechanical radiation reaction.

Interdisciplinary Overlap

  • Mathematics: Functional Differential Equations, Delay Differential Equations, Non-Linear Partial Differential Equations, Functional Analysis, Stochastic Processes, Numerical Analysis
  • Physics: Classical Electrodynamics, Quantum Electrodynamics, Many-Body Systems, Quantum Chemistry, Quantum Information, Foundations of Physics

Popular Description of our Research

As our name suggests, the central topic of our research group is the interaction between light and matter, more precisely the interaction between electrodynamic fields and charged elementary particles.

Apart from gravitation, this type of interaction governs the biggest part of our daily experiences. We owe to it everything we see and feel. It has long been the subject of study with traditional treatises as old as the writings of Aristotle. Yet in many aspects its foundational understanding still lies on the frontier of human knowledge, rendering its study at the same time traditional and novel. The methodical study of electromagnetic phenomena began with Isaac Newton's prism experiments in 1672 and was extended by Wolfgang Goethe's "Theory of Colors". Thereafter followed a very active period with influential contributions of scientists like Carl Gauss, Alexander von Humboldt, and Wilhelm Weber which peaked with the works of James Clerk Maxwell and Henry Lorentz in the last third of the 19th century. Along the way the close interplay between physics and mathematics triggered many revolutions.

In the beginning of the 20th century the insights of Hermann Minkowski and Albert Einstein allowed the unified description of electric and magnetic forces within the theory of special relativity, nowadays called the theory of "classical electrodynamics". Thanks to the insights of Paul Dirac, Vladimir Fock, Werner Heisenberg, and Wolfgang Pauli, amongst others, it was possible to reformulate this theory such that it would hold true not only on scales of our world of experience but also on the smallest scales of matter, defined by the elementary particles. In the 60s Freeman Dyson, Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga, and others built from these foundations the so-called theory of "quantum electrodynamics", which together with models of the weak and strong nuclear force were compiled to the so-called "standard model" describing all elementary particles and their interactions -- except gravitation. In its modern form, electrodynamic theory has led to many crucial predictions, and those have been verified with remarkable accuracy in particle collider experiments such as the CERN. In these experiments elementary particles typically clash violently into each other, interact shortly, and scatter apart. The analysis of recorded cross sections then gives insights into the fundamental structure of matter.

Despite its success, however, the mathematical foundation of the theory of electrodynamics is plagued by ill-defined equations of motion which generate infinities that cause any straight-forward computation of measurands to fail. To extract predictions, physicists and mathematicians developed formal computation recipes known as "perturbative renormalization theory". Though a mathematical rigorous understanding of these methods is lacking, they seem to work well in regimes where, e.g., colliding particles do not have much time to interact before they scatter apart. In other regimes where, e.g., charged particles are subject to ultra-strong laser fields for longer times, both theory as well as experiment indicate that such conventional methods are likely to fail to produce satisfactory predictions.

In the next two decades, however, a new generation of experiments will explore nature far beyond such scattering situations. This is due to recent advances in laser technology that will allow to probe electrodynamic phenomena in much more controlled environments and may shrink experimental setups like the CERN with a 26km circumference to the size of laboratory tables. For planning, prediction, and analysis of such experiments, new mathematically rigorous so-called "non-perturbative" methods have to be developed. Beside the foundations of electrodynamics, our objective is to contribute in this direction with:

  • A description of the controlled creation of light-induced electron-positron pairs from the vacuum, e.g., in view of planned experiments within the European project "Extreme Light Infrastructure".
  • A description of radiation-induced friction of accelerated charges which is, e.g., a crucial ingredient for the design of near-future wake field accelerator technology.
Beyond its value in the fundamental understanding of nature, the field of research we are embedded in has a far outreach. It has provided the foundations and repeatedly revolutionized the development of high-technology in clinical diagnostics, nuclear pharmacology, and oncology.