# Mathematical Physics

Mathematical physics spans complex quantum and classical systems. Some examples are:

### Systems with many or infinitely many degrees of freedom

Understanding the behaviour of systems with many or infinitely many degrees of freedom as they arise naturally in quantum field theory and quantum many-body theory, remains a challenging issue today still. Significant advances have been made in the last two decades through the application of constructive field theory methods, in particular fermionic renormalization group flows. Another area where much progress has been made in the last few years is in the emergence of irreversible behaviour from reversible dynamics, such as variants of the Boltzmann and diffusion equation from long-time quantum dynamics. Heidelberg scientists contribute actively to these fields.

### Renormalization group and physics on all scales

Often the same macrophysics can arise from either an underlying classical or quantum many-body system, not only in qualitative, but also in quantitative ways. An example of this is given by the universal properties of quantum or classical many-body systems near second-order phase transitions described by renormalization group fixed points, which are characterized by the same universality class. Research in this field incorporates understanding systems far from equilibrium, and devising techniques to study the exact dynamics of these systems.

### String Theory

String theory establishes a remarkable link between cutting-edge mathematics and a broad variety of topics in theoretical physics. For instance, in the context of string compactifications the topology and geometry of the compactification space directly encodes the physical properties of the associated string vacuum, including aspects of non-perturbative string dynamics. The group in Heidelberg studies this connection in a wide range of string and M-theory compactifications. Further research topics include the mathematics and physics of conformal field theory and other fundamental aspects of quantum field theory. Active collaborations take place with the Department of Mathematics.