Quantum Field Theory in Curved Spacetime

In this talk, I discuss the theory of the covariant Klein Gordon equation, (□ g + m 2)∅ = 0, on the class of globally hyperbolic spacetimes (M, g). (Here globally hyperbolic means there exists a diffeomorphism δ : M → R × C where C is a three manifold and each of the surfaces δ -1({t} × C) is a Cauchy surface i.e. an achronal 3-surface which is cut precisely once by every inextendible causal curve in M.) My aim is to give an impression of what has been achieved in clarifying the conceptual and mathematical foundations of quantum field theory in curved spacetime and to raise some open issues of importance for the further development of the subject. Lack of space enforces the omission of many important topics (especially, the back-reaction problem and the definition of a quantum energy momentum tensor T μν , higher spin fields, interacting fields, and non-globally hyperbolic spacetimes) and there will be space neither for an adequate bibliography nor to always provide adequate motivation and details. The remedy for some of these deficiencies will be found in the references (see especially Section 3 in [1]).