Johannes Gutenberg University Mainz > Faculty 08 > Physics > Physics Research > Research Areas > Mathematical Physics
Quantum field theory is a framework that merges field theory with the principles of quantum mechanics. It underpins all modern aspects of phenomenology ranging from the Standard Model of particle physics to the dynamics of quasiparticles in condensed matter theory. Mathematical physics seeks to improve the overarching framework of quantum field theory in order to enable theoretical predictions of new phenomena, make connections between separate theoretical regimes, and use physical insights to develop formal mathematics.
In Mainz, conceptual studies of quantum field theory focus on the mathematical structure of perturbation theory and of the renormalization group flow, the description of non-perturbative phenomena, the study of effective quantum field theories, the properties of topological and supersymmetric quantum field theories, the structure of conformal field theories and their holographic gravitational formulation, the description of thermal quantum field theories, and the formulation of dualities among quantum field theories.
Scattering amplitudes are a lively research field at the intersection of high energy particle physics, gravitational physics, and mathematics. Scattering amplitudes are related to the probability with which a certain scattering process occurs. Perturbation theory offers – in theory, at least – a systematic way to calculate scattering amplitudes through Feynman diagrams. However, any practitioner in the field will soon realize that an approach based on Feynman diagrams is feasible only for the simplest processes. The complexity of the calculation increases with the number of external particles and with the number of internal loops. Furthermore, the final answer is very often much shorter than any intermediate expression. This is an indication that not all structures and symmetries of the problem have been identified. The aim of this research field is to identify and exploit mathematical structures, which are not obvious from the Lagrangian. These mathematical structures are responsible for short and compact final results. One goal is to devise algorithms which ensure that all intermediate expressions have only the minimal required complexity. This is clearly relevant for applications in the phenomenology of particle physics. But there is an additional benefit: Identifying hidden structures in perturbative (quantum) field theory will give us a hint on extended theories beyond the Standard Model of particle physics.
String theory provides a framework for ultraviolet complete theories of quantum gravity, which is capable of describing most ingredients of phenomenologically realistic models of particle physics, such as the Standard Model and its numerous extensions. Similarly, progress has been made in the description of aspects of cosmology within string theory.
The formulation of such an overarching theory remains challenging. It necessarily demands many mathematical tools, which are often not even developed yet. Therefore, progress in string theory is tied to advances in mathematics.
String theory at JGU Mainz focuses on mathematical physics with an emphasis on string and field theories. Apart from studying the underlying mathematical structures of string theory and their physical consequences, we also aim to discover novel applications in mathematics motivated from physics.
