Scope and empirical status of the photons and radiation modes page

What the page adopts. The page adopts the radiation extension of the framework, Radiation extension of the framework: transverse mode and photon assignments: a fixed propagation-mode one-photon polarisation model with a two-dimensional transverse internal space, a quantised free transverse radiation sector with bosonic Fock space, and the mode assignments $\omega =c\mathrm{\mid }k\mathrm{\mid }$, $E=\mathrm{\hslash }\omega$ and $p=\mathrm{\hslash }k$ in a declared inertial chart. These are additional model assumptions of the framework; they are not derived from P1-P7, not derived from classical Maxwell theory or special relativity, and not obtained as a limit of the massive one-particle model. The page also adopts the preparation and apparatus discipline P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions) and the observable and probability postulate P2 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes) for every reading of a detector outcome.

What the page proves conditionally. The fixed-mode polarisation model of the framework and The radiation sector of the framework: modes and the number, energy and momentum assignments fix the model vocabulary, units and domains. The analyser outcome law of the fixed-mode polarisation model: Born probabilities and the Malus ratio proves the Born outcome law ${\cos }^2(\theta -\phi )$, ${\sin }^2(\theta -\phi )$ of the declared ideal analyser, its affinity over mixtures, its phase invariance and the conditional update of a repeated ideal analyser. Number, energy and momentum laws of a one-photon preparation proves that a normalisable one-photon preparation has the sharp number readout one, the frequency and energy laws given by the push-forwards of $\mathrm{\parallel }f(k){\mathrm{\parallel }}^2{d}^{3}k$ under $c\mathrm{\mid }k\mathrm{\mid }$ and $\mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }$ (a unit packet coordinate $f(k)$ has units ${\mathrm{m}}^{3\mathrm{/}2}$ under $d^{3}k$), their means under the first-moment condition, and the corresponding momentum-component laws and means under absolute first-moment integrability. Frequency, energy and momentum variances are strictly positive and finite on the corresponding multiplier domains; the singleton probabilities vanish for every normalisable unit vector without a moment-domain hypothesis. Number and energy statistics of a declared mode: number states, coherent states and the continuum thermal limitation proves the sharp count of the $n$-quanta mode state, the Poisson count law and moments of the mode-coherent preparation, the energy means and variances of those preparations, and that $e^{-\beta H_f}$ is not trace class for the full continuum sector when $\beta >0$ has unit ${\mathrm{J}}^{-1}$. Thus its trace-normalized full-sector Gibbs density is unavailable; other preparations are outside the theorem. All these are conditional consequences of P1, P2 and the adopted radiation model in the declared measurement models. The repeated ideal-analyser conditional update additionally adopts P3 (Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes); its use is restricted to outcomes of positive probability. Their units are recorded in Dimensions, SI unit conventions and quantity types of the framework and in the assigned observables of the model.

What the page does not claim. The page does not derive the radiation extension from Maxwell's equations, does not derive a Lorentz transformation law for mode labels, polarisations or energies, and does not assert Galilean covariance of the radiation sector; the two polarisation states of the photon are adopted, and their relativistic description by Lorentz-group representations is outside this non-relativistic extension. It does not introduce a photon position wave function, a photon localisation operator or a general photon-position distribution; the sharp-mode label $k$ is a spectral parameter and not a state, and no normalisable one-photon vector has sharp frequency, energy or momentum. It does not supply a light-matter Hamiltonian, a dipole coupling, selection rules, absorption or stimulated emission, and it does not supply vacuum spontaneous emission; those belong to the later pair on light-matter and spectral transitions, where the coupling and environment assumptions are declared separately.

What the page's idealisations are. The declared analyser, the declared total-number detector and the declared energy-window detector are ideal projection-valued measurements of the model (The declared ideal counting, energy-window and analyser detection models of the radiation sector). No laboratory polariser, counter or spectrometer is identified with them; no efficiency, dark count, dead time, background, calibration or resolution is declared, and a destructive absorption detector would require its own instrument. A single measurement law is not a frequency: repeated trials require a separately declared preparation and sampling model in the sense of P7, which this page does not supply.

Empirical status. The page records no experiment, no measured count, no calibration, no uncertainty budget and no statistical test, and none is invented. The Born ratio ${\cos }^2(\theta -\phi )$ of the declared analyser coincides with the intensity ratio of the declared classical transverse-field model of the companion page; the coincidence is a property of the two declared models and does not establish field quantisation, and classical electromagnetic waves also interfere. Whether a given source, polariser or detector is described by the declared fixed-mode model is a tested identification with uncertainty under P7; agreement of the model's conditional predictions with reported data would be evidence about the combined model and apparatus assumptions, not a deductive proof of the model or of the framework.

Mathematical hypotheses. The conditional deductions summarized here retain the choice hypotheses of their mathematical suppliers; in particular, the transverse free-field Fock supplier assumes Countable Choice, Dependent Choice and Choice. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain and The Axiom of Choice record these assumptions.