Specified free state and observable interpretation

Use the fields, Fock spaces, vacua, coordinate conventions, and natural/SI normalizations adopted in Specified bosonic scalar and CAR Dirac model adoption. For an open spacetime region $O\subseteq {\mathbb{R}}^4$, a scalar test is a real $f\in C_c^{\mathrm{\infty }}(O;\mathbb{R})$ and a Dirac test is a $F\in C_c^{\mathrm{\infty }}(O;{\mathbb{C}}^4)$. These are compactly supported smooth test functions on the fixed inertial Minkowski chart.

For scalar tests use the bounded Weyl operators $W_{\mu }(f)=W_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(\sqrt{2}{K}_{\mu }f),$ where $K_{\mu }$ is the future-shell restriction and $W_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}$ is the unitary generated by the closed Segal field, with the normalization and phase convention of Smeared free scalar field and its Poincare representation. Define the scalar bounded observable algebra ${\mathfrak{A}}_s(O)$ to be the norm closure of the unital $\ast$-algebra generated by these $W_{\mu }(f)$. The real smeared field is symmetric on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ and its self-adjoint closure has the exact multiplier domain supplied by Domain-qualified free observables and point-field obstruction and Self-adjoint Segal fields and explicit Weyl unitaries.

For Dirac tests, $\mathrm{\Psi }(F)$ is the bounded smeared CAR field of Smeared free Dirac CAR fields and graded locality. Define ${\mathfrak{A}}_D(O)$ as the norm closure of the unital algebra generated by products containing an even number of these field operators and their adjoints, with all test supports in $O$. “Even” here means even field parity, or equivalently an even number of odd CAR field generators in each product. The even algebra is the selected local Dirac observable algebra; an odd charged field is not included as a commuting local observable.

The selected vacuum states on bounded operators are the vector states ${\omega }_s(A)=⟨{\mathrm{\Omega }}_s,A{\mathrm{\Omega }}_s⟩$ and ${\omega }_D(B)=⟨{\mathrm{\Omega }}_D,B{\mathrm{\Omega }}_D⟩$. For the scalar Fock space, define the total number, Hamiltonian, and physical momentum components by $N_s=d{\mathrm{\Gamma }}_s(1),H_s=\mathrm{\hslash }cd{\mathrm{\Gamma }}_s({\omega }_{\mu }),P_{s,j}=\mathrm{\hslash }d{\mathrm{\Gamma }}_s(k_j)\mathrm{.}$ Here $d{\mathrm{\Gamma }}_s$ acts on the $n$-particle sector by summing the displayed one-particle multiplier over its $n$ arguments. Their maximal domains are the square-integrability domains of those sector sums; in particular, $D(N_s)=\{\mathrm{\Psi }=({\mathrm{\Psi }}_n):{\sum }_{n\ge 1}{n}^2\mathrm{\parallel }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ These physical components have one-particle momentum $+\mathrm{\hslash }{k}_j$, as in Quantum field quantities and compatible SI normalizations. The mathematical spatial generators in Smeared free scalar field and its Poincare representation and Domain-qualified free observables and point-field obstruction use $G_{s,j}=d{\mathrm{\Gamma }}_s(-k_j)$ with the positive-exponent convention; thus $P_{s,j}=-\mathrm{\hslash }{G}_{s,j}$. For a translation increment $a=(a^0,\mathbf{a})$ in length coordinates, the scalar translation unitary is the sectorwise phase $\exp (i(a^0{H}_s\mathrm{/}(\mathrm{\hslash }c)-{\sum }_j{a}^j{P}_{s,j}\mathrm{/}\mathrm{\hslash }))$. Changing the sign does not change the maximal domain, since the squared sector multiplier is unchanged. This also follows directly from Second-quantized multipliers, joint spectral measure and exact domains with multiplier $k_j$. For the Dirac Fock space, let $H$ and $P_j$ be the mathematical translation generators of Fermionic Dirac translation generators and positive spectrum. Their one-particle multipliers are respectively $({\omega }_{\mu },\mathbf{k})$ for particles and $({\omega }_{\mu },-\mathbf{k})$ for antiparticles, and the sector multiplier is their sum. Define the physical Dirac Hamiltonian and momentum by $H_D=\mathrm{\hslash }cH$ and $P_{D,j}=\mathrm{\hslash }{P}_j$, using the positive SI constants from Quantum field quantities and compatible SI normalizations. Their exact domains are unchanged by these nonzero scale factors; their multipliers have units joules and momentum, respectively.

For any of these self-adjoint operators $A$, a normalized state $\mathrm{\Psi }\in D(A)$ has model mean $⟨\mathrm{\Psi },A\mathrm{\Psi }⟩$ and variance ${\mathrm{Var}}_{\mathrm{\Psi }}(A)=\mathrm{\parallel }(A-⟨\mathrm{\Psi },A\mathrm{\Psi }⟩)\mathrm{\Psi }{\mathrm{\parallel }}^2$. This requires the second spectral moment but does not require $\mathrm{\Psi }\in D(A^2)$. Products of unbounded operators use their composition domains.

If a signed charge parameter $q_{\mathrm{S}\mathrm{I}}\in \mathbb{R}$ measured in coulombs is supplied, the intended Dirac charge-number operator is $Q_{\mathrm{S}\mathrm{I}}=q_{\mathrm{S}\mathrm{I}}(N_p-N_{\bar{p}})$. Here ${\mathrm{\Pi }}_p$ and ${\mathrm{\Pi }}_{\bar{p}}$ are the orthogonal projections onto ${\mathcal{H}}_{+}$ and $\overline{{\mathcal{H}}_{-}}$; on the $n$-particle sector, $N_p$ and $N_{\bar{p}}$ are the sums of the corresponding projections over the $n$ tensor factors. Its candidate maximal direct-sum domain is $D(Q_{\mathrm{S}\mathrm{I}})=\{\mathrm{\Psi }=({\mathrm{\Psi }}_n):{\sum }_{n\ge 0}{\parallel q_{\mathrm{S}\mathrm{I}}{\sum }_{r=1}^n({\mathrm{\Pi }}_p-{\mathrm{\Pi }}_{\bar{p}}{)}^{(r)}{\mathrm{\Psi }}_n\parallel }^2<\mathrm{\infty }\}\mathrm{.}$ This is a candidate definition only; no self-adjointness or proved-domain claim is asserted here.

Measurements in this model are assigned only when a state and a normalized finite bounded Kraus instrument have actually been constructed. For such an instrument, the adopted outcome probabilities and positive-probability conditional states are exactly those of Physical interpretation of a supplied constructed quantum model. They are model probabilities, not observed outcomes. The field definitions are test-smeared: neither ${\mathrm{\Phi }}_{\mu }(x)$ nor $\mathrm{\Psi }(x)$ is a pointwise operator supplied here, and the smearing results do not define coincident products or a diagonal restriction.

The current Domain-qualified free observables and point-field obstruction Statement and Proof establish the scalar and Dirac field and translation-generator claims above, but do not state or prove the planned charged-number operator or its domain. The research module research/extended-frameworks-2026-10-03/quantum-field-theory/workers/free-fields/proof-modules.md#FF7 contains a species-count domain argument, but that argument is not a current MF04 item carrier or readiness input. The displayed charge expression is therefore a candidate definition conditional on that missing mathematical interface; no self-adjointness or proved-domain claim is made here.