Abstract: We review quantum electrodynamics in external potentials, in particular the definition of the current density, in view of the insights gained in quantum field theory on curved space-times in the last two decades. We point out some deficiencies of popular definitions of the current density and present new results, in particular on vacuum polarization in static potentials and the vacuum current in homogeneous time-dependent fields.

 Abstract: Concepts, familiar from pure states in quantum mechanics, such as “superposition” and “transition probabilities”, are shown to be also meaningful for generic states of infinite systems of any von Neumann type, which are defined on funnels of type I algebras. In the physically significant case of states of type III_1, these concepts have also an operational interpretation in terms of primitive observables which extend the standard framework of observables. (Joint work with Erling St\ormer)

 Abstract: The Unruh effect asserts that a pointlike detector system which is coupled to a quantum field in the inertial vacuum state, and which is uniformly accelerated, will be found in a Gibbs-state in the limit of large times and weak couplings, with the detector temperature proportional to the acceleration. We argue that the temperature indicated by the detector should not be taken as the physical Tolman temperature of the vacuum state, as this leads to contradictions. Moreover we discuss if the Unruh detector measures the temperature of the vacuum state at all. The talk is based on joint work with Detlev Bucholz (arXiv:1412:5892).

 Abstract: Topological field theories – despite their name and their applications in physics – constitute a rigorous piece of mathematics, with deep links to low-dimensional topology and to representation theory. We give an introduction to defects in topological field theories and explain their relation to structures of independent interest in representation theory.

 Abstract: In a recent work, Haber, Gell-Redman and Vasy constructed a Feynman propagator for the wave equation as the inverse of the wave operator on Sobolev spaces, whose order increase along the Hamilton flow. The result is valid for a class of spacetimes including perturbations of the radial compactification of Minkowski space and relies on propagation estimates complemented by a study of the wave equation as a b-differential operator. I will give an introduction to the framework, including results on propagation of the b-wave front set, and explain how the propagators relate to Duistermaat & Hoermander’s parametrices. I will then show how a natural symplectic space of solutions is defined in this setting and outline some related progress in characterizing quantum fields by their asymptotic data.

 Abstract: We use the method of homological quantum reduction to construct a deformation quantization of possibly singular phase spaces which can be obtained as symplectic quotients of Hamiltonian systems, where the coefficients of the moment map define a complete intersection. Several examples are discussed, among others where the quotient space is the orbit space of a linear torus action and an example where the singularity type is worse than an orbifold singularity. The talk is based on joined work with Bordemann, Herbig, and Iyengar.

Abstract: In a large class of factorizing scattering models, also not from a Lagrangian, we construct candidates for the local energy density on the one-particle level starting from first principles, namely from the abstract properties of the energy density. We find that the form of the energy density on the one-particle level can be fixed up to a polynomial function of the rapidity. On the level of one-particle states, we also prove the existence of lower bounds for local averages of the energy density, and show that such inequalities can fix uniquely the form of the energy density in certain models.

Abstract: Well-posedness, that is, the existence of unique solutions depending continuously on given data, is a fundamental requirement for a PDE to be considered `good’. It is furthermore a necessary requirement for numerical treatment. I will review local well-posedness of evolution PDEs. I will then explain what effect gauge freedom has upon this classification for constrained Hamiltonian systems. Starting from the well-known harmonic formulation I will demonstrate how all of this works for GR. Subsequently I will examine the situation for a modification of GR called dynamical Chern-Simons gravity, where the equations are a good deal more complicated.

Abstract: Observations of the cosmic microwave background radiation, most recently by Planck mission, confirm the inflationary Lambda cold dark matter model, especially at angular scales below the degree scale. The larger angular scales show some unexpected features, which individually are not extremely unlikely (at about 3 sigma), but pose some puzzling questions when considered together. In this talk I’ll give a review of the observations and discuss several (failed) attempts to explain them.

Abstract: Locally covariant quantum field theory (LCQFT) has been proposed by Brunetti, Fredenhagen and Verch as an axiomatic setting for describing quantum field theories on generic (globally hyperbolic) Lorentzian manifolds. In my talk I will give a brief introduction to LCQFT and present our results on the construction of (mostly Abelian) gauge theories in this framework. The most elegant and transparent construction of Abelian U(1)-gauge theory and its higher analogs given by connections on (higher) gerbes is obtained by making use of techniques from differential cohomology. The mathematical structure of the resulting quantum field theories can be analyzed in full detail and I will show that these theories violate some important axioms of LCQFT. Trying to understand and resolve the incompatibility between gauge theories and LCQFT naturally leads us to develop a homotopy theoretic generalization of LCQFT. I will present some first results in this direction, which in particular indicate that gauge field observables should be described by a homotopy cosheaf.