Free observable domains and limits of point notation

The adopted physical systems are exactly the two selected Fock representations in [F2], and [F1] fixes the spacetime and unit convention. The test spaces and meanings of ${\mathfrak{A}}_s(O)$ and ${\mathfrak{A}}_D(O)$ are those defined in [F4]; thus all fields used here are smeared and all Dirac local observables are even products.

Apply [F6] to the listed real scalar smearings and sector-sum generators. It supplies their exact maximal domains and, for every normalized $\mathrm{\Psi }\in D(A)$, finite $⟨\mathrm{\Psi },A\mathrm{\Psi }⟩$ and finite variance. The same statement gives the product-domain condition, so no claim about $D(A^2)$ or an everywhere-defined product follows.

The region hypotheses in the Statement are exactly the spacelike-support hypotheses of [F8] and [F10]. The first gives commutativity of the scalar Weyl algebras; the second gives commutativity of the bounded even Dirac local algebras. If either region is empty, its test space is $\{0\}$ and its unital generated algebra is $\mathbb{C}I$, which commutes with the other algebra as well.

Let $({\epsilon }_n)$ be any sequence of positive widths tending to zero. For every Schwartz $g$, changing variables $y=(x-x_0)\mathrm{/}{\epsilon }_n$ gives $\int {\rho }_{{\epsilon }_n,x_0}(x)g(x)d^{4}x=\int \rho (y)g(x_0+{\epsilon }_{n}y)d^{4}y\mathrm{.}$ The integrand tends pointwise to $\rho (y)g(x_0)$ and is bounded by the integrable function $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\rho (y)$, so [F11] gives convergence to $g(x_0)$. For every spatial Schwartz spinor $h$, the analogous change of variables gives $\int u_{{\epsilon }_n,x_0}(\mathbf{x}{)}^{†}h(\mathbf{x})d^3\mathbf{x}=\int \sigma (y)z^{†}h({\mathbf{x}}_0+{\epsilon }_{n}y)d^{3}y\mathrm{.}$ Its integrand is bounded by $\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}\sigma (y)$ and converges pointwise to $\sigma (y)z^{†}h({\mathbf{x}}_0)$; [F11] and normalization of $\sigma$ give limit $z^{†}h({\mathbf{x}}_0)$. Since the width sequence was arbitrary, both families converge to the stated point masses on Schwartz tests.

Let $({\epsilon }_n)$ again be any positive sequence tending to zero. Define $\widehat{\rho }(\xi )=\int e^{-i\xi \cdot y}\rho (y)d^{4}y$. Positivity and normalization give $\mathrm{\mid }\widehat{\rho }(\xi )\mathrm{\mid }\le 1$, and [F11] gives $\widehat{\rho }(\xi )\to 1$ as $\xi \to 0$. The shell formula in [F2] yields $K_{\mu }{\rho }_{{\epsilon }_n,x_0}(p)=e^{ip\circ x_0}\widehat{\rho }({\epsilon }_n(-p^0,\mathbf{p}))$. Fix $r_0=\max (1{\mathrm{m}}^{-1},\mu )$ and, for each finite $R\ge r_0$, let $A_R=\{p\in H_{\mu }^{+}:r_0\le \mathrm{\mid }\mathbf{p}\mathrm{\mid }\le R\}$. This annulus has finite measure, and dominated convergence [F11] gives ${\int }_{A_R}\mathrm{\mid }{K}_{\mu }{\rho }_{{\epsilon }_n,x_0}{\mathrm{\mid }}^{2}d{\nu }_{\mu }\to {\nu }_{\mu }(A_R)$. In radial coordinates $d{\nu }_{\mu }=r^2(4{\pi }^2\sqrt{r^2+{\mu }^2}{)}^{-1}dr$; for $r\ge r_0$ its density is at least $r\mathrm{/}(4\sqrt{2}{\pi }^2)$, so ${\nu }_{\mu }(A_R)\to \mathrm{\infty }$ as $R\to \mathrm{\infty }$. Since the full shell norm dominates each annular integral and these limiting lower bounds are arbitrarily large, $\mathrm{\parallel }{K}_{\mu }{\rho }_{{\epsilon }_n,x_0}\mathrm{\parallel }\to \mathrm{\infty }$. The scalar field formula and vacuum in [F2] give $\mathrm{\parallel }{\mathrm{\Phi }}_{\mu }({\rho }_{{\epsilon }_n,x_0})\mathrm{\Omega }\mathrm{\parallel }=\mathrm{\parallel }{K}_{\mu }{\rho }_{{\epsilon }_n,x_0}\mathrm{\parallel }$. Arbitrariness of $({\epsilon }_n)$ proves the claimed scalar norm divergence.

Changing variables gives $\mathrm{\parallel }{u}_{\epsilon ,x_0}{\mathrm{\parallel }}_2={\epsilon }^{-3\mathrm{/}2}\mathrm{\parallel }\sigma {\mathrm{\parallel }}_2\to \mathrm{\infty }$, since $0<\mathrm{\parallel }\sigma {\mathrm{\parallel }}_2<\mathrm{\infty }$. At fixed physical time $t$, let $A_{\epsilon }={\mathrm{\Psi }}_t^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(u_{\epsilon ,x_0})={\mathrm{\Psi }}_{ct}(u_{\epsilon ,x_0})$, the bounded equal-time field of [F9] at length-coordinate increment $ct$. Its CAR give $A_{\epsilon }{A}_{\epsilon }^{\ast }+A_{\epsilon }^{\ast }{A}_{\epsilon }=\mathrm{\parallel }{u}_{\epsilon ,x_0}{\mathrm{\parallel }}_2^{2}I$. By the operator norm definition and adjoint identity, $\mathrm{\parallel }{A}_{\epsilon }^{\ast }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\parallel }{A}_{\epsilon }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$; the operator-norm triangle inequality and submultiplicativity then imply $\mathrm{\parallel }{u}_{\epsilon ,x_0}{\mathrm{\parallel }}_2^2\le 2\mathrm{\parallel }{A}_{\epsilon }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2$, hence $\mathrm{\parallel }{A}_{\epsilon }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\to \mathrm{\infty }$. Thus the Dirac family has no finite operator-norm limit.

The selected definition [F5] supplies no other point field or diagonal product. The norm obstructions in steps 1.5 and 1.6 are specific to the displayed representations and topologies.

When the state and finite instrument required by [F3] are supplied, the Born postulate assigns the stated model probabilities. No such state, instrument, apparatus, or observation is introduced by this theorem, so it establishes no empirical outcome.

By [F3], for each compactly supported smooth real variation $h$ and compactly supported smooth spinor variation $\chi$, the first variations of the displayed actions are the stated integrals against $(\mathrm{\square }-{\mu }^2){\varphi }_{\mathrm{n}}$ and $(i{\gamma }^a{\mathrm{\partial }}_a-\mu ){\psi }_{\mathrm{n}}$. The lemma proves both necessity and sufficiency of the displayed pointwise equations for stationarity under all such variations; this step uses the commuting classical spinor convention of [F2].

Since $\mathrm{\square }-{\mu }^2=-({\mathrm{\partial }}_0^2-\mathrm{\Delta }+{\mu }^2)$, [F4] and linearity give $K_{\mu }((\mathrm{\square }-{\mu }^2)f)=0$ for every Schwartz $f$. The differential operator has real coefficients, so the same identity holds for $\bar{f}$. The scalar field formula in [F5] applies $a$ and $a^{†}$ only to $K_{\mu }\bar{f}$ and $K_{\mu }f$; both vanish for $f$ replaced by $(\mathrm{\square }-{\mu }^2)f$. Hence the constructed scalar field satisfies the stated weak test equation.

By [F7], the one-particle vector of $(i{\mathrm{\partial }}_0-h_D(-i\mathrm{\nabla }))F$ is zero. Substitution in the bounded CAR field formula in [F6] makes both its creation and annihilation terms zero, so the corresponding smeared field is zero. From [F2], $\beta (i{\mathrm{\partial }}_0-h_D(-i\mathrm{\nabla }))=i\beta {\mathrm{\partial }}_0+i\beta {\alpha }^j{\mathrm{\partial }}_j-\mu =i{\gamma }^a{\mathrm{\partial }}_a-\mu ,$ using ${\beta }^2=I_4$. Thus this is exactly the tested Dirac differential expression in the classical equation of [F3].

The quantum representations used in steps 1.2 and 1.3 enter only through the physical adoption [F2]; steps 1.1–1.3 establish equation agreement under that adoption and do not construct or derive the representation from the classical action. The conclusion is therefore conditional on the selected free model and carries no empirical or general quantization claim.