Positive transverse free photon model

Adopt the following specified free-photon model in the oriented, time-oriented affine Minkowski spacetime and length coordinates of Quantum field quantities and compatible SI normalizations, with metric $(-,+,+,+)$ and $x^0=ct$. The classical electromagnetic input is the smooth source-free branch of The microscopic vacuum Maxwell equations: smooth and weak branches; the electromagnetic two-form and its potential components are those of The electromagnetic two-form, its observer decomposition and the four-current, and a supplied potential is subject to Potential representations, gauge transformations and the Lorenz and Coulomb conditions. These EM suppliers are physical input and conventions: they do not by themselves construct a quantum representation.

For almost every $\mathbf{k}\in {\mathbb{R}}^3\setminus \{0\}$, set $\mathbf{n}=\mathbf{k}\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }$ and $P_T(\mathbf{k})=I_3-\mathbf{n}{\mathbf{n}}^{\mathsf{T}}$. The mathematical definition Positive transverse photon one-particle space specifies the positive one-photon Hilbert space

${\mathcal{H}}_T=\{u\in L^2({\mathbb{R}}^3,d^3\mathbf{k};{\mathbb{C}}^3):\mathbf{k}\cdot u(\mathbf{k})=0\text{ for almost every }\mathbf{k}\},$

with the inherited pairing $⟨u,v{⟩}_2=\int \overline{u(\mathbf{k})}\cdot v(\mathbf{k})d^3\mathbf{k}$, linear in the second variable as in Schwartz/distribution spaces and Hilbert Fock conventions. The value assigned to $P_T$ at $\mathbf{k}=0$ is immaterial because that point has Lebesgue measure zero; no continuous polarization frame over all momentum directions is chosen. Adopt the bosonic Fock representation ${\mathcal{F}}_s({\mathcal{H}}_T)$ and the free Hamiltonian

$H_{\gamma }=\mathrm{\hslash }cd\mathrm{\Gamma }(\mathrm{\mid }\mathbf{k}\mathrm{\mid }),$

where on the $n$-photon sector $d\mathrm{\Gamma }(\mathrm{\mid }\mathbf{k}\mathrm{\mid })$ is multiplication by ${\sum }_{r=1}^n\mathrm{\mid }{\mathbf{k}}_r\mathrm{\mid }$ and is zero on the vacuum sector. Its exact self-adjoint domain and the unitary mass-cone action are those in the mathematical supplier Photon quotient Lorentz representation and positive dynamics. The actual smeared field-strength distributions and bounded Weyl operators are those constructed in Bounded local photon Weyl observable algebra; a pointwise potential or field-strength operator is not asserted to exist. These mathematical suppliers are draft carriers with mixed current Step 1 status: the transverse-space, null-quotient, covariance-spectrum, framework-units, and C01 measurement carriers have current ready records; the local-Weyl carrier is escalated through its field-strength prerequisite. Readiness records are not independent approval, and every use remains conditional on the exact statement and proof supplied.

For the physical interpretation of an actually constructed state and a supplied normalized measurement instrument, use Physical interpretation of a supplied constructed quantum model: Born probabilities are assigned only to that specified model and instrument. This postulate records the model's free Maxwell quanta as photons. It does not assert that the model has been empirically confirmed, and it introduces no interacting four-dimensional QED Hilbert space, scattering operator, LSZ limit, asymptotic completeness, or nonperturbative continuum limit. In calculations using an indefinite four-polarization auxiliary space, only the positive transverse model or its separately constructed null quotient is physical; four auxiliary components are not four positive photon states.