Notice
Christian Gérard - Introduction to field theory on curved spacetimes (Part 1)
- document 1 document 2 document 3
- niveau 1 niveau 2 niveau 3
Descriptif
The aim of these lectures is to give an introduction to quantum field theory on curved spacetimes, from the point of view of partial differential equations and microlocal analysis. I will concentrate on free fields and quasi-free states, and say very little on interacting fields or perturbative renormalization. I will start by describing the necessary algebraic background, namely CCR and CAR algebras, and the notion of quasi-free states, with their basic properties and characterizations. I will then introduce the notion of globally hyperbolic spacetimes, and its importance for classical field theory (advanced and retarded fundamental solutions, unique solvability of the Cauchy problem). Using these results I will explain the algebraic quantization of the two main examples of quantum fields ona manifold, namely the Klein-Gordon (bosonic) and Dirac (fermionic) fields.In the second part of the lectures I will discuss the important notion of Hadamardstates , which are substitutes in curved spacetimes for the vacuum state in Minkowskispacetime. I will explain its original motivation, related to the definition of therenormalized stress-energy tensor in a quantum field theory. I will then describethe modern characterization of Hadamard states, by the wavefront set of their twopointfunctions, and prove the famous Radzikowski theorem , using the Duistermaat-Hörmander notion of distinguished parametrices . If time allows, I will also describe the quantization of gauge fields, using as example the Maxwell field.
Documentation
Liens
Dans la même collection
-
Philippe G LeFloch - Weakly regular spacetimes with T2 symmetry
LeFlochPhilippe G.I will discuss the initial value problem for the Einstein equations and present results concerning the existence and asymptotic behavior of
-
Andras Vasy - Quasilinear waves and trapping: Kerr‐de Sitter space
VasyAndrásIn this talk I will describe recent work with Peter Hintz on globally solving quasilinear wave equations in the presence of trapped rays,
-
Claudio Dappiaggi - On the role of asymptotic structures in the construction of quantum states for …
DappiaggiClaudioIn the algebraic approach to quantum field theory on curved backgrounds, there exists a special class of quantum states for free fields,
-
Andras Vasy - Microlocal analysis and wave propagation (Part 2)
In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In
-
Jérémie Joudioux - Hertz potentials and the decay of higher spin fields
JoudiouxJérémieThe study of the asymptotic behavior of higher spin fields has proven to be a key point in understanding the stability properties of
-
Thomas Backdahl - Symmetry operators, conserved currents and energy momentum tensors
BackdahlThomasConserved quantities, for example energy and momentum, play a fundamental role in the analysis of dynamics of particles and fields. For
-
Jérémie Szeftel The resolution of the bounded L2 curvature conjecture in General Relativity (Part 1)
SzeftelJérémieIn order to control locally a space time which satisfies the Einstein equations, what are the minimal assumptions one should make on its
-
Lionel Mason - Perturbative formulae for scattering of gravitational wave
MasonLionel J.The Christodoulou Klainerman proof of existence of asymptotically simple space-times shows that it is reasonable to consider the scattering of
-
Andras Vasy - Microlocal analysis and wave propagation (Part 4)
In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In
-
Alexander Strohmaier - Workshop
StrohmaierAlexanderI will explain how one can formulate and formalize the Gupta Bleuler framework for the Quantization of the electromagnetic field in an
-
-
Andras Vasy - Microlocal analysis and wave propagation (Part 1)
VasyAndrásIn these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on
Avec les mêmes intervenants et intervenantes
-
Christian Gérard - Aspects de la théorie quantique des champs en espace-temps courbe
GérardChristianLa théorie quantique des champs est formulée d'habitude sur l'espace-temps plat de Minkowski. L'extension de ce cadre à des espaces-temps généraux permet de mettre en lumière de nouveaux
Sur le même thème
-
"Le mathématicien Petre (Pierre) Sergescu, historien des sciences, personnalité du XXe siècle"
HerléaAlexandreAlexandre HERLEA est membre de la section « Sciences, histoire des sciences et des techniques et archéologie industrielle » du CTHS. Professeur émérite des universités, membre effectif de l'Académie
-
Webinaire sur la rédaction des PGD
LouvetViolaineRédaction des Plans de Gestion de Données (PGD) sous l’angle des besoins de la communauté mathématique.
-
Alexandre Booms : « Usage de matériel pédagogique adapté en géométrie : une transposition à interro…
« Usage de matériel pédagogique adapté en géométrie : une transposition à interroger ». Alexandre Booms, doctorant (Université de Reims Champagne-Ardenne - Cérep UR 4692)
-
R. Bamler - Compactness and partial regularity theory of Ricci flows in higher dimensions
BamlerRichard H.We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow.
-
P. Burkhardt - Pointwise lower scalar curvature bounds for C0 metrics via regularizing Ricci flow
Burkhardt-GuimPaulaWe propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second
-
C. Li - Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions
LiChaoIn this talk, I will discuss some recent developments on the topology of closed manifolds admitting Riemannian metrics of positive scalar curvature. In particular, we will prove if a closed PSC
-
T. Ozuch - Noncollapsed degeneration and desingularization of Einstein 4-manifolds
OzuchTristanWe study the noncollapsed singularity formation of Einstein 4-manifolds. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result
-
D. Tewodrose - Limits of Riemannian manifolds satisfying a uniform Kato condition
TewodroseDavidPresentation of a joint work with G. Carron and I. Mondello where we study Kato limit spaces.
-
Y. Lai - A family of 3d steady gradient Ricci solitons that are flying wings
LaiYiWe find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at
-
A. Mondino - Time-like Ricci curvature bounds via optimal transport
MondinoAndreaThe goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the
-
M. Lesourd - Positive Scalar Curvature on Noncompact Manifolds and the Positive Mass Theorem
LesourdMartinThe study of positive scalar curvature on noncompact manifolds has seen significant progress in the last few years. A major role has been played by Gromov's results and conjectures, and in
-
R. Perales - Recent Intrinsic Flat Convergence Theorems
PeralesRaquelThéorèmes récents de convergence plane intrinsèque