Scalar future mass-shell measure and test restriction

On $\mathbf{k}\mathrm{\ne }0$ if $\mu =0$, and on all $\mathbf{k}$ if $\mu >0$, $F_{\mu }$ is a smooth chart of the shell. On every compact subset of this chart the density $[(2\pi {)}^{3}2{\omega }_{\mu }{]}^{-1}$ is bounded; for $\mu =0$ near the omitted apex it is proportional to $\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^{-1}$, whose integral over a radius-$r$ ball is proportional to ${\int }_0^{r}sds<\mathrm{\infty }$. Compact annuli, and for $\mu >0$ compact balls, give a countable cover by finite-measure sets; hence $d{\nu }_{\mu }$ is locally finite and sigma-finite.

Fix $\mathrm{\Lambda }$ and write $F=F_{\mu }$, $F({\mathbf{k}}^{\mathrm{\prime }})=\mathrm{\Lambda }F(\mathbf{k})$. Because $\mathrm{\Lambda }$ preserves the future shell, this defines a smooth diffeomorphism $\mathbf{k}\mapsto {\mathbf{k}}^{\mathrm{\prime }}$ on the chart; let $J=D{\mathbf{k}}^{\mathrm{\prime }}\mathrm{/}D\mathbf{k}$. Put $q_0=(1,0,0,0)$. The columns of $[q_0,DF(\mathbf{k})]$ have determinant $1$. Differentiating $\mathrm{\Lambda }F(\mathbf{k})=F({\mathbf{k}}^{\mathrm{\prime }})$ gives $\mathrm{\Lambda }DF(\mathbf{k})=DF({\mathbf{k}}^{\mathrm{\prime }})J$, and therefore $1=\det [\mathrm{\Lambda }{q}_0,DF({\mathbf{k}}^{\mathrm{\prime }})]\det J\mathrm{.}$ For any $u\in {\mathbb{R}}^4$, direct expansion of the determinant gives $\det [u,DF({\mathbf{k}}^{\mathrm{\prime }})]=-\eta (u,F({\mathbf{k}}^{\mathrm{\prime }}))\mathrm{/}{\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})$. Take $u=\mathrm{\Lambda }{q}_0$. Metric preservation and $F({\mathbf{k}}^{\mathrm{\prime }})=\mathrm{\Lambda }F(\mathbf{k})$ imply $-\eta (\mathrm{\Lambda }{q}_0,F({\mathbf{k}}^{\mathrm{\prime }}))=-\eta (q_0,F(\mathbf{k}))={\omega }_{\mu }(\mathbf{k})$. Thus $\det J={\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})\mathrm{/}{\omega }_{\mu }(\mathbf{k})>0$.

The change-of-variables theorem [F3] and the Jacobian from 2.1 give $\frac{d^3{\mathbf{k}}^{\mathrm{\prime }}}{{\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})}=\frac{\det J{d}^3\mathbf{k}}{{\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})}=\frac{d^3\mathbf{k}}{{\omega }_{\mu }(\mathbf{k})}\mathrm{.}$ For the pullback $g({\mathrm{\Lambda }}^{-1}p)$ in the Statement, apply the same calculation to ${\mathrm{\Lambda }}^{-1}$, which is again proper orthochronous. Multiplication by the constant $[2(2\pi {)}^3{]}^{-1}$ then proves invariance for nonnegative Borel functions. The massless apex is a single omitted point and was never in the chart; the locally finite density has no extra point mass there.

In the chart $\mathbf{k}\mapsto p_{\mu }(\mathbf{k})$, write $w_{\mu }(\mathbf{k})=[2(2\pi {)}^3{\omega }_{\mu }(\mathbf{k}){]}^{-1}$. By [F3] and the fixed invertible reflection and rescaling in Scalar future mass shell, invariant measure and test restriction, ${\mathcal{F}}_{+}f$ is Schwartz for every $f\in \mathcal{S}({\mathbb{R}}^4)$. Thus $q_2(f):={\sup }_{q\in {\mathbb{R}}^4}(1+\mathrm{\mid }q\mathrm{\mid }{)}^2\mathrm{\mid }{\mathcal{F}}_{+}f(q)\mathrm{\mid }<\mathrm{\infty }\mathrm{.}$ On the shell, $\mathrm{\mid }{\mathcal{F}}_{+}f({\omega }_{\mu }(\mathbf{k}),\mathbf{k})\mathrm{\mid }\le q_2(f)(1+\mathrm{\mid }\mathbf{k}\mathrm{\mid }{)}^{-2}$, so $\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}_{H_{\mu }}^2\le \frac{q_2(f{)}^2}{2(2\pi {)}^3}{\int }_{{\mathbb{R}}^3}\frac{(1+\mathrm{\mid }\mathbf{k}\mathrm{\mid }{)}^{-4}}{{\omega }_{\mu }(\mathbf{k})}{d}^3\mathbf{k}<\mathrm{\infty }\mathrm{.}$ The radial integrand is $O(r)$ near zero if $\mu =0$, $O(r^2)$ near zero if $\mu >0$, and $O(r^{-3})$ at infinity. Therefore every $K_{\mu }f$ lies in $H_{\mu }$, so the claimed range is a subset of the Hilbert space. By [F1], real-valued $L^2({\mathbb{R}}^3)$ has a countable dense subset $S_{\mathbb{R}}$. For complex-valued $h=u+iv\in L^2({\mathbb{R}}^3;\mathbb{C})$, its real and imaginary parts lie in real-valued $L^2$. For any $\epsilon >0$, choose $s,t\in S_{\mathbb{R}}$ with $\mathrm{\parallel }u-s{\mathrm{\parallel }}_2,\mathrm{\parallel }v-t{\mathrm{\parallel }}_2<\epsilon \mathrm{/}\sqrt{2}$. Then $\mathrm{\parallel }h-(s+it){\mathrm{\parallel }}_2^2=\mathrm{\parallel }u-s{\mathrm{\parallel }}_2^2+\mathrm{\parallel }v-t{\mathrm{\parallel }}_2^2<{\epsilon }^2$. Enumerate $S_{\mathbb{R}}$ by $(s_j{)}_{j\in {\mathbb{N}}_0}$, repeating terms if it is finite. The Cantor pairing map $(j,k)\mapsto (j+k)(j+k+1)\mathrm{/}2+k$ enumerates ${\mathbb{N}}_0^2$, so $S_{\mathbb{R}}+i{S}_{\mathbb{R}}$ is countable and dense in complex-valued $L^2({\mathbb{R}}^3)$. Multiplication by $\sqrt{w_{\mu }}$ is an isometry from $H_{\mu }$ onto $L^2({\mathbb{R}}^3\setminus \{0\},d^3\mathbf{k};\mathbb{C})$ (for $\mu =0$) or $L^2({\mathbb{R}}^3,d^3\mathbf{k};\mathbb{C})$ (for $\mu >0$), with inverse division by $\sqrt{w_{\mu }}$. In the massless case, extend any function on the punctured space by zero at the null point, approximate it by the countable dense family just constructed, and restrict those approximants back to the punctured space; restriction does not increase the norm. Thus both chart spaces, and therefore $H_{\mu }$, are separable.

Let $\mathrm{\Omega }={\mathbb{R}}^3$ for $\mu >0$ and $\mathrm{\Omega }={\mathbb{R}}^3\setminus \{0\}$ for $\mu =0$. For ${\mathrm{\Omega }}_n=\{1\mathrm{/}n<\mathrm{\mid }\mathbf{k}\mathrm{\mid }<n\}$, these sets increase to $\mathrm{\Omega }\setminus \{0\}$; its complement in $\mathrm{\Omega }$ is null for the shell measure. For $h\in L^2(\mathrm{\Omega },w_{\mu }{d}^3\mathbf{k})$, monotone convergence [F5] gives ${\int }_{{\mathrm{\Omega }}_n}\mathrm{\mid }h{\mathrm{\mid }}^2{w}_{\mu }{d}^3\mathbf{k}⟶{\int }_{\mathrm{\Omega }}\mathrm{\mid }h{\mathrm{\mid }}^2{w}_{\mu }{d}^3\mathbf{k}<\mathrm{\infty }\mathrm{.}$ Subtracting these finite integrals from the total shows $\mathrm{\parallel }h{\mathbf{1}}_{\mathrm{\Omega }\setminus {\mathrm{\Omega }}_n}{\mathrm{\parallel }}_{L^2(w_{\mu })}\to 0$. On each ${\mathrm{\Omega }}_n$, $w_{\mu }$ is smooth and bounded above and below by positive constants. Approximate $h{\mathrm{\mid }}_{{\mathrm{\Omega }}_n}$ in ordinary $L^2({\mathrm{\Omega }}_n)$ by a function in $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_n)$ using [F2]; equivalence of the weighted and ordinary norms on ${\mathrm{\Omega }}_n$ gives weighted approximation. Thus $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is dense in the shell space.

Fix $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Choose $\rho \in C_c^{\mathrm{\infty }}(\mathbb{R})$ with $\rho (0)=1$, using [F4], and put $G(q^0,\mathbf{q})=\psi (\mathbf{q})\rho (q^0-{\omega }_{\mu }(\mathbf{q}))\mathrm{.}$ Because $\psi$ has compact support away from the massless apex, ${\omega }_{\mu }$ is smooth on a neighborhood of its support, so this is a smooth compactly supported function on ${\mathbb{R}}^4$. By the Fourier automorphism [F3] there is $f\in \mathcal{S}({\mathbb{R}}^4)$ with ${\mathcal{F}}_{+}f(q^0,\mathbf{q})=G(q^0,\mathbf{q})$ in the convention of Scalar future mass shell, invariant measure and test restriction. On the shell, $K_{\mu }f(p_{\mu }(\mathbf{k}))=G({\omega }_{\mu }(\mathbf{k}),\mathbf{k})=\psi (\mathbf{k})$. Hence the test restriction range contains $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, which is dense by 2.1.

Define the continuous Schwartz seminorm $q_2(f)={\sup }_{p\in {\mathbb{R}}^4}(1+\mathrm{\mid }p\mathrm{\mid }{)}^2\mathrm{\mid }{\mathcal{F}}_{+}f(p)\mathrm{\mid }\mathrm{.}$ Continuity follows from [F3,F4] and [F2]. On shell, $\mathrm{\mid }{\mathcal{F}}_{+}f(p_{\mu }(\mathbf{k}))\mathrm{\mid }\le q_2(f)(1+\mathrm{\mid }\mathbf{k}\mathrm{\mid }{)}^{-2}$ because $\mathrm{\mid }{p}_{\mu }(\mathbf{k})\mathrm{\mid }\ge \mathrm{\mid }\mathbf{k}\mathrm{\mid }$. Hence $\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}_{H_{\mu }}^2\le \frac{q_2(f{)}^2}{2(2\pi {)}^3}{\int }_{{\mathbb{R}}^3}\frac{(1+\mathrm{\mid }\mathbf{k}\mathrm{\mid }{)}^{-4}}{{\omega }_{\mu }(\mathbf{k})}{d}^3\mathbf{k}\mathrm{.}$ The integral is finite: in polar coordinates its radial integrand is $r^2(1+r{)}^{-4}\mathrm{/}\sqrt{r^2+{\mu }^2}$, which is $O(r)$ near $0$ when $\mu =0$, is $O(r^2)$ near $0$ when $\mu >0$, and is $O(r^{-3})$ at infinity. Therefore $K_{\mu }f\in H_{\mu }$, and the displayed seminorm bound proves continuity and linearity.

By [F4], applying ${\mathrm{\partial }}_0^2-\mathrm{\Delta }+{\mu }^2$ before ${\mathcal{F}}_{+}$ multiplies by $-(p^0{)}^2+\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2+{\mu }^2$. On the shell this multiplier is $-{\omega }_{\mu }(\mathbf{k}{)}^2+\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+{\mu }^2=0$. Restriction to the shell therefore gives the asserted Klein–Gordon identity.

Since $U$ is unitary, $U^{\otimes n}$ is unitary on the completed tensor power by its isometric action on elementary tensors and its inverse $(U^{-1}{)}^{\otimes n}$. It commutes with every tensor-factor permutation, because both compositions send an elementary tensor to $U{h}_{\pi (1)}\otimes \cdots \otimes U{h}_{\pi (n)}$. It therefore commutes with the symmetric averaging projection of [F2] and restricts to a unitary $U_s^{(n)}$ on its range. The zeroth sector is the identity on $\mathbb{C}$.

The direct sum of the sector unitaries preserves the sum of squared sector norms and has inverse ${⨁}_n(U^{-1}{)}_s^{(n)}$, so it is unitary. It fixes the zeroth-sector vacuum and preserves finite sums of sectors in both directions. On an elementary symmetric tensor, applying the Fock creation formula before or after this direct sum gives the symmetrization of $Uh$ together with the images of the original tensor factors; hence the two creation maps intertwine. The annihilation formula is its adjoint contraction and the same tensor calculation replaces each contracted vector $h$ by $Uh$. Linearity, boundedness on each fixed sector, and density of elementary tensors extend both identities to ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$.

Fix $n$. For an elementary tensor, $\mathrm{\parallel }{U}_g^{\otimes n}(h_1\otimes \cdots \otimes h_n)-U_{g_0}^{\otimes n}(h_1\otimes \cdots \otimes h_n)\mathrm{\parallel }\le {\sum }_{r=1}^n{\prod }_{s\mathrm{\ne }r}\mathrm{\parallel }{h}_s\mathrm{\parallel }\mathrm{\parallel }(U_g-U_{g_0})h_r\mathrm{\parallel },$ by the telescoping expansion of the tensor difference and unitary norm preservation. Strong continuity of $U_g$ makes the right side tend to zero. Density of elementary tensors and the uniform norm bound $1$ extend continuity to the completed sector and its closed symmetric subspace. Finite sector sums are dense by [F1]; approximating an arbitrary Fock vector by such a sum and using $\mathrm{\parallel }{\mathrm{\Gamma }}_s(U_g)\mathrm{\parallel }=1$ proves strong continuity on the whole direct sum.

The affine pullback $f_g(x)=f({\mathrm{\Lambda }}^{-1}(x-a))$ preserves Schwartz space. Repeated chain rule [F6] expresses each derivative as a finite linear combination of derivatives of $f$ at ${\mathrm{\Lambda }}^{-1}(x-a)$. Also $1+\mathrm{\mid }x\mathrm{\mid }\le C(1+\mathrm{\mid }{\mathrm{\Lambda }}^{-1}(x-a)\mathrm{\mid })$ for a finite constant $C$ depending on $a,\mathrm{\Lambda }$. Thus every polynomially weighted derivative remains bounded by the corresponding finite sum of Schwartz seminorms.

Multiplication by $e^{ip\circ a}$ preserves the $H_{\mu }$ norm. Composition with ${\mathrm{\Lambda }}^{-1}$ also preserves that norm by [F2]; it preserves null-equivalence classes for the same reason. Therefore $U_{\mu }(g)$ is a well-defined isometry. Its inverse is the corresponding formula for $g^{-1}$, so it is unitary. Direct substitution and Lorentz invariance of $p\circ x$ give $U_{\mu }(a,\mathrm{\Lambda })U_{\mu }(b,\mathrm{\Gamma })=U_{\mu }(a+\mathrm{\Lambda }b,\mathrm{\Lambda }\mathrm{\Gamma })$.

In the Fourier integral for $K_{\mu }{f}_g$, change variables $x=a+\mathrm{\Lambda }y$; [F5] gives $\mathrm{\mid }\det \mathrm{\Lambda }\mathrm{\mid }=1$ and [F7] justifies the change of variables. The integrand is absolutely integrable because $f$ is Schwartz; apply [F7] to the positive and negative parts of its real and imaginary components to obtain this change of variables for the complex integral. Lorentz invariance of the phase then gives $K_{\mu }{f}_g(p)=e^{ip\circ a}{K}_{\mu }f({\mathrm{\Lambda }}^{-1}p)\mathrm{.}$ Both sides belong to $H_{\mu }$ by [F4], so this equality is also equality of Hilbert-space vectors. This is the action formula used in step 1.2.

Let $h\in C_c^{\mathrm{\infty }}$ in the shell chart, with support away from the massless apex. If $g\to g_0$ in the Poincare group, $U_{\mu }(g)h$ converges pointwise to $U_{\mu }(g_0)h$. In a compact neighborhood of $g_0$, the supports lie in one compact chart subset $C$ away from the apex; ${\nu }_{\mu }(C)<\mathrm{\infty }$ by [F2]. The squared pointwise difference is bounded by $4\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}^2{\mathbf{1}}_C$, so [F8] gives norm continuity for this dense family. Density follows from [F3]. For arbitrary $v\in H_{\mu }$, choose such $h$ with $\mathrm{\parallel }v-h\mathrm{\parallel }<ϵ$; unitarity gives $\mathrm{\parallel }{U}_{\mu }(g)v-U_{\mu }(g_0)v\mathrm{\parallel }\le 2ϵ+\mathrm{\parallel }{U}_{\mu }(g)h-U_{\mu }(g_0)h\mathrm{\parallel }\mathrm{.}$ First take $g\to g_0$ and then $ϵ\downarrow 0$. This proves strong continuity on all of $H_{\mu }$.

Since $K_{\mu }f,K_{\mu }\bar{f}\in H_{\mu }$ by [F9], the field formula is defined on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. The first term $a(K_{\mu }\bar{f})$ is complex-linear in $f$ because complex conjugation is followed by the conjugate-linear dependence of $a$; the second is linear because both $a^{†}$ and $K_{\mu }$ are linear. Both terms preserve ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ by [F1,F2]. For real $f$, the shell vectors coincide, so ${\mathrm{\Phi }}_{\mu }(f)=a(K_{\mu }f)+a^{†}(K_{\mu }f)=\sqrt{2}{\mathrm{\Phi }}_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(K_{\mu }f){\mathrm{\mid }}_{{\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}}$. It is symmetric on the common finite-particle domain by [F2]. Substituting $h=\sqrt{2}{K}_{\mu }f$ and $k=\sqrt{2}{K}_{\mu }g$ into [F3]'s Weyl relation gives the stated scalar phase, and its vacuum Gaussian gives $⟨\mathrm{\Omega },W_{\mu }(f)\mathrm{\Omega }{⟩}_2=e^{-\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}_2^2\mathrm{/}2}$.

For $g=(a,\mathrm{\Lambda })$, [F10] gives the one-particle representation and $U_{\mu }(g)K_{\mu }f=K_{\mu }{f}_g$. Since the affine action is real, $U_{\mu }(g)K_{\mu }\bar{f}=K_{\mu }\overline{f_g}$. The Fock lift [F7] intertwines creation and annihilation, so the field formula in step 1.1 gives ${\mathrm{\Gamma }}_s(U_{\mu }(g)){\mathrm{\Phi }}_{\mu }(f){\mathrm{\Gamma }}_s(U_{\mu }(g){)}^{-1}={\mathrm{\Phi }}_{\mu }(f_g)$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. The explicit Segal Weyl formula [F3] and the same lift give bounded Weyl covariance. Strong continuity of the Fock lifts follows from [F7] and [F10].

Since $a(K_{\mu }\bar{f})\mathrm{\Omega }=0$ and $⟨\mathrm{\Omega },a^{†}(h)\mathrm{\Psi }{⟩}_2=0$, the vacuum one-point function vanishes. Expanding the two fields, only $a(K_{\mu }\bar{f})a^{†}(K_{\mu }g)\mathrm{\Omega }$ contributes to the vacuum pairing; CCR [F2] makes that value $⟨K_{\mu }\bar{f},K_{\mu }g{⟩}_2$. Taking the first test to be $\bar{f}$ and the second $f$ gives $\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}^2\ge 0$; for real $f$ this is the ordinary diagonal. For an ordered product of $n$ real smeared fields, write the first field as creation plus annihilation. The creation part vanishes against the left vacuum; commute the annihilation part successively past each later field using [F2]. Each commutator is the corresponding two-point pairing, and the remaining annihilator kills the right vacuum. This yields the recursive sum over its possible partner and proves the Wick pairing rule by induction on $n$.

Cauchy–Schwarz gives $\mathrm{\parallel }{C}_h\mathrm{\parallel }\le \mathrm{\parallel }h\mathrm{\parallel }$ for $C_h(v)=⟨h,v{⟩}_2$. The tensor-extension bound and orthogonal symmetrizer in [F11] therefore give $\mathrm{\parallel }{a}_n(h)\mathrm{\parallel }\le \sqrt{n}\mathrm{\parallel }h\mathrm{\parallel }$ and $\mathrm{\parallel }{a}_n^{†}(h)\mathrm{\parallel }\le \sqrt{n+1}\mathrm{\parallel }h\mathrm{\parallel }$, the latter also using $\mathrm{\parallel }h\otimes v\mathrm{\parallel }=\mathrm{\parallel }h\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }$ from the product tensor pairing. Let $\mathrm{\Theta }$ have sectors only through $N$. These bounds and orthogonality of distinct output sectors imply $\mathrm{\parallel }{\mathrm{\Phi }}_{\mu }(f)\mathrm{\Theta }\mathrm{\parallel }\le (\sqrt{N}\mathrm{\parallel }{K}_{\mu }\bar{f}\mathrm{\parallel }+\sqrt{N+1}\mathrm{\parallel }{K}_{\mu }f\mathrm{\parallel })\mathrm{\parallel }\mathrm{\Theta }\mathrm{\parallel }$. Continuity of $K_{\mu }$ [F9] and conjugation [F12], followed by Hilbert Cauchy–Schwarz, makes every stated matrix coefficient continuous; complex linearity is step 1.1. Mutual adjoints [F2] give $⟨{\mathrm{\Phi }}_{\mu }(f)\mathrm{\Psi },\mathrm{\Theta }{⟩}_2=⟨\mathrm{\Psi },{\mathrm{\Phi }}_{\mu }(\bar{f})\mathrm{\Theta }{⟩}_2$ for finite-particle vectors. Thus every such $\mathrm{\Theta }$ lies in the adjoint domain with the asserted restricted adjoint action. Finite-particle invariance permits reversing every finite polynomial product by this identity, so for $\omega (B)=⟨\mathrm{\Omega },B\mathrm{\Omega }{⟩}_2$ one has $\omega (B^{\ast }B)=\mathrm{\parallel }B\mathrm{\Omega }{\mathrm{\parallel }}^2\ge 0$ and $\omega (I)=1$.

For a Schwartz test $\phi (z)$, define $G_{\phi }(p)=\int \phi (z)e^{-ip\circ z}{d}^{4}z$; for $F(x,y)\in \mathcal{S}({\mathbb{R}}^8)$ define $G_F(p)=\int F(x,y)e^{-ip\circ x}{e}^{ip\circ y}{d}^{4}x{d}^{4}y$. By [F12], these are continuous Schwartz transforms, the second followed by the injective linear frequency restriction $L$ determined by the displayed phases. Injectivity in finite dimensions gives $\mathrm{\mid }Lp\mathrm{\mid }\ge c\mathrm{\mid }p\mathrm{\mid }$ for some $c>0$; each derivative of a restricted transform is a fixed finite linear combination of its ambient derivatives at $Lp$. These bounds control every weighted restricted derivative by finitely many ambient Schwartz seminorms, proving the claimed restriction continuity and decay. In either case $\mathrm{\mid }G(p)\mathrm{\mid }\le q_4(G)(1+\mathrm{\mid }p\mathrm{\mid }{)}^{-4}$, with $q_4$ a continuous Schwartz seminorm. The shell integral of $(1+\mathrm{\mid }p\mathrm{\mid }{)}^{-4}$ is finite by [F4]: its radial behavior is $O(r)$ at the massless apex, $O(r^2)$ near zero when $\mu >0$, and $O(r^{-3})$ at infinity. Therefore the integrals of $G_{\phi }$ and $G_F$ against $d{\nu }_{\mu }$ define continuous complex-linear tempered distributions. On product tests $F(x,y)=f(x)g(y)$ the inner integral is $\overline{K_{\mu }\stackrel{‾}{f}(p)}{K}_{\mu }g(p)$; its shell integral is exactly the covariance from step 2.2. This proves the relative and bidistribution interfaces without assigning point operators.

The one-particle translation phase is $e^{i({\omega }_{\mu }(\mathbf{k})a^0-\mathbf{k}\cdot \mathbf{a})}$, so its real generator vector is $\varpi (\mathbf{k})=({\omega }_{\mu }(\mathbf{k}),-\mathbf{k})$. It is finite and measurable. The measure is sigma-finite by [F5], so [F8] applies and gives the joint PVM and vacuum at zero. On an $n$-particle sector the joint multiplier is ${\sum }_{r=1}^n({\omega }_{\mu }({\mathbf{k}}_r),-{\mathbf{k}}_r)$. Since $\mathrm{\mid }{\sum }_r{\mathbf{k}}_r\mathrm{\mid }\le {\sum }_r\mathrm{\mid }{\mathbf{k}}_r\mathrm{\mid }\le {\sum }_r{\omega }_{\mu }({\mathbf{k}}_r)$, every such value lies in the closed future cone; the vacuum value does too. Thus the joint spectral measure is supported there, as claimed.

Assume Countable Choice as required by [F4]. For a spacetime test $\phi \in \mathcal{S}({\mathbb{R}}^4)$ set $q_{\phi }(t,\mathbf{k})={\int }_{{\mathbb{R}}^3}{e}^{i\mathbf{k}\cdot \mathbf{x}}\phi (t,\mathbf{x})d^3\mathbf{x}$. Integration by parts in the spatial variables gives $(1+\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2{)}^2{q}_{\phi }(t,\mathbf{k})=\int e^{i\mathbf{k}\cdot \mathbf{x}}(1-{\mathrm{\Delta }}_{\mathbf{x}}{)}^2\phi (t,\mathbf{x})d^3\mathbf{x}\mathrm{.}$ Let $P(\phi )$ be the maximum, over spatial multi-indices $\mathrm{\mid }\beta \mathrm{\mid }\le 4$, of ${\sup }_{t,\mathbf{x}}(1+\mathrm{\mid }t\mathrm{\mid }{)}^4(1+\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}^2{)}^2\mathrm{\mid }{\mathrm{\partial }}_{\mathbf{x}}^{\beta }\phi (t,\mathbf{x})\mathrm{\mid }$. By [F9] this is bounded by a finite sum of the defining Schwartz seminorms. Since ${\int }_{{\mathbb{R}}^3}(1+\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}^2{)}^{-2}{d}^3\mathbf{x}<\mathrm{\infty }$, $\mathrm{\mid }{q}_{\phi }(t,\mathbf{k})\mathrm{\mid }\le CP(\phi )(1+\mathrm{\mid }t\mathrm{\mid }{)}^{-4}(1+\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2{)}^{-2}\mathrm{.}$ For every $\mu \ge 0$, $\mathrm{\mid }\sin ({\omega }_{\mu }t)\mathrm{/}{\omega }_{\mu }\mathrm{\mid }\le \mathrm{\mid }t\mathrm{\mid }$ off the massless apex. Consequently the joint pairing $⟨{\mathrm{\Delta }}_{\mu },\phi ⟩=(2\pi {)}^{-3}{\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^3}\frac{\sin ({\omega }_{\mu }(\mathbf{k})t)}{{\omega }_{\mu }(\mathbf{k})}{q}_{\phi }(t,\mathbf{k})d^3\mathbf{k}dt$ is absolutely convergent and has modulus at most $C^{\mathrm{\prime }}P(\phi )$, because $\int \mathrm{\mid }t\mathrm{\mid }(1+\mathrm{\mid }t\mathrm{\mid }{)}^{-4}dt$ and $\int (1+\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2{)}^{-2}{d}^3\mathbf{k}$ are finite. It is complex-linear and continuous on $\mathcal{S}({\mathbb{R}}^4)$, so it defines the claimed spacetime tempered distribution [F5]. The estimate also supplies the required polynomial growth in time and proves that this pairing is the integral of its fixed-time spatial pairings. The apex value can be assigned arbitrarily without changing any integral; the separate $1\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }$ shell factor is locally integrable in three dimensions. For the fixed-time family, derivatives of $\sin ({\omega }_{\mu }t)\mathrm{/}{\omega }_{\mu }$ have at most polynomial growth in $\mathbf{k}$, uniformly on bounded time intervals. Taking enough spatial Schwartz weights therefore permits all time derivatives under its spatial pairing. It follows both for this family and, by integration by parts in $t$, for the joint distribution that $({\mathrm{\partial }}_t^2-{\mathrm{\Delta }}_{\mathbf{x}}+{\mu }^2){\mathrm{\Delta }}_{\mu }=0$, ${\mathrm{\Delta }}_{\mu }(0,\cdot )=0$, and ${\mathrm{\partial }}_t{\mathrm{\Delta }}_{\mu }(0,\cdot )={\delta }_0$; the last identity is Fourier inversion [F4].