The declared ideal counting, energy-window and analyser detection models of the radiation sector

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the ideal conditional-measurement postulate P3 and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments and the radiation sector definition The radiation sector of the framework: modes and the number, energy and momentum assignments, in the measurement vocabulary of Declared ideal measurements and instruments of the framework. Assume 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, The Axiom of Choice), including the hypotheses inherited from the radiation postulate and the cited measurement mathematics.

(b) Declared ideal total-number detector. The declared ideal total-number detector of the radiation sector is the declared ideal sharp measurement with outcome set $\mathbb{N}$, outcome projection $P_n:={\mathrm{\Pi }}_n(n\in \mathbb{N}),$ the sector projections of the Fock space $F(\mathfrak{h})$; the family $({\mathrm{\Pi }}_n)$ consists of mutually orthogonal orthogonal projections with strong sum $I$ (Fock sector projections clauses (a),(b),(c)). Its count law in a declared radiation preparation $\rho$ is the reading of clause (ii) of P2, $p_n(\rho )=\mathrm{tr}(\rho {\mathrm{\Pi }}_n),{\sum }_{n\in \mathbb{N}}{p}_n(\rho )=1,$ and its number readout is the assigned total-number observable $(N,1)$ of Number operator domain read through the atom law of P2. The conditional post-count state and the discarded-outcome state are the Lueders updates of the measurement definition (The selective Lueders update by an orthogonal projection, The non-selective Lueders map and the discarded-outcome ensemble). The word "count" refers to the quanta of the declared model; no laboratory counter, no absorption process, no detection efficiency, no dark count, no dead time and no background is declared here.

(c) Declared ideal energy-window detector. For a Borel set $B\subseteq [0,\mathrm{\infty })$ read in joules the declared ideal energy-window detector of window $B$ is the declared ideal sharp measurement with outcome structure $(E_{H_f}(B),I-E_{H_f}(B)),$ where $E_{H_f}$ is the spectral projection-valued measure of the assigned free-field energy observable $(H_f,\mathrm{J})$; its window law in the preparation $\rho$ is $p_B(\rho )=\mathrm{tr}(\rho E_{H_f}(B))$ and its complement probability is $\mathrm{tr}(\rho (I-E_{H_f}(B)))$, the reading of P2. A finite-resolution detector is described by such a window or by a declared finite or countably infinite family $(B_a{)}_{a\in A}$ of pairwise disjoint Borel windows covering $[0,\mathrm{\infty })$, with outcome projections $(E_{H_f}(B_a){)}_{a\in A}$. Countable additivity of the spectral PVM and $E_{H_f}([0,\mathrm{\infty }))=I$ give their strong sum $I$, so this is an admissible declared ideal sharp measurement. The statement that a laboratory spectrometer realizes a declared window, with what resolution and calibration, is an identification under P7.

(d) Declared analyser detection. The declared polarisation-analyser detection of the fixed propagation mode is the analyser family $\{P_{\theta },P_{\theta }^{\perp }\}$ of The fixed-mode polarisation model of the framework, read as a declared ideal sharp measurement of the polarisation factor; its law and its conditional update are proved on this page. It detects the polarisation alternative of the model and is not a photon counter.

(e) Repetition requires a separate sampling model. The declaration of a detector gives the outcome law of a single declared measurement. A prediction for repeated trials, a count rate, or any statistical comparison with data requires in addition a declared preparation-and-apparatus model and a sampling model in the sense of P7 -- for instance an identically-and-independently repeated preparation with a declared number of trials -- which this page does not supply. A single measurement outcome law is not a frequency and no frequency is asserted.

(f) Destructive detectors are outside the declared class. A photoabsorption detector that destroys the absorbed quantum, and any detector whose post-outcome state of the radiation field is not the Lueders update, requires its own declared instrument, that is its own outcome effects together with its own conditional updates (clause (v) of P3, clause (d) of the measurement definition). No such instrument is declared on this page, and the "blocked" outcome of the analyser is used only through its discarded-outcome law.

(g) Status. Clauses (b)-(d) fix vocabulary and read the cited postulates; they add no physical assumption beyond P1-P3, P7 and the radiation model, and they assert no detection mechanism, no response function and no empirical result. In particular, the declaration of an ideal counting model is not a claim that any physical device counts photons, and the energy-window model is not a claim that any physical spectrometer measures exact energy.