Seminarium Teorii Względności i Grawitacji

On a large class of globally hyperbolic one can define four natural Green functions of the Klein-Gordon equation. As is well-known, we have the forward propagator and the backward propagator. It is less known that usually one can also define the (distinguished) Feynman and anti-Feynman propagator, useful in Quantum Field Theory, but also well-motivated by operator theory.One of possible definitions of the two latter propagators is the boundary value of the resolvent of the Klein-Gordon operator. Note that this definition presupposes the (essential) self-adjointness of the Klein-Gordon operator, a question which sounds bizarre and academic at the first sight, but is surprisingly physical relevant.In some rare but important cases these four propagator satisfy a very useful property: Forward+Backward=Feynman+anti-Feynman. I will discuss consequences of this property and, time permitting, examples where it is satisfied. These examples include: Minkowski space, 1-dimensional spaces with reflectionless potentials, deSitter and and anti-deSitter spaces.