Copied conventions

$M={\mathbb{R}}^4$, $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\square ={{\partial }_t}^2-\Delta$, $P_m=\square +m^2$, $m\ge 0$. $\mathcal{D}=\mathcal{D}(M)={C_c}^{\infty }(M;\mathbb{C})$ with its LF topology. The retarded and advanced Green operators ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}:\mathcal{D}\to C^{\infty }(M)$ satisfy

and $({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{)}^{\mathsf{T}}={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, i.e.\ $⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,d⟩=⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}d⟩$ for $c,d\in \mathcal{D}$. Put ${\Delta }_S={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$; then ${{\Delta }_S}^{\mathsf{T}}=-{\Delta }_S$, and ${\Delta }_S\mathcal{D}$ is exactly the space of smooth solutions of $P_{m}u=0$ with spatially compact support BGP (Thm. 3.4.7). The sign conventions of X-B are

A Hadamard bisolution ${\Delta }_{+}\in {\mathcal{D}}^{\prime }(M\times M)$ is fixed once, with

where in addition the first covector lies in one closed half of the null cone Radzikowski (Thm. 5.1); in particular no element of $\mathrm{WF}({\Delta }_{+})$ has a zero covector in either slot.

${\mathcal{A}}_{\mathrm{off}}$ is the algebra of polynomial functionals of $\phi \in C^{\infty }(M)$ with compact derivative kernels, with the Wick product $F\star G={\sum }_{n\ge 0}\frac{{\hslash }^n}{n!}⟨F^{(n)},{\Delta }_{+}^{\otimes n}{G}^{(n)}⟩$. For $c\in \mathcal{D}$, $F_c(\phi )=\int c\phi {\mathrm{d}}^{4}x$ and ${\mathcal{L}}_{\mathrm{off}}=\{F_c\}\cong \mathcal{D}$. ${\mathfrak{J}}_0$ is the pointwise ideal generated by $\{F_{P_{m}u}:u\in \mathcal{D}\}$; it is a star ideal; ${\mathfrak{A}}_0={\mathcal{A}}_{\mathrm{off}}/{\mathfrak{J}}_0$ and $\hat{\varphi }(c)=[F_c]$. Elements of ${\mathfrak{J}}_0$ vanish on smooth free solutions. Field degree is the polynomial degree in $\phi$; ${{\mathcal{A}}_{\mathrm{off}}}^{(n)}$ denotes the homogeneous part of degree $n$, and ${{\mathfrak{A}}_0}^{(1)}$ the image of ${{\mathcal{A}}_{\mathrm{off}}}^{(1)}$ in ${\mathfrak{A}}_0$, i.e.\ ${{\mathfrak{A}}_0}^{(1)}=\{\hat{\varphi }(d):d\in \mathcal{D}\}$.