Mathematische Physik und Operatoralgebren

Auf diesen Seiten finden sich Informationen zu den Personen, Forschungsinteressen, Lehrveranstaltungen und Projekten der Arbeitsgruppen Quantenfeldtheorie/Operatoralgebren (Gandalf Lechner) und Quantenmechanik/ (Hermann Schulz-Baldes).

QFT-Gruppe (Lechner)
QM-Gruppe (Schulz-Baldes)

Professoren

GL

Lehrstuhl für Mathematik (Operatoralgebren) (Heisenberg-Professur)

Professorinnen und Professoren

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Social Media und Webportale

HS

Professur für Mathematik (Mathematische Physik)

Leitung

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Sekretariat

Department Mathematik

Verwaltungspersonal

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Vertretungsprofessor

MF

Lehrstuhl für Mathematik (Operatoralgebren) (Heisenberg-Professur)

Wissenschaftliche Mitarbeitende

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Wissenschaftliche Mitarbeitende

TF

Professur für Mathematik (Mathematische Physik)

Wissenschaftliche Mitarbeitende

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

JG

Lehrstuhl für Mathematik (Operatoralgebren) (Heisenberg-Professur)

Wissenschaftliche Mitarbeitende

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Professur für Mathematik (Mathematische Physik)

Wissenschaftliche Mitarbeitende

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Ian Koot

Lehrstuhl für Mathematik (Operatoralgebren) (Heisenberg-Professur)

Promovierende

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Social Media und Webportale

DM

Professur für Mathematik (Mathematische Physik)

Wissenschaftliche Mitarbeitende

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Masterstudierende

  • Moritz Lanz
  • Andreas Rothemund
  • Benedikt Wenzel

Emeritus

Andreas Knauf

Naturwissenschaftliche Fakultät

Professorinnen und Professoren im Ruhestand

Adresse

Cauerstraße 11 91058 Erlangen

Kontakt

Publikationen am Lehrstuhl Mathematische Physik und Operatoralgebren

  • , :
    Anomalous transport in quasicrystals
    5th International Conference on Quasicrystals (Avignon, 22.05.1995 - 26.05.1995)
    In: Proceedings of the 5th International Conference on Quasicrystals, Singapore:
    BibTeX: Download

Projekte am Lehrstuhl Mathematische Physik und Operatoralgebren

Laufzeit: 16.08.2026 - 15.08.2028
Mittelgeber: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)

Die Tomita-Takesaki-Theorie der modularen Flüsse in von-Neumann-Algebren fand in den letzten Jahren vielfältige Anwendungen in der Hochenergie- und Festkörperphysik, als ein Werkzeug zur Untersuchung von Energie, Entropie und Verschränkung in Quantenfeldtheorie (QFT) und semiklassischer Gravitation. Das zentrale Objekt der Theorie, der einem Quantenzustand und einer Raumzeitregion zugeordnete modulare Hamiltonoperator, ist jedoch nur in wenigen Beispielen bekannt. Diese beinhalten hauptsächlich…

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Laufzeit: 01.10.2025 - 30.09.2026
Akronym: RE:Math
Projektleitung: ,

Das Vorhaben Re:Math unterstützt die Lehreinheit Mathematik und Data Science, die Studieneingangsphase qualitätsvoll weiterzuentwickeln, die seit Jahren als kritische Phase in mathematischen Studiengängen beschrieben wird. Angesichts der steigenden Bedeutung mathematischer Kompetenzen für berufliche und gesellschaftliche Teilhabe in einer Kultur der Digitalität, sinkenden Studieninteressiertenzahlen und dem wachsenden Lehrkräftemangel muss daher die Studieneingangsphase dringend zukunftsweisend…

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Laufzeit: 01.04.2023 - 31.03.2026
Mittelgeber: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
Projektleitung:

Topological invariants and their index theory, the bulk-boundary correspondence and the more recently introduced spectral localizer are well-established mathematical concepts for disordered topological insulators and are also influential for numerical studies of such materials. This proposal is about extending prior results and techniques to systems with crystalline defects, disordered semimetals and topological metals, as well as non-hermitian topological systems stemming from (leaky and driven…

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Laufzeit: 01.09.2021 - 31.08.2026
Mittelgeber: DFG-Einzelförderung / Heisenberg-Programm (EIN-HEI)
Projektleitung:

Das Ziel des Heisenberg-Programms ist es, herausragenden Wissenschaftlerinnen und Wissenschaftlern, die alle Voraussetzungen für die Berufung auf eine Langzeit-Professur erfüllen, zu ermöglichen, sich auf eine wissenschaftliche Leitungsfunktion vorzubereiten und in dieser Zeit weiterführende Forschungsthemen zu bearbeiten. In der Verfolgung dieses Ziels müssen nicht immer projektförmige Vorgehensweisen gewählt und realisiert werden. Aus diesem Grunde wird bei der Antragstellung und auch später b…

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Laufzeit: 01.04.2020 - 31.03.2023
Mittelgeber: Deutsche Forschungsgemeinschaft (DFG)
Projektleitung:

The first goal of index theory is to relate topological invariants to indices of Fredholm operators. The most famous result in this direction is the Atiyah-Singer index theorem, but there exist far reaching non-commutative generalizations. While there is a general theory, such index theorems have to be established case by case in applications. The second goal of index theory is to connect invariants and indices of problems related via exact sequences. For example, this allows to read off the top…

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Laufzeit: 01.07.2018 - 31.12.2021
Mittelgeber: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
Projektleitung: ,

The theoretical limits of distributed compressive sensing are studied by tools from both information theory and statistical physics. The investigations cover both noise-free and noisy distributed compressive sensing. The theoretical insights are utilized to design approximate message passing algorithms for joint recovery of large distributed compressive sensing networks with feasible computational complexity. These algo- rithms enable us to verify the non-rigorous results obtained by the replica…

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Laufzeit: 01.03.2016 - 28.02.2019
Mittelgeber: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
Projektleitung: ,

Das Projekt wird die Leistungsgrenzen von komprimierender Abtastung untersuchen und praxistaugliche Algorithmen entwerfen, die diesen Leistungsgrenzen nahe kommen.Das Projekt zielt auf hohe Kompressionsraten, bei dem Regularisierung des Problems mit Hilfe der L1-Norm suboptimal ist.Komprimierende Abtastung wird aus der Sicht der statistischen Physik untersucht und hierin als der Sonderfall eines Spinglassystems behandelt werden.Sowohl die mittlere als auch die Minimaxverzerrung wird als Zielfunk…

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Laufzeit: 01.01.2016 - 01.10.2019
Mittelgeber: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
Projektleitung:

The first goal of index theory is to relate topological invariants to indices of Fredholm operators. The most famous result in this direction is the Atiyah-Singer index theorem, but there exist far reaching non-commutative generalizations. While there is a general theory, such index theorems have to be established case by case in applications. The second goal of index theory is to connect invariants and indices of problems related via exact sequences. For example, this allows to read off the top…

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Laufzeit: 01.06.2010 - 31.03.2016
Mittelgeber: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
Projektleitung:

Das Forschungsvorhaben kann in zwei Problembereiche unterteilt werden:I Untersuchung der Topologie, der Dynamik und der Spektren von ungeordneten zeitumkehrinvarianten Systemen mit ungeradem Spin II Topologische und spektrale Aspekte der Streutheorie in Medien mit einem periodischen Hintergrundpotential oder an HyperflächenDas Hauptziel im Problembereich I ist eine detaillierte mathematische Analysis von Spin- Randströmen, die zum Beispiel in Graphenschichten auftreten und in Zukunft so genannte…

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Laufzeit: 01.02.2006 - 30.09.2009
Mittelgeber: Deutsche Forschungsgemeinschaft (DFG)
Projektleitung:

Im Rahmen diese Projektes wurden verschiedene Fragestellungen aus dem Bereich der Fest-körperphysik ungeordneter Systeme rigoros analysiert. Die meisten Fortschritte wurden beider kontrollierten Störungstheorie für quasi-eindimensionale zufällige Medien gemacht, die esnun erlauben, auch so genannte Anomalien zu untersuchen. Dies erlaubt insbesondere auchVerbindungen zur Theorie der vollen Zufallsmatrizen herzustellen. Es wurde Delokalisierungfür bestimmte quasi-eindimensionale ungeordnete System…

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Publikationen Gruppe Prof. Andreas Knauf (ab 1987)

Planung der Lehre in Funktionalanalysis, mathematischer Physik, Operatoralgebren in den nächsten Semestern

Zur Information von allen Studierenden, die sich näher mit Funktionalanalysis, Operatoralgebren und/oder mathematischer Quantenphysik (QM/QFT) befassen möchten, gibt es hier eine Übersicht über geplante Lehrveranstaltungen in diese Richtung. Der typische Einstieg in diese Forschungsrichtung verläuft über die drei aufeinander aufbauenden Vorlesungen

  • Funktionalanalysis 1 (Sommersemester)
    Basis für alle weiterführenden Vorlesungen an diesem und anderen Lehrstühlen. Kann direkt nach den Grundvorlesungen in Analysis/Lineare Algebra belegt werden. Themen: Normierte und andere topologische Vektorräume (Banachräume, Hilberträume), Operatoren auf diesen Räumen, Grundprinzipien der Funktionalanalysis.
  • Funktionalanalysis 2 (Wintersemester)
    Baut auf Funktionalanalysis 1 auf und führt sie fort. Themen: Spektraltheorie, ggf. Einführung in Operatoralgebren.
  • Mathematische Quantentheorie (Sommersemester)
    Baut auf den Funktionalanalysis-Kursen auf und präsentiert die mathematische Struktur der nichtrelativistischen Quantenphysik. Keine Vorkenntnisse in Physik erforderlich.

Weitere sehr empfehlenswerte Vorlesungen sind Topologie, Funktionentheorie, und Darstellungstheorie. Anschließend können spezialisiertere Vorlesungen und Seminare besucht werden, zB zu Vielteilchenquantensystemen, Quantenfeldtheorie, von Neumann Algebren, etc.

Die folgende Liste enthält die grobe Lehrplanung für die nächsten Semester. Einerseits enthält sie nicht alle Vorlesungen, die vom Lehrstuhl Mathematische Physik angeboten werden, und andererseits enthält sie auch thematisch passende Veranstaltungen von anderen Lehrstühlen.

Informationen zu weiteren nicht-spezialisierten Veranstaltungen am Lehrstuhl, Vorlesungsskripten, und vergangenen Veranstaltungen finden sich auf den Seiten der einzelnen Dozenten.

Winter 26/27

  • Vorlesung Funktionalanalysis 2. 4+2 SWS. (Lechner/Große)
    Spektraltheorie und Operatoralgebren.
  • Seminar Spektraltheorie. 2 SWS (Schulz-Baldes)
  • Querschnittsmodulseminar Funktionalanalysis. 2 SWS (Lechner)
    Dieses Seminar vertieft und erweitert die Inhalte von Funktionalanalysis 1 in Vorträgen der TeilnehmerInnen.
  • Seminar Matrixalgebren und Quanteninformationstheorie. 2 SWS (Fröb)
    Kann direkt nach den Grundvorlesungen Analysis/Lineare Algebra im Bachelor belegt werden.
  • Seminar Dispersive PDEs, 2 SWS (Boßmann/Wiedemann)

Sommer 27 (noch unvollständig)

  • Vorlesung Mathematische Quantentheorie. 4+2 SWS (Schulz-Baldes)
  • Vorlesung Funktionalanalysis 1. 4+2 SWS (Boßmann)
  • Vorlesung Nets of Operator Algebras. 4+2 SWS (Lechner)
  • Seminar Spektraltheorie. 2 SWS (Schulz-Baldes)

Winter 27/28 (noch unvollständig)

  • Vorlesung Funktionalanalysis 2. 4+2 SWS (Boßmann)
  • Vorlesung Operatoralgebren. 4+2 SWS (Schulz-Baldes)
  • Querschnittsmodulseminar Funktionalanalysis. 2 SWS (Boßmann)
  • Seminar Spektraltheorie. 2 SWS (Schulz-Baldes)

Sommer 28 (noch unvollständig)

  • Vorlesung Mathematische Quantentheorie. 4+2 SWS (Boßmann)
  • Vorlesung Funktionalanalysis 1. 4+2 SWS (Schulz-Baldes)
  • Seminar Spektraltheorie. 2 SWS (Schulz-Baldes)

Winter 27/28 (noch unvollständig)

  • Vorlesung Funktionalanalysis 2. 4+2 SWS (Schulz-Baldes)
  • Seminar Spektraltheorie. 2 SWS (Schulz-Baldes)

Frühere Semester (seit Sommer 2026)

Sommer 26

  • Vorlesung Mathematische Quantentheorie. 4+2 SWS. (Lechner)
  • Seminar Entropie. 2 SWS. (Lechner)
  • Vorlesung Fourieranalysis. 4+2 SWS. (Fröb)

Research seminar Mathematical Physics

The research seminar on mathematical physics meets on Thursday 4pm ct in Übung 1 during term times, and is jointly run by

If you want to be kept up to date with seminar announcements, you can write to one of us and we will put you on our mailing list.

Winter 2026/27

Past talks

Abstract:

Tim Rotheneder (FAU) Positivity Domains and Coverings of Compactly Causal Symmetric Spaces Abstract: Given a compactly causal symmetric Lie algebra $(\mathfrak{g},\tau,C)$ and an Euler element $h\in \mathfrak{g}$ as infinitesimal data and an associated causal symmetric space $(G,H,\tau,C)$ with $M:= G/H$, we discuss how the topology of the positivity region $W_M^+(h)$ depends on the global choice of $G$ and $H$. Eventually, we apply our theoretical results to the anti-de Sitter example, i.e. $\mathfrak{g} = \mathfrak{so}_{2,d}(\R)$, $\tau$ is conjugation with $I_{1,d+1}$ and $C$ is the forward light cone.

Abstract:

Lauritz van Luijk (Perimeter Institute) Entanglement in von Neumann Algebraic Quantum Information Theory Abstract: I will present results from a recent series of works with Alexander Stottmeister, Reinhard F. Werner, and Henrik Wilming in which we study bipartitions of quantum systems with an infinite number of degrees of freedom from an information-theoretic perspective. Our goal is to understand the interplay between information-theoretic properties of the physical systems and algebraic properties of the von Neumann algebras describing them. I will present two developments: (1) A bijective correspondence between the classification of factors into types and subtypes and a set of operational entanglement properties. For instance, Connes‘ classification of type III factors corresponds to the smallest achievable error when trying to ‚embezzle‘ entanglement from the system. (2) A result from my PhD thesis showing that the von Neumann algebraic description of subsystems can be derived from a set of model-independent operational axioms.

Abstract:

Malte Leimbach (Bonn) Compact quantum metric spaces and spectral truncations Abstract: Starting from Connes‘ distance formula, Rieffel generalized Lipschitz constants of functions to so-called Lip-norms of bounded operators. Lip-norms induce metrics on state spaces and a crucial requirement is that these metrics induce the weak* topology, in which case one speaks of a compact quantum metric space. There are various quantum versions of Gromov–Hausdorff distance allowing for considering questions about convergence of compact quantum metric spaces. In this talk I will focus on compact quantum metric spaces modeled on operator systems and Kerr–Li’s operator Gromov–Hausdorff distance. I will discuss some convergence results related to the spectral truncations put forward by Connes–van Suijlekom.

Abstract:

Christian Sadel (FAU) General transfer matrix techniques and absolutely continuous spectrum Abstract:

Abstract:

Réamonn Ó Buachalla (Prag) Noncommutative Kahler Structures Abstract: We begin by reviewing the general framework of differential calculi over *-algebras. Building on this, we introduce the notion of a complex structure, understood as a noncommutative analogue of the decomposition of complexified differential forms on a complex manifold. We then proceed to the concept of a (positive definite) noncommutative Kähler structure and outline several fundamental results that arise from its existence, including the Lefschetz decomposition, the Kähler identities, the Hodge decomposition, and the refinement of de Rham cohomology into Dolbeault cohomology. If time permits, we will conclude with a discussion of the motivating example of the standard quantum sphere and its Dolbeault-Dirac spectral triple.

Abstract:

David Damanik (Rice University) Return to Wonderland Abstract: Barry Simon’s Wonderland Theorem exhibited the genericity and ubiquity of singular continuous spectrum in the study of Schrödinger operators. In this talk we revisit this issue and in particular the following two questions: Is there a version of the Wonderland Theorem for dynamically defined potentials? and What is the generic spectral type of a Schrödinger operator with a bounded potential? We will explain the answers to these questions and how they arise. This is joint work with Artur Avila.

Abstract:

Evgeny Korotyaev (Northeast Normal University, Changchun, China) Spectral invariants for vector periodic NLS Abstract. Firstly, we discuss the various properties of periodic Zkharov-Shabat operator, associated with scaler periodic NLS. We describe the main results and techniques. Secondly, we discuss first order operators with a periodic 3×3 matrix potential on the real line. This operator is the Lax operator for the periodic vector NLS equation. The spectrum of the operator covers the real line and it is union of the spectrum of multiplicity 3, separated by intervals (gaps) of multiplicity 1. We prove the following: · The corresponding 2 or 3-sheeted Riemann surface is described. · Necessary and sufficient conditions are given when the Riemann surface is 2-sheeted. In the case of the 2-sheeted Riemann surface the solution of vector NLS equation is determined in terms of solutions of scalar NLS equations. · One constructs an entire function, which is negative on the spectrum of multiplicity 3 and is positive on its gaps. · The conformal mapping of the upper half plane on the domain on the upper half plane is constructed and the main properties are This conformal mapping has asymptotics at high energy where coefficients are constants of motion. As a corollary we obtain the estimate of potentials in terms of gap lengths. · Finally the Borg type result is obtained.

Abstract:

Dragan Marković (FAU) Chaos and operator dependent anomalous transport in semiclassical Bose-Hubbard chains Abstract: We investigate anomalous transport and chaotic dynamics in the semiclassical one-dimensional Bose–Hubbard chain using the Truncated Wigner Approximation with quantum corrections. At early times, the system exhibits robust superdiffusion characterized by universal, quantized exponents governed primarily by the initial state, largely independent of model parameters or the strength of chaos. This anomalous regime persists even in long chains and reflects a special scaling symmetry of the Hamiltonian. At later times, a crossover to normal diffusion emerges, which is most stable when nonintegrability is strong, leading to homogeneous states; for weaker nonintegrability, long-lived oscillations and inhomogeneities survive despite strong chaos. The coexistence of fast (angle) and slow (action) variables provides a natural framework to interpret the transport properties through free energy functionals: while quenched ensembles fail to equilibrate, annealed ensembles capture the effective late-time behavior. Together, these results highlight anomalous diffusion as a universal, early-time phenomenon distinct from prethermalization, smoothly evolving into conventional hydrodynamic transport at long times. 1. Superdiffusion, normal diffusion and chaos in semiclassical Bose-Hubbard chains, Phys. Rev. E 112, 024211; arXiv: 2502.19584 2. Chaos and anomalous transport in a semiclassical Bose-Hubbard chain, Phys. Rev. E 109, 034213; arXiv: 2308.14720 3. Thermodynamic formalism and anomalous transport in 1D semiclassical BH chain, arXiv preprint: 2312.17008

Abstract:

Chris Bourne (Nagoya) Interfaces of discrete quantum systems Abstract: Loosely speaking, an interface describes a spatial mixing of several systems described by a C*-algebra of observables such that these mixed dynamics are not felt ‘at infinity’. Building from work by Măntoiu, we give a mathematical description of (discrete) interfaces using C*-modules and (discrete) crossed products. Several examples will be given as well as spectral and index-theoretic properties.

Abstract:

Francesca Arici (Leiden) KK duality for Temperley—Lieb subproduct systems Abstract: The notion of KK-duality is a noncommutative analogue of the Spanier–Whitehead duality. It induces natural isomorphisms between the K-theory and K-homology of the dual C∗-algebras. Notable examples of noncommutative C*-algebras satisfying KK-duality are Cuntz—Krieger algebras. In this talk, we will describe a quantum analogue of the result of Kaminker and Putnam on Cuntz—Krieger algebras. Specifically, we consider the Cuntz— Pimsner algebras of subproduct systems defined by Temperley–Lieb polynomials, as defined by Habbestad—Neshveyev. These algebras can be thought of as algebras of functions on algebraic subsets of noncommutative spheres. Joint work with D. Gerontogiannis and S. Neshveyev.

Abstract:

Ian Koot Relative positions of Half-sided Modular Inclusions Abstract: Half-sided Modular Inclusions are inclusions of standard subspaces where the modular groups of the two standard subspaces satisfy some ’nice‘ relations. Recently we proved that inclusions of two standard subspaces that lie as Half-sided Modular Inclusions in a larger standard subspace can be fully characterized by inclusions of associated complex subspaces. We use these results, together with the representation theory of the Canonical Commutation Relations and the theory of Hardy spaces, to clarify how two half-sided modular inclusions in the same standard subspace can relate to each other.

Abstract:

Simone Rademacher (LMU München) Large deviations for Bose-Einstein condensates Abstract: Bose-Einstein condensation (BEC) is a special phenomenon of trapped Bose gases at low temperatures where a macroscopic fraction of the particles occupies the same one-particle quantum state, called condensate. This talk concerns a probabilistic approach to BEC: We consider the ground state of an interacting Bose gas on the three-dimensional unit torus, known to exhibit BEC. For weak interactions in the mean-field regime, we show that bounded one-particle operators satisfy large deviation estimates and compute the rate function up to second order. For singular interactions in the Gross-Pitaevskii regime, we prove an upper bound for the tails of the quantum depletion, the operator counting the number of particles outside the condensate, based on an explicit asymptotic formula its generating function. The talk is based on joint works with Nils Behrmann, Christian Brennecke and Phan Thanh Nam.

Abstract:

Alexander Schenkel (Nottingham) C*-categorical prefactorization algebras for superselection sectors and topological order Abstract: I will present a geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice Z^n. I will show that, under certain assumptions which are implied by Haag duality, the monoidal C*-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of Z^n. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder R^1 x S^{n-1}. While the sphere S^{n-1} arises geometrically as the angular coordinates of cones, the origin of the line R^1 is analytic and rooted in Haag duality. The usual braided (for n=2) or symmetric (for n>2) monoidal C*-categories of superselection sectors are recovered by removing a point of the sphere and using the equivalence between E_n-algebras and locally constant prefactorization algebras defined on open disks in R^n. The non-trivial homotopy groups of spheres induce additional algebraic structures on these E_n-monoidal C*-categories, which in the simplest case of Z^2 is given by a braided monoidal self-equivalence arising geometrically as a kind of `holonomy‘ around the circle S^1. This talk is based on joint work with Marco Benini, Victor Carmona and Pieter Naaijkens [arXiv:2505.07960].

Abstract:

Nicolò Drago (Genua) Classical and quantum KMS states on spin lattice systems Abstract: We study the classical and quantum KMS conditions within the context of spin lattice systems. Specifically, we define a strict deformation quantization (SDQ) for a S^2-valued spin lattice system over Z^d generalizing the renown Berezin SDQ for a single sphere. This allows to promote a classical dynamics on the algebra of classical observables to a quantum dynamics on the algebra of quantum observables. We then compare the notion of classical and quantum thermal equilibrium by showing that any weak*-limit point of a sequence of quantum KMS states fulfils the classical KMS condition. In short, this proves that the semiclassical limit of quantum thermal states describes classical thermal equilibrium, strengthening the physical interpretation of the classical KMS condition. Finally we provide two sufficient conditions ensuring uniqueness of classical and quantum KMS states: The latter are based on a version of the Kirkwood-Salzburg equations adapted to the system of interest. As a consequence we identify a mild condition which ensures uniqueness of classical KMS states and of quantum KMS states for the quantized dynamics for a common sufficiently high temperature. Joint work with L. Pettinari and C.J.F. van de Ven.

Abstract:

Phan Thành Nam (LMU) Bogoliubov-Lee-Yang theory for the mean-field Bose gas at positive temperatures Abstract: I will discuss recent joint work with Andreas Deuchert and Marcin Napiórkowski on the rigorous justification of a version of Bogoliubov theory proposed by Lee and Yang in 1958 to describe the interacting Bose gas at positive temperatures, comparable to the critical temperature of the Bose-Einstein phase transition. We consider the homogeneous mean-field Bose gas on the 3D torus and prove a trace-norm approximation for the grand canonical Gibbs state, which also yields precise formulas for correlation functions. Our result goes beyond the standard quasi-free approximation, and a key ingredient of our proof is an abstract correlation inequality based on Stahl’s theorem.

Abstract:

Matthias Keller (Potsdam) On Landis conjecture for positive Schrödinger operators on graphs Abstract: Landis conjecture is concerned with growth bounds for harmonic functions of Schrödinger operator whose potential in absolute value is bounded by one. We study this problem for real potentials and positive Schrödinger operators on general graphs and in the Euclidean lattice in particular. (This is joint work with Ujjal Das and Yehuda Pinchover.)

Abstract:

Jean-Bernard Bru (BCAM) Exchange Interactions and Cuprate Superconductivity Abstract: In this talk, we will explain the effect of quantum interactions exchanging different types of particles. We will consider a system made of two fermions and one boson, in order to study the effect of such an off-diagonal interaction term. We will in particular show the existence of exponentially localized dressed bound fermion pairs. We will give particular attention to the regime of very large on-site (Hubbard) repulsions, because this situation is relevant for cuprate superconductors. Indeed, in the meantime, we will explain the high-temperature superconductivity of cuprates and apply our model to such physical systems.

Abstract:

Gandeeb Bhattarai (Tübingen) Functional Central Limit Theorem (FCLT) in Quantum many body systems Abstract:

Abstract:

Markus Fröb (Leipzig) How to compute modular Hamiltonians Abstract: The Tomita-Takesaki theory of modular automorphisms of von Neumann algebras has attracted renewed interest in recent years in connection with the definition of entropies in quantum field theory. I will present a short introduction to the theory, in which the main object of interest is the modular Hamiltonian. I then explain how one can compute this Hamiltonian explicitly in various situations, in particular for relativistic fermions in 1+1 dimensions. Based on arXiv:2312.04629, arXiv:2406.19360, arXiv:2411.09696 and arXiv:2501.09669.

Abstract:

Anne van Grinsven (LMU) Mean-Field Approximation of the Ground State Energy of a Quasi-2D Bose Gas Abstract: We discuss a rigorous derivation of the mean-field approximation for the ground state energy of a quasi-two-dimensional Bose gas. This system consists of a trapped Bose gas with a potential that is highly confining in one direction, leading to effective two-dimensional behavior. Working in the mean-field regime, we consider systems where particles interact weakly but frequently. The argument builds on the method developed by Lewin, Nam and Rougerie for three-dimensional trapped systems. The main focus is the adaptation of this approach to account for the strong confinement and the associated dimensional reduction. We conclude by commenting on possible extensions, including the convergence of reduced density matrices and the incorporation of dipole-dipole interactions.

Abstract:

Ian Koot (FAU) Inclusions in Half-Sided Modular Inclusions Abstract:

Abstract:

Yasu Kawahigashi (Tokyo) Tensor networks in two-dimensional topological order and operator algebras Abstract: Certain 4-tensors appear in mathematical studies of 2-dimensional topological order recently. We identify them with bi-unitary connections that appear in subfactor theory in operator algebras and are variants of quantum 6j-symbols. We then translate various notions in the two studies in both directions and have deeper understanding of their mathematical structures.

Abstract:

Lars Koekenbier (FAU) Transfer matrix analysis of non-hermitian Hamiltonians: asymptotic spectra and topological eigenvalues Abstract: Transfer matrix techniques are used to provide a new proof of Widom’s results on the asymptotic spectral theory of  nite block Toeplitz matrices. Furthermore, a rigorous treatment of the skin e ect, spectral outliers, the generalized Brillouin zone and the bulkboundary correspondence in such systems is given. This covers chiral Hamiltonians with topological eigenvalues close to zero, but no line-gap.

Abstract:

Wilhelm Kroschinsky (Bonn) The time-stability of Bose-Einstein Condensates in the Gross-Pitaevskii regime Abstract: We revisit the problem of time-stability of Bose-Einstein condensate (BEC) of an initially trapped 3D system of bosons in the Gross-Pitaevskii regime. We prove that the system still exhibits BEC when the trap is switched off, and the effective dynamics is governed by the solution of the time- dependent Gross-Pitaevskii equation. Our main strategy is to control renormalized excitation number operators with respect to the Schrödinger dynamics 𝑡 ↦ → 𝑒−𝑖𝐻𝑁 𝑡 , rather than controlling the number of excitations with respect to a suitable excitation dynamics 𝑡 ↦ → 𝑒−𝐵𝑡 𝑒−𝑖𝐻𝑁 𝑡 .

Abstract:

Gerardo Franco Cordova Analytic Properties of the Scattering Matrix of Discrete Schrödinger Operators Abstract: We develop a scattering theory for discrete Schrödinger operators. We represent the scattering matrix in terms of special solutions to the eigenvalue equation, known as Jost solutions. This representation facilitates the analytic extension of certain coefficients of the scattering matrix, enabling a detailed study of their analytic properties. Furthermore, we connect these analytic properties of the scattering matrix to the spectral properties of the Hamiltonian, ultimately deriving a version of the Levinson formula.

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Ram Band The Dry Ten Martini Problem for Sturmian Schrödinger Operators Abstract: Are all gaps there?, asked Mark Kac in 1981 during a talk at the AMS annual meeting, and offered ten Martinis for the answer. This led Barry Simon to coin the names the Ten Martini Problem (TMP) and the Dry Ten Martini Problem for two related problems concerning the Almost-Mathieu operator. The TMP is about showing that the spectrum of the Almost-Mathieu operator is a Cantor set. The Dry TMP is about the values that the integrated density of states (IDS) attains at the spectral gaps. The gap labelling theorem predicts the possible set of values which the IDS may attain at the spectral gaps. The Dry TMP is whether or not all these values are attained, or equivalently, are all gaps there? We present an affirmative solution to the Dry Ten Martini Problem for Sturmian Hamiltonians – this is proven in a joint work with Siegfried Beckus and Raphael Loewy. In a joint work with Gilad Sofer, we study families of Sturmian operators on metric and discrete graphs. We show how their gap labels are characterized and what would be the corresponding Dry Ten Martini Problem for those.

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Christiaan van de Ven Large deviations in mean-field quantum spin systems Abstract: Continuous fields of C*-algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this talk, I consider the continuous C*-bundle generated by increasing symmetric tensor powers of the complex (\ell\times\ell) matrices, which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework I discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function. To this end, it has proved necessary to demonstrate the existence of a semiclassical analog of the Baker-Campbel-Hausdorff formula, defined in terms of a series of nested Poisson brackets.

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Tom Stoiber, (Yeshiva University) Higher-order Topological Insulators and K-theory Abstract: Topological insulators are materials that are insulating in their interior but exhibit protected surface states, a phenomenon that is mathematically well-understood through the use of C*-algebras and operator K-theory. In higher-order topological insulators, all faces of a crystal are insulating, while surface states appear only at boundaries of higher codimension, such as hinges or corners. For so-called intrinsic topological insulators the existence of these states is independent of boundary conditions but requires spatial symmetries like mirror, rotational, or inversion symmetry. Since the topological protection emerges only in the infinite volume limit, we need to construct C*-algebras that describe symmetric infinite crystals with various boundary configurations. These algebras naturally admit cofiltrations based on the codimensions of the boundaries, leading to spectral sequences in equivariant K-theory. We then demonstrate that the classification and phenomenology of higher-order topological insulators align seamlessly with this formalism. Specifically, the higher differentials in the spectral sequence identify precisely the classes of intrinsic higher-order topological insulators and their associated surface states.

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Michael Heins (Würzburg) A Holomorphic Perspective of Strict Deformation Quantization Abstract: Classical mechanical systems may be modelled by means of Poisson algebras, which physically arise as the observables corresponding to the system. The quantization scheme called Deformation Quantization establishes a second product on the observable algebra, a so-called star product. Imposing a classical and semiclassical limit condition with respect to a positive parameter ℏ then intertwines the classical with the quantum multiplication. To facilitate the construction, it is both customary and advisable to take a step back and pass to formal power series with coefficients in the classical algebra and indeterminant ℏ. One thus speaks of a formal deformation quantization. This talk is about overcoming the formal character of the resulting theory, which is a necessity for physically meaningful statements, and accounts for the fact that ℏ is a physical constant and not a free parameter. As the formal star product is, in particular, itself given by a power series, this naturally leads into the realm of holomorphic functions, both in finite and infinite dimensions. After briefly reviewing the formal situation, we discuss a recently proposed notion of strict deformation quantization and investigate how one can use established results from complex analysis to think about these objects. Along the way, we shall illustrate the abstract ideas in the setting of standard ordered quantization on the cotangent bundle of the real line.

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Thomas Creutzig (FAU) Towards logarithmic conformal field theory Abstract: A vertex operator algebra is the algebraic notion of chiral (and anti-chiral) algebra in two-dimensional conformal field theory. VOAs are an established subject of mathematics in its own right, but connecting to the physics of two-dimensional CFT is still difficult. This difficulty arises as VOA is of algebraic nature while the observables in CFT, the correlation functions, are of analytic nature. Representation theory of modern VOAs almost exclusively is non semi-simple, which should lead to logarithmic CFT, that is CFTs whose correlation functions can have logarithms. I will define full logarithmic VOAs, which have exactly the features that one wants a genus zero logarithmic CFT to have, and will give an example.

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LQP 49 Workshop Dates: 7-9 November 2024 Topics: This workshop is dedicated to mathematical quantum physics and more specifically quantum field theory. The topics covered here will hence be concentrated on quantum field theory (on Minkowski space or curved spacetimes, axiomatic, constructive, rigorous, Euclidean versions), as well as more general mathematical quantum physics, operator algebras (in particular, von Neumann algebras and subfactors), quantum information theory, and noncommutative geometry.

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Lea Boßmann (FAU) Focusing dynamics for 2d Bose gases in the instability regime Abstract: We consider the dynamics of a 2d Bose gas with singular attractive interactions in the instability regime, where the corresponding focusing nonlinear Schrödinger equation (NLS) has a blow-up. We show that the evolution of the condensate is effectively described by this NLS for all times before the blow-up. Moreover, we prove the validity of the Bogoliubov approximation for the fluctuation dynamics, resulting in a norm approximation of the many-body dynamics. This is joint work with Charlotte Dietze and Phan Thành Nam.

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Azam Jahandideh (Poznan) Stochastic quantization of two-dimensional $P(\Phi)$ Quantum Field Theory Abstract: We give a simple and self-contained construction of the $P(\Phi$ Euclidean Quantum Field Theory in the plane and verify the Osterwalder-Schrader axioms: translational and rotational invariance, reflection positivity and regularity. In the intermediate steps of the construction, we study measures on spheres. In order to control the infinite volume limit, we use the parabolic stochastic quantization equation and the energy method. To prove the translational and rotational invariance of the limit measure we take advantage of the fact that the symmetry groups of the plane and the sphere have the same dimension. Joint work with Paweł Duch and Wojciech Dybalski.

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Diana Taschetto (Utrecht) The Dual Dynamical Foundation of Orthodox Quantum Mechanics Abstract: In his groundbreaking The Mathematical Foundations of Quantum Mechanics von Neumann asked the question, why quantum mechanics has a dual dynamics (unitary evolution and collapse postulate)? He could not find a proper solution to it, i.e., a solution based on dynamical principles. In this talk, we shall present such a solution, certified by the golden rule of physics that the dynamics of a theory must follow from the action principle. The reason why quantum mechanics has this “peculiar dual dynamics” is that the theory is based, as we shall demonstrate, on a “peculiar” dual action.

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Luca Giorgetti (Rom) Crossing symmetry and Fourier transforms Abstract: Crossing symmetry is an invariance property of scattering amplitudes involving (pairs of) particles and their TCP conjugates. Recently, see arXiv:2212.02298,crossing symmetry has been reformulated as an analytic extension property ofcertain functions defined using Tomita-Takesaki modular theory. In the talk, I will introduce a linear map on bounded operators on the tensor square of a given complex Hilbert space, called the “crossing map”, whose fixed points describe crossing symmetry. I describe its basic properties and, among its many surprising relations with other mathematical objects, I will describe how it turns out to be a special case of the subfactor theoretical Fourier transform (which generalizes, e.g., the ordinary Fourier transform for finite abelian groups). Joint work with R. Correa da Silva and G. Lechner, arXiv: 2402.15763

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Melchior Wirth (IST Wien) Operator-Valued Twisted Araki-Woods Algebras Abstract: I will introduce the class of operator-valued twisted Araki-Woods algebras, which are second quantization von Neumann algebras built on certain Hilbert bimodules over a base von Neumann algebra. When the base algebra is the field of complex numbers, this class includes the q-Gaussian algebras and free Araki-Woods factors, and for arbitrary base algebras Shlyakhtenko’s von Neumann algebras generated by operator-valued semicircular variables fall into this class. In the case when the base algebra is a type I factor, I will present how a disintegration theory for the underlying Hilbert bimodules leads to a decomposition as a tensor product of the base algebra and a scalar-valued twisted Araki-Woods algebra. Moreover, operator-valued twisted Araki-Woods algebras come with a natural weight, and I will discuss the associated modular theory as well as some sufficient criteria for factoriality. This is joint work with Rahul Kumar R.

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Daniela Cadamuro (Leipzig) The massive modular Hamiltonian in the case of bosons and fermions Abstract: The Tomita-Takesaki modular operator for local algebras plays an important role in quantum field theory, and more recently in the study of relative entropy. However, the explicit expression of this operator, except for the case of wedges, is difficult to describe mathematically. We have obtained numerical results for the form of the modular Hamiltonian for a double cone in a massive scalar free field in (1+1)- and (3+1)-dimensional Minkowski space, which shows how it differs from the wedge case, in particular regarding the dependence of the modular Hamiltonian on the mass of the field. We also obtained results for the free massive Majorana fermions in 1+1 dimensions in the cases of a single and two double cones, and point out the differences with the bosonic case. Jakob Hedicke (Toronto) On spaces of light rays and contact structures Abstract: As first observed in the works of Penrose and Low, in many cases the space of light rays of a Lorentzian spacetime can be naturally equipped with the structure of a smooth contact manifold. In these cases the contact geometry of the space of light rays has strong connections to the causality of the underlying Lorentzian manifold. After an introduction to the relevant notions from contact- and Lorentzian geometry, we will discuss criteria that ensure the space of light rays to be a contact manifold and we will determine its contact structure in several examples.

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Fernando Lledó (Madrid) Examples of finite dimensional approximations in C*-algebras Abstract: Motivated by the shocking Banach-Tarski paradox and, in particular, by Foelner type approximations of amenable groups, I will present finite dimensional matrix approximations in two classes of operator algebras: – The resolvent algebra introduced by Buchholz and Grundling in 2008 to give an alternative bounded operator approach to the canonical commutation relations (CCR) in quantum mechanics. – The uniform Roe algebras of an inverse semigroup, where the inverse semigroup is viewed as a metric space. In this talk I will emphasize the importance of examples. The results presented are included in the recent publications [1] F. Lledó and D. Martínez, *A note on commutation relations and finite dimensional approximations,* Expositiones Mathematicae *40* (2022) 947–960. [2] F. Lledó and D. Martínez, *The uniform Roe algebra of an inverse semigroup*, Journal of Mathematical Analysis and Applications *499 *(2021) 124996. [3] P.Ara, F. Lledó and D. Martínez, Amenability and paradoxicality in semigroups and C*-algebras, J. Funct. Anal. *279* (2020) 108530

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Dietmar Bisch (Vanderbilt) New subfactors with small Jones index Abstract: Since Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Not much is known about this set. My student Julio Caceres and I have recently shown that certain intersting indices between 4 and 5 are realized by new hyperfinite subfactors with Temperley-Lieb-Jones standard invariant. This leads to a conjecture regarding Jones‘ problem. Our construction involves commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors. I will give some background to make the talk accessible to non-experts.

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Yuto Moriwaki (Riken, Tokyo) Conformal block, braided operad and factorization Abstract: From representations of a vertex operator algebra, D-modules (conformal blocks) on configuration spaces of the Riemann sphere are constructed. We show that compositions of conformal blocks are consistent with the operad structure of the configuration spaces. This gives an alternative proof of the result of Huang-Lepowky that the representation category of a regular vertex operator algebra has a structure of braided tensor category.

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Yoh Tanimoto (Rom) Lattice construction of QFT and formal mathematics Abstract: I give an overview of the strategy of Balaban-Dimock and our recent attempts of constructing Euclidean field theory. In relation with it, I talk about my recent experience with proof assistant.

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Hermann Schulz-Baldes (FAU) Transfer matrix analysis of non-hermitian Hamiltonians: asymptotic spectra and topological eigenvalues Abstract: Transfer matrix techniques are used to provide a new proof of Widom’s results on the asymptotic spectral theory of finite block Toeplitz matrices. Furthermore, a rigorous treatment of the skin effect, spectral outliers, the generalized Brillouin zone and the bulk-boundary correspondence in such systems is given. This covers chiral Hamiltonians with topological eigenvalues close to zero, but no line-gap. This is joint work with Lars Koekenbier.

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Edoardo D’Angelo (Genua) Functional Renormalization and the Nash-Moser theorem Abstract: The Renormalization Group (RG) Equation determines the flow of the effective action under changes in an artificial energy scale, which roughly corresponds to the scale of the system under consideration. I report on a rigorous construction of a non-perturbative RG flow for the effective action in Lorentzian manifolds. I give the main ideas of a proof of local existence of solutions for the RG equation, when a suitable Local Potential Approximation is considered. The proof is based on an application of the renown Nash-Moser theorem. Time permitting, I also discuss an application of the RG equation to the non-perturbative renormalizability of quantum gravity.

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Davide Lonigro (FAU) Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics Abstract: The multitime probability distributions obtained by repeatedly probing a quantum system via the measurement of an observable generally violate Kolmogorov’s consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory. We show that, nonetheless, they do result from the sampling of one pair of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario. Based on arXiv:2402.01218. Joint work with F. Sakuldee, Ł. Cywiński, D. Chruściński, and P. Szańkowski. —

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Thomas Thiemann (FAU Physik) On representations of the hypersurface deformation algebroid of quantum gravity Abstract: Quantum Gravity (QG), understood as a Quantum Field Theory (QFT) of General Relativity (GR), faces mathematical challenges that go beyond those of QFT in curved spacetime (CST). In this talk we focus on the hypersurface deformation algebroid (HDA) which in the classical theory encodes the Einstein equations. Implementing the quantum dynamics leads to the analysis of representations thereof by operators in QG. After an introduction to those concepts, we show that non-trivial representations of the HDA exist in a weak coupling limit of GR in 3+1 dimensions with 2 propagating, self-interacting polarisations.

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Christian Sadel (Chile) Transfer matrices and absolutely continuous spectrum for l-channel unitary operators Abstract: Based on the work on scattering zippers we develop a unitary analogue to one- and l-channel Hermitian operators with 2l by 2l transfer matrices. Then we obtain an analogue of Carmona’s spectral averaging formula and obtain criteria for absolutely continuous spectrum. Joint work with O. Bourget, G. Moreno and A. Taarabt.

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Romeo Brunetti (Trento)) Algebraic structures and relations in the dynamical algebras approach to quantum field theory Abstract: In this talk I wish to present some established and new ideas in the C*-Dynamical Algebra formalism of Buchholz and Fredenhagen. In particular, I’ll concentrate on new relations induced by solutions of interacting equations of motion (work in progress with Klaus Fredenhagen).

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Martin Ravn (München) Spectral Estimates for Fermionic n-Body Operators – Old and New Abstract: We consider bounds on the eigenvalues of the n-body operators associated to fermionic N-particle states. We review the classic norm estimates of Yang, and present a recent estimate on the Hilbert-Schmidt norm of 2-body operators and their truncated versions. The Hilbert-Schmidt estimate is interesting since it is of the same order as the optimal norm estimate, O(N). This implies that although a fermionic 2-body operator can have eigenvalues of size ~N, it can not have too many large eigenvalues.

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Daan Janssen (York) Characteristic renormalization and applications to black hole formation Abstract: We present a regularization prescription for Hadamard two-point functions defined on the boundary of a lightcone, which can be used to analyze renormalized quantities for linear scalar fields in a curved background. This can be applied to formulate the semi-classical Einstein equations as a characteristic initial value problem. Furthermore, we shall discuss how these tools may be used to estimate backreaction effects of quantum fields near gravitationally collapsing bodies and study formation of horizons and singularities in semi-classical gravity.

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Adriano Chialastri (SISSA, Trieste) On entanglement measures in quantum field theory: von Neumann entropy and Bell’s inequalities Abstract: Entanglement as a topic has its origins in Einstein, Podolsky and Rosen’s 1935 work, which sparked a decades-long debate with its theoretical turning point in Bell’s 1964 work, where the so-called Bell’s inequalities were born. However, throughout the years, there has been a continuous interest in entanglement measures within the context of quantum field theory (QFT), where different quantities like von Neumann entropy and relative entropy have long been studied. In this talk, we will discuss about entanglement measures within the formalism of algebraic QFT, which provides a solid mathematical background for them. In particular, we will analyze von Neumann entropy, with all its problems in QFT, and its relations with maximal Bell correlation, which is an entanglement measure quantifying the violation of Bell’s inequalities. This talk is based on a joint work with S. Carpi (University of Rome Tor Vergata).

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Arnold Neumaier (Wien) Coherent Quantization II: Causal groups and quantum fields Abstract: This is the second of three lectures on coherent quantization and field theory to be given 11.-18.12.2023 in Erlangen (Germany). Causal groups are a new class of mathematical objects abstracted from the concept of dynamical C^*-algebras for quantum field theories introduced by Buchholz and Fredenhagen in their paper Comm. Math. Phys. 377 (2020), 947-969. Unlike these authors (who obtain their dynamical C^*-algebras from an analysis of causal perturbatiion theory) we motivate causal groups in a fully nonperturbative way from the consideration of classical discrete-time dynamical systems. This gives an intuitive understanding of the properties later assumed axiomatically for causal groups over causal spaces (generalizing Minkowski spacetime). Each causal group over Minkowski space gives rise to a particular dynamical C^*-algebra. The dynamical C^*-algebras of Buchholz and Fredenhagen arise from abstract causal groups defined by generators and relations. We show how to construct causal groups over causal spaces having a Tomonaga-Schwinger structure associated with an appropriate classical many-fingered time dynamics. This gives a clear geometric meaning to the new concept. Their unitary representations give nonperturbative constructions of nonrelativistic and relativistic quantum field theories. Conditions are given under which the Haag-Kastler axioms or the Wightman axioms can be established. This reduces the rigorous construction of realistic quantum field theories such as QED or QCD to the (still unsettled) construction of unitary representations of causal groups with the properties defining QED or QCD. A constructed QFT can be identified with a particular perturbatively defined one (such as QED or QCD) by performing aposteriori causal perturbation theory to lowest (1 loop) order. This lecture is essentially independent of the first lecture. The slides of the lecture will probably become available – at least three hours before the lecture – at https://arnold-neumaier.at/cohErlangen2023.html where one can also find background material (abstracts and some references) for all lectures.

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Leonardo Sangaletti (Leipzig) An L4 quantum energy inequality in the thermal sector Abstract: Energy density and its positivity properties represent a fundamental subject in classical and quantum physics. In this talk, we will investigate this topic in the thermal representation of a free massive quantum scalar field. After a brief review of the fundamental mathematical tools at the base of this work, we will construct the GNS representation of our QFT induced by a state at thermal equilibrium (KMS). Therein, we will identify the generator of the time evolution and its spatial density. The symmetry between the particles and holes makes evident the impossibility for a lower bound of the expectation value of the energy density in this representation. In order to tackle this problem, we will investigate and extend some results of modular theory and non-commutative Lp spaces. In this way, we obtain a general result concerning the expectation value of an operator affiliated to a von Neumann algebra. Finally, the proven results are used to derive an L4 state dependent non-trivial QEI.

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Nicolai Rothe (Technische Universität Berlin) Cosmological solutions to the semiclassical Einstein equation with Minkowski-like vacua Abstract: We will discuss some newly found solutions to the full massless semiclassical Einstein equation (SCE) in a cosmological setting. After a short introduction to the relevant notions we present the SCE in a particular shape which allows for the construction of certain vacuum states for a free, massless scalar field. These states may be viewed as as the least possible generalization of the Minkowski vacuum to general (cosmological) space-times. In this setting, solving the SCE breaks down into solving a certain ODE which can be approached numerically and, at least generically, we obtain solutions that well fit physical expectations. Moreover, these solutions indicate dark energy as a quantum effect back-reacting on cosmological metrics. Since in our model m=Λ=0, this may not be traced back to the usual, obvious dark-energy/cosmological constant effect of a quantum field. Also we will shortly discuss some more physical problems that can be solved by our model.

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Matthias Lesch (Bonn) Rearrangement Lemma, divided differences and the multivariate holomorphic functional calculus Abstract: The so called Rearrangement Lemma is a technical device in the context of heat trace expansions on noncommutative spaces resp. for operators with noncommutative leading symbol. A couple of years ago I gave a systematic treatment and discussed the link to the classical divided difference formalism. In this talk I would like to recast the issue from the point of view of the multivariate holomorphic functional calculus. The latter has a rich history and is of some interest in its own as it is by no means just a straightforward generalization of the usual one-variable case. I will review this history and give some elementary applications to noncommutative versions of Newton interpolation and Taylor formulas as well as a holomorphic version of the rearrangement lemma. The talk is based on an ongoing project with Luiz Hartmann from Sao Carlos, Brazil.

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Alexander Cerjan An Operator-Based Approach to Topological Physics: Band Structures and Bloch Eigenstates not Required Abstract: Over the past two decades, the study of topological properties in physical systems has generated significant excitement, as such systems can realize robust boundary-localized states that have a wide range of applications. However, the theoretical frameworks that have been previously used to understand these phenomena are inextricably tied to band theory, usually requiring a system’s Bloch eigenstates or a projection onto the occupied subspace. Thus, the many successes of topological band theory also serve to highlight the current fundamental challenges facing the field, such as the difficulties in studying topology in aperiodic systems, non-linear and interacting systems, metallic systems, and quantitatively accounting for finite size effects. In this talk, I will present an operator-based framework for topological physics that makes use of a system’s real-space description without the need to calculate its band structure or Bloch eigenstates. Instead, this framework is based on the system’s spectral localizer, and provides a set of local markers, protected by local gaps, for every symmetry class in every physical dimension. I will discuss how this operator-based framework can be used to identify topology in non-interacting metals and gapless heterostructures, and show recent experimental observations of bulk-boundary correspondence in a topological acoustic metal metamaterial. Moreover, I will discuss how this framework can be directly applied to nonlinear systems and realistic photonic systems (i.e., Maxwell’s equations). This work is part-funded by Sandia National Laboratories (SNL). SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525. References: [1] W. Cheng*, A. Cerjan*, S.-Y. Chen, E. Prodan, T. A. Loring, and C. Prodan, Revealing topology in metals using experimental protocols inspired by K-theory, Nature Communications 14, 3071 (2023). [2] A. Cerjan and T. A. Loring, An operator-based approach to topological photonics, Nanophotonics 11, 4765 (2022). [3] S. Wong, T. A. Loring, and A. Cerjan, Probing topology in nonlinear topological materials using numerical K-theory, arXiv:2307.08374.

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Luca Giorgetti (Rom) Fusion category and hypergroup actions on conformal nets Abstract: Groups of automorphisms of a conformal net (the operator algebraic description of chiral CFT) naturally give rise to subnets (subtheories), the so-called group orbifolds. Moreover, automorphisms can be regarded as the “ordinary” global symmetries in the conformal net setting. We describe a generalization of these by means of unital completely positive (UCP) maps of the net (a sort of “generalized” non-invertible global symmetries of the net) obtained in a series of works in collaboration with M. Bischoff and S. Del Vecchio. In the talk I will restrict to completely rational conformal nets, where hypergroups of UCP maps (instead of groups of automorphisms) are capable of describing arbitrary subnets of a given net. I will recall some background of conformal nets, their representations, and report on some new explicit formulas for such UCP actions, relating hypergroups of UCP maps with certain unitary fusion category actions on conformal nets.

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Giuseppe Di Giulio (Uni Würzburg) Symmetry-resolved modular correlation functions in free fermionic theories Abstract: Recently there has been a huge research activity on the interplay between symmetries and entanglement, exploiting the block-diagonal structure of the reduced density matrix (RDM) in each charge sector. The goal of this talk is to study how the presence of a global U(1) charge affects the modular flow, a central object in the algebraic description of quantum field theory. Roughly speaking, the modular flow is given by a generalized time evolution induced by a RDM of a given spatial region. I will discuss the symmetry resolution of the modular flow and the modular correlation function of U(1)-invariant operators. I will provide a consistent definition of symmetry-resolved modular flow defined for a local algebra of operators associated with a sector with a fixed charge. I will also discuss the symmetry-resolved modular correlation functions, showing that they satisfy the KMS condition in each symmetry sector. In order to complement this analysis with an example, I will provide a toolkit for computing the symmetry-resolved modular correlation function of the charge density operator in free fermionic theories. I will show that, in a 1 + 1-dimensional free massless Dirac field theory, this quantity is independent of the charge sector at leading order in the ultraviolet cutoff expansion. This feature can be regarded as an equipartition of the modular correlation function.

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Lars Koekenbier (FAU) Spectral localizer for line-gapped non-hermitian systems Abstract: Short-ranged and line-gapped non-hermitian Hamiltonians have strong topological invariants given by an index of an associated Fredholm operator. It is shown how these invariants can be accessed via the signature of a suitable spectral localizer. This numerical technique is implemented in an example with relevance to the design of topological photonic systems, such as topological lasers.

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Danilo José Polo Ojito Interface currents and corner states in magnetic quarter-plane systems Abstract:The purpose of this talk is to study the propagation of currents along the interface of two 2-d magnetic systems, where one of them occupies the first quadrant of the plane. By considering the tight-binding approximation model and K-theory, we prove that, for an integer number that is given by the difference of two bulk topological invariants of each individual system, such interface currents are quantized. We further state the necessary conditions to produce corner states for these kinds of underlying systems, and we show that they have topologically protected asymptotic invariants.

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Evgeny Korotyaev (Changchun, China) Inverse resonance scattering on rotationally symmetric manifolds Evgeny Korotyaev jointly with Hiroshi Isozaki Academy for Advance interdisciplinary Studies, Northeast Normal University Changchun, China Abstract: We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M = (0,\infty) x Y whose rotation radius is constant outside some compact interval. Here Y is a compact m-dimensional Riemannian manifold. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrodinger operators with compactly supported potentials on the half-line. We prove 1) Asymptotics of counting function of resonances at large radius. 2) The rotation radius is uniquely determined by its eigenvalues and resonances. 3) There exists an algorithm to recover the rotation radius from its eigenvalues and resonances. The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces.

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Valter Moretti (Trento) On the Relativistic Spatial Localization for massive real scalar Klein-Gordon quantum particles Abstract: I will present some recent achievements about the long-standing issue of relativistic spatial localization of a quantum (massive scalar) particle. I will focus in particular on problems related to local causality. A new notion of POVM localization due to D.Terno will be discussed by pointing out some new physically meaningful features of it, focussing onthe interplay with the Newton-Wigner localization, the Hegerfeldt theorem and the causal localization rquirement introduced by D.P.L. Castrigiano. Talk Based on arXiv:2304.02133 (Lett. Math. Phys. 2023 in print).

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Paolo Meda (Trento) The Semiclassical Einstein Equations and the Stability of Linearized Solutions Abstract: The semiclassical formulation of gravity is discussed in the framework of algebraic quantum field theory in curved spacetimes. The main topic of the talk is the Semiclassical Einstein Equations, which describe the backreaction of a quantum field on spacetime geometry. In the first part of the talk, it is shown that an initial-value problem for local solutions of the Semiclassical Einstein Equations can be formulated in cosmological spacetimes, after fixing four initial data on the scale factor of the Universe. In the second part of the talk, it is studied the problem of stability of linearized solutions, using a toy model which mimics the semiclassical equations in cosmological spacetimes. In this case, it is proved that, if the quantum field driving the backreaction is massive, then there are choices of renormalization constants for which linear perturbations with compact spatial support decay for large times, thus indicating stability of the underlying theory [arXiv:2007.14665, 2201.10288].

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Ian Koot (FAU) Relative Entropy and the (Quantum) Method of Types Abstract: The Method of Types is a set of results in probability theory and information theory that connect (relative) entropy to the probabilities of outcomes in repeated measurement. There does not seem to exist a noncommutative generalization of the Method of Types, even though some results and concepts have noncommutative versions. We give an overview of the Method of Types, emphasizing the role that (relative) entropy plays in it, and discuss possibilities and obstacles in trying to generalize the method to the noncommutative setting. The aim is to get a better understanding of (relative) entropy, in the commutative setting but especially in the noncommutative setting.

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Jan Mandrysch (Leipzig) Energy inequalities in integrable quantum field theories Abstract: Many results in general relativity rely crucially on classical energy conditions inflicted on the stress-energy tensor. Quantum matter, however, violates these conditions since the energy density can fluctuate and in particular become arbitrarily negative at a point. Nonetheless quantum matter should have some reminiscent notion of stability, which can be captured by so-called quantum (weak) energy inequalities (QEIs). These are lower bounds of the smeared quantum-stress-energy tensor and could be proven in many types of free quantum field theories on both flat and curved spacetimes. In models with self-interaction though only few results exist. We are here presenting results in a wide class of integrable QFT models in 1+1d including for instance the O(N)-nonlinear-sigma model. A preprint is available under https://arxiv.org/abs/2302.00063

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Daniel Burgarth (FAU) Bounds for the Jaynes-Cummings Hamiltonian Abstract: The Jaynes-Cummings Hamiltonian is one of the most commonly used model for the interaction of light and matter. It arrises through a more fundamental model by an approximation known as the „rotating wave approximation“ (RWA). Although commonly used in physics, the approximation error is hard to control due to the unboundedness of the operators involved. Even in finite dimensional systems such bounds are often only perturbative. I will demonstrate a simple yet very powerful way to find bounds for the Jaynes-Cummings Hamiltonian and related models. PS: This will be a blackboard talk – feel free to ask me anything and I’ll be happy to go off on tangents instead. The real purpose of this talk is to find common ground for potential collaborations between mathematics and physics at Erlangen

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Joris De Moor (FAU) Footprint of a topological phase transition on the density of states Abstract: For a one-dimensional random discrete Schrödinger operator, the energies at which all transfer matrices commute and have their spectrum off the unit circle are called critical hyperbolic. Disorder driven topological phase transitions in such models are characterized by a vanishing Lyapunov exponent at the critical energy. It is shown that the density of states away from a transition has a pseudogap with an explicitly computable Hölder exponent, while it has a logarithmic divergence (Dyson spike) at the transition points. The proof is based on renewal theory for the Prüfer phase dynamics and the optional stopping theorem for martingales of suitably constructed comparison processes.

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Prof. Kang Li (FAU) An introduction to AF algebras Abstract: In this talk, I will discuss the basic properties of AF algebras and introduce AF-embeddability problem.

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Christian Jäkel (Sao Paulo) Modular Hamiltonians on the de Sitter Space Abstract: While the main aim of this work is to contribute to the ongoing investigation of modular Hamiltonians, this work is also a case study. It aims to illustrate how to decipher the laws of nature and cast them in a mathematical language. Once the tools of Tomita-Takesaki modular theory are added to Wigner’s group theoretic approach, the far-reaching and unexpected consequences of the selection of a highly symmetric space-time, like de Sitter space, gain visibility.

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Christoph Setescak (Regensburg) Disordered Topological Insulators: From Experiment to K-theory Abstract: Topological Insulators are semiconductors that exhibit a topological non-triviality in their mathematical description, leading to the emergence of a gap-closing surface state at their boundary with certain highly sought-after properties. Roe C*-algebras provide a mathematical framework to describe topological insulators in the presence of disorder, whereby the topological phase corresponds to a non-trivial class in K-theory. It will be illustrated how atomically resolved experimental data can be reproduced within this framework. Moreover, these invariants can be calculated numerically via the index of a Fredholm operator. For the two-dimensional Kane-Mele model we were able to reproduce the phase diagram previously reported in the literature, but in the case of the three-dimensional topological insulator Bi2Se3, no result recreating previous work was obtained.

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Yoh Tanimoto (Rom) Unitary vertex algebras and Wightman conformal field theories Abstract: We report recent progress on the axiomatic approaches to two-dimensional conformal field theory. We prove the equivalence between unitary vertex operator algebras and Moebius-covariant Wightman fields with some analytic conditions. We will discuss possible generalizations to full 2d CFT.

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Onirban Islam (Potsdam) Feynman propagators on curved spacetimes Abstract: It is a classic result that any wave operator on a globally hyperbolic spacetime admits unique advanced and retarded propagators. With the advent of quantum field theory, a new type of propagator emerges—the Feynman propagator. These propagators are an essential ingredient of quantum field theory and are intimately connected with quantum states. Moreover, they arise naturally in global and spectral analyses on Lorentzian manifolds. In contrast to the advanced and retarded propagators, Feynman propagators are not unique unless the spacetime admits time-translation symmetry. In this talk, I shall present a construction of Feynman propagators satisfying a positivity property based on the idea of microlocalisation which, in a sense, is a tool to connect a first-order pseudodifferential operator to the partial derivative. This, in turn, provides a new construction of Hadamard states. I shall first introduce the required notions from microlocal analysis and then explain microlocalisation in a geometric fashion. (Joint work with Alexander Strohmaier based on arXiv:2012.09767 [math.AP])

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Claus Köstler (Cork) Distributional symmetries and invariance principles in noncommutative probability Abstract: Distributional symmetries and invariance principles provide deep structural results in classical probability. For example, the de Finetti theorem characterizes an infinite sequence of random variables to be conditional independent and identically distributed if and only if its joint distribution is invariant under permuting these random variables. Recently significant progress was made in transferring such de Finetti type results to an operator algebraic setting of noncommutative probability. My talk will introduce to and overview some of these newer developments. Selected References: [1] Köstler, Claus. A noncommutative extended de Finetti theorem. J. Funct. Anal. 258 (2010), no. 4, 1073–1120. [2] Gohm, Rolf; Köstler, Claus. Noncommutative independence from the braid group B∞. Comm. Math. Phys. 289 (2009), no. 2, 435–482. [3] Köstler, Claus; Speicher, Roland. A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation. Comm. Math. Phys. 291 (2009), no. 2, 473–490. [4] Dykema, Kenneth J.; Köstler, Claus; Williams, John D. Quantum symmetric states on free product C∗-algebras. Trans. Amer. Math. Soc. 369 (2017), no. 1, 645–679. [5] Evans, D. Gwion; Gohm, Rolf; Köstler, Claus. Semi-cosimplicial objects and spreadability. Rocky Mountain J. Math. 47 (2017), no. 6, 1839–1873. [6] Köstler, Claus; Krishnan, Arundhathi; Wills, Stephen J. Markovianity and the Thompson Monoid F+. ePrint arXiv:2009.14811, to appear in J. Funct. Anal. [7] Köstler, Claus; Krishnan, Arundhathi. Markovianity and the Thompson Group F. SIGMA Symmetry Integrability Geom. Methods Appl. 18 (2022), Paper No. 083.

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Patricia Ribes Metidieri (Nijmegen) Entanglement: ubiquitous, but measurable? Abstract: It is well known that entanglement is ubiquitous in quantum field theory: even the simplest states within the simplest field theories are highly entangled. The foundation of this statement rests on two results: (1) the Reeh-Schlieder theorem, which shows that all field variables in any one region of spacetime are entangled with variables in other regions, and (2) the calculations of entanglement entropy between a region and its complement, which show that entanglement between adjoining spacetime regions is not just large but UV divergent. In this talk, I will argue that these results do not provide much information about the entanglement between individual local degrees of freedom. I will then present a way of quantifying such entanglement, involving only a finite number of degrees of freedom, finite regions of space, and quantities that are directly measurable. I will summarize our understanding both in (1+D)-dimensional Minkowski spacetime and de Sitter spacetime, paying special attention to the consequences of the latter in our ability to detect quantum effects from the inflationary era.

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Claudio Dappiaggi Stochastic Partial Differential Equations and Renormalization à la Epstein-Glaser Abstract: We present a novel framework for the study of a large class of non-linear stochastic partial differential equations, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use specific techniques proper of microlocal analysis. These allow us to deal with renormalization using an Epstein-Glaser perspective, hence without resorting to any specific regularization scheme. As a concrete example we shall use this method to discuss both the stochastic $\Phi^3_d$ model and the non-linear Schroedinger equation. Talk based on [1] C.D., Nicolò Drago, Paolo Rinaldi & Lorenzo Zambotti, Commun. Contemp. Math. 24 (2022), no. 7, Paper No. 2150075, [2] Alberto Bonicelli, C.D. & Paolo Rinaldi, https://arxiv.org/pdf/2111.06320

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Ivan Romualdo de Oliveira Localizability of Quantum Mechanical Systems on Curved Spacetimes Abstract:

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Pieter Naaijkens Classification of topologically ordered phases of matter in 2D Abstract: In this talk I will consider topological phases of matter which have what is called long range entanglement (LRE). Ground states with LRE in a non-trivial phase cannot be converted into a product state using only (sufficiently) local operations. An interesting aspect is that such states can support quaisparticles with braided statistics, called anyons. I will compare different approaches to extracting the algebraic properties of these anyons from first principles, and explain how this relates to the classification of topologically ordered phases. I will then highlight some recent developments and current challenges in the field.

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Karl-Henning Rehren String-localized QFT in action: Application to the Abelian Higgs model Abstract: String-localized QFT improves perturbation theory with massive vector bosons. It keeps the perturbation theory renormalizable without introducing indefinite metric and ghosts. The ensuing string-dependence can be systematically eliminated in terms of higher-order corrections. The latter serve as physical predictions. A prototype is the Abelian Higgs model: Interactions of a massive vector field can be renormalized without spontaneous symmetry breaking. But it must couple to a scalar field with a double-well potential self-coupling. A benefit compared to the standard BRST method is that off-shell interacting charged fields can be constructed on the Hilbert space. Joint work with Jens Mund and Bert Schroer (arXiv:2209.06133)

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Ko Sanders (FAU) Quantum energy inequalities and their applications Abstract: Quantum energy inequalities can be used to express the stability of a quantum field theory in analogy to the classical energy conditions in GR. In particular, they are expected to be a suitable assumption to prove quantum analogs of e.g. the classical singularity theorems of Penrose and Hawking, where energy conditions played a key role. After a brief general review of these quantum energy inequalities, I will argue that they also have another important application: they can be used to control the high energy behaviour of quantum fields. This is in analogy to Hamiltonian bounds for quantum fields in Minkowski space, replacing the Hamiltonian operator by an averaged energy density. This potentially opens up the way to manipulate pointwise quantum fields and operator product expansions in a rigorous way, also for quantum fields in curved spacetimes, where in general no Hamiltonian operator is available.

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Richard Montgomery (Santa Cruz) Scattering and Compactifying in Newton’s three body problem Abstract: One approach to scattering theory is to add a manifold at inifinty as a place for things „to go to“ as they leave the manifold. These things can be classical trajectories or waves (a la the Melrose school). I will outline the approach, state a theorem and a few open problems.

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Prof. Alexander Bendikov (Breslau) Hierarchical Schrödinger-type operators: the case of potentials with local singularities Abstract: The goal of this work (joint with A. Grigoryan and S. Molchanov) is twofold. We prove that the operator H=L+V , the perturbation of the Taibleson-Vladimirov multiplier L=D^{α} by the potential V(x)=b‖x‖^{-α}, b≥b_{∗}, is essentially self-adjoint and non-negative definite (the critical value b_{∗} depends on α and will be specified in the paper). While the operator H is non-negative definite the potential V(x) may well take negative values, e.g. b_{∗}<0 for all 0<α<1. The equation Hu=v admiits the Green function g_{H}(x,y), the integral kernel of the operator H⁻¹. We obtain sharp lower- and upper bounds on the ratio of the functions g_{H}(x,y) and g_{L}(x,y). Examples illustrate our exposition.

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Nora Doll (FAU) Abstract:

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Dr. Simon Wood Lie theory beyond category O in conformal field theory and vertex operator algebras Abstract: Affine Lie algebras provide the means for constructing some of the best understood conformal field theories or vertex operator algebras. If the level of the affine Lie algebra is non-negative integral, then the integrable modules form a modular tensor category (among many other properties, this implies an action of the modular group on characters). In this talk I will give an overview of why this is highly desirable from the perspective of conformal field theory and some new results on modular properties and their consequences at certain non-integral levels, called admissible levels. No prior knowledge of vertex operator algebras or conformal field theory will be assumed. I will do my best to motivate everything through Lie theory.

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Roberto Conti (Rom) Automorphisms of the Cuntz algebras and their Weyl groups. A case study. Abstract: Multiplets of isometries with orthogonal ranges summing up to 1 were systematically used by Doplicher and Roberts in the early 70’s in their study of the superselection structure of Quantum Field Theory. Nowadays, the C*-algebra generated by any such multiplet is known as Cuntz algebra O_n, where n is the cardinality of the given multiplet. Since their introduction, Cuntz algebras have been the subject of endless investigations, from many different points of view. They are probably the most studied class of C*-algebras ever. Notably, the study of their automorphisms reveals many challenging facets, where operator algebras meet Lie theory, dynamical systems and combinatorics. We will present an overview of recent results, along with some open questions.

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Workshop LQP46 Weitere Infos auf: https://en.www.math.fau.de/mathematical-physics/gandalf-lechner/events10056/workshop-lqp46/

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Joris De Moor Partially hyperbolic random dynamics on Grassmannians Abstract: A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure showing that on average the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.

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Dr. Alexander Stottmeister (Hannover) Anyon braiding and renormalization Abstract: A braiding operation defines a real-space renormalization group for anyonic chains. The resulting renormalization group flow can be used to define a quantum scaling limit by operator-algebraic renormalization. I will illustrate how this works for the Ising chain, also known as transverse-field Ising model. In this case, the quantum scaling limit results in the vacuum state of the well-known Ising CFT. Distinguishing between the braiding and its inverse is directly related to the chiral sectors of the Ising CFT.

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Prof. Rainer Verch (Leipzig) Temperature and entropy-area relation of quantum matter near spherically symmetric outer trapping horizons Abstract: We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. These spacetimes possess in general no timelike Killing vector field, but admit a Kodama vector field which provides a replacement. Spherically symmetric spacelike cross-sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time-parameter of the Kodama flow, where the inverse temperature is related to the surface gravity of the horizon cross-section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunneling probability in the scaling limit between in- and outgoing Fourier modes with respect to the the Kodama time shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling-limit-theory have relative entropies behaving proportional to the cross-sectional horizon area. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking. (Joint work with F. Kurpicz and N. Pinamonti, arXiv:2102.11547, Letters in Mathematical Physics 111 (2021) No 110)

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Manuel Quaschner (FAU) Noncollision singularities with external forces Abstract: The existence of noncollision singularities in the $n$-body problem was already conjectured by Painlevé in 1895. Even before the existence was proven in the 1990s, the question came up, whether the set of all initial conditions leading to noncollision singularities is a set of measure 0. A first result of this kind was proven for $n=4$ particles in $d\geq 2$ dimensions by Saari (1977). Using the so called Poincaré surface method, Fleischer (2018) could improve this for $n=4$ particles in $d \geq 2$ dimensions by extending the result to a wider class of potentials. But the problem is still open for more than four particles. After an overview of these works, we add some sufficiently small external forces to the well understood case of $n=4$ particles. In order to apply the Poincaré surface method, we need to prove good estimates on the shape of the orbits, which are close to the movement on a straight line. In the end we will show how one could use this results to prove the improbability of larger systems that can be suitably decomposed into diverging systems of up to four particles each.

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Prof. Felix Finster (Regensburg) An introduction to causal fermion systems and the causal action principle Abstract: The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. Moreover, causal fermion systems provide a general framework for modelling and analyzing non-smooth spacetime structures. The dynamics of a causal fermion system is described by a nonlinear variational principle, the causal action principle. The aim of the talk is to give a simple introduction. I will proceed chronologically and explain step by step how the underlying concepts and objects evolved from 1990 to today. This will lead us to the abstract definitions. The underlying physical principles will be discussed. At the end of the talk, I will briefly outline how to get the connection to a quantum state in the algebraic formulation.

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Ian Koot (FAU) The propagation number of operator systems Abstract: It is well known that every C*-algebra is isomorphic to a norm-closed, self-adjoint subalgebra of the bounded operators on some Hilbert space. We can generalize this to norm-closed, self-adjoint subspaces of bounded operators; these structures are called operator systems. The spectral triple-formulation of Noncommutative Geometry is based on C*-algebras, and in their recent efforts to generalize this to operator systems, Connes and Van Suijlekom defined a property of operator systems which they called the propagation number. We invastigate the behaviour of the propagation number under the minimal tensor product of operator systems, where an exact expression for the propagation number of the tensor product in terms of the propagation number of its factors can be found. We also discuss the possibility of an expression for the propagtion number of the dual operator system.

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Marcello Seri (Groningen) Contact mechanics and numerical integration Abstract: It is well known that an analogue to Hamiltonian systems can be defined on contact manifolds, the odd dimensional cousins to the even symplectic manifold. Their odd dimensionality introduces an extra freedom that allows the energy to vary and which is not necessarily restricted to a time-dependence. In the talk, we will present some geometric integrators for contact Hamiltonian systems, numerical schemes that guarantee the the preservation of the contact geometric structure. The derivation of the numerical integrators will be used as way to review some of the most relevant properties of contact Hamiltonian systems and to provide some intriguing examples that motivated the recent surge of interest in their analytical and numerical study.

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Johannes Kellendonk (Lyon I) Ellis semigroup in symbolic dynamics Abstract: Given a group G of homeomorphisms of a compact space X, the Ellis semigroup of G is the closure of G in the topology of point wise convergence. Its topological and algebraic properties characterise the G action on X. This semigroup has been introduced by Robert Ellis in the 60’s. We will provide a short overview of this theory and present some recent results which arise in the context of symbolic dynamics, that is, for G = Z and X a space of symbolic sequences.

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Dr. Konstantin Merz (Braunschweig) Eigenvalue estimates for Schrödinger operators using Fourier analysis Abstract: Estimating the location and accumulation rate of eigenvalues of Schrödinger operators is a classic problem in spectral theory and mathematical physics. For short-range potentials these problems can often be effectively treated using Fourier analytic methods like the Tomas-Stein restriction theorem. As an example we derive eigenvalue asymptotics for Schrödinger-type operators whose kinetic energy vanishes on a codimension one submanifold. Time permitting, we discuss another example: locating eigenvalues of ordinary Schrödinger operators with randomized, long-range, complex-valued potentials using a randomized version of the Tomas-Stein theorem by Bourgain. The talk is based on joint work with Jean-Claude Cuenin.

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Stefan Waldmann (Würzburg) Convergence of star products Abstract: In my talks I will report on recent progress in the understanding of the convergence problem in formal deformation quantization. While the existence and classification of formal star products is well-understood in the general case of Poisson manifolds, the questions whether such formal star products have reasonable convergence properties, as required by physical applications, is wide open. I will give several classes of examples where the situation can be analyzed explicitly.

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Ricardo Correa da Silva (FAU) Modular Structures on a von Neumann Algebra on Hilbert-Schmidt Operators and Applications to Thermodynamical Equilibrium States of Infinite Degenerated Systems After an introduction to Tomita-Takesaki Modular Theory and KMS states, we will characterize all cyclic and separating vectors for a well-known von Neumann algebra acting on the Hilbert-Schmidt operators aiming to study the thermal equilibrium states (KMS states) for this algebra. Then, we discuss the description of infinitely degenerate Hamiltonians in this algebra, in particular, the example of the Landau levels showing that there is no cyclic and separating vector such that the modular Hamiltonian corresponds to the Landau Hamiltonian, as argued in other work. Finally, we try to reproduce the thermodynamical limit of the system confined in a finite box, for physical reasons, to better understand how the degeneracy grows as the box radius goes to infinity.

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Maximilian Duell (LMU München) N-Particle Scattering in Wedge-local Quantum Field Theories Abstract: Wedge-local quantum field theories (wQFT) are a generalization of relativistic local QFTs, in which observables can be localized in very large wedge-shaped regions in space-time. It is long known that the wedge-local perspective sometimes has theoretical advantages. Only more recently, (w)QFT models describing non-trivial scattering reactions have been constructed directly in a wedge-local framework by Lechner, Buchholz, Summers, and others. Such constructions have provided non-trivial models also on four-dimensional Minkowski space-time, whereas the corresponding construction problem for interacting local QFT is still open. On the other hand, Scattering theory in a wedge-local setting was previously only developed up to the two-particle level and a generalization to higher particle numbers was not expected for geometric reasons. In my talk I will explain how to construct N-particle scattering states for massive particle in wedge-local QFTs. Based on this construction, N-particle S-matrices, collision cross sections, and other quantities of physical interest can be calculated directly in wQFTs. To conclude I will also discuss more recent work, where this scattering theory is applied to wQFTs obtained by a general deformation construction of Buchholz, Lechner and Summers. (Based on my PhD thesis and joint work with W. Dybalski)

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Tom Stoiber Callias-type operators associated to spectral triples Abstract: Callias-type (or Dirac-Schroedinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.

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Koen van den Dungen The index of generalised Dirac-Schrödinger operators Abstract: We study the relation between spectral flow and index theory within the framework of (unbounded) KK-theory. In particular, we consider a generalised notion of ‚Dirac-Schrödinger operators‘, consisting of a self-adjoint elliptic first-order differential operator D with a skew-adjoint ‚potential‘ given by a (suitable) family of unbounded operators on an auxiliary Hilbert module. We show that such Dirac-Schrödinger operators are Fredholm, and we prove a relative index theorem for these operators (which allows cutting and pasting of the underlying manifolds). Furthermore, we show that the index of a Dirac-Schrödinger operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of D. We prove this result without assuming that the potential is differentiable; instead, we assume that the ‚variation‘ of the potential is sufficiently small near infinity. In the special case of the real line, we recover the well-known equality of the index with the spectral flow of the potential.

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Mathematicians meet Fermi surfaces: a semiclassical approach to the dynamics of Bloch electrons Marcello Seri (Groningen) Abstract: The Fermi surface is an important concept in solid state physics and in the theory of transport of electron in metals. While physically this has Been thoroughly investigated in decades of experiments, the mathematics has lagged behind. Only one class of examples has been thoroughly studied as part of a program started by Novikov and developed by his topological school. In this talk I will start from the physical definition of Fermi surface and a review of recent results by Novikov and Maltsev to justify the need for a new approach. I will then suggest a rigorous definition of the physical Fermi surface in presence of perturbations in the semiclassical regime inspired by recent development in the mathematics of topological insulators. The talk is based on a joint work with Max Lein and Giuseppe De Nittis.

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Flat bands of edge states via weak invariants of semimetals Hermann Schulz-Baldes (joint work with Tom Stoiber), FAU

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Gap-filling via coarse geometry Matthias Ludewig, Regensburg Abstract: We explain a mathematical framework which uses methods from coarse geometry in order to model topological insulators. In this framework, the boundary behavior is described in terms of exact sequences in operator K-theory. In particular, spectral gap-filling is implied by the non-triviality of a certain K-theoretic boundary map. As an application, this allows to show that the Landau Hamiltonian on a hyperbolic half-plane (and also on more general imperfect half-spaces) has no spectral gaps.

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Stability of a topological index under weak spin symmetry breaking Joris de Moor, FAU

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THE NONCOMMUTATIVE GEOMETRY OF THE LANDAU HAMILTONIAN: METRIC ASPECTS Maximiliano Sandoval, Santiago, Chile Abstract: This work provides a first step towards the construction of a noncommutative geometry for the Quantum Hall Effect in the continuous. Taking inspiration from the ideas developed by Bellissard during the 80’s we build a spectral triple for the C*-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard’s theory (the so-called first Connes’ formula) is proved.

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Non-deterministic billiards Manuel Quaschner Abstract: In [Kn18], a simple non-deterministic model with three particles was introduced in order to approximate the orbits of solutions of a Hamiltonian equation with admissible potential and at most four bodies that do not have an asymptotic velocity. In these non-deterministic models, near-collisions are substituted by collisions without conservation of kinetic energy. In order to approximate the orbits some properties of non-deterministic models with three particles had to be derived such as the exponential growth (in the number of collisions) of the total kinetic energy and of the moment of inertia or a bound on the angular momenta of each particle in terms of the total angular momentum. All these properties fail to hold in general, if one considers non-deterministic models with more than three particles. Our aim is to investigate these models and derive similar results to the results in the case of three particles. Therefor we have to define another number of collisions in which the growth of the above quantities is exponential by considering a reduced ordered systems of particles moving on a line.

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The Cayley transform and K-theory Dr. Chris Bourne (RIKEN, Tohoku) Abstract: The Cayley transform gives a passage between unitary operators and self-adjoint (unbounded) operators on a Hilbert space. I will review this construction and its generalisation to Hilbert C*-modules, where the Cayley transform furnishes an isomorphism on (odd) K-theory. Next, we consider a graded analogue of these results and their application to bulk and boundary invariants of topological phases. This is joint work with Johannes Kellendonk and Adam Rennie.

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Vortrag faellt aus wg. Erkrankung! (Non-deterministic billiards) Manuel Quaschner Abstract: In [Kn18], a simple non-deterministic model with three particles was introduced in order to approximate the orbits of solutions of a Hamiltonian equation with admissible potential and at most four bodies that do not have an asymptotic velocity. In these non-deterministic models, near-collisions are substituted by collisions without conservation of kinetic energy. In order to approximate the orbits some properties of non-deterministic models with three particles had to be derived such as the exponential growth (in the number of collisions) of the total kinetic energy and of the moment of inertia or a bound on the angular momenta of each particle in terms of the total angular momentum. All these properties fail to hold in general, if one considers non-deterministic models with more than three particles. Our aim is to investigate these models and derive similar results to the results in the case of three particles. Therefor we have to define another number of collisions in which the growth of the above quantities is exponential by considering a reduced ordered systems of particles moving on a line.

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Amanda Young (TUM) On the Stability of Gapped Ground State Phases of Frustration-Free Quantum Spin Systems Abstract: Gapped quantum spin systems with topologically ordered ground states have been of interest due to their potential for developing fault-tolerant quantum codes. A key feature of such models is that the spectral gap remains open in the presence of small, local perturbations. In [1-3], a new approach utilizing Hasting’s quasi-adiabatic continuation proved spectral gap stability for frustration-free quantum spin models with periodic boundary conditions and ground states that satisfy a finite-volume topological order condition. Imperative to these results is that the quasi-adiabatic continuation (also known as the spectral flow) satisfies a Lieb-Robinson type locality estimate. In this talk, we first review quasi-locality estimates for quantum lattice systems with specific focus on the quasi-adiabatic continuation. We then discuss how to generalize the stability result from [3] in several directions: including to quantum spin systems with more general boundary conditions, quantum spin systems with discrete symmetry breaking, and (time permitting) lattice fermion models. This talk is based on joint work with Bruno Nachtergaele and Robert Sims. [1] S. Bravyi, M. Hastings and S. Michalakis, Topological quantum order: stability under local perturbations, J. Math. Phys. 51 093512 (2010) [2] S. Bravyi and M. Hastings, A short proof of stability of topological order under local perturbations, Commun. Math. Phys. 307, pp. 609-627 (2011) [3] S. Michalakis and J. Zwolak, Stability of Frustration-Free Hamiltonians, Commun. Math. Phys. 322, pp. 277-302 (2013)

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Lei Zhao (Augsburg) J^+-Invariants for two-center Stark-Zeeman systems Abstract: For immersed curves in the plane, Arnold has defined three invariants in which the J^+-invariant has been adapted by Cieliebak-Frauenfelder-van Koert in 2017 to two invariants for periodic orbits of (one-center) Stark-Zeeman systems which includes the two-center problem and the restricted three-body problem as examples. These invariants are invariant under homotopy along generic family of periodic orbits of Stark-Zeeman systems. In this talk I will recall their construction, as well as explain an extension, jointly with K. Cieliebak and U. Frauenfelder, for Stark-Zeeman systems with two Newtonian/Coulombian singularities, in which case we obtain four J^+-like invariants, based on Levi-Civita and Birkhoff regularizations.

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Florian Dorsch Pseudo-gaps for random hopping models arXiv-Artikel

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Dominik Schroeder (ETH) Edge Universality for non-Hermitian Random Matrices arXiv-Artikel

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Andreas Knauf Morse-Theorie für Niveaumengen arXiv-Artikel mit Nikolay Martynchuk Abstract: Classical Morse theory proceeds by considering sublevel sets f -1((-∞, a]) of a Morse function f: M → ℝ, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f -1(a) and give conditions under which their topology changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M. When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the configuration space. (Counter-)examples and applications to celestial mechanics are also discussed.

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The KO-valued spectral flow for skew-adjoint Fredholm operators Alan Carey (Canberra) Abstract: Talk about http://de.arxiv.org/abs/1907.04981

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Douglas Lundholm Exchange and exclusion for non-abelian anyons Anyons are effective particles arising in certain 2D quantum systems and having properties intermediate to bosons and fermions. Roughly speaking, they may be defined by representing an exchange of coordinates of the many-body wave function by an arbitrary complex phase (abelian anyons) or, more generally, a unitary matrix (non-abelian anyons). I will discuss how to extend many-body methods of exchange and exclusion to the case of non-abelian anyons, focusing on two of the most popular such models: the Fibonacci and the Ising anyon models. This is joint work with Viktor Qvarfordt.

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Scattering Theory on Resonances in Quantum Field Theory Miguel Ballesteros (UNAM)

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Das inverse Henderson-Problem Sabine Jansen (LMU Muenchen) The inverse Henderson problem of statistical mechanics concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles. Roughly stated, it asks for the interaction potential given the equilibrium pair correlation function of the system. In 1974 Henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system. Here we provide a rigorous proof of a slightly more general version of the latter statement using Georgii’s version of the Gibbs variational principle.

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Transfer matrices for Hermitian operators of higher dimensions Christian Sadel (Santiago de Chile) Abstract: We show how a formula by Carmona for the spectral measure of a half line Jacobi operator generalizes to general Hermitian operators of locally finite hopping type. We use products of sets of rectangular, radial transfer matrices for the spectral analysis. As corollaries we receive criteria for proving a.c. spectrum. These criteria are applied to certain models with disorder.

Abstract:

Vojkan Jaksic, McGill University, Montreal On a mathematical theory of repeated quantum measurements The statistics of the (finite alphabet) outcomes of repeated quantum measurements is studied by methods of thermodynamic formalism. Viewed as one-dimensional spin sys- tems with long range interactions, repeated quantum measurements exhibit very rich (and sometimes very singular) thermodynamic behaviour. We will describe general thermodynamical formalism of these systems and illustrate its unexpected features on a number of examples.

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Projective dynamics, first integrals and symmetries Referent: Alain Albouy, Paris

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An introduction to PT symmetric problems on a wedge shaped contour Referent: Prof. Carsten Trunk, TU Ilmenau

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