Scope of the photons and radiation modes examples page

What the three items establish. Malus and Jones ratios of the declared analyser: the same cos^2 from the Born law and from a classical transverse amplitude shows that the Born probability ${\cos }^2\theta$ of the declared ideal analyser on the preparation along $e_1$ and the transmitted intensity fraction ${\cos }^2\theta$ of the declared elementary classical transverse-amplitude model coincide for every declared angle, so that this ratio alone does not establish field quantisation. Number statistics of a single declared mode: sharp, Poisson and geometric preparations computes the count laws and their means and variances for three declared preparations of one mode -- sharp $n$, Poisson for dimensionless coherent-state label $z$ with mean and variance $\mathrm{\mid }z{\mathrm{\mid }}^2$, and the geometric law of a declared effective-energy preparation with ${\epsilon }_0>0$ joules and $\beta >0$ in ${\mathrm{J}}^{-1}$, whose mean is $1\mathrm{/}(e^{\beta {\epsilon }_0}-1)$ -- and shows different fluctuations for the coherent and geometric preparations at equal positive mean. A number state joins that equal-mean comparison when the common mean is a positive integer and $n$ is chosen equal to it. A two-bin comparison of a declared classical field model and the quantised model: identical normalised laws do not discriminate proves that the declared classical and quantised models assign the same normalised two-bin law, with the same interference cross-term, for every declared amplitude datum, and concludes that a two-bin fringe law of that form is not a discriminator between the two declared models.

What they rest on. All three items are conditional on the state and composition postulate P1, the observable and probability postulate P2, the adopted radiation extension and the declared fixed-mode or radiation-sector vocabulary of the main page; the counting results use only the declared ideal total-number detector of The declared ideal counting, energy-window and analyser detection models of the radiation sector. The classical models appearing in the first and third items are explicit elementary declared models -- dimensionless amplitude coordinates relative to a fixed reference amplitude, with relative intensity equal to squared amplitude norm -- and are not Maxwell's theory; the effective mode energy ${\epsilon }_0$ of the second item is a declared parameter, not a sharp energy of a normalisable mode.

Different objects. A one-photon preparation is a unit ray of the transverse one-particle space read in the one-photon sector; a mode-coherent preparation is a unit ray of the radiation Fock space with no sharp number when $z\mathrm{\ne }0$, while $z=0$ is the vacuum with sharp number zero; and the declared classical intensity datum is a complex transverse amplitude with its declared intensity. The three are distinct mathematical objects of the declared models, and no identification of one with another is made.

What is not claimed. The page records no experiment, no apparatus, no calibration, no measured count, no frequency, no efficiency and no uncertainty budget, and none is invented. It does not claim that any laboratory polariser, counter or interferometer realizes the declared ideal models; that claim is a tested identification with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). It does not assert thermal equilibrium or a temperature for the geometric preparation, and it does not assert that the coincidence of the classical and quantum ratios is evidence for field quantisation. Single-photon polarisation and counting experiments remain research candidates for a separate reported-experiment item; a finite sample of detections is a random object under any declared sampling model, visible fringes can fail to appear with nonzero probability under a correct coherent model, and a $p$-value or confidence level is not the probability that a model is false. Comparing any of these conditional predictions with data would test the combined model and apparatus assumptions, not prove the model or 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.