# 39th LQP Workshop

"Foundations and Constructive Aspects of QFT"

**Sonderforschungsbereich 878 "Groups, Geometry & Actions"**

**Münster, 20-21 January, 2017 **

### Announcement

This series of workshops is devoted to talks in the areas of quantum
field theory, quantum statistical mechanics, quantum information,
non-commutative geometry, quantum gravity and cosmology, or
related topics of mathematical physics and mathematics.

By tradition, young researchers are particularly encouraged
and invited to report on their M.Sc. or PhD thesis results.

The workshop will start on Friday, January 20,
12h00 with a special lecture. The regular programme starts at 14h15 and
finishes on Saturday, January 21, at about 15h30.

### Location & Accomodation

The workshop will take place in the M5 lecture hall of
the Mathematics department, Einsteinstraße 62.

To use the wireless network please configure eduroam at your
home institution before coming to Münster.

For participants registered before December 15 we booked, on their request, a room in one of these hotels:

- Hotel Jellentrup, Hüfferstr. 52 (map), at 650m walk from the workshop venue,
- Haus Niemann, Bentelerstr. 4 (map), at 1500m walk from the workshop venue.

Friday evening there is a conference dinner at
I Galletti, An der
Germania Brauerei 4 (map).
The restaurant is located 1800m away from the
workshop venue, 2500m from hotel Jellentrup and 2000m from Haus Niemann.

Next to the restaurant there is a small ice rink (maybe 500m^{2}),
open until 22h00. You can save 6 EUR by bringing your own skates.

### Registration

To register, please send an email to (raimar -at- math.uni-muenster.de) indicating the following:

Affiliation:

### Program

Note: The talk by Detlev Buchholz will take place at the Institute for Theoretical Physcis,
Wilhelm-Klemm-Straße 9, lecture hall 404 (map).

All other talks will take place at the
Mathematical Institute, Einsteinstraße 62, lecture hall M5 (map).

### Abstracts

Infrared problems appear in the setting of quantum field theory with long range forces at various levels: by divergences in perturbation theory, the failure of ordinary scattering theory and the inapplicability of fundamental results of the axiomatic approach. Whereas the first two problems found well-established solutions, the general (model independent) understanding of basic features, such as the existence of charge conjugation and of particle statistics remained to be longstanding open problems. In this talk solutions are presented which are based on the insight that experimental limitations (the arrow of time) provide a natural Lorentz invariant infrared regularization. Based on it simple charges of electric type and their conjugates, the statistics of the corresponding sectors and their covariance and energetic properties can unambiguously be determined and classified. (Joint work with the late John E. Roberts)

The following normative principle was expressed by the renowned philosopher of science John Earman in 2004: "While idealizations are useful and, perhaps, even essential to progress in physics, a sound principle of interpretation would seem to be that no effect can be counted as a genuine physical effect if it disappears when the idealizations are removed." Despite its obvious validity, it is remarkable how often idealizations violate these principles. For example, all rigorous theories of spontaneous symmetry breaking in quantum statistical mechanics and in quantum field theory strictly apply to infinite systems only, since ground states of finite quantum systems are typically unique (and hence symmetric), whilst thermal equilibrium states of such systems (i.e. KMS states) are even always unique. The "Swiss" approach to the measurement problem based on superselection rules (Jauch, Hepp, Emch) faces a similar problem. We show how to deal with this principle, using the operator-algebraic language familiar to the Local Quantum Physics community.

When the *-product is expressed relative to the time-ordered product, the properties of the Møller operators become clearer. This simplifies the perturbative computation of the interacting *-product. The result is a formula that makes sense nonperturbatively.

I will present a construction of multi-particle scattering states which is suitable for a large class of wedge-local QFTs, including e.g. Grosse-Lechner-type models. A corresponding scattering theory up to the two-particle level has been developed previously, but a generalization to higher particle numbers was not expected for geometric reasons. Lifting this restriction allows us to formulate and discuss the question of asymptotic completeness in the wedge-local setting.

Recently Fredenhagen and Lindner have constructed an
interacting KMS state for a massive scalar field theory over Minkowski
Spacetime in the framework of perturbative algebraic quantum field theory.
In this talk, based on a joint work with N. Drago and N. Pinamonti (ArXiv:[
1609.01124

We consider the adiabatic limit of Hadamard states for free quantum Klein-Gordon fields, when the background metric and the field mass are slowly varied from their initial to final values. If the Klein-Gordon field stays massive, we prove that the adiabatic limit of the initial vacuum state is the (final) vacuum state, by extending to the symplectic framework the adiabatic theorem of Avron-Seiler-Yaffe. In cases when only the field mass is varied, using an abstract version of the mode decomposition method we can also consider the case when the initial or final mass vanishes, and the initial state is either a thermal state or a more general Hadamard state.

It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (Lévy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by Baumann, based on the Dell'Antonio–Robinson–Greenberg theorem, for a relativistic quantum field in positive metric to be a free field. The talk is based on joint work with Sergio Albeverio.

Together with Batu Güneysu and Jacob Schach Møller we recently studied stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics (QED). In this talk we discuss differentiability properties of the corresponding stochastic flow. Furthermore, we present a Bismut-Elworthy-Li type formula for the derivatives of the semi-group generated by the non-relativistic QED Hamiltonian revealing the smoothing properties of the semi-group. Finally, we explain how to prove the smoothness of an associated Fock space operator-valued semi-group kernel.

I will discuss relations of linear symplectic dynamics and their quantum counterparts in the case of an infinite number of degrees of freedom. In particular, I will describe conditions guaranteeing the existence of the (possibly, renormalized) quantum Hamiltonian and the formulas for its "vacuum energy". If time permits, I will discuss some examples from local QFT, which illustrate various difficulties in defining quantum dynamics: 1) mass-dependent perturbation of neutral bosons (dynamics exists in Fock space but an infinite renormalization of the Wick ordered Hamiltonian is necessary); 2) charged bosons in an external electromagnetic field (dynamics does not exists in Fock space, however the vacuum energy after an infinite renormalization is well defined).

We will start by explaining what we mean by spectrum of a dynamical system from the point of view of resonances in dynamical systems and show how this concept relates to the instantonic theories studied by Frenkel-Losev-Nekrasov. Then we describe some work in collaboration with Gabriel Rivière (Lille 1 university) where we calculate such spectras for generic gradient flows. If time permits, we shall give potential applications to topological field theories.

Measurable quantities that have positive values in classical dynamical systems need not to be positive in quantum theory. For example, consider a free quantum mechanical particle in 1 dimension. There are quantum states in which the particle's velocity is positive with probability 1, but where the probability flux for its position is locally negative; that is, while its velocity points to the right, the particle travels to the left. These effects are however small and limited in space and time by certain lower bounds, which are called "quantum inequalities". Similar effects also appear for a particle whose motion is governed by a Schroedinger equation with a certain class of potentials. The talk will present some recent results and work in progress on this topic.

The Spin-Boson model is a model from QFT which describes a two level sys- tem coupled to a scalar field. In this talk, I will present new results about the strong interaction limit of the massive Spin-Boson model. As the interac- tion approaches infinity, one can under very general assumptions describe the asymptotics of the resolvent (in a suitable sense), the ground state energy and the ground state eigenvector. One application of these results is to show the existence of a non degenerate exited state in the strong interaction limit and prove that the energy of the excited state converges to the energy of the ground state. If time allows, I will also talk about why the results makes it hard to renormalise the Spin-Boson model and the strategy to prove the above mentioned results. This is joint work with Jacob Schach Møller.

### Participants

Name | Affiliation |
---|---|

Dorothea Bahns | Mathematisches Institut, Universität Göttingen |

Karsten Bohlen | Universität Wuppertal |

João Braga de Góes e Vasconcellos | University of Genova |

Detlev Buchholz | Universität Göttingen |

Daniela Cadamuro | Universität Göttingen |

Joachim Cuntz | WWU Münster |

Jonas Dahlbæk | Aarhus University, Department of Mathematics |

Thomas Norman Dam | Aarhus University |

Jan Derezinski | University of Warsaw |

Nicoló Drago | Genoa University |

Maximilian Duell | Technische Universität München |

Steven Duplij | Münster |

Wojciech Dybalski | Technische Universität München |

Siegfried Echterhoff | WWU Münster |

Michał Eckstein | Jagiellonian University, Kraków |

Federico Faldino | University of Genova |

Francesco Fidaleo | Università Tor Vergata, Roma |

Christian Fleischhack | Universität Paderborn |

Klaus Fredenhagen | II. Institut für Theoretische Physik, Uni Hamburg |

Alexander Frei | LMU/TU München |

Hanno Gottschalk | Universität Wuppertal |

Eli Hawkins | University of York |

Michael Kiss | University of York |

Klaas Landsman | Radboud Universiteit Nijmegen |

Max Lewandowski | Universität Potsdam |

Oliver Matte | Aarhus Universitet |

Gernot Münster | WWU Münster |

Dang Nguyen Viet | Institut Camille Jordan Lyon 1 |

Carlos Pérez Sánchez | WWU Münster |

Karl-Henning Rehren | Universität Göttingen |

Kasia Rejzner | University of York |

David Sabonis | TU München / University of Copenhagen |

Jan Schlemmer | WWU Münster |

Devashish Singh | Department of Mathematics, University of Genova |

Ruben Stienstra | Radboud University Nijmegen |

Michael Stiller | II. Institut für Theoretische Physik, Uni Hamburg |

Alexander Stottmeister | Università Tor Vergata, Roma |

Rainer Verch | Universität Leipzig |

Raimar Wulkenhaar | WWU Münster |

Jins de Jong | WWU Münster |

#### Organizers

Carlos Pérez Sánchez (c_pere03 [-AT-] uni-muenster.de)Jan Schlemmer (Jan.Schlemmer [-AT-] math.uni-muenster.de)

Jins de Jong (jinsdejong [-AT-] hotmail.com)

Raimar Wulkenhaar (raimar [-AT-] math.uni-muenster.de)