Minkowski Fourier transform and forward-cone support convention

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the Fourier transform suppliers. With $p\cdot z$ from Proper-orthochronous Poincaré action and test pullback, for $\psi \in \mathcal{S}({\mathbb{R}}^4)$ set ${\mathcal{F}}_M\psi (z):={\int }_{{\mathbb{R}}^4}{e}^{-ip\cdot z}\psi (p)d^{4}p\mathrm{.}$ To see that this is a continuous automorphism of Schwartz space, write ${\mathcal{F}}_{2\pi }$ for the Euclidean transform ${\mathcal{F}}_{2\pi }\psi (x)=\int e^{-2\pi ix\cdot p}\psi (p)d^{4}p$, as in Fourier transform of a tempered distribution. For $u,v\in {\mathbb{R}}^4$, write $u{\cdot }_{\mathrm{E}}v:={\sum }_{\mu =0}^3{u}^{\mu }{v}^{\mu }$. Since $p\cdot z=(\eta z){\cdot }_{\mathrm{E}}p$, the same defining integral gives ${\mathcal{F}}_M\psi (z)=({\mathcal{F}}_{2\pi }\psi )(\eta z\mathrm{/}(2\pi ))\mathrm{.}$ The map $z\mapsto \eta z\mathrm{/}(2\pi )$ is an invertible real-linear substitution, so output composition with it is a continuous automorphism of Schwartz space by Invertible linear substitutions preserve Schwartz space, and ${\mathcal{F}}_{2\pi }$ is a topological automorphism by Fourier transform is a topological automorphism of Schwartz space.

For $D\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^4)$ define its Minkowski Fourier transform by bilinear transposition, $⟨{\widehat{D}}_M,\psi ⟩:=⟨D,{\mathcal{F}}_M\psi ⟩(\psi \in \mathcal{S}({\mathbb{R}}^4))\mathrm{.}$ For a closed set $C\subseteq {\mathbb{R}}^4$, the statement $\mathrm{supp}{\widehat{D}}_M\subseteq C$ means $⟨{\widehat{D}}_M,\psi ⟩=0$ for every compactly supported smooth $\psi$ whose support is disjoint from $C$; the inclusion $C_c^{\mathrm{\infty }}({\mathbb{R}}^4)\hookrightarrow \mathcal{S}({\mathbb{R}}^4)$ is Test function inclusion in schwartz space is continuous. In particular, the forward-cone condition means $C={\bar{V}}_{+}$ with the definition in Proper-orthochronous Poincaré action and test pullback. No joint self-adjoint energy-momentum generators are included in this convention.