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.