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)

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Two physical photon degrees and positive free dynamics

Physical scope

The selected flat-space source-free photon model only: positive transverse one-photon states, the auxiliary null quotient, Lorentz-covariant free dynamics, and its exact Hamiltonian domain; no interacting gauge model or empirical photon state is asserted.

Statement

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the selected Fock-space construction, with the positive constants and physical energy units of Quantum field quantities and compatible SI normalizations. In the adopted model Positive transverse free photon model, for $k \neq 0$ set $n = k / ∣ k ∣$ and $P_{T} (k) = I_{3} - n n^{T}$ . Then for almost every nonzero momentum, $P_{T} (k)$ is a positive orthogonal projection of rank two, and the positive one-photon Hilbert space is its closed range $H_{T} \subset L^{2} (R^{3}, d^{3} k; C^{3})$ . This description chooses no global continuous polarization frame.

If an auxiliary four-polarization space $K = L^{2} (R^{3}, d^{3} k; C^{4})$ with indefinite form $[ϵ, δ] = \int ϵ^{†} η δ d^{3} k$ is used, the physical subsidiary space $K_{0} = {ϵ : - ϵ^{0} + n \cdot ϵ = 0 a.e.}$ has null radical $N = {b (k) (1, n) : b \in L^{2}}$ . The quotient $K_{0} / N$ is isometric to $H_{T}$ with positive inner product. The physical Fock Hamiltonian $H_{γ} = ℏ c d Γ (∣ k ∣)$ is self-adjoint, nonnegative, and has the exact domain and strongly continuous Lorentz-covariant mass-cone action in Photon quotient Lorentz representation and positive dynamics. The vector-valued auxiliary space is not a positive state space, and none of its negative directions is assigned a negative Born probability.

Facts & Assumptions

Given: Countable Choice; the selected positive-transverse free-photon model; the exact transverse-space definition; and the inspected draft quotient and covariance carriers.

[A1]

Positive transverse free photon model: the adopted state space is the bosonic Fock space over the positive transverse one-photon Hilbert space, with free dynamics and smeared field-strength observables. The postulate has a current escalated Step 1 record because its local-Weyl input is escalated through the field-strength prerequisite. Its framework-units and C01 measurement inputs have current ready records. This is readiness evidence, not independent approval; the model is used conditionally.

[A2]

Countable Choice is assumed for the mathematical Fock construction. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Positive transverse photon one-particle space: for almost every $k \neq 0$ , $P_{T} = I - n n^{T}$ is an orthogonal projection with two-dimensional range; its multiplier range $H_{T}$ is closed and positive, and no global continuous polarization frame is selected. The draft carrier has a current ready Step 1 record for its item hash; this is not independent mathematical approval.

[F2]

Quantum field quantities and compatible SI normalizations: $ℏ, c > 0$ are the exact SI defining constants, $x^{0} = c t$ , and physical energy is $E = ℏ c ω$ for temporal wave number $ω$ . Its draft carrier has a current ready Step 1 record for its item hash; this is readiness evidence, not independent mathematical approval.

[L1]

Lorenz auxiliary subsidiary null quotient: on $K_{0} = {ϵ : - ϵ^{0} + n \cdot ϵ = 0}$ , $T ϵ = ϵ - n ϵ^{0}$ is bounded and onto $H_{T}$ , $[ϵ, δ]_{η} = ⟨ T ϵ, T δ ⟩_{2}$ , and its radical is $N = {b (k) (1, n) : b \in L^{2}}$ . Thus $K_{0} / N$ is a positive Hilbert space isometric to $H_{T}$ . The draft carrier has a current ready Step 1 record for its item hash; this is readiness, not independent mathematical approval.

[L2]

Photon quotient Lorentz representation and positive dynamics: the future-null-cone quotient action $[s (p)] \mapsto [Λ s (Λ^{- 1} p)]$ is a strongly continuous unitary representation; its bosonic Fock lift has energy multiplier $\sum_{r} ∣ k_{r} ∣$ with the exact maximal sector domain and vacuum energy zero, and physical four-momentum spectrum the closed future cone. The draft carrier has a current ready Step 1 record for its item hash; this is readiness, not independent mathematical approval.

[F3]

Second-quantized multipliers, joint spectral measure and exact domains: the bosonic sector-sum multiplier $d Γ (∣ k ∣)$ is self-adjoint on its exact maximal square-integrability domain, with the vacuum sector at zero. Its draft carrier has a current ready Step 1 record for its item hash; this is readiness evidence, not independent mathematical approval.

Proof

Given: [A1]–[A2], [F1]–[F3], and the current conditional mathematical interfaces [L1]–[L2].

1.1A1F1algebra

For $k \neq 0$ , $∣ n ∣^{2} = 1$ , so $P_{T}^{*} = P_{T}$ and $P_{T}^{2} = (I - n n^{T})^{2} = I - 2 n n^{T} + n (n^{T} n) n^{T} = P_{T}$ . Its kernel is $span {n}$ , so its range is the two-dimensional plane $n^{⊥}$ . As a measurable bounded orthogonal projection, its multiplication operator is idempotent on $L^{2}$ . If $P_{T} u_{r} \to u$ with $P_{T} u_{r} = u_{r}$ , boundedness gives $P_{T} u = \lim_{r} P_{T} u_{r} = \lim_{r} u_{r} = u$ ; hence the range is closed. A change at $k = 0$ changes no $L^{2}$ vector because that singleton has Lebesgue measure zero. This gives a positive rank-two transverse fiber almost everywhere without selecting a polarization frame.

1.2L1given

By [L1], each $ϵ \in K_{0}$ maps to a transverse vector $T ϵ = ϵ - n ϵ^{0}$ , and $[ϵ, ϵ]_{η} = ∥ T ϵ ∥_{H_{T}}^{2} \geq 0$ . The kernel is exactly $N$ , so the induced norm is positive on nonzero quotient classes. The supplier proves that $T$ is onto and that the quotient is isometric to the complete Hilbert space $H_{T}$ ; therefore no negative-norm auxiliary direction becomes a physical state in this quotient. This conclusion is conditional on the current draft interface.

2.1A1A2F2F3L2givenalgebra∎

The free Fock construction is understood under Countable Choice [A2]. By [F3], the unscaled multiplier $d Γ (∣ k ∣)$ is self-adjoint on its exact maximal sector-sum domain. The physical energy operator in the selected model is $H_{γ} = ℏ c d Γ (∣ k ∣)$ , so it has the same domain. In sector $n$ it is multiplication by $H_{n} = ℏ c \sum_{r = 1}^{n} ∣ k_{r} ∣$ . Every summand is nonnegative and $ℏ c > 0$ by [F2], so $H_{n} \geq 0$ ; on the vacuum sector the empty sum is zero. The strongly continuous unitary Lorentz action follows from [L2]. These are properties of the selected free model and do not assert an interacting theory.

Two physical photon degrees and positive free dynamics

Statement

Facts & Assumptions

Proof

Depends on

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