Appears in collection : Summer School 2014 - Asymptotic Analysis in General Relativity

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 andCAR algebras, and the notion of quasi-free states, with their basic properties andcharacterizations. 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 on a 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.