Abstracts & slides

Christian Bär (Universität Potsdam)

Boundary value problems on Riemannian and Lorentzian manifolds

Abstract:  Boundary value problems for the Dirac operator on a Riemannian manifolds are rather well understood. In particular, one has a general description of admissible boundary conditions. The Lorentzian case has been studied only recently and it turns out that there are similarities but also fundamtental differences to the Riemannian case. I will describe both situations and contrast them. This is joint work with Werner Ballmann, Sebastian Hannes and Alexander Strohmaier.

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Nicolas Besset (Université Grenoble Alpes)

Decay of the local energy and scattering for the charged Klein-Gordon equation in the exterior De Sitter-Reissner-Nordström spacetime

Abstract:  We show decay of local energy of solutions of the charged Klein-Gordon equation in the exterior De Sitter-Reissner-Nordström spacetime by means of a resonances expansion of the local propagator. This is an extension of the well-known result of Bony and Häfner for the wave equation in the De Sitter-Schwarzschild case. The existence of the charge of the Klein-Gordon scalar field gives rise to some new difficulties which are dealt with a perturbative approach, assuming that the product of the black hole charge and the field's one is sufficiently small. Obtained estimates will be then sufficient to deduce asymptotic completeness. 

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Claudio Dappiaggi (Università di Pavia)

Boundary conditions and ground states for a scalar field theories on BTZ spacetime

Abstract:  We consider the prototypical example of a scalar field on the exterior region of BTZ spacetime and we individuate all admissible boundary conditions that can be applied on the conformal boundary, being at the same time compatible with background isometries. These are nothing but Robin boundary conditions and we analyse them in detail observing that, in some instances, "bound state" mode solutions are present. Subsequently, for any choice of boundary condition where no bound state is present, we construct the two-point function of the ground state and we discuss its mathematical properties. Finally we comment on some applications, e.g. for the analysis of Hawking radiation. Joint work with Francesco Bussola, Hugo Ferreira and Igor Khavkine.

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Nguyen Viet Dang (Université Claude Bernard Lyon 1)

Renormalization of quantum field theory on Riemannian manifolds

Abstract:  I will present an overview of results obtained with Bin Zhang (Sichuan University). We will start by giving a simple example which shows the necessity to subtract infinities in quantum field theory. Then we introduce the notion of Feynman amplitudes and study their singularities using a zeta regularization with several parameters. Finally we will apply this method to perturbative renormalization of quantum field theories on Riemannian manifolds.

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Thierry Daudé (Université de Cergy-Pontoise)

Inverse scattering at fixed energy in black hole spacetimes

Abstract:  In this talk, we first describe a class of axisymmetric, electrically charged, spacetimes with positive cosmological constant, called Kerr-Newmann-de-Sitter black holes, which are exact solutions of the Einstein equations. The main question we address is the following: can we determine the metrics of such black holes by observing waves at the "infinities" of the spacetime? Precisely, the considered waves will be massless Dirac fields evolving in the outer region of Kerr-Newman-de-Sitter black holes. We shall define the corresponding scattering matrix, the object that encodes the far field behavior of these Dirac fields from the point of view of static observers. We finally shall show that the metrics of such black holes is uniquely determined by the knowledge of this scattering matrix at a fixed energy. This result was obtained in collaboration with François Nicoleau (Nantes).

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Jan Dereziński (University of Warsaw)

Balanced geometric Weyl quantization with applications to QFT on curved spacetimes

Abstract:  First I will describe a new pseudodifferential calculus for (pseudo-)Riemannian spaces, which in our opinion (my, D.Siemssen's and A.Latosiński's) is the most appropriate way to study operators on such a manifold. I will briefly describe its applications to computations of the asymptotics  the heat kernel and Green's operator on Riemannian manifolds. Then I will discuss analogous applications to Lorentzian manifolds, relevant for QFT on curved spaces.  I will mention an intriguing question of the self-adjointness of the Klein-Gordon operator. I will describe the construction of the (distinguished) Feynman propagator on asymptotically static spacetimes.  I will show how our pseudodifferential calculus can be used to compute the full asymptotics around the diagonal of various inverses and bisolutions of the Klein-Gordon operator.

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Alexis Drouot (Columbia University, New York)

A quantitative version of Hawking radiation

Abstract:  Hawking radiation predicts that stars collapsing to black holes approach a thermal equilibrium (as measured by outside observers). I will review its mathematical formulation, due to A. Bachelot. I will then show that for spherical black holes with positive cosmological constants, the convergence to the thermal equilibrium is exponentially fast.

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Maximilian Duell (TU München)

The Reeh-Schlieder property and a construction of scattering states

Abstract:  The Reeh-Schlieder property is a paradoxical feature of generic Quantum Field Theories, according to which local observables within any one fixed bounded region suffice to create from the vacuum a dense set of physical states. I will begin my talk by reviewing some mathematical and physical aspects of the Reeh-Schlieder property. In the second part of the talk the resulting vacuum correlations will be used to construct Haag-Ruelle scattering states in the context of theories with infrared problems. 

Partially based on CMP 352, 935-966 (2017)

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Stephen A. Fulling (Texas A&M)

Acceleration radiation and the Equivalence Principle

Abstract:  In relativity a uniformly accelerated entity is equivalent to a stationary entity located in a gravitational field.  This equivalence principle does not say that “acceleration is relative” in the sense that velocity is, but it suggests that there should be some qualitative reciprocity between inertial and uniformly accelerated systems.  For the simple and fundamental problem of radiation from an accelerated point charge, the equivalence principle has long given rise to apparent paradoxes that are still not completely resolved to everyone's satisfaction.  In quantum field theory Unruh and Wald showed that an accelerated detector, such as a multistate atom, emits radiation from the viewpoint of a stationary observer.  The qualitative equivalence principle suggests that a stationary atom, or one in free fall into a black hole, radiates with respect to an accelerated observer (M. Scully, A. Svidzinsky, et al., papers in preparation).  With J. Wilson I have convincingly shown a similar effect for mirrors in 2-dimensional space-time, building on work with P. Davies in the 1970s.

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Oran Gannot (Northwestern University)

Logarithmic decay of waves on spacetimes bounded by Killing horizons

Abstract:  I will describe the decay of linear waves on a class of stationary spacetimes bounded by non-degenerate Killing horizons. Without any assumptions on the trapped set, solutions of the wave equation exhibit logarithmic energy decay. This is analogous to well known results in other geometric settings. The proof follows from high frequency bounds on the resolvent.

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Christian Gérard (Université Paris-Sud)

Wick rotation on curved spacetimes

Abstract:  We will present some results obtained recently with Michał Wrochna on the Wick rotation on curved spacetimes. In a first part we will sketch a proof of the fact that pure analytic Hadamard states exist on any globally hyperbolic, analytic spacetime having an analytic Cauchy surface. The importance of analytic Hadamard states comes from the fact that they satisfy the Reeh-Schlieder property. The proof relies on the Wick rotation in Gaussian normal coordinates and on a standard tool in elliptic boundary value problems, called the Calderón projector.

In a second part we will use the Wick rotation to prove the existence and Hadamard property of the Hartle-Hawking state on a spacetime having a stationary, bifurcate Killing horizon, thereby extending a result of Sanders which dealt with the static case.

Bernard Kay (University of York)

Instability of enclosed horizons

Abstract:  I outline a number of results concerning the Klein-Gordon equation in the region of 1+1 dimensional Minkowski space to the left of a uniformly accelerating mirror in the right Rindler wedge, in the presence of vanishing boundary conditions on the mirror. These include the result [1] by myself and Umberto Lupo, that, for the quantum version of this system, there exists no stationary Hadamard state (when the notion of “Hadamard” is suitably defined); and [2] for the classical version of this system and in the presence of vanishing boundary conditions also on an image mirror in the left Rinder wedge, suitable compactly supported arbitrarily small initial data on a suitable initial surface for the region between the two mirrors will develop an arbitrarily large stress-energy scalar near where the two Rindler horizons cross. We conjecture that analogous results hold in 1+3 dimensions for a Kruskal black hole, where the mirror is replaced by a reflecting spherical box with constant Schwarzschild radius. I discuss the possible physical significance of these results and conjectures and mention a possible connection (explained in [2]) with my “matter-gravity entanglement hypothesis” [3].

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Peter Hintz (University of California, Berkeley)

Stability of Minkowski space and asymptotics of the metric

Abstract:  I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of ⁴ to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.

Stefan Hollands (Universität Leipzig)

Perturbative renormalization of YM₄ on curved Lorentzian spacetimes

Abstract:  Around the year 2000, it was proven by Brunetti and Fredenhagen and by Hollands and Wald that the ϕ⁴₄ quantum field theory can be consistently renormalized to all orders in perturbation theory on any globally hyperbolic Lorentz manifold. By "consistently", one means that the (formal) series for the interacting quantum field operators in this theory have the desired properties such as Einstein causality (term-by-term), and that a renormalization prescription exists which depends "locally and covariantly" on the spacetime geometry. These works were a culmination of a long development and not only involved new tools (e.g. from microlocal analysis) in a crucial way, but also raised renormalization theory to a new level at the conceptual level.

Some years later, I generalized these results to non-abelian Yang-Mills gauge fields (YM₄). A crucial new feature appearing in this theory is gauge-invariance. In fact, it is non-trivial both to formulate this invariance at the quantum level, and to show that a renormalization prescription exists which is compatible with it and the other features mentioned before. In this talk, I sketch the essential ideas of renormalization theory in curved spacetime including some new developments relating it to Fedosov quantization by Collini and myself, and also related results due to Taslimitehrani and Zahn.

Lionel Mason (University of Oxford)

Scattering amplitudes on plane wave space-times and the double copy

Abstract:  Perturbatively around flat space, the scattering amplitudes of gravity are related to those of Yang-Mills by colour-kinematic duality, under which gravitational amplitudes are obtained as the 'double copy' of the corresponding gauge theory amplitudes. We consider the question of how to extend this relationship to curved scattering backgrounds, focusing on certain 'sandwich' plane waves. We calculate the 3-point amplitudes on these backgrounds and find that a notion of double copy remains in the presence of background curvature: graviton amplitudes on a gravitational plane wave are the double copy of gluon amplitudes on a gauge field plane wave. This is non-trivial in that it requires a non-local replacement rule for the background fields and the momenta and polarization vectors of the fields scattering on the backgrounds. It must also account for new 'tail' terms arising from scattering off the background. These encode a memory effect in the scattering amplitudes, which naturally double copies as well.

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Valter Moretti (Università di Trento)

Hadamard states from light-like hypersurfaces

Abstract:  I review some of the results obtained in collaboration [1] with C. Dappiaggi and N. Pinamonti on Hadamard quasifree states and their properties constructed out the structure of null horizons or null infinity or cosmological horizons.

[1] C. Dappiaggi, V. Moretti, N. Pinamonti, Hadamard States from Light-like Hypersurfaces, SpringerBriefs in Mathematical Physics (2017)

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Jean-Philippe Nicolas (Université de Brest)

Peeling of scalar fields on Kerr spacetime

Abstract:  The peeling is an asymptotic behaviour of massless fields along outgoing null geodesics in asymptotically flat spacetimes, initially observed by Sachs at the beginning of the 1960's, then reformulated in very simple terms by Penrose in 1965 using conformal geometry. The question of its genericity, especially when talking about the peeling of the Weyl tensor of an Einstein spacetime, was controversial for several decades after Penrose's paper. For Einstein's equations, the question is now essentially settled, but given an Einstein spacetime, it is not clear whether there is a large class of Cauchy data giving rise to solutions with a good peeling. Lionel Mason and the speaker answered the question for fields of spin 0, 1/2 and 1 on Schwarzschild's spacetime in 2009 and 2012. We extended recently the results to linear and non linear scalar fields on the Kerr geometry in a joint work with Pham Truong Xuan. We shall recall the history of the subject, describe the principles of the approach developed with Lionel Mason and talk about the specific features of our work for Kerr metrics.

Ko Sanders (Dublin City University)

On the local temperature of a quantum field

Abstract:  Temperature is a local quantity that can vary from point to point. However, for quantum fields we normally identify it as the parameter β in the KMS condition. This parameter is constant in space and time and relies essentially on global structures: a stationary spacetime and a KMS state.

In this talk I will review a local definition temperature for massless free scalar fields, proposed by Buchholz, Ojima and Roos. Under suitable conditions it can be shown that a state has a well-defined local temperature. Moreover, for KMS states the local temperature expresses qualitatively similar information as the global parameter β. To prove these statements one needs to obtain local information (about the expectation value of the Wick square) from a globally defined state. This mathematically difficult problem is solved by invoking the Positive Mass Theorem of Schoen and Yau.

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Alexander Strohmaier (University of Leeds)

Local and global index theory for globally hyperbolic spacetimes

Abstract:  I will discuss some aspects of index theory for the Dirac operator on globally hyperbolic spacetimes. If the spacetime is spatially compact the Dirac operator can be shown to be Fredholm on a suitable domain of distributions. This Fredholm property is due to propagation of singularities rather than local elliptic regularity. I will discuss what a local index theorem in this context looks like and will explain another approach of the proof of the index formula in the Lorentzian context. If time permits I will make some remarks about the relation to the chiral anomaly.

András Vasy (Stanford University)

Feynman propagators, positive propagator differences and essential self-adjointness

Abstract:  I will discuss global Fredholm frameworks for Feynman (as well as causal) propagators in a variety of analytic and geometric settings. I will then discuss how this relates to positive propagator differences in these settings, as well as essential self-adjointness of the wave operator in geometries modelled on generalizations of Minkowski space at infinity. Parts of this work are joint with Jesse Gell-Redman, Nick Haber and Michał Wrochna.