Tools of quantum field theory over curved backgrounds Lectures given

These lectures are thought as a short introduction to the main tools of quantum field theory over curved backgrounds. The topics are thus introduced assuming that a potential reader has no previous specific knowledge on any of the used concepts except for quantum field theory over Minkowski spacetime. The main goal will be to show that, whenever the spacetime is not flat, the most effective approach to quantization is the algebraic one; to this avail, we will also consider the more common techniques and we shall show their limits of applicability. Particularly our desire is to put a reader in the condition of reading most if not all the papers dealing with concrete applications of the algebraic approach to quantum field theory over curved backgrounds without forcing him to browse through dozens of other publications in order to understand the techniques used. Contents 1 A walk through the geometry of curved spacetimes 3 1.1 Manifolds and differentiable structure: what is a smooth field?................. 4 1.1.1 Vectors and Tangent space: what is the derivative of a field?.............. 6 1.1.2 The metric structure: what plays the role of ηµν?.....................