LQP 40 Foundations and Constructive Aspects of Quantum Field Theory

Forschungsprofilbereich "Mathematische und computergestützte Wissenschaften − Mathematical and Computational Sciences"

Zentrum für Mathematische Physik - Center for Mathematical Physics

Leipzig, 23-24 June, 2017

Location & Accomodation

The workshop will take place in the Max-Planck institute for Mathematics in the Sciences, Inselstr. 22.

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Registration

To register, please send an email to thomas-paul.hack@itp.uni-leipzig.de indicating the following:

Name:
Affiliation:

Applications for giving and talk and / or receiving financial support are not longer possible.

Program

 FRIDAY Time Speaker Talk 14:00-14:15 Welcome address 14:15-14:45 Zhirayr Avetisyan On the elementary particle states in homogeneous cosmology (slides) 14:50-15:20 Mathias Hänsel Qualitative analysis of solutions of the semiclassical Einstein Equations in FLRW spacetimes (slides) 15:20-15:50 Coffee break 15:50-16:20 Hugo Ferreira On the algebraic quantization of a massive scalar field in anti-de Sitter spacetime (slides) 16:25-17:00 Maximilian Duell Scattering in QFT without Mass Gaps and Strengthened Reeh-Schlieder Condition (slides) 17:10-17:25 Break 17:25-17:55 Paweł Duch Existence of weak adiabatic limit in almost all models of perturbative QFT (slides) 18:00-18:45 Alessandro Pizzo Bose Particles in a Box: Convergent Expansion of the Ground State in the Mean Field Limiting Regime (slides) 20:00 Social dinner at Auerbachs Keller SATURDAY Time Speaker Talk 09:15-09:45 Carlos I. Pérez Sánchez Correlation functions for colored tensor models and their Schwinger Dyson Equations (slides) 09:50-10:20 Alexander Efremov Renormalization of SU(2) Yang-Mills with Flow Equations (slides) 10:20-10:50 Coffee break 10:50-11:20 Onirban Islam Entanglement entropy of the Dirac field in static spacetime (slides) 11:25-12:10 Gandalf Lechner Yang-Baxter Representations of the Infinite Symmetric Group (slides) 12:10-14:00 Lunch break 14:00-14:30 Simone Del Vecchio Infinite index extensions of local nets and defects (slides) 14:35-15:20 Karl-Henning Rehren String-localized fields of higher spin: massless limit and stress-energy tensor (slides)

Abstracts

• Zhirayr Avetisyan: "On the elementary particle states in homogeneous cosmology"
In ordinary quantum mechanics there are two related notions of an idealized elementary particle state of the system. The first one is the (often improper) energy eigenstates associated to the free particle Hamiltonian operator (Laplacian), and the second (due to Wigner) is the unitary irreducible representations of the Poincare group on the system's Hilbert space. These two definitions are equivalent thanks to the well known fact from classical harmonic analysis that the improper eigenfunctions of the Laplacian are generalized matrix elements of the Wigner representations. In curved spacetimes and with more general Hamiltonians the picture becomes much more intricate. In homogeneous cosmology one has an isometry group acting transitively on spatial hypersurfaces, which gives rise to the Wigner-type notion of an instantaneous symmetry elementary particle state. On the other hand, if the system admits a dynamical formulation than Hamiltonians at different times commute and possess a common system of energy eigenstates, i.e., dynamical elementary particle states. Now the relation between the two notions runs into an open problem in non-commutative harmonic analysis. The situation is even more perplexed for quantum fields in homogeneous cosmological spacetimes. In this talk we will discuss the mathematical problems arising on this way, some conjectures and recent advancements.

• Mathias Hänsel: "Qualitative analysis of solutions of the semiclassical Einstein Equations in FLRW spacetimes"
Dynamical system methods are used to analyse the qualitative behavior of solutions to the semiclassical Einstein equations in Friedmann-Lemai tre-Robertson-Walker (FLRW) spacetimes. We specialise to the case of the conformally coupled, massless quantum scalar field and the electromagnetic field. If the quantum fieelds are in the conformal vacuum state we obtain a two-dimensional dynamical system. For this system we can idendify the structurally stable cases, obtain bifurcation diagrams and investigate the (Lyapunov-) stability of the Minkowski and deSitter equilibrium points. Finally we are able to compare the qualitative behavior of solutions to other matter models and in particular to the CDM model of cosmology.

• Hugo Ferreira: "On the algebraic quantization of a massive scalar field in anti-de Sitter spacetime"
We discuss the algebraic quantization of a real, massive scalar ﬁeld in the Poincaré patch of the (d+1)-dimensional anti-de Sitter spacetime, with arbitrary boundary conditions. By using the functional formalism, we show that it is always possible to associate to such system an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. In addition, we characterize the wavefront set of the ground state associated to this system. As a consequence, we are able to generalize the deﬁnition of Hadamard states and construct a global algebra of Wick polynomials.

• Maximilian Duell: "Scattering in QFT without Mass Gaps and Strengthened Reeh-Schlieder Condition"
In the non-perturbative setting of algebraic QFT we give a mathematically rigorous construction of the scattering matrix for massive Wigner particles in presence of massless excitations. In contrast to previous approaches we do not impose any Herbst-type technical assumptions on the spectrum of the mass operator near the particle masses. Instead we base our approach on features of the relativistic vacuum state which are similar to the well-established Reeh-Schlieder property. The method should apply, in particular, to scattering of stable particles in abelian gauge theories. A concrete example are hydrogen atoms from the point of view of Quantum Electrodynamics (based on CMP 352, 935-966, 2017).

• Paweł Duch: "Existence of weak adiabatic limit in almost all models of perturbative QFT"
The Wightman and Green functions are one of the basic objects of the relativistic quantum field theory in the Minkowski spacetime. The perturbative definition of the Green functions in a large class of models was given using the momentum-space approach. In the causal perturbation theory developed by Epstein and Glaser both the Wightman and Green functions can be defined by taking the so-called weak adiabatic limit. However, its existence has been shown so far only in purely massive theories, the quantum electrodynamics and the massless $\phi^4$ theory. The aim of the talk is to present the proof of the existence of the weak adiabatic limit in most of the interacting models including an arbitrary model with the interaction vertices of the canonical dimension equal 4. The developed method can be also used to show the existence of the central splitting solution in the quantum electrodynamics.

• Alessandro Pizzo: "Bose Particles in a Box: Convergent Expansion of the Ground State in the Mean Field Limiting Regime"
I shall report on a novel multi-scale technique to study many-body quantum systems. The method is based on the Feshbach-Schur map and the scales are represented by occupation numbers of particle states. Consider an interacting Bose gas at zero temperature, constrained to a finite box and in the mean field limiting regime, where the $N$ gas particles interact through a pair potential of positive type and with an ultraviolet cut-off. The (nonzero) Fourier components of the potential are assumed to be sufficiently large with respect to the corresponding kinetic energies of the modes. For this system, we provide a convergent expansion of the ground state of the Hamiltonian in terms of the bare operators. In the limit $N\to\infty$ the expansion is up to any desired accuracy.

• Carlos I. Pérez Sánchez: "Correlation functions for colored tensor models and their Schwinger Dyson Equations"
Tensor models are a random geometry framework that generalizes, to arbitrary dimension, matrix models. They are used to model quantum gravity but, lately, some applications to AdS/CFT were discovered. Colored tensor models generate random orientable geometries. We scrutinize their correlation functions and briefly give their geometric interpretation in terms of bordisms. Based on a Ward-Takahashi identity, we give the Schwinger-Dyson Equations they obey. We work non-perturbatively.

• Alexander Efremov: "Renormalization of SU(2) Yang-Mills with Flow Equations"
We give a proof of perturbative renormalizability of SU(2) Yang–Mills theory in four-dimensional Euclidean space. The main motivation is to present a proof which does not make appear mathematically undefined objects (as for example dimensionally regularized generating functionals). A large part of the proof is dedicated to bounds on massless correlation functions and to the restoration of the Slavnov-Taylor identities.

• Onirban Islam: "Entanglement Entropy of Dirac Field"
We compute an upper bound for the relative entanglement entropy of the ground state of the massive Dirac field on a static spacetime. This entanglement measure is bounded by an exponential decay for apart regions in a spacelike compact Cauchy hypersurface.

• Gandalf Lechner: "Quantum backflow and scattering"
The Yang-Baxter equation (YBE) lies at the heart of many subjects, including quantum statistical mechanics, QFT, knot theory, braid groups, subfactors, quantum groups, quantum information ... . In this talk, I will consider involutive solutions of the YBE ("R-matrices"). Any such R-matrix defines a representation and an extremal character of the infinite symmetric group as well as a corresponding tower of subfactors. Using these structures, I will describe how to find all R-matrices up to a natural notion of equivalence inherited from applications in QFT (given by the character and the dimension), how to completely parameterize the set of solutions, and how to decide efficiently whether two given R-matrices are equivalent.

• Simone Del Vecchio: "Infinite index extensions of local nets and defects"
Extensions of a (properly infinite) von Neumann algebra can be described by Q-systems, introduced by Longo. If the extension is ""small"" (it has finite Jones index), the theory of Q-systems with its application to Quantum Field Theory is worked out by Longo and Rehren. We present a way to suitably extend some results to the case of infinite Jones index. In particular we show applications to extensions and defects in Quantum Field Theory. Joint work with Luca Giorgetti.

• Karl-Henning Rehren: "String-localized fields of higher spin: massless limit and stress-energy tensor."
Lifting the massless limit of Wigner representations of higher spin to the associated local quantum fields, encounters several well-known obstructions due to conflicts between Hilbert space positivity, covariance and causality. In a unified setting using "string-localization", these conflicts can be resolved, and details of the decoupling of the degrees of freedom can be studied for any spin.

Participants

No. Name Affiliation
1 Avetisyan, Zhirayr UCL Department of Mathematics, London
2 Belhassen, Mohamed Universität Leipzig
3 Borasi, Luigi Universität Bonn
4 Braga Vasconcellos, João University of Genova
5 Brunetti, Romeo University of Trento
6 Buchholz, Detlev Universität Göttingen
8 Chao, En-Hung Universität Leipzig
9 Ciolli, Fabio Università di Roma Tor Vergata
10 Dappiaggi, Claudio Dipartimento di Fisica -- Università di Pavia
11 de Jong, Jins WWU Münster
12 Del Vecchio, Simone Dipartimento di Matematica, Universita' di Tor Vergata, Roma
13 Derezinski, Jan University of Warsaw
14 Diehl, Joscha MPI MIS Leipzig
15 Drago, Nicolò Dipartimento di Fisica -- Università di Pavia
16 Drawert, Sebastian Universität Leipzig
17 Duch, Paweł Jagiellonian University Kraków
18 Duell, Maximilian TU München
19 Dwivedi, Saurav Department of Mathematics, Masaryk University, Brno
20 Dybalski, Wojciech TU München/LMU
21 Efremov, Alexander CPHT, Ecole Polytechnique
22 Faldino, Federico University of Genova
23 Ferreira, Hugo INFN Pavia / Pavia University
24 Fidaleo, Francesco Dipartimento di Matematica, Universita' di Tor Vergata, Roma
26 Fredenhagen, Klaus Universität Hamburg
27 Giri, Praveen Universität Leipzig
28 Gonzo, Riccardo Università di Padova / Universität Göttingen
29 Gottschalk, Hanno University of Wuppertal
30 Gransee, Michael Universität Leipzig
31 Hack, Thomas-Paul Universität Leipzig
32 Hollands, Stefan Universität Leipzig
33 Hänsel, Mathias Universität Leipzig
34 Islam, Onirban Universität Leipzig
35 Kaminski, Wojciech University of Warsaw
36 Kozoň, Marek Universität Leipzig
37 Kurpicz, Felix Universität Leipzig
38 Lechner, Gandalf Cardiff University
39 Morinelli, Vincenzo Università di Roma Tor Vergata
40 Murro, Simone Universität Regensburg
41 Pizzo, Alessandro Dipartimento di Matematica, Universita' di Tor Vergata, Roma
42 Pottel, Steffen Universität Leipzig
43 Pérez Sánchez, Carlos I. University of Münster
44 Rehren, Karl-Henning Universität Göttingen
45 Rudolph, Gerd Universität Leipzig
46 Sahlmann, Hanno Department Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg
47 Sanders, Ko Dublin City University
48 Schmidt, Matthias Universität Leipzig
49 Schroer, Bert Centro Brasileiro de Pesquisas Fısicas, Rio de Janeiro
50 Sibold, Klaus Universität Leipzig
51 Siemssen, Daniel University of Warsaw
52 Stiller, Michael II Institute for Theoretical Physics, Uni Hamburg
53 Taslimitehrani, Mojtaba MPI MIS Leipzig
54 Thurmair, Moritz Universität Leipzig
55 Tolksdorf, Jürgen MPI MIS Leipzig
56 Verch, Rainer Universität Leipzig
57 Wrochna, Michał Université Grenoble Alpes
58 Zahn, Jochen Universität Leipzig

Last updated: June 29