Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Classical equations and adopted exact free quantum representation

Physical scope

The selected flat four-dimensional free real scalar and Dirac models, with the stated (-+++) signature, inverse-length mass parameter and chosen Fock representations. The result matches compact-variation classical equations with weak test equations of the adopted representations. It does not derive quantization, a general spin-statistics rule, interacting dynamics, empirical validity, or a measurement outcome.

Statement

Assume Countable Choice and the selected free-field model of Specified bosonic scalar and CAR Dirac model adoption, using the metric, length coordinates, inverse- length parameter $μ \geq 0$ , field dimensions and Clifford convention fixed by Quantum field quantities and compatible SI normalizations. The classical scalar field $ϕ_{n} : R^{4} \to R$ is smooth and real; the classical Dirac field $ψ_{n} : R^{4} \to C^{4}$ is smooth with commuting components. Their adopted free actions are the scalar and Dirac functionals stated in Specified bosonic scalar and CAR Dirac model adoption. Action symbols are used only through compactly supported smooth variations, so no global integrability of the classical background fields is assumed.

The compact-variation stationarity equations are $(□ - μ^{2}) ϕ_{n} = 0, (i γ^{a} \partial_{a} - μ) ψ_{n} = 0,$ with $□ = - \partial_{0}^{2} + Δ_{x}$ . In the separately adopted scalar and Dirac Fock representations, the corresponding weak test equations are, for $f \in S (R^{4})$ and $F \in S (R^{4}; C^{4})$ , $Φ_{μ} ((□ - μ^{2}) f) = 0, Ψ ⁣ ((i \partial_{0} - h_{D} (- i \nabla)) F) = 0.$ Here “weak test equation” means precisely that smearing the constructed field with the displayed differential operator gives the zero operator on its stated common domain. With $h_{D} (k) = α \cdot k + β μ$ , $γ^{0} = β$ , and $γ^{j} = β α^{j}$ as adopted in Specified bosonic scalar and CAR Dirac model adoption, the Dirac test operator is equivalent to $i γ^{a} \partial_{a} - μ$ .

Thus the classical action equations and the tested equations of the selected free representations have the same differential expressions at these normalizations. The Fock representation, species/statistics assignments and vacua are additional physical model assumptions; they are not deductions from the classical actions. No empirical outcome or measurement is asserted.

Facts & Assumptions

Given: ${A C}_{ω}$ , the fixed QFT unit and Clifford conventions, the adopted classical actions and free representations, and the current draft mathematical carriers for the variation and test-restriction arguments.

[A1]

Countable Choice, written ${A C}_{ω}$ , means every sequence of nonempty sets has a choice function. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Quantum field quantities and compatible SI normalizations fixes $η = diag (- 1, 1, 1, 1)$ , $x^{0} = c t$ , $□ = η^{a b} \partial_{a} \partial_{b}$ , inverse-length $μ$ , ${γ^{a}, γ^{b}} = - 2 η^{a b} I_{4}$ , and the length dimensions of the scalar and Dirac fields.

[F2]

Specified bosonic scalar and CAR Dirac model adoption adopts the displayed scalar and commuting Dirac actions, the selected scalar bosonic and Dirac particle/antiparticle CAR Fock representations, their vacua, and the explicit $h_{D}, β, γ^{a}$ conventions. These are physical model assumptions, not mathematical consequences of the actions.

[F3]

Classical free action variations and normalization assumes smooth real scalar and smooth commuting complex spinor fields, with compactly supported smooth variations. It states the exact first-variation identities, that no global action integrability is assumed, and that stationarity under all such variations is equivalent to the pointwise Klein–Gordon and Dirac equations. Its current Step 1 receipt is ready for this draft mathematical carrier; independent review is still pending.

[F4]

Continuity and Klein–Gordon annihilation of scalar shell restriction states, for every $μ \geq 0$ and Schwartz test $f$ , $K_{μ} ((\partial_{0}^{2} - Δ + μ^{2}) f) = 0$ . Its current Step 1 receipt is ready for this draft mathematical carrier; the containing QFT-MF02 page remains a draft prerequisite.

[F5]

Smeared free scalar field and its Poincare representation defines $Φ_{μ} (f) = a (K_{μ} \overline{f}) + a^{†} (K_{μ} f)$ on the finite-particle domain and its test-function convention. The field carrier remains draft and its upstream QFT-MF01 dependencies are unresolved, so this representation is used conditionally.

[F6]

Smeared free Dirac CAR fields and graded locality defines the bounded CAR field $Ψ (F) = c (P_{+} u_{F}) + c^{†} (\overline{P_{-} u_{F}})$ on its stated Fock space. Its current Step 1 decision is escalated through QFT-MF01/MF02 dependencies, so this field map is used conditionally.

[F7]

Smeared free Dirac CAR fields and graded locality proves $u_{(i \partial_{0} - h_{D} (- i \nabla)) F} = 0$ for Schwartz spinor tests. This tested equation is conditional on the escalated field and upstream carriers.

Proof

Given: [A1] and [F1]–[F7].

1.1A1F2F3given

By [F3], for each compactly supported smooth real variation $h$ and compactly supported smooth spinor variation $χ$ , the first variations of the displayed actions are the stated integrals against $(□ - μ^{2}) ϕ_{n}$ and $(i γ^{a} \partial_{a} - μ) ψ_{n}$ . 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].

1.2F1F4F5algebra

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

1.3F1F2F3F6F7algebra

By [F7], the one-particle vector of $(i \partial_{0} - h_{D} (- i \nabla)) F$ 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], $β (i \partial_{0} - h_{D} (- i \nabla)) = i β \partial_{0} + i β α^{j} \partial_{j} - μ = i γ^{a} \partial_{a} - μ,$ using $β^{2} = I_{4}$ . Thus this is exactly the tested Dirac differential expression in the classical equation of [F3].

2.1F2step 1.1step 1.2step 1.3given∎

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.

Classical equations and adopted exact free quantum representation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources