Foundational and structural aspects of gauge theories

25 May 2017, 17:30 → 2 Jun 2017, 18:00 Europe/Berlin

02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

Gauge symmetry represents in modern theoretical physics one of the main pillars in several successful frameworks. The standard model of elementary particles is probably the most famous example and, especially in the realm of quantum physics, many models implementing the gauge principle have been formulated and thoroughly studied. Yet, it is widely accepted, that our present understanding of the structural aspects of these theories as well as their role in a foundational and axiomatic framework for quantum field theory is not as developed as our current knowledge of their phenomenological consequences. This topical workshop is a timely event which represents an important occasion to review recent achievements and to start a fruitful discussion on the open issues and on the directions of future research.

Executive Summary (PDF) Gauge4.pdf

Participants (PDF) Indico_FSAGT.pdf

Monday 29 May ¶

1 Welcome Address¶

2 Viability of the Asymptotic Safety scenario beyond renormalizability?¶

The talk will include a general introduction to the main ideas of the Asymptotic Safety approach to quantum gravity. This program aims at finding an acceptable quantum field theory of gravity and spacetime geometry which complies with a number of indispensable physical principles such as Hilbert space positivity and background independence in addition to (nonperturbative) renormalizability. We discuss some of these requirements in a simplified two-dimensional setting where the tools of conformal field theory are available.

Speaker: Prof. Martin Reuter (Universität Mainz)

Slides 1_-_Reuter.pdf

3 Coffee Break¶

4 Weak signals of quantum spacetime from electromagnetic interactions of neutral dark matter¶

On DFR quantum spacetime a U(1) gauge theory turns into a gauge theory based on the noncommutative group of unitaries of the DFR C*-algebra, and this implies that classical electrodynamics becomes an interacting theory and that also neutral particles can have non-trivial interaction with the electromagnetic field. This effect can serve therefore as an observable signature of quantum spacetime with relatively low background. We analyze this interaction and give some order of magnitude estimates for the radiation emitted by possible compact astrophysical objects of neutral dark matter. (Joint work with S. Doplicher, K. Fredenhagen, N. Pinamonti.)

Speaker: Prof. Gerardo Morsella (University of Rome "Tor Vergata")

Slides 2_-_Morsella.pdf

Lunch Break¶

5 An IDEAL characterization of FLRW spacetimes¶

An IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric $g_0$ consists of a set of tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$, its Riemann curvature and their derivatives, that are satisfied if and only if $g$ is locally isometric to the reference spacetime metric $g_0$. The same notion can be extended to include matter fields, where the equations $T[g,phi]=0$ are allowed to also depend on the matter fields $phi$. We will present the first IDEAL characterization of cosmological FLRW spacetimes, with and without a dynamical scalar field. The solution of this problem also has implications for the construction of local gauge invariant observables for cosmological perturbations. Namely, by construction, the linearization $dotT[g_0,ph{i}_0;deltag,deltaphi]$ about $(g,phi)mapsto(g_0+deltag,ph{i}_0+deltaphi)$ gives a complete set of local gauge invariant field combinations for cosmological perturbations. This is joint work with G. Canepa.

Speaker: Dr Igor Khavkine (University of Milan)

Slides 3_-_Khavkine.pdf

6 Coffee Break¶

7 Hyperbolic Differential Complexes¶

Complexes of differential operators are a concept of great importance in several areas of mathematics. Particularly, elliptic differential complexes play a fundamental role in the theory of the index of elliptic operators put forward by Atiyah, Singer and others in the sixties, whose relevance in analysis, geometry and topology is well known. Much less developed is a similar theory for complexes of hyperbolic partial differential operators, which are of utmost importance for formulating the dynamics of relativistic field theories with constraints and/or gauge symmetries. In this talk, we shall present a few steps towards such a theory, partly based on a former proposal by MacKichan (1975). Our main focus will be how to properly formulate and prove well-posedness of the Cauchy problem for such complexes, and how this theory elegantly encodes both constraints and gauge symmetries. Connections to the BV-BRST formalism for gauge- theoretic field models and possible extensions to nonlinear systems will also be discussed, if time allows. (joint work with Michael Forger)

Speaker: Prof. Pedro Lauridsen Ribeiro (University of Sao Paolo)

Slides 4_-_Ribeiro.pdf

8 Non-existence of natural states for Abelian Chern-Simons theory¶

We give an elementary proof that Abelian Chern-Simons theory, described as a functor from oriented surfaces to C*-algebras, does not admit a natural state. Non-existence of natural states is thus not only a phenomenon of quantum field theories on Lorentzian manifolds, but also of topological quantum field theories formulated in the algebraic approach.

Speaker: Dr Simone Murro (Universität Regensburg)

Slides 5_-_Murro.pdf

Tuesday 30 May ¶

9 From factorization algebras to twisted functorial field theories - the topological case¶

After recalling functorial field theories a la Atiyah-Segal I will explain a natural generalization thereof, called "twisted" field theories by Stolz-Teichner and closely related to Freed-Teleman's "relative" boundary field theories. Focussing on the topological “extended” case, I will explain how to obtain some examples, in particular some arising from factorization algebras.

Speaker: Dr Claudia Scheimbauer (University of Oxford)

10 Coffee Break¶

11 BV formalism in functorial QFT in Lorentzian and Euclidean signature¶

In this talk I will outline how the Batalin-Vilkoviski (BV) formalism can be used to quantize gauge theories and effective quantum gravity in Lorentzian signature, in the framework of locally covariant QFT. I will also present recent results on relations of this framework to the approach of Costello that operates in Euclidean signature.

Speaker: Dr Kasia Rejzner (University of York)

Lunch Break¶

12 Towards homotopical algebraic quantum field theory¶

An algebraic quantum field theory is an assignment of algebras to spacetimes. These algebras should be interpreted as quantizations of the algebras of functions on the moduli spaces of a classical field theory. In many cases of interest, especially in gauge theories, these moduli spaces are not conventional spaces but `higher spaces' called stacks. Consequently, functions on such spaces do not form an algebra but a `higher algebra' which one may describe by homotopical algebra. This motivates us to study assignments of `higher algebras' to spacetimes, which is what I call homotopical algebraic quantum field theory. In this talk I will clarify the above picture and explain its advantages compared to traditional algebraic quantum field theory. For this I will also present simple toy-models related to Abelian gauge theory and homotopy Kan extensions.

Speaker: Dr Alexander Schenkel (University of Nottingham)

Slides 3_-_Schenkel.pdf

13 Coffee Break¶

14 Background-independence in gauge theories¶

Classical Yang-Mills theory is background-independent in the sense that splitting the gauge connection into a background connection and a dynamical vector potential is a symmetry of the theory. I will talk about a definition of background independence in (perturbative) quantum YM theory, that is the preservation of this symmetry at the quantum level. In a geometrical formulation, we define background-independent observables as flat sections of an algebra bundle over the manifold of background configurations, with respect to a flat connection which implements background variations. It turns out that background independence at the quantum level is in general violated by potential obstructions. I will discuss such obstructions for YM theory and will remark on perturbative quantum gravity. (Joint work with Jochen Zahn).

Speaker: Mr Mojtaba Taslimitehrani (Universität Leipzig)

Slides 4_-_Taslimitehrani.pdf

15 A perturbative renormalizability of SU(2) Yang-Mills theory in four-dimensional Euclidean space which is based on the Flow Equations of the renormalization group¶

Speaker: Mr Alexander Efremov (University of Paris)

Slides 5_-_Efremov.pdf

Wednesday 31 May ¶

16 The locally convex Weyl algebra¶

In my talk I will report on recent progesses in understanding the convergence properties of certain examples of star products. For the Weyl-Moyal star products on a Poisson vector space and its variations one obtains a quantization for a large class of real-analytic functions leading to a locally convex algebra containing elements with canonical commutation relations specified by the constant Poisson structure. Depending on the analytic properties of the Poisson vector space, the resulting algebra has many nice properties which I will point out. The analytic structure of the algebra then helps to prove self-adjointness of linear and quadratic elements in all GNS representations in a very systematic way. Applications to quantum field theory arise when the underlying Poisson vector space is the space of solutions to linear wave equations.

Speaker: Prof. Stefan Waldmann (Universität Würzburg)

17 Coffee Break¶

18 Topological charges and spacelike linearity in QED¶

We present some recent results on the universal C*-algebra of the electromagnetic quantum field which is represented in any theory describing the electromagnetic field. We discuss the appearance of a new kind of topological charges described by a pair of fields localized in certain topologically non-trivial and spacelike separated regions.

Speaker: Dr Giuseppe Ruzzi (University of Rome "Tor Vergata")

Slides 2_-_Ruzzi.pdf

Lunch Break¶

19 Non-relativistic QED in different gauges¶

One consequence of local gauge symmetry in QED is the conservation of the spacelike asymptotic flux of the electric field. It is an old conjecture that this quantity depends on the gauge fixing in the quantization procedure. In this talk I will discuss this problem in a non-relativistic model of QED. I will show how to pass from the usual Coulomb gauge to the axial gauge, compute the flux in both cases and give arguments in favour of the above conjecture.

Speaker: Dr Wojciech Dybalski (Technische Universität München)

Slides 3_-_Dybalski.pdf

20 Coffee Break¶

21 Scattering of atoms and non-locality of the vacuum in QED¶

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 https://arxiv.org/abs/1603.07512, to appear in CMP)

Speaker: Mr Maximilan Duell (Technische Universität München)

Slides 4_-_Duell.pdf

22 Extending the algebraic construction of integrable QFTs¶

arious integrable models of QFT have been recently constructed in a mathematical satisfactory way by means of operator-algebraic techniques. Some of these, such as the O(N)-invariant nonlinear sigma models, can be seen as a toy model for 4-dimensional nonabelian gauge theories. It would be physically even more interesting to consider gauge theories with bound states, where moreover we allow fusions among particles. Examples of interesting integrable models in this respect are the sine-Gordon and the Thirring models. This talk will report on ongoing work regarding the mathematical construction of such models, and explore their relevance for physics.

Speaker: Dr Daniela Cadamuro (Technische Universität München)

Thursday 1 June ¶

23 Higher structures in Dijkgraaf-Witten theories¶

Dijkgraaf-Witten theories are fully extended topological field theories that are constructed using gauge-theoretic principles for a gauge theory based on a finite group. They provide a laboratory linking questions in gauge theory, representation theory and (higher) category theory. We present some aspect of these theories, emphasizing defects of any codimension

Speaker: Prof. Christoph Schweigert (Universität Hamburg)

Slides 1_-_Schweigert.pdf

24 Coffee Break¶

25 Supersymmetric gauge theories and geometric representation theory¶

Speaker: Prof. Richard Szabo (Heriot-Watt University)

Slides 2_-_Szabo.pdf

Lunch Break¶

26 Hopf algebra gauge theories on ribbon graphs¶

We explain how the concept of a lattice gauge theory with values in a group can be generalised to a gauge theory with values in a Hopf algebra on a graph embedded into a surface. We give an axiomatic description of Hopf algebra gauge theories and show that they include the quantum algebra of observables obtained by the combinatorial quantisation of Chern-Simons theory. We relate Hopf algebra gauge theories to lattice models from condensed matter physics. More specifically, we show that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to a Hopf algebra gauge theory for its Drinfeld double D(H).

Speaker: Prof. Catherine Meusburger (Universität Erlangen)

Slides 3_-_Meusburger.pdf

27 Coffee Break¶

28 Weyl quantization for gauge theories in terms of projective limits of graphs¶

Weyl quantization and an adapted pseudo-differential calculus may serve as powerful tool to discuss the semi-classical limit of quantum system. We will present results regarding the construction of a Weyl quantization for gauge theories defined via projective limits of (finite) graphs. Moreover, we will approach the problem of defining associated symbol spaces and their pseudo-differential calculus.

Speaker: Dr Alexander Stottmeister (University of Rome "Tor Vergata")

Slides 4_-_Stottmeister.pdf

29 Stability of thermal states in perturbative algebraic quantum field theory¶

In this talk we discuss the stability properties shown by KMS states for interacting massive scalar fields propagating over Minkowski spacetime. These states have been recently constructed in the framework of perturbative algebraic quantum field theories.

Speaker: Mr Federico Faldino (University of Genoa)

Slides 5_-_Faldino.pdf

Friday 2 June ¶

30 A noncommutative approach to the quantisation of lattice gauge theories¶

We will describe the quantisation of gauge theories on a lattice/graph in terms of their algebras of observables and of the Hilbert space on which the algebra is represented. The algebra of observables for the quantum system admits a natural geometric realization as a groupoid C*-algebra. We will study the behaviour of such algebras under lattice refinements and the resulting continuum limit of the theory. Based on joint work with R. Stienstra and W. van Suijlekom.

Speaker: Dr Francesca Arici (University of Nijmegen)

Slides 1_-_Arici.pdf

31 Coffee Break¶

32 Massive vector bosons: an indication for non-compatibility of the Higgs mechanism with the renormalization group flow¶

Part I. The Higgs mechanism is not needed: this is shown by giving a consistent quantization of massive vector bosons without using the Higgs mechanism. Part II. Compatibility of the Higgs mechanism with the RG-flow: the Higgs mechanism implies that the prefactors of the various interaction terms are uniquely determined functions of the coupling constant(s) and masses. We investigate whether these functions are stable under the RG-flow. Using the framework of Epstein-Glaser renormalization, we find that the answer is 'no', if the renormalization mass scale(s) are chosen in a way corresponding to the minimal subtraction scheme. This result is derived for the U(1)-Higgs model to 1-loop order.

Speaker: Dr Michael Dütsch (Universität Göttingen)