# 31st LQP Workshop "Foundations and Constructive Aspects of QFT"

## Announcement

 The workshop has the working title » Local Quantum Physics, Gravity, Mathematical Structures « and is sponsored by the Research Academy Leipzig and PbF2 (institutions of the University of Leipzig) and the Ev. Studienwerk Villigst. As is now tradition for the LQP workshop series, the workshop will focus on the areas of quantum field theory, quantum statistical mechanics, quantum information, non-commutative geometry, quantum gravity and cosmology, and related topics of mathematical physics and mathematics. Also in keeping with the LQP workshop tradition, young researchers are particularly invited to report on their theses' results. Thanks to our sponsors, we can support a limited number of participants towards travel and accomodation. The workshop will start at about 2 pm on Friday, 23 Nov., and end between 3 pm and 4 pm on Saturday, 24 Nov. For details, see below.

## Location & Accomodation

 As this seems to be the largest workshop so far in the LQP series, we had to relocate and will now have the workshop at the Max Planck Institute for Mathematics in the Sciences, Inselstr. 22. From the main station, you can easily walk to the MPI (about 1 km). For detailed locations of the MPI and two of the hotels, please refer to this map. At the institute, there will be signs guiding you to the lecture room G10, where the workshop will take place.On Friday evening, we will have dinner at the restaurant Thüringer Hof.

## Registration

Registration is now closed.

## Program

 FRIDAY Time Speaker Talk Slides 14:00-14:10 Welcome address 14:10-14:50 Raimar Wulkenhaar Exact solution of a non-local four-dimensional quantum field theory [slides] 14:55-15:35 Daniela Cadamuro Locality in integrable QFTs - characterization and explicit examples [slides] 15:40-16:00 coffee break 16:00-16:40 Jan Schlemmer Thermal states of deformed Quantum Field Theories [slides] 16:45-17:25 Falk Lindner On the equivalence of the approaches to thermo field theory in the context of perturbative AQFT [slides] 17:30-17:50 break 17:50-18:30 Gennaro Tedesco Multilocal Fermionization [slides] 18:35-19:15 Michal Eckstein Spectral Action on Quantum Spheres [slides] SATURDAY Time Speaker Talk Slides 9:15-9:45 Daniel Paulino The DFR-Algebra for Poisson Vector Bundles [slides] 9:50-10:20 Albert Much Effective Quantum Plane from Quantum Physics [slides] 10:25-10:55 Benjamin Eltzner Quantization of Perturbations in Inflation 11:00-11:25 coffee break 11:25-11:55 Marco Benini Quantum field theories on affine bundles [slides] 12:00-12:40 Thomas-Paul Hack A toy model for quantum field theory on curved supermanifolds [slides] 12:40-14:10 lunch break 14:10-14:50 Manuel Hohmann Observer space geometry of Finsler spacetimes [slides] 14:55-15:25 Reza Safari Coherent states on curved spacetime and the semi-classical Einstein equation [slides]

## Abstracts

• Raimar Wulkenhaar: "Exact solution of a non-local four-dimensional quantum field theory"
We consider a toy model for a Euclidean quantum field theory which can be expressed as a matrix model with partition function $Z(E,J,\lambda)=\int dM \exp({\rm trace}(- \frac{1}{2}EM^2+JM-\frac{\lambda}{4} M^4))$. The integral is over the space of Hermitean $N\times N$-matrices, $E$ is an external matrix which for $N\to\infty$ has the same spectrum as the Laplace operator in 4 dimensions, $\lambda>0$ is a scalar coupling constant and $J$ is used to generate correlation functions. We compute this partition function exactly in the limit $N\to \infty$ and after renormalisation of $E,\lambda$. The exact solution holds for $0\leq \lambda \leq 64\pi$ and describes an interacting non-local (Euclidean) quantum field theory in 4 dimensions. The solution involves Schwinger-Dyson techniques and the theory of Carleman type singular integral equations. Higher correlation functions satisfy an algebraic recursion formula which connects to non-crossing chord diagrams counted by the Catalan numbers.
• Daniela Cadamuro: "Locality in integrable QFTs - characterization and explicit examples"
In quantum field theory, the construction of local observables in the presence of nontrivial interaction is a difficult task due to their complicated structure. In a class of integrable quantum field theories, we give an explicit characterization of these local observables using the properties of the coefficient functions in an expansion by interacting creators and annihilators. Some results on the operator domains of these local observables are given. Using these, we constructed explicit examples of local observables in the quantum Ising model.
• Jan Schlemmer: "Thermal states of deformed Quantum Field Theories"
The analysis of thermal states of quantum field theories complements the investigation of their observational implications by scattering theory. For the wedge-local quantum field theories obtained by deformations, an additional motivation to studying such states is the fact that they lead to realizations of deformed QFTs not satisfying a spectrum condition. In this talk I will report on work in progress concerning the status of thermal states of such theories.
• Falk Lindner: "On the equivalence of the approaches to thermo field theory in the context of perturbative AQFT"
In the attempt to describe thermal equilibrium states in perturbative QFT in Minkowski spacetime, there exist 2 major formalisms, which depart from the standard approach to perturbative QFT. The main difference is the introduction of unobservable quantities, which are interpreted as degrees of freedom of the thermal bath. I derive the formalisms as extensions from a state independent formulation of perturbative QFT and show the equivalence in the observable sector. The existence problem of the KMS-state is sketched.
• Gennaro Tedesco: "Multilocal Fermionization"
We present a simple isomorphism between the algebra of one real chiral Fermi field and the algebra of $n$ real chiral Fermi fields. This isomorphism preserves the vacuum state. This is possible by a "change of localization", and gives rise to new multilocal symmetries generated by the corresponding multilocal current and stress-energy tensor. The result gives a common underlying explanation of several remarkable recent results on the representation of the free Bose field in terms of free Fermi fields, and on the modular theory of the free Fermi algebra in disjoint intervals.
• Michal Eckstein: "Spectral Action on Quantum Spheres"
In 2003, Dąbrowski and Sitarz provided an interesting example of a spectral triple on the algebra of the quantum (Podleś) sphere with a Dirac operator depending explicitly on the deformation parameter $q$. The resulting geometry is rather degenerate suffering from the dimension drop and non-regularity. In my talk I will show that despite these problems it is possible to construct physical models on quantum spheres via the spectral action.
• Daniel Paulino: "The DFR-Algebra for Poisson Vector Bundles"
In this talk we present the construction of a general family of $C^*$-algebras that includes, as a special case, the quantum space-time algebra'' first introduced by Doplicher, Fredenhagen and Roberts. To this end, we briefly review, within the $C^*$-algebra context, the Weyl-Moyal quantization procedure on a fixed Poisson vector space. We then show how to extend this construction to a Poisson vector bundle over a general manifold $M$, giving rise to a fibration of $C^*$-algebras over that manifold. As examples of such construction we recover the original DFR algebra, as well as covariant versions of the time commutative and euclidean noncommutative spacetimes.
• Albert Much: "Effective Quantum Plane from Quantum Physics"
By using the novel tool warped convolutions, we construct in the realm of quantum physics quantized spacetimes.
• Benjamin Eltzner: "Quantization of Perturbations in Inflation"
The theory of linear cosmological perturbations employs a semi-classical approach to general relativity to derive the spectrum of temperature fluctuations of the cosmic microwave background. This approach does not use the semi-classical Einstein equation, but instead geometric as well as matter quantities are split up into a background part and a perturbation, the latter being quantized in the treatment of the inflationary universe. In this talk, the quantization procedure and its implications are investigated in some detail. Questions considered include locality of the fields, canonical choice of states and the interchange of quantization and reduction.
• Marco Benini: „Quantum field theory on affine bundles“
A general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes is discussed. Using the language of category theory, we prove that these models satisfy the generally covariant locality principle. As a by-product, this construction provides a new class of locally covariant quantum field theories, which are a mild generalization of the linear ones, including for example the case of inhomogeneous field equations. We also show the possibility of a functorial assignment of linear quantum field theories to affine ones both at the geometric and the algebraic level. This assignment enables us to induce states for affine quantum field theories satisfying the microlocal spectrum condition by pulling back quasi-free Hadamard states for the corresponding linear theories via the choice of suitable homomorphisms at the algebraic level.
Based on a joint work with Claudio Dappiaggi and Alexander Schenkel [arXiv:1210.3457 [math-ph]].
• Thomas-Paul Hack: „A toy model for quantum field theory on curved supermanifolds“
Locally supersymmetric QFTs can in principle be formulated by means of the available tools of locally covariant QFT in curved spacetimes. In this case local supersymmetry would be an "accidental" gauge symmetry depending on the parameters and fields in the Lagrangean. In the superspace approach to supersymmetric theory, however, supersymmetry transformations arise as geometric transformations and supersymmetric field theories can be formulated in terms of superfields which transform covariantly with respect to these geometric transformations. Following this idea we construct the free Wess-Zumino model in three dimensions as a toy model for locally supercovariant QFT on curved supermanifolds.
• Manuel Hohmann: „Observer space geometry of Finsler spacetimes“
Two different ideas to generalize our classical picture of spacetime in an observer dependent fashion will be briefly reviewed. The first idea is Finsler geometry, where the Lorentzian metric of spacetime depends on the four-velocity of an observer. The second idea is a lift of general relativity from a Lorentzian manifold to the space of observers, i.e., future timelike unit tangent vectors. Possible relationships between these two ideas will be discussed, and it will be shown how the two constructions can be linked.
• Reza Safari: „Coherent States and the Semi-Classical Einstein equation“
The semi-classical Einstein equation will be solved by using coherent states for the Klein-Gordon field. Coherent states are defined with respect to a (generalised) ground state and characterised by a classical solution $f$ of the Klein-Gordon equation. On a given spacetime, we compute the energy density of a general coherent state and try to find an $f$ such that the semi-classical Einstein equation is satisfied in the related coherent state. This general idea will be applied in three cases. First the expectation value of the ground state energy density in the 3D torus spacetime - the Casimir effect - will be calculated and will be used in order to solve the semi-classical Einstein equation in this spacetime. Then we consider states of low energy and their corresponding coherent states in de Sitter spacetime and try to solve the semi-classical Einstein equation by means of them; in this case it turns out that a solution does not exist. Finally, the semi-classical Einstein equation in general Robertson-Walker spacetimes will be solved by means of coherent states with respect to states of low energy under the assumption that the energy density of the latter is negligible.

## Participants

List of registered participants (66):

 Alazzawi, Sabina Universität Wien Andersson, Andreas Universität Leipzig / MPI-MIS Avetisyan, Zhirayr Universität Leipzig / MPI-MIS Banos Ros, Gisela Maria Universität Leipzig Benini, Marco University of Pavia Bischoff, Marcel Universität Göttingen Brum, Marcos Universität Hamburg Buchholz, Detlev Universität Göttingen Cadamuro, Daniela Universität Göttingen Chan-Lang, Sion Universität Hamburg Dappiaggi, Claudio University of Pavia Dierigl, Markus LMU München Diez, Tobias Universität Leipzig Dütsch, Michael MPI-MIS Eckstein, Michal Jagiellonian University Eltzner, Benjamin Universität Leipzig Fernandez, Mikel CSIC Madrid Fiedler, Leander Universität Hannover Fleischhack, Christian Universität Paderborn Franco, Nicolas Copernicus Center, Kraków Fredenhagen, Klaus Universität Hamburg Giorgetti, Luca Universität Göttingen Gransee, Michael Universität Leipzig / MPI-MIS Hack, Thomas-Paul Universität Hamburg Hanisch, Florian Universität Potsdam Hänsel, Mathias Universität Leipzig Hanusch, Maximilian Universität Paderborn Hohmann, Manuel University of Tartu Imani, Paniz Universität Hamburg Knauer, Kevin Universität Leipzig Köhler, Christian Universität Wien Lechner, Gandalf Universität Leipzig Lindner, Falk Universität Hamburg Ludwig, Thomas Universität Leipzig / MPI-MIS Martinetti, Pierre Università di Napoli Much, Albert Universität Leipzig / MPI-MIS Naaijkens, Pieter Universität Hannover Paulino, Daniel Universität Hamburg / Sao Paulo Plaschke, Matthias Universität Wien Pottel, Steffen Universität Leipzig / MPI-MIS Rack, Tatjana Universität Hamburg Rainer, Martin Universität Potsdam/Würzburg Rehren, Karl-Henning Universität Göttingen Reichold, Adam Universität Leipzig Rejzner, Katarzyna University of Rome Tor Vergata Rudolph, Gerd Universität Leipzig Safari, Reza Universität Hamburg Sahlmann, Hanno Universität Erlangen-Nürnberg Schenkel, Alexander Universität Wuppertal Schlemmer, Jan Universität Wien Schmidt, Matthias Universität Leipzig Schroer, Bert FU Berlin Sibold, Klaus Universität Leipzig Stephan, Christoph Universität Potsdam Stiller, Michael Universität Hamburg Stottmeister, Alexander Universität Erlangen-Nürnberg Taslimitehrani, Mojtaba Universität Hamburg Tedesco, Gennaro Universität Göttingen Tolksdorf, Jürgen MPI-MIS Uhlmann, Armin Universität Leipzig Verch, Rainer Universität Leipzig Wöhr, Andreas Universität Tübingen Wrochna, Michal Universität Göttingen Wulkenhaar, Raimar Universität Münster Zähle, Johannes Universität Leipzig Zschoche, Jan Universität Leipzig / MPI-MIS