# 18th Colloquium on Mathematics and Foundations of Quantum Theory

## April 12, 2024

## About

This event is organized alternatingly each seminar by the groups of Dirk - André Deckert (LMU), Wojciech Dybalski (U Poznań), Felix Finster (U Regensburg), and Peter Pickl (U Tübingen).## Location

**Lecture hall:** A027

**Address:**
Department of Mathematics,
LMU Munich,
Theresienstr. 37/39,
80333 Munich

## Program

Time
| Event |
Description |

12:00-12:50 | Talk | Janik Kruse Mourre theory and asymptotic observables in local relativistic quantum field theory |

12:50-13:10 | Discussion | |

13:10-13:20 | Break | |

13:20-14:10 | Talk | Viet Hoang Effective polaron dynamics for an impurity particle interacting with a Fermi gas |

14:10-14:30 | Discussion | |

14:30-14:40 | Break | |

14:40-15:30 | Talk | Paweł Duch Stochastic quantization of fractional $\Phi^4_3$ model of Euclidean quantum field theory |

15:30-15:50 | Discussion | |

15:50-16:00 | Break |

## Talks

### Mourre theory and asymptotic observables in local relativistic quantum field theory

**Speaker:** Janik Kruse (U Poznań)

**Abstract:** The problem of asymptotic completeness in scattering theory,
fundamental for interpreting quantum theories in terms of particles, has seen
significant advancements in non-relativistic quantum mechanics throughout the
20th century. Yet, asymptotic completeness in local relativistic quantum field
theory (QFT) remains a formidable challenge. Conceptual and technical barriers,
including the superselection structure of QFT and the lack of understanding in
systems with relativistic dispersion relations, hinder progress. In
non-relativistic quantum mechanics, many proofs of asymptotic completeness rely
on the convergence of asymptotic observables. In QFT, Araki-Haag detectors
serve as natural observables, yet their convergence on arbitrary states has
remained unsolved for decades. Relatively recently, Dybalski and Gérard (2014)
have made progress in this area by translating quantum mechanical propagation
estimates to QFT, addressing products of multiple detectors sensitive to
particles with distinct velocities. However, the convergence of a single
detector was not treated due to the absence of a low velocity propagation
estimate. Typically, such an estimate is proved through Mourre’s conjugate
operator method—a technique thus far resistant to extension from quantum
mechanics to QFT. In this talk, I present a recent publication
(https://arxiv.org/abs/2311.18680), in which we closed this gap. By extending
Mourre theory from quantum mechanics to QFT, we achieve strong convergence of
a single Araki-Haag detector on states of bounded energy below the
three-particle threshold.

### Effective polaron dynamics for an impurity particle interacting with a Fermi gas

**Speaker:** Viet Hoang (U Tübignen)

**Abstract:** We study the quantum dynamics of a homogeneous ideal Fermi gas
coupled to an impurity particle on a 3-dimensional box with periodic
boundaries. For large Fermi momentum \kF, we prove that the effective
dynamics is generated by a Fröhlich-type polaron Hamiltonian, which linearly
couples the impurity particle to an almost-bosonic excitation field. Our method
applies to the case of an interaction coupling \lambda=1 and time
scales of order \kF^{-1}, allowing us to compare our results with
experimental observations.

### Stochastic quantization of fractional $\Phi^4_3$ model of Euclidean quantum field theory

**Speaker:** Paweł Pawel (U Poznań)

**Abstract:** We present a construction of the fractional $\Phi^4$
model of Euclidean quantum field theory in three-dimensions. The measure
of the model is obtained as a perturbation of the Gaussian measure with
covariance given by the inverse of a fractional Laplacian. Since the
Gaussian measure is supported in the space of Schwartz distributions and
the quartic interaction potential of the model involves pointwise
products, to construct the measure it is necessary to solve the
so-called renormalization problem. To this end, we study the stochastic
quantization equation, which is a nonlinear parabolic PDE driven by the
white noise. We prove a certain a priori estimate for solutions of this
equation using the flow equation approach to singular stochastic PDEs
and the maximum principle. We consider the entire range of powers of the
fractional Laplacian for which the model is super-renormalizable. Based
on a joint work with M. Gubinelli and P. Rinaldi.