Abstract: Integrable models provide simplified examples of quantum field theories with self-interaction. As often in relativistic quantum theory, their local observables are difficult to control mathematically. One either tries to construct pointlike local quantum fields, leading to possibly divergent series expansions, or one defines the local observables indirectly via wedge-local quantities, losing control over their explicit form. We propose a new, hybrid approach: We aim to describe local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with the net of local von Neumann algebras known from the wedge-local approach. This is shown to work at least in the Ising model.

Abstract: Since Dirac’s theory of the positron, physicists have been interested in the effects of vacuum polarization on e.g. the binding energies of electrons in atoms. In this talk, we will present early approaches to vacuum polarization and illustrate why a numerical approach with modern techniques from QFT is desirable.

Abstract: The “inverse spectral transformation”, a method developed in the context of soliton theory, can be used to solve boundary value problems of the stationary and axially symmetric Einstein and Einstein-Maxwell equations. Applying this technique, the talk shows in detail how the Kerr-Newman solution can be constructed as the unique asymptotically flat solution to the black hole boundary value problem (for a single connected horizon) in a straightforward manner. In this way, a proof of the “no-hair theorem” including the case of a degenerate horizon is given.

Abstract: We outline an approach to rigorously implement the Wilson-Kadanoff renormalization group for Hamiltonian lattice systems by operator-algebraic methods. We try to convey the main ideas by discussing some working examples such as lattice gauge theory in 1+1 and 1+2 dimensions and discretized (free) scalar fields. Finally, we point out potential connections with tensor networks and the multi-scale entanglement renormalization ansatz.

Abstract: We prove the split property for any finite helicity free quantum fields. Finite helicity Poincaré representations extend to the conformal group and the conformal covariance plays an essential role in the argument: the split property is ensured by the finiteness of the trace of e^{-\beta L_0} where L_0 is the conformal Hamiltonian of the Möbius covariant restriction of the net on the time axis. It extends the argument for the scalar case presented by Buchholz d’Antoni and Longo in 2007. We provide the direct sum decomposition into irreducible representations of the conformal extension of any helicity-h representation to the subgroup of transformations fixing the time axis. Our analysis provides new relations among finite helicity representations and suggests a new construction for representations and free quantum fields with non-zero helicity. Joint work with R. Longo, F. Preta, K.-H. Rehren

Abstract: In the last few years, the methods of constructive Fermionic Renormalization Group have successfully been applied to the study of the scaling limit of several two-dimensional statistical mechanics models at the critical point, including: weakly non-planar 2D Ising models, Ashkin-Teller, 8-Vertex, and close-packed interacting dimer models. In this talk, I will focus on the illustrative example of the interacting dimer model and review some of the universality results derived in this context. In particular, I will discuss a proof of the massless Gaussian free field (GFF) behavior of the height fluctuations, in the generic case of non-zero average tilt for the height profile. It turns out that GFF behavior is connected with a remarkable identity (`Kadanoff’ or `Haldane’ relation) between an amplitude and an anomalous critical exponent, characterizing the large distance behavior of the dimer-dimer correlations. Based on joint works with V. Mastropietro and F. Toninelli.

Abstract: Beyond defining particles in terms of unitary irreducible representations (UIR) of the isometry group of spacetime, a purely group theoretical approach can also provide information about localization properties. Here, we illustrate the usage of this tool with a discussion of massless particles on the two-dimensional de Sitter space. The representation theory ensures the existence of a causal quantum theory of arbitrarily well-localized observables, despite the absence of point-like fields. Huygens’ principle is valid in the quantum theory, despite that fact that it is violated in the classical theory. The description of free massless particles gives rise to a representation of the Virasoro algebra. The generators of this algebra, which do not respect the n-particle subspaces in Fock space, implement the geometric action of the Witt algebra on the currents. They arise by decomposing the stress-energy tensor in its Fourier coefficients. Joint work with U. Wiedemann.

Abstract: I will discuss some issues related to the standard interpretation of Hermitian elements of a *-algebra as observables within the GNS construction and referring to the so-called moment problem. These issues are particularly relevant when no C*-algebra structure is assumed on the *-algebra as it happens in some natural formulations of QFT. I will prove that an extended notion of observable is generally necessary, based on Positive Operator Valued Measures (POVMs) instead of Projection Valued Measures (PVMs). Joint work with Nicolo Drago.

Abstract: We establish a remarkable relationship between a quantum spin system (the quantum Curie-Weiss model) and a discretized Schroedinger operator with a double well potential, with special emphasis to spontaneous symmetry breaking (SSB). Moreover, we introduce the concept of deformation quantization of a bundle of C*-algebras, and apply this to model the so-called classical limit of quantum systems. From this point of view, a link between classical and quantum theory is provided.