Quantum Mathematical Physics: A Bridge between Mathematics and Physics
Book Preface
The present volume is based on the international conferenceQuantumMathematical Physics – A Bridge between Mathematics and Physics that was held at the University of Regensburg (Germany) from September 29 to October 2, 2014. This conference was a successor of similar international conferences which took place at the HeinrichFabri Institute (Blaubeuren) in 2003 and 2005, at the Max Planck Institute for Mathematics in the Sciences (Leipzig) in 2007 and at the University of Regensburg in 2010. The basic intention of this series of conferences is to bring together mathematicians and physicists to discuss profound questions in quantum field theory and gravity. More specifically, the series aims at discussing concepts which underpin different mathematical and physical approaches to quantum field theory and gravity.
Since the invention of general relativity and quantum mechanics at the beginning of the twentieth century, physicists made an enormous effort to incorporate gravity and quantum physics into a unified framework. In doing so, many approaches have been developed to overcome the basic conceptual and mathematical differences between quantum theory and general relativity.Moreover, both quantum theory and general relativity have their own problems and shortcomings. It turns out that many of these problems are related to each other and to the problem of the unification of quantum theory and gravity. The aim of the conference was to shed light on these problems and to indicate possible solutions.
On one hand, general relativity describes systems on large scales (like the solar system, galaxies, and cosmological phenomena). This is reflected in the fact that in general relativity, spacetime has locally the simple structure of Minkowski space, whereas gravitational effects usually show up in the largescale geometry. Under generic assumptions, there are phenomena like black holes and cosmological singularities which are not yet understood in a physically satisfying way. Quantumtheory, on the other hand, usually describes systems on small scales (like atoms, nuclei, or elementary particles). Indeed, on small scales the Heisenberg uncertainty principle becomes relevant and quantum effects come into play. One of the open problems is that there is no satisfying mathematical description of interacting quantum fields.
One of the fundamental difficulties in combining gravity with quantum physics lies in the fact that general relativity is a theory on the dynamics of spacetime itself, whereas quantum theory usually aims to describe the dynamics of matter within a given spacetime background (in the simplest case by Minkowski space). Moreover, the geometric description of general relativity makes it necessary to describe objects locally in an arbitrary small neighborhood of a point. But localizing quantum mechanical wave functions to such a small neighborhood, the Heisenberg uncertainty principle gives rise to large energy fluctuations. Considering these energy fluctuations as a gravitational source, one obtains a contradiction to the above picture that gravity comes into play only on large scales. Thus, although both theories are experimentally well confirmed, they seem to conceptually contradict each other. This incompatibility also becomes apparent in the mathematical formulation: From a mathematical perspective, general relativity is usually regarded as a purely geometric theory. However, quantum physics is described mathematically in an algebraic and functional analytic language.
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March 7, 2016 
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