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Quantum gravity is the search for a theory that may unify general relativity with quantum mechanics and ultimately with the standard model of elementary particle physics. A key obstacle to this unification program lies in the problem that the two theories in question are built on very different and a priori incompatible conceptual foundations. Concrete approaches to quantum gravity (such as loop quantum gravity or string theory) often ignore or postpone the solution of this problem, which thus sheds doubt on their physical interpretation. This motivates my interest in this problem and more generally in the foundations of quantum theory. In particular, I am working on an extension of the standard formulation of quantum mechanics, precisely to make it compatible with general relativistic concepts and hence overcome the obstacle described. It turns out that this is best done directly in the context of quantum field theory rather than non-relativistic quantum mechanics. This program is called the general boundary formulation of quantum theory.

I am also working on more direct approaches to quantum gravity which are both non-perturbative and background-independent (in contrast e.g. to string theory). Surprisingly many approaches have converged on spin foam models, for example the canonical quantization known as loop quantum gravity as well as covariant path integral quantizations. Spin foams can be thought of as encoding quantum space-time. Remarkably, they exhibit a fundamental discrete structure. Current problems include the proper interpretation as a generally covariant quantum theory (including the problem of time) and understanding the summation over space-times or histories. The further goal is to recover classical general relativity and fixed-background quantum field theory as limiting cases.

Quantum groups emerged as generalized symmetries in the study of integrable systems. Ever since, they are of increasing importance in both physics and mathematics. They play a role in topological quantum field theory and knot invariants, in quantum gravity, noncommutative geometry, conformal field theory, systems with anyonic particles, supersymmetry, the renormalization of quantum field theory as well as other areas. I am interested in many of these connections as they lead to new and often surprizing links between the different fields. Furthermore, it appears now that they form a key ingredient in many approaches at understanding the structure of space-time at the Planck-scale (loop quantum gravity, noncommutative space-time models, phenomenological models). I am also interested in the purely mathematical development of quantum groups and their role as symmetry objects in noncommutative geometry.

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Last updated 12 January 2022.