Theorem: Local Lorentzian Hadamard form

Every regular Lorentzian point has a sufficiently small developing neighbourhood on which

where $H_F$ is smooth and locally analytic. It is obtained by continuing the time difference of $H_{F,E}$.

Proof. For positive Euclidean separation the spectral kernel is $e^{-a\sqrt{B_F}}/(2\sqrt{B_F})$. Its boundary value at $a=ϵ+\mathrm{i}(t-t^{\prime })$ is $W_F$, by the spectral theorem and Lemma lem:infrared. The analytic remainder has a holomorphic extension near the regular diagonal. After shrinking the neighbourhood it continues smoothly in the time difference. The direct term gives the displayed boundary value. The local geometry and connection are flat and the field is massless, so the local logarithmic Hadamard coefficient vanishes DecaniniFolacci.