Wahrscheinlichkeitstheorie, Funktionalanalysis und konvexe Analysis auf unendlich-dimensionalen Räumen sowie der Bezug zur mathematischen Physik
Porträt von Jobst Ziebell
Foto: Jobst Ziebell
Über mich
Kreidetafel
Illustration: KI-generiert
Ich bin ein mathematischer Physiker, der an der Schnittstelle zwischen konvexer Analysis und Wahrscheinlichkeitstheorie arbeitet. Meine Forschung beschäftigt sich insbesondere mit Flüssen von konvexen Funktionalen in Quantenfeldtheorien, die als Wahrscheinlichkeitsmaße auf unendlich-dimensionalen Distributionsräumen modelliert werden.
Mich interessieren auch topologische Aspekte von asymmetrischen lokalkonvexen Räumen, insbesondere in Verbindung mit Gamma-Konvergenz.
8 Publikationen filtern
Existence theorem on the UV limit of Wilsonian RG flows of Feynman measures
AutorenA. László, Z. Tarcsay, J. ZiebellErscheinungsjahrErschienen in:Journal of physics: a journal recognized by the European Physical Society
In nonperturbative formulation of Euclidean signature quantum field theories, the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman measures. Such an RG flow is a family of Feynman measures on the space of ultraviolet (UV) regularized fields, linked by the Wilsonian RG equation. In this paper we show that under mild conditions, a Wilsonian RG flow of Feynman measures extending to arbitrary regularization strengths has a factorization property: there exists an ultimate Feynman measure (UV limit) on the distribution sense fields, such that the regularized instances in the flow are obtained from this UV limit via taking the marginal measure against the regulator. Existence theorems on the flow and UV limit of the corresponding action functional are also discussed.
Tikhonov well-posedness and differentiability on asymmetrically normed spaces
AutorenJ. Fischer, J. ZiebellErscheinungsjahrErschienen in:Quaestiones mathematicae : QM ; journal of the South African Mathematical Society
On normed vector spaces there is a well-known connection between the Tikhonov well-posedness of a minimisation problem and the differentiability of an associated convex conjugate function. We show how this duality naturally generalises to the setting of asymmetrically normed spaces and prove a universal differentiability property of the convex conjugate of the cumulant-generating function of a mean-zero measure on a locally convex space.
AutorJ. ZiebellErscheinungsjahrErschienen in:Letters in mathematical physics : a journal for the rapid dissemination of short contributions in the field of mathematical physics
A simple condition is given that is sufficient to determine whether a measure that is absolutely continuous with respect to a Gaußian measure on the space of distributions is reflection positive. It readily generalises conventional lattice results to an abstract setting, enabling the construction of many reflection positive measures that are not supported on lattices.
A Rigorous Derivation of the Functional Renormalisation Group Equation
AutorJ. ZiebellErscheinungsjahrErschienen in:Communications in mathematical physics
The functional renormalisation group equation is derived in a mathematically rigorous fashion in a framework suitable for the Osterwalder–Schrader formulation of quantum field theory. To this end, we devise a very general regularisation scheme which retains reflection positivity as well as the infinite degrees of freedom including smoothness. Furthermore, it is shown how the classical limit is altered by the regularisation process leading to an inevitable breaking of translation invariance. We also give precise conditions for the convergence of the obtained theories upon removal of the regularisation.
Super-Gaussian decay of exponentials: A sufficient condition
AutorenB. Hinrichs, D. Janssen, J. ZiebellErscheinungsjahrErschienen in:Journal of mathematical analysis and applications
In this article, we present a sufficient condition for the exponential exp(−f) to have a tail decay stronger than any Gaussian, where f is defined on a locally convex space X and grows faster than a squared seminorm on X. In particular, our result proves that exp(−p(x)²⁺ε+αq(x)²) is integrable for all α,ε>0 w.r.t. any Radon Gaussian measure on a nuclear space X, if p and q are continuous seminorms on X with compatible kernels. This can be viewed as an adaptation of Fernique's theorem and, for example, has applications in quantum field theory.
Pauli-term-induced fixed points in d-dimensional QED
AutorenH. Gies, K. Tam, J. ZiebellErscheinungsjahrErschienen in:European Physical Journal C
We explore the fixed-point structure of QED-like theories upon the inclusion of a Pauli spin-field coupling. We concentrate on the fate of UV-stable fixed points recently discovered in d= 4 spacetime dimensions upon generalizations to lower as well as higher dimensions for an arbitrary number of fermion flavors Nf. As an overall trend, we observe that going away from d= 4 dimensions and increasing the flavor number tends to destabilize the non-Gaussian fixed points discovered in four spacetime dimensions. A notable exception is a non-Gaussian fixed point at finite Pauli spin-field coupling but vanishing gauge coupling, which also remains stable down to d= 3 dimensions and for small flavor numbers. This includes also the range of degrees of freedom used in effective theories of layered condensed-matter systems. As an application, we construct renormalization group trajectories that emanate from the non-Gaussian fixed point and approach a long-range regime in the conventional QED ₃ universality class that is governed by the interacting (quasi) fixed point in the gauge coupling.
Preparing for a Future with Quantum Technologies: An Innovative Approach to Accessible Quantum Education
AutorenY. Pathak, R. Sollapur, J. Ziebell, R. Geiß, T. Kaiser, F. Setzpfandt, A. Tünnermann, T. PertschErscheinungsjahrErschienen in:Conference on Education and Training in Optics and Photonics: ETOP 2023
The functional renormalisation group, its mathematics and applications to asymptotic safety
AutorJ. ZiebellErscheinungsjahr
We present a regularisation scheme for scalar Quantum Field theories that enables a flexible and mathematically consistent formulation of interacting theories in arbitrary dimensions. In contrast to a lattice approach, it retains the smooth features of spacetime and the infinite de- grees of freedom such that, in particular, the rotational symmetry can be left unbroken. In this framework, we give a mathematically rigorous derivation of the Wetterich equation as well as sufficient conditions for the passage to the limit of vanishing regularisation. We also introduce an iterative construction procedure for exact solutions to the Wetterich equation that works by producing higher-order correlation functions form the renormalisation group flow of lower order correlators. Then a generalisation of Quantum Electrodynamics is considered in the asymptotic safety framework and particular solutions are found that reproduce physical results in a low-energy regime. Finally, the applicability of the introduced regularisation scheme to the ϕ4 theory is proved. It follows from an integrability statement that can be thought of as a generalisation of Fernique’s theorem on exponential tails of Gaußian measures.
