Schedule for: 18w5015 - Physics and Mathematics of Quantum Field Theory

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The theory of random tensors is expanding rapidly in scope. At the combinatorial (zero dimensional) level the tensorial $1/N$ expansion has a much more complicated structure than 'tHooft $1/N$ expansion for matrices, but it is led by a simpler class of graphs, called melonic. Adding time to the picture gives quantum mechanical tensor models which generalize in a promising way the Sachdev-Ye-Kitaev model of near $AdS_2/CFT_1$ holographic correspondence. We shall review this fast growing topic.

The Feynman propagator is at the heart of quantum field theory. However, in quantum field theory in curved spacetimes, no locally covariant notion of a distinguished Feynman propagator exists.
Instead, often a distinguished class of Feynman propagators is considered, which share a common parametrix. Nevertheless, certain classes of spacetimes possess distinguished Feynman propagators.

First, I will give an in-depth introduction to propagators (Green functions) on curved spacetimes and their role in quantum field theory. In particular, I will highlight the importance of the so-called Hadamard states – an appropriate generalization of the Poincaré invariant vacuum state. Then, I will show that the free Klein–Gordon field on asymptotically static spacetimes comes equipped with a natural Feynman propagator (albeit globally constructed and generally not related to a state). Finally, I will argue that this Feynman propagator is closely related to the question of the self-adjointness of the Klein–Gordon operator on $L^2$(spacetime) and the boundary value of its resolvent.

I will review several results obtained in collaboration with Philippe Gravejat, Christian Hainzl and Eric Séré on a mean-field model in QED. I will particularly discuss charge renormalization and the derivation of the Euler-Heisenberg effective Lagrangian. I will also mention the advantages and disadvantages of the model, as well as open problems.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I shall discuss a class of nonequilibrium states in CFT in one space dimension and how heat transport in such states connects to the representation theory of the group of circle diffeomorphisms.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

C*-dynamical systems are known to be a powerful tool in the analysis of infinite spin and Fermionic systems. It has been argued that this approach cannot work for Bosons, however, where singular states describing unrestrained accumulations of matter do appear. In this talk it is shown that this fact does not constitute a mathematical obstruction if one deals with the (particle number preserving) observables in the resolvent algebra generated by a Bose field.

For, in contrast to the Weyl algebra, this algebra contains ideals which vanish in singular states. A slight extension of this algebra thereby admits an automorphic action of dynamics for a large class of twobody interactions; it allows to proceed to corresponding C*-dynamical systems. If time permits, some standard problems in many body theory are also briefly addressed from this novel point of view.

(Joint work with Ajay Chandra and Gianluca Guadagni)

An outstanding problem in the area of rigorous renormalization group theory is to develop a Wilsonian formalism which can handle space-dependent couplings in the Euclidean setting. I will present such a method in the simpler hierarchical model case and explain how this allowed us to prove a 46 years old prediction by Wilson regarding the anomalous scaling of the square/energy field in a hierarchical ferromagnet.

The model we studied is a hierarchical version of a 3d ferromagnet with long-range interactions in a range of parameters which puts it slightly below the upper critical dimension. We constructed the scaling limit for the joint law of the elementary/spin field together with the square/energy field as well as all mixed correlation functions. The Euclidean version of this scaling limit is conjectured to be a conformal field theory in three dimensions according to recent work by physicists in the conformal bootstrap program. There is a natural analogue of conformal invariance in the hierarchical model which thus provides an ideal testing ground for renormalization group-based attempts at rigorously proving conformal invariance. The idea is to feed the space-dependent renormalization group space-dependent ultraviolet cutoffs. I will also mention emerging connections to the AdS/CFT correspondence.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

(Joint work with A. Pizzo)

I will discuss some recent progress on infrared problems in non-perturbative approaches to QED. First, I will explain the general distinction between the `infraparticle’ and `infravacuum’ picture of the electron which pertains to the existence resp. non-existence of the spacelike asymptotic flux of the electric field. On this basis the problem of electron-photon scattering will be discussed both in models of non-relativistic QED and in the setting of algebraic QED.

On the infraparticle side, we will revisit the Faddeev-Kulish approach, theory of particle weights and the problem of velocity superselection. On the infravacuum side the talk will cover Kraus-Polley- Reents representations and the Buchholz-Roberts approach. If time allows, relations to a current trend in high-energy physics triggered by the works of A. Strominger et al. on the Weinberg's soft-photon theorems will be discussed.

(Joint work with W. Dybalski)

I plan to show an unexpected connection between infrared problems and the theory of noncommutative recurrence relations. This theory turns out to be a robust method to prove infrared regularity of physical quantities which suffer from superficial infrared divergencies even after implementation of multi-scale techniques.

(Part I of joint talk with Y. Tanimoto; joint work with H. Bostelmann)

A recent viewpoint in the construction of interacting quantum field theories in the operator algebraic setting considers observables localized in unbounded wedge-shaped regions rather than compact regions, allowing simpler expressions of certain observables. With this approach, we construct two-dimensional interacting massive scalar Haag-Kastler nets with factorizing S-matrices containing poles. We exhibit the concrete expression of certain observables in wedges, discuss their self-adjointness and prove the existence of observables in double cones.

(Part II of joint talk with D. Cadamuro)

The non-abelian Yang-Mills self-interaction is marginal in four dimensions in the usual terminology of the renormalization group, and thus requires renormalization to all orders in an expansion (in either the coupling or a loop expansion). It is well-known in principle how to reconcile UV-renormalization with local gauge invariance, but a fully rigorous and satisfactory treatment taking also into account the problems posed by the complicated IR-behavior of the theory on non-compact manifolds has only been given relatively recently. In this talk, I describe how a combination of the Epstein-Glaser method/method of renormalization group flow equations and cohomological techniques a la Batalin-Vilkovisky can be used to establish the existence of correlation n-point functions of arbitrary gauge invariant local operators to arbitrary orders in perturbation theory. The second method of proof comes by construction with rather refined bounds for these objects capturing their short and long distance properties. Time permitting, I furthermore explain how the operator product arises in this context.

Originally conceived by Wilson in 1969, the operator product expansion (OPE) has found a multitude of applications in high-energy physics, conformal field theory and condensed matter systems. In the context of perturbative Euclidean quantum field theory, we review a couple of recent results showing that the OPE is much better behaved than generally expected:

- instead of being just an asymptotic expansion, it converges for arbitrary finite separations of the operators,
- the OPE coefficients factorise for suitable spacetime configurations,
- there is an explicit formula for the recursive construction of the OPE coefficients, which is UV- and IR-finite.
We also show how most of these results generalise to gauge theories, including suitable Ward identities for the OPE coefficients, and comment on how the above results could be used for a non-perturbative construction of quantum field theories.

In searches for new phenomena in particle physics, we are often interested in observing tiny deviations from the Standard Model (SM) predictions. In consequence, the SM predictions must be known very precisely. This is often a challenge even in situations when purely perturbative calculations are sufficient. At present, the most powerful methods amount to expressing the observables of interest in terms of so-called Master Integrals (MIs). The MIs are not being evaluated directly but rather via solving systems of differential equations. In the process of finding the MIs via the Laporta algorithm, large numbers (often billions) of linear equations need to be generated and solved, with simplifications of complicated rational functions at each step. In consequence, even the most powerful present-day computer clusters are sometimes insufficient. The situation could radically improve if a clever mathematical solution of the considered algebraic problem could be found in general.

hep-ph/0102033, arXiv:0804.3008, arXiv:1004.4199, arXiv:1412.2296.

Proposal: Walk to the peak of Tunnel mountain, ca 3km one way, elevation gain 250m, no refreshments on the top, but rewarding views. Meeting: 9:00am in front of Corbett Hall. Appropriate footwear, water bottle, rain protection recommended.

Feel free to make your own plans!

Conformal anomalies in quantum field theories of different spins coupled to gravity will be discussed. The contributions of spins 3/2 and 2 are gauge dependent and a criterion to fix this ambiguity will be presented. The arguments pointing to the necessity of the cancellation of the total contribution to the conformal anomaly and the existence of theories satisfying this requirement will be discussed.

The first half of this talk will survey recent progress on the stochastic quantization of Phi4 models, emphasizing the fundamental work of Martin Hairer and his theory of regularity structures. The second half will discuss some new applications of this theory to parabolic stochastic PDE's of a quasi-linear nature.

(Joint work with R.Verch)

Algebraic quantum field theory is founded on the idea of algebras of observables associated with local regions of spacetime. However, not much attention has been given to how these observables can actually be measured. On the other hand, quantum measurement theory provides an operational understanding of measurement schemes, in which a probe system is used to measure a quantum observable of the system of interest. However these discussions are not usually framed in a spacetime context. This talk will describe a generally covariant formalism of measurement schemes adapted to quantum field theory in curved spacetimes, illustrated by a specific model that can be analysed in detail.

I will give an overview of how the Batalin-Vilkovisky (BV) formalism can be incorporated into perturbative algebraic quantum field theory in full generality and I will discuss two examples: gauge theories and effective quantum gravity. In the latter case, there is an additional difficulty in constructing gauge-invariant observables. I will explain how this is addressed in our approach.

(Joint work with Ch. Gerard)

In QFT on curved spacetimes, a central difficulty is how to reconcile local properties of states (short-distance behaviour of N-point functions) with global ones (positivity, spacetime or gauge symmetries, asymptotic convergence to vacuum or thermal state, boundary conditions, etc.). In the last few years, these difficulties have been overcome in a wide range of situations, leading in particular to rigorous descriptions of thermal effects on black hole spacetimes. The aim of the talk will be to present the new state-of-the-art and discuss the main conjectures, with a particular focus on:

- distinguished states on black hole spacetimes and their thermal properties
- Hadamard states and (generalized) Feynman propagators in scattering situations
- the Reeh-Schlieder property
- states with good holographic properties on AdS spacetimes

We analyze some properties shown by extremal 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 by Fredenhagen and Lindner. In particular, we discuss the validity of the return to equilibrium property when the interaction Lagrangian has compact spatial support. If the adiabatic limit is considered, the return to equilibrium is in general not valid. This implies that an equilibrium state under the adiabatic limit for a perturbative interacting theory evolved with the free dynamics does not converge to the free equilibrium state. Actually, we show that the ergodic mean of this state converges to a non-equilibrium steady state (NESS) for the free theory. We thus compute the relative entropy among equilibrium states for different evolutions showing that such an extent is compatible with perturbation theory. We then analyze the entropy production in the NESS discussed above to estimate how far from equilibrium is this state.

I will discuss different types of the adiabatic limit in perturbative quantum field theory in the Minkowski space in the Epstein-Glaser approach: the algebraic adiabatic limit, the weak adiabatic limit and the strong adiabatic limit.

(1) The algebraic adiabatic limit is the construction of the net of local algebras of interacting fields and interacting observables. This construction is well-understood even in the case of models with gauge symmetries and is applicable also to theories defined on curved spacetime.
(2) The weak adiabatic limit allows to construct the Wightman and Green functions. I will present the recent results about its existence in most physically relevant models of quantum field theory and show that these results can be used to define a Poincaré-invariant state on the algebra of interacting fields in the algebraic adiabatic limit. The state obtained in this way can be interpreted as an interacting vacuum state.
(3) The strong adiabatic limit is used to define the scattering matrix and the interacting fields as operators acting on a Hilbert space. This limit is under control only in the case of purely massive models. Because of the infrared problem it does not exist in most models with massless particles. The notable example is quantum electrodynamics for which even the first order correction to the scattering matrix is ill-defined. Based on the ideas of Dollard, Kulish and Faddeev, I will propose a definition of a modified scattering matrix and modified interacting fields in a model with the infrared problem and show the existence of the strong adiabatic limit in the modified sense in low orders of perturbation theory.

Starting with proposals by Schuster and Toro (2013), the massless "continuous-spin" or "infinite-helicity" irreducible representations of the Poincaré group have been the subject of several recent investigations. Their status as string-localizable particles was shown in principle by Mund, Schroer and Yngvason (2006). From the viewpoint of Wigner's equations of motion for such (as yet unobserved) particles, we endeavour to give a first-quantized approach to them, based on a background of classical elementary systems.

One of the unintuitive features of quantum mechanics is "quantum backflow": Particles can, in certain circumstances, move in the direction opposite to their momentum. More technically, for a one-dimensional particle with positive momentum, the (averaged) probability flux operator may have negative expectation values, although to a limited amount. This effect has been known for quite some time for free particles. We recently verified that the effect is stable under interaction in the sense of scattering theory: particles with *asymptotically* positive momentum can exhibit a limited amount of negative probability flux in the interaction region. This mathematical result, somewhat unexpectedly for us, resulted in a brief flurry of coverage in the international press.

The conformal bootstrap seeks to use conformal symmetry, associativity of the operator product expansion, and other physical consistency conditions to constrain nontrivial conformal field theories (CFTs). I will review recent progress in the conformal bootstrap, including Rattazzi, Rychkov, Tonni, and Vichi's algorithm for deriving bounds on CFT data, bounds on the 3d critical Ising model, and some analytic results about the bootstrap equations.

(Joint work with Jens Mund)

We establish the Haag-Kastler axioms for a class of interacting quantum field theories on the twodimensional de Sitter space, which satisfy finite speed of light. The $P (\varphi)_2$ model constructed by Barata, Jäkel & Mund, describing massive scalar bosons with polynomial interactions, provides an example.

We have two different axiomatic approaches to chiral conformal field theory (CFT). The conformal net approach is based on the theory of operator algebras on Hilbert spaces (C^*-algebras and von Neumann algebras) and it is the chiral CFT version of algebraic quantum field theory (AQFT). On the other hand, the vertex operator algebra (VOA) approach is based on an algebraic reformulation of the relevant properties of conformal invariant quantum fields on the circle.

I will give an overview of the present status of understanding in relation to the connections between these two approaches.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.