Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes

Abstract

According to Radzikowski's celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of Hadamard form iff they are given by a linear combination of distinguished parametrices $\frac{i}{2}({\tilde{G}}_{aF}-{\tilde{G}}_F+{\tilde{G}}_A-{\tilde{G}}_R)$ in the sense of Duistermaat-Hörmander. Inspired by the construction of the corresponding advanced and retarded Green operator $G_A,G_R$ as done in Bär, Ginoux, Pfäffle 2007, we construct the remaining two Green operators $G_F,G_{aF}$ locally in terms of Hadamard series. Afterwards, we provide the global construction of $\frac{i}{2}({\tilde{G}}_{aF}-{\tilde{G}}_F)$, which relies on new techniques like a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e. the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.

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