# AQFTUK

## Meeting 6th-7th July 2022, York

### 6th - 7th July at the University of York

(Department of Mathematics, G/N/020)

For further information please contact Chris Fewster or Sam Crawford

Talks will be available to view online via Zoom. Use this link to attend.

## Schedule

### Wednesday 6th July

11:00 - 12:00 Kasia Rejzner (York): *Renormalisation group in AQFT*

12:00 - 12:30 Edoardo D'Angelo (Genoa): *Wetterich equation on lorentzian manifolds*

12:30 - 14:00 **Lunch**

14:00 - 15:00 Daniela Cadamuro (Leipzig): *Relative entropy of coherent states on general CCR algebras*

15:00 - 15:30 Sam Hannah (Cardiff): *Algebra objects in categorical structures*

15:30 - 16:00 **Coffee**

16:00 - 17:00 Robert Allen (Bristol): *Grothendieck-Verdier Duality*

### Thursday 7th July

09:30 - 10:30 Onirban Islam (Potsdam): *Feynman propagators for wave-type and Dirac-type operators on a curved spacetime*

10:30 - 11:00 **Coffee**

11:00 - 12:00 Alex Schenkel (Nottingham): *BV and BFV formalism beyond perturbation theory*

12:00 - 12:30 Victor Carmona (Seville): *Strictifying the time-slice axiom*

12:30 - 14:00 **Lunch**

14:00 - 15:00 Rainer Verch (Leipzig): *The D-CTC Condition is Generically Fulfilled in Classical (Non-quantum) Statistical Systems*

15:00 - 15:10 **Coffee**

15:10 - 15:40 Jan Mandrysch (Leipzig): *Quantum energy inequalities in integrable QFT models*

15:40 - 16:10 Maximilian Ruep (York): *Time-orderable prefactorization algebras of local and causal quantum operations for additive AQFTs*

16:10 - **Departure**

## Abstracts

### Kasia Rejzner: *Renormalisation group in AQFT*

*Renormalisation group in AQFT*

In this talk I will review recent progress on describing renormalisation using the C*-algebraic framework proposed by Buchholz and Fredenhagen in their paper from 2019. The recent progress involves the formulation of the time-slice axiom and the renormalisation group. This is based on a join paper with Brunetti, Duetsch and Fredenhagen.

### Edoardo D'Angelo: *Wetterich equation on lorentzian manifolds*

*Wetterich equation on lorentzian manifolds*

The Wetterich equation governs the renormalization flow of the effective action under the scaling of an IR cutoff. In this talk, I introduce a generalization of the Wetterich approach to scalar field theories in generic, Hadamard states on globally hyperbolic, Lorentzian spacetimes, using techniques developed in the perturbative Algebraic Quantum Field Theory framework. The main novel ingredient is the use of a local regulator, which preserves locality and unitarity. As a proof of concept, I also briefly discuss the application of the theory to thermal states in Minkowski and to the Bunch-Davies vacuum in de Sitter space.

### Daniela Cadamuro: *Relative entropy of coherent states on general CCR algebras*

*Relative entropy of coherent states on general CCR algebras*

In QFT the total entropy of a state is generically infinite, so one considers the relative entropy between two states with reference to a subalgebra of the observables, such as the von Neumann algebra associated with a double cone or a spacelike wedge. Such entropy can be computed using Tomita-Takesaki modular theory. In this talk, we study the relative entropy for a subalgebra of a generic CCR algebra between a general (possibly mixed) quasifree state and its coherent excitations, and give a formula for this entropy in terms of single-particle modular data. We also investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces, and study lower estimates for the second derivative of the relative entropy along this family, which replace the usual notion of convexity of the entropy. Our main input is a regularity condition for the family of subspaces (“differential modular position”) which generalizes the notion of half-sided modular inclusions. Examples include thermal states for the conformal U(1)-current.

### Sam Hannah: *Algebra objects in categorical structures*

*Algebra objects in categorical structures*

TBC

### Robert Allen: *Grothendieck-Verdier Duality*

*Grothendieck-Verdier Duality*

Rigidity is a form of duality which generalises the notion of the vector space dual. Grothendieck-Verdier duality is a structure which arises naturally in many disparate fields of mathematics, when requiring rigidity is too restrictive. For example, the representation theory of a vertex algebra which admits a contragredient dual is not necessarily rigid, but is always Grothendieck-Verdier. The notion of a Grothendieck-Verdier category is nicely compatible with the introduction of a braiding and a twist, leading to the definition of a ribbon Grothendieck-Verdier category. I will motivate and introduce ribbon Grothendieck-Verdier categories and present some examples and consequences.

### Onirban Islam: *Feynman propagators for wave-type and Dirac-type operators on a curved spacetime*

*Feynman propagators for wave-type and Dirac-type operators on a curved spacetime*

It is a classic result that any normally hyperbolic operator on a globally hyperbolic spacetime admits unique advanced and retarded Green's operators. With the advent of quantum field theory, a new type of Green's operator emerges: the Feynman propagator. These propagators are an essential ingredient of perturbative quantum field theory and are intimately connected with quantum states. They also play a pivotal role in spectral geometry. For instance, such propagators appear naturally in Lorentzian generalisations of the index theorem and the Duistermaat-Guillemin-Gutzwiller trace formula. By far, there are several constructions of these propagators. I shall talk about a global microlocal construction of Feynman propagators for normally hyperbolic operators on a globally hyperbolic spacetime by generalising the classic Duistermaat-Hörmander construction into a vector bundle setting. It will then be explained that, for normally hyperbolic operators which are symmetric with respect to a hermitian bundle metric, the Feynman propagators can be constructed to satisfy a positivity property that reflects the existence of Hadamard states on curved spacetimes.

In the later part of this talk, I shall present a more direct construction of the Feynman propagators for Dirac-type operators. Albeit the natural bundle metric on spinors is not positive-definite, in this case, a direct microlocal construction of these propagators satisfying positivity can be given. (Joint work with A. Strohmaier based on arXiv:2012.09767 [math.AP]).

### Alex Schenkel: *BV and BFV formalism beyond perturbation theory*

*BV and BFV formalism beyond perturbation theory*

Modern approaches to quantum field theory, such as factorization algebras and homotopical AQFT, are based on cohomological methods in field theory that have been developed over the past decades in the context of the BV and BFV formalisms. While these techniques work pretty well for perturbative quantum field theories, they have intrinsic limitations to describe global features, such as the topology of the gauge group or moduli of bundles.

Derived algebraic geometry is a powerful geometric framework in which one can attempt to globalize the BV and BFV formalisms. I will start this talk with a very basic introduction to derived algebraic geometry, focussing in particular on its more concrete and computational aspects. I will then illustrate the power and potential of this framework for new developments in mathematical physics by studying two applications: 1.) The non-perturbative BV formalism for a function f : [X/G] —> k on a quotient stack, and 2.) the quantization of a derived cotangent stack T*[X/G] over a quotient stack, which is a global version of BFV quantization.

This talk is based on joint works with Benini and Safronov [2104.14886] and Benini and Pridham [2201.10225].

### Victor Carmona: *Strictifying the time-slice axiom*

*Strictifying the time-slice axiom*

There are several approaches in the literature to axiomatize what should be a Quantum Field Theory. Among them, Algebraic Quantum Field Theories are a very flexible notion that encompasses several approaches studied before. Concisely, they are assignations of "operator algebras" to space-time regions satisfying Einstein causality. An important additional axiom that one would want for an AQFT to satisfy is the time-slice axiom, which accounts for uniqueness of solutions for time evolution of observables.

When considering homotopical phenomena in Quantum Field Theory, e.g. gauge theories, there are two reasonable ways in which an AQFT can satisfy the time-slice axiom: strictly or in a homotopical fashion. In some sense, the strict version is easier to deal with in applications such as relative Cauchy evolution (recently discussed in [2108.10592] for linear homotopy AQFTs), but the homotopical version is more natural and is fulfilled in examples. In this talk, we compare both the strict and the homotopical time-slice axiom and show that a surprising feature of AQFTs coming from Lorentzian settings is that both notions coincide. Our approach is based on the operadic formulation of AQFTs due to Benini-Schenkel and collaborators and provides general criteria to address similar questions. As a concrete consequence of our main results one has: given an AQFT satisfying the homotopy time-slice axiom, one can produce an equivalent AQFT that satisfy this axiom strictly. We will discuss several examples for which this holds and an important open problem. This is joint work with Marco Benini and Alexander Schenkel.

### Rainer Verch: *The D-CTC Condition is Generically Fulfilled in Classical (Non-quantum) Statistical Systems*

*The D-CTC Condition is Generically Fulfilled in Classical (Non-quantum) Statistical Systems*

The D-CTC condition, introduced by David Deutsch as a condition to be fulfilled by analogues for processes of quantum systems in the presence of closed timelike curves, is investigated for classical statistical (non-quantum) bi-partite systems. It is shown that the D-CTC condition can generically be fulfilled in classical statistical systems, under very general, model-independent conditions. The central property used is the convexity and completeness of the state space that allows it to generalize Deutsch's original proof for q-bit systems to more general classes of statistically described systems. The results demonstrate that the D-CTC condition, or the conditions under which it can be fulfilled, is not characteristic of, or dependent on, the quantum nature of a bi-partite system. (Joint work with Juergen Tolksdorf, arXiv:1912.02301, Foundations of Physics 51, no 93 (2021))

### Jan Mandrysch: *Quantum energy inequalities in integrable QFT models*

*Quantum energy inequalities in integrable QFT models*

Many results in general relativity rely crucially on classical energy conditions imposed on the stress-energy tensor. Quantum matter, however, violates these conditions since its energy density can become arbitrarily negative at a point. Nonetheless quantum matter should have some reminiscent notion of stability, which can be captured by the so-called quantum (weak) energy inequalities (QEIs), lower bounds of the smeared quantum-stress-energy tensor. QEIs have been proven in many free quantum field theory (QFT) models on both flat and curved spacetimes. In interacting theories only few results exist. We are here presenting analytical results on QEIs in interacting integrable QFT models in 1+1 dimension: A state-independent result for constant diagonal scattering functions and a result at 1-particle-level including the O(N)-nonlinear-sigma model and the Bullough-Dodd model.

### Maximilian Ruep: Time-orderable prefactorization algebras of local and causal quantum operations for additive AQFTs

In this talk I will argue that any net of local and causal quantum operations associated to a given algebraic quantum field theory can be endowed in a natural way with a structure close to that of a time-orderable prefactorization algebra of Benini, Perin and Schenkel. For additive AQFTs, I will give a concrete example of a time-orderable prefactorization algebra of local and causal operations that in particular contains the update maps derived from the measurement schemes of Fewster and Verch. Finally, I will argue that Buchholz and Fredenhagen’s C*-approach may then be regarded as specifying a net of (pure) local and causal operations, followed by finding an associated AQFT generated by observables.

This talk is based in parts on work together with H. Bostelmann and C. J. Fewster.

## How to reach the venue

Some general info https://www.york.ac.uk/about/transport-maps-parking/

The talks will take place in G/N/020 in the Department of Mathematics, YO10 5DD, indicated on the map below.