Scalar future mass shell, invariant measure and test restriction

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Here $\mathcal{S}({\mathbb{R}}^4)$ is the Schwartz space of smooth complex functions whose coordinate-weighted derivatives are all bounded, as defined in Schwartz space and its seminorms. Fix an oriented, time-oriented affine Minkowski space and an inertial coordinate chart as in Oriented time-oriented affine Minkowski space (the spacetime primitives) and Inertial frames and inertial charts of the affine Minkowski model. Use coordinates $x=(x^0,\mathbf{x})$ and metric $\eta =\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(-1,1,1,1)$, matching the $(-+++)$ convention of the SR suppliers. Some QFT sources use its negative $(+---)$; the Lorentz group is unchanged, and the phase is fixed explicitly below. Coordinates $x$ have length units, momenta have inverse-length units, and the Fourier phase below is dimensionless. A parameter $\mu \ge 0$ has inverse-length units and is only a mathematical shell parameter; it is not identified with a physical rest mass. For $\mathbf{k}\in {\mathbb{R}}^3$, set ${\omega }_{\mu }(\mathbf{k})=\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+{\mu }^2},p_{\mu }(\mathbf{k})=({\omega }_{\mu }(\mathbf{k}),\mathbf{k})\mathrm{.}$ The future mass shell is $H_{\mu }^{+}=\{(p^0,\mathbf{p}):(p^0{)}^2-\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2={\mu }^2,p^0>0\}\mathrm{.}$ When $\mu =0$, its apex is omitted; thus $\mathbf{k}\mathrm{\ne }0$ in the massless parameterization. The scalar shell pairing is $p\circ x:=p^0{x}^0-\mathbf{p}\cdot \mathbf{x}=-\eta (p,x)\mathrm{.}$ Define the measure and one-particle Hilbert space by $d{\nu }_{\mu }(p_{\mu }(\mathbf{k}))=\frac{d^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mu }(\mathbf{k})},H_{\mu }=L^2(H_{\mu }^{+},d{\nu }_{\mu };\mathbb{C}),$ where the omitted massless apex has measure zero. For $f\in \mathcal{S}({\mathbb{R}}^4)$ define its pointwise shell Fourier restriction by $K_{\mu }f(p)={\int }_{{\mathbb{R}}^4}f(x)e^{ip\circ x}{d}^{4}x\mathrm{.}$ The Fourier convention is exactly the displayed integral, with no prefactor. If $\widehat{f}({\xi }^0,\mathit{\xi })=\int f(x)e^{-2\pi i(x^0{\xi }^0+\mathbf{x}\cdot \mathit{\xi })}{d}^{4}x$ denotes the standard transform in Fourier transform is a topological automorphism of Schwartz space, then ${\mathcal{F}}_{+}f(p^0,\mathbf{p}):=\int f(x)e^{i(p^0{x}^0-\mathbf{p}\cdot \mathbf{x})}{d}^{4}x=\widehat{f}\text{\hspace{0.17em}⁣}(-\frac{p^0}{2\pi },\frac{\mathbf{p}}{2\pi })\mathrm{.}$ Its inverse is $f(x)=(2\pi {)}^{-4}{\int }_{{\mathbb{R}}^4}{\mathcal{F}}_{+}f(p^0,\mathbf{p})e^{-i(p^0{x}^0-\mathbf{p}\cdot \mathbf{x})}d{p}^0{d}^3\mathbf{p}\mathrm{.}$ Thus this convention is a time reflection and rescaling of the cited Schwartz automorphism and has the stated reciprocal normalization. The proper orthochronous Lorentz group consists of linear maps preserving $\eta$, orientation and the chosen future cone. A Poincare map is $x\mapsto a+\mathrm{\Lambda }x$ with $a\in {\mathbb{R}}^4$ and $\mathrm{\Lambda }$ in that group.

This is a mathematical free scalar one-particle construction. It adopts the chosen coordinates, measure and Fourier convention; it asserts no experimental observation, physical mass identification or measurement outcome.