Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes

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 $\frac{i}{2}\big(\widetilde{G}_{aF} - \widetilde{G}_{F} + \widetilde{G}_A - \widetilde{G}_R\big)$ in the sense of Duistermaat-H\"ormander. Inspired by the construction of the corresponding advanced and retarded Green operator $G_A,G_R$ as done in B\"ar, Ginoux, Pf\"affle 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 $\frac{i}{2}\big(\widetilde{G}_{aF} - \widetilde{G}_{F}\big)$, which relies on new techniques like a well-posed Cauchy problem for bisolutions and a patching argument using \v{C}ech 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.