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Qm Photons And Radiation Modes

1 · Prerequisites

2 · Summary

This page develops the radiation extension of the framework: the fixed-mode polarisation model, the quantised transverse radiation sector and the number, energy and momentum readings of its preparations. The extension is an explicitly adopted model assumption, Radiation extension of the framework: transverse mode and photon assignments: a declared propagation mode carries a two-dimensional transverse polarisation space; the free radiation is the bosonic Fock space of the transverse one-particle space $h = {f \in L^{2} (R^{3}; C^{3}) : k \cdot f (k) = 0 a.e.}$ ; and in a declared inertial chart the mode label $k$ is assigned the frequency $ω = c ∣ k ∣$ , the photon energy $E = ℏ ω$ and the photon momentum $p = ℏ k$ , with $c$ and $ℏ$ carrying the SI values of Dimensions, SI unit conventions and quantity types of the framework. The postulate records explicitly that these assignments are not derived from the massive model, from Maxwell's theory or from special relativity, that no Lorentz transformation law for mode labels is claimed and that the extension does not claim Galilean covariance of the radiation sector.

The supporting local mathematics precedes the physical readings. Analyzer projections of a two-dimensional polarisation space and their overlaps computes the finite-dimensional analyser projections and the overlap $⟨ e_{θ}, ψ_{φ} ⟩ = \cos (θ - φ)$ . Restricted momentum and energy multipliers of the transverse one-photon space shows that the restrictions of the momentum and energy multipliers to the transverse one-particle space are self-adjoint with the push-forward spectral laws, have no eigenvalues, and give every unit one-photon vector in the corresponding multiplier domain finite strictly positive variance; in particular no normalisable vector has sharp frequency, energy or momentum. The fixed-mode isometry of Fock space and the transfer of number and ladder operators identifies the Fock space of a single declared mode inside $F (h)$ , transfers the number and ladder operators and computes the mode energy moments. Total momentum multipliers on transverse symmetric Fock sectors constructs the total signed momentum multipliers on their maximal symmetric transverse sector domains, their self-adjoint Fock direct sums and spectral measures, including the vacuum atom at zero.

The radiation sector of the framework: modes and the number, energy and momentum assignments fixes the sector, mode and observable vocabulary, with the number observable read in the dimensionless unit $1$ , the free-field energy observable $H_{f} = ⨁_{n} T_{n}$ with $ε (k) = ℏ c ∣ k ∣$ read in joules and the total momentum components and their one-photon restrictions read in $k g m s^{- 1}$ . On the two-dimensional polarisation factor, The fixed-mode polarisation model of the framework declares the pure polarisation preparations and the ideal analyser family, and The analyser outcome law of the fixed-mode polarisation model: Born probabilities and the Malus ratio proves the Born law $\cos^{2} (θ - φ)$ , $\sin^{2} (θ - φ)$ , 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 the sharp one-photon number law, the frequency and energy laws as push-forwards of $∥ f (k) ∥^{2} d^{3} k$ (a unit packet coordinate $f (k)$ has units $m^{3 / 2}$ under this measure convention), their means whenever the first moment is finite, and the corresponding momentum-component laws and means under absolute first-moment integrability. The declared ideal counting, energy-window and analyser detection models of the radiation sector declares the ideal total-number and energy-window detectors and the analyser detection, with the explicit limits that no counter response, efficiency, background or destructive instrument is supplied and that repeated trials require a separate sampling model. 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 the failure of the unregularised continuum free-field Gibbs operator $e^{- β H_{f}}$ to be trace class for $β > 0$ in $J^{- 1}$ , so that this trace-normalised full-sector Gibbs density is unavailable; other thermal preparations are outside the theorem and require separate definitions. Scope and empirical status of the photons and radiation modes page collects the scope and empirical status of the page.

Every physical statement on the page is conditional on the cited postulates and mathematics, and the page records no experiment: there is no apparatus, calibration, trial count, uncertainty budget or measured value anywhere in it, and none is invented. The identification of a laboratory source, polariser, counter or spectrometer with the declared models is a tested identification with uncertainty under Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Analyzer projections of a two-dimensional polarisation space and their overlaps

Statement

Let $V$ be a two-dimensional complex inner product space with inner product linear in the first argument (Real and complex inner product spaces, with the inner product linear in the first argument, Real and complex inner-product spaces and their induced length), and let $(e_{1}, e_{2})$ be an orthonormal basis of $V$ (Orthonormal families, complete orthonormal systems and Hilbert bases). For $θ \in R$ put $e_{θ} : = \cos θ e_{1} + \sin θ e_{2}, e_{θ}^{⊥} : = - \sin θ e_{1} + \cos θ e_{2},$ and for $x \in V$ define $P_{θ} x : = ⟨ x, e_{θ} ⟩ e_{θ}, P_{θ}^{⊥} x : = ⟨ x, e_{θ}^{⊥} ⟩ e_{θ}^{⊥} .$ Then:

(a) for every $θ$ the maps $P_{θ}$ and $P_{θ}^{⊥}$ are orthogonal projections on $V$ with ranges $C e_{θ}$ and $C e_{θ}^{⊥}$ (The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive), one has $P_{θ} + P_{θ}^{⊥} = I$ and $P_{θ} P_{θ}^{⊥} = 0$ , and $P_{θ} P_{θ} = P_{θ} = P_{θ}^{*}, P_{θ} P_{θ}^{⊥} = P_{θ}^{⊥} P_{θ} = 0;$

(b) for all real $φ, θ$ the unit vectors $ψ_{φ} : = \cos φ e_{1} + \sin φ e_{2}$ satisfy $⟨ e_{θ}, ψ_{φ} ⟩ = \cos (θ - φ), ⟨ ψ_{φ}, P_{θ} ψ_{φ} ⟩ = ∥ P_{θ} ψ_{φ} ∥^{2} = ∣ ⟨ e_{θ}, ψ_{φ} ⟩ ∣^{2} = \cos^{2} (θ - φ),$ and $⟨ ψ_{φ}, P_{θ}^{⊥} ψ_{φ} ⟩ = \sin^{2} (θ - φ)$ ;

(c) the assignments $θ \mapsto P_{θ}$ and $θ \mapsto P_{θ}^{⊥}$ are $2 π$ -periodic, $P_{θ + π} = P_{θ}$ and $P_{θ + π / 2} = P_{θ}^{⊥}$ , and for every $z \in C$ and $x \in V$ one has $P_{θ} (z x) = z P_{θ} x$ and $∥ P_{θ} (z ψ_{φ}) ∥^{2} / ∥ z ψ_{φ} ∥^{2} = \cos^{2} (θ - φ)$ whenever $z \neq 0$ ; in particular the number $\cos^{2} (θ - φ)$ depends only on the ray $[ψ_{φ}]$ , not on the representative;

(d) the pair $(P_{θ}, P_{θ}^{⊥})$ is a two-element family of mutually orthogonal projections on $V$ with $P_{θ} + P_{θ}^{⊥} = I$ , the spectral values $1$ and $0$ of the bounded self-adjoint operator $P_{θ}$ have projections $P_{θ}$ and $P_{θ}^{⊥}$ , respectively. These two alternatives may be labelled $+ 1$ and $- 1$ in a declared analyser; those labels are not the spectral values of $P_{θ}$ . The eigenspace of $P_{θ}$ at eigenvalue $1$ is $C e_{θ}$ and the eigenspace at eigenvalue $0$ is $C e_{θ}^{⊥}$ .

No physical interpretation of $V$ , of the pair $(e_{1}, e_{2})$ or of the projections $P_{θ}$ is asserted.

Facts & Assumptions

Given: the two-dimensional complex inner product space $V$ with inner product linear in the first argument, the orthonormal basis $(e_{1}, e_{2})$ of $V$ , the vectors $e_{θ}, e_{θ}^{⊥}, ψ_{φ} \in V$ and the maps $P_{θ}, P_{θ}^{⊥} : V \to V$ .

[F1]

Inner product and norm. For all $x, y \in V$ and $a, b \in C$ , $⟨ a x + b y, z ⟩ = a ⟨ x, z ⟩ + b ⟨ y, z ⟩$ and $⟨ x, a y + b z ⟩ = \overline{a} ⟨ x, y ⟩ + \overline{b} ⟨ x, z ⟩$ ; $⟨ x, x ⟩ \geq 0$ with equality only for $x = 0$ ; and $∥ x ∥^{2} = ⟨ x, x ⟩$ (Real and complex inner product spaces, with the inner product linear in the first argument, Real and complex inner-product spaces and their induced length).

[F2]

Orthonormality. $⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ for $i, j \in {1, 2}$ (Orthonormal families, complete orthonormal systems and Hilbert bases).

[F3]

Finite expansion. For an orthonormal basis $(u_{1}, u_{2})$ of $V$ , write $x = a_{1} u_{1} + a_{2} u_{2}$ . Taking inner products with $u_{j}$ gives $a_{j} = ⟨ x, u_{j} ⟩$ by [F1] and orthonormality, and expanding $⟨ x, x ⟩$ gives $∥ x ∥^{2} = ∣ a_{1} ∣^{2} + ∣ a_{2} ∣^{2}$ . Thus the finite expansion and norm identity hold without invoking the countable-choice Hilbert-space version of Parseval.

[F4]

Finite-dimensional orthogonal projection. The projection onto a subspace $M$ of $V$ assigns the unique $M$ -component of $x$ in the decomposition $V = M \oplus M^{⊥}$ ; equivalently, its value lies in $M$ and the residual lies in $M^{⊥}$ . This is the finite-dimensional agreement clause of The Hilbert orthogonal projection onto a closed subspace.

Proof

technique · direct; compute the overlaps of the displayed vectors from the orthonormality of $(e_{1},e_{2})$ and verify the projection identities

Given: The data of the statement and [F1]-[F4].

1.1F1F2algebra

[The rotated pairs are orthonormal] By [F1] and [F2], $⟨ e_{θ}, e_{θ} ⟩ = \cos^{2} θ + \sin^{2} θ = 1$ , $⟨ e_{θ}^{⊥}, e_{θ}^{⊥} ⟩ = \sin^{2} θ + \cos^{2} θ = 1$ and $⟨ e_{θ}, e_{θ}^{⊥} ⟩ = - \cos θ \sin θ + \sin θ \cos θ = 0$ ; hence $(e_{θ}, e_{θ}^{⊥})$ is an orthonormal family of the two-dimensional space $V$ , therefore an orthonormal basis of $V$ .

1.2F1F2algebra

[Overlaps and the cosine identity] By [F1] and [F2], $⟨ e_{θ}, ψ_{φ} ⟩ = \cos θ \cos φ + \sin θ \sin φ = \cos (θ - φ)$ and $⟨ e_{θ}^{⊥}, ψ_{φ} ⟩ = - \sin θ \cos φ + \cos θ \sin φ = \sin (φ - θ)$ . Then $⟨ ψ_{φ}, P_{θ} ψ_{φ} ⟩ = ⟨ ψ_{φ}, ⟨ ψ_{φ}, e_{θ} ⟩ e_{θ} ⟩ = ∣ ⟨ ψ_{φ}, e_{θ} ⟩ ∣^{2} = \cos^{2} (θ - φ)$ because $\cos (θ - φ)$ is real, and $∥ P_{θ} ψ_{φ} ∥^{2} = ∣ \cos (θ - φ) ∣^{2} ∥ e_{θ} ∥^{2} = \cos^{2} (θ - φ)$ ; similarly $⟨ ψ_{φ}, P_{θ}^{⊥} ψ_{φ} ⟩ = \sin^{2} (θ - φ)$ . This is (b).

2.1F1F3step 1.1F4algebra

[The projections are orthogonal projections] For $x \in V$ , $P_{θ} x = ⟨ x, e_{θ} ⟩ e_{θ}$ , so $P_{θ} (P_{θ} x) = ⟨ ⟨ x, e_{θ} ⟩ e_{θ}, e_{θ} ⟩ e_{θ} = ⟨ x, e_{θ} ⟩ ⟨ e_{θ}, e_{θ} ⟩ e_{θ} = P_{θ} x$ by [F1] and step 1.1: $P_{θ}^{2} = P_{θ}$ . For the adjoint, $⟨ P_{θ} x, y ⟩ = ⟨ ⟨ x, e_{θ} ⟩ e_{θ}, y ⟩ = ⟨ x, e_{θ} ⟩ ⟨ e_{θ}, y ⟩ = ⟨ x, ⟨ y, e_{θ} ⟩ e_{θ} ⟩ = ⟨ x, P_{θ} y ⟩$ for all $x, y$ , so $P_{θ}^{*} = P_{θ}$ ; the same computation with $e_{θ}^{⊥}$ in place of $e_{θ}$ gives $P_{θ}^{⊥ 2} = P_{θ}^{⊥} = P_{θ}^{⊥ *}$ . For either unit vector $u = e_{θ}$ or $u = e_{θ}^{⊥}$ , $P_{u} x = ⟨ x, u ⟩ u$ lies in $C u$ and $⟨ x - P_{u} x, u ⟩ = 0$ , so the residual is orthogonal to every multiple of $u$ . Hence [F4] identifies $P_{u}$ as the orthogonal projection onto $C u$ ; its range is exactly that line because $P_{u} u = u$ . By the finite norm identity in [F3], $∥ P_{u} x ∥^{2} = ∣ ⟨ x, u ⟩ ∣^{2} \leq ∥ x ∥^{2}$ , so it is bounded. The line is closed: if $a_{r} u \to v$ , then $a_{r} = ⟨ a_{r} u, u ⟩ \to ⟨ v, u ⟩$ (use the finite coordinate norm identity), and therefore $v = ⟨ v, u ⟩ u$ .

3.1F1F3step 1.1step 2.1algebra

[Complementarity] By step 1.1 and [F3] applied to the orthonormal basis $(e_{θ}, e_{θ}^{⊥})$ , every $x \in V$ satisfies $x = ⟨ x, e_{θ} ⟩ e_{θ} + ⟨ x, e_{θ}^{⊥} ⟩ e_{θ}^{⊥} = P_{θ} x + P_{θ}^{⊥} x$ ; hence $P_{θ} + P_{θ}^{⊥} = I$ . Applying $P_{θ}$ to the same expansion and using $P_{θ} e_{θ}^{⊥} = ⟨ e_{θ}^{⊥}, e_{θ} ⟩ e_{θ} = 0$ gives $P_{θ} P_{θ}^{⊥} = 0$ , and likewise $P_{θ}^{⊥} P_{θ} = 0$ . Since $P_{θ} = I - P_{θ}^{⊥}$ and $C e_{θ} \cap C e_{θ}^{⊥} = {0}$ by orthogonality of the nonzero vectors $e_{θ}, e_{θ}^{⊥}$ , the eigenspace of $P_{θ}$ at eigenvalue $1$ is $C e_{θ}$ and at eigenvalue $0$ is $C e_{θ}^{⊥}$ ; this proves (a) and (d).

4.1F1step 1.2algebra∎

[Periodicity and phase invariance] Since $e_{θ + π} = - e_{θ}$ and $e_{θ + π / 2} = e_{θ}^{⊥}$ , the identities $P_{θ + π} = P_{θ}$ and $P_{θ + π / 2} = P_{θ}^{⊥}$ follow from the definition of $P_{θ}$ and [F1]; in particular $P_{θ}$ is $2 π$ -periodic. For $z \neq 0$ and the unit vector $ψ_{φ}$ one has $∥ z ψ_{φ} ∥^{2} = ∣ z ∣^{2}$ , and $P_{θ} (z ψ_{φ}) = z P_{θ} ψ_{φ}$ by linearity of $x \mapsto ⟨ x, e_{θ} ⟩$ in the first argument of [F1]; hence $∥ P_{θ} (z ψ_{φ}) ∥^{2} / ∥ z ψ_{φ} ∥^{2} = ∥ P_{θ} ψ_{φ} ∥^{2} = \cos^{2} (θ - φ)$ by step 1.2. This proves (c) and completes the proof.

Remarks

What the lemma is not. The lemma is finite-dimensional linear algebra: no dynamics, no measurement postulate and no physical identification of $e_{θ}$ with an apparatus direction is asserted. The physical reading of the projections as the outcome structure of a polarisation analyser is fixed in The fixed-mode polarisation model of the framework.
The identity $\cos^{2} (θ - φ)$ is classical and quantum at once. The same projection formula governs the classical transverse field amplitude and the quantised polarisation amplitude; the comparison, and the reason that the coincidence of the numbers does not establish field quantisation, is the subject of the companion examples page of this pair, which is not a prerequisite of this lemma.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Restricted momentum and energy multipliers of the transverse one-photon space

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the cited multiplication and spectral items. On $L^{2} (R^{3}; C^{3})$ (The space $L^{p} (μ)$ as the quotient by null functions) put $h : = {f : k \cdot f (k) = 0 for almost every k},$ a closed subspace of $L^{2} (R^{3}; C^{3})$ , and write $Π (k) : = i d - ∣ k ∣^{- 2} k k^{T}$ for $k \neq 0$ , $Π (0) : = 0$ , so that $h = ran M_{Π}$ with $M_{Π}$ the bounded multiplication by $Π$ (The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive). Let $c, ℏ > 0$ with $c$ in $m s^{- 1}$ and $ℏ$ in $J s$ , and for $j \in {1, 2, 3}$ and $m \in {ℏ k_{j}, ℏ c ∣ k ∣}$ define $D (M_{m}^{h}) : = {f \in h : \int_{R^{3}} m (k)^{2} ∥ f (k) ∥_{C^{3}}^{2} d^{3} k < \infty}, M_{m}^{h} f : = m f .$ Then for each such $m$ :

(a) $M_{m}^{h}$ is a densely defined self-adjoint operator on the Hilbert space $h$ (Hilbert space, Symmetric, self-adjoint and essentially self-adjoint operators), and it is the restriction to $h$ of the self-adjoint multiplier $M_{m}$ on $L^{2} (R^{3}; C^{3})$ of Real multiplication operators and free plus bounded-potential Schrödinger operators;

(b) its spectral projection-valued measure $E_{m}$ is given by $E_{m} (B) = M_{1_{m^{- 1} (B)}} ∣_{h} (B \subseteq R Borel),$ and for every unit $f \in h$ the outcome law $B \mapsto ∥ E_{m} (B) f ∥^{2}$ is the push-forward of the finite measure $∣ f ∣^{2} d^{3} k ∣_{h}$ under $k \mapsto m (k)$ ; in particular for $m = ℏ k_{j}$ it is the push-forward under $k \mapsto ℏ k_{j}$ and for $m = ℏ c ∣ k ∣$ it is the push-forward under $k \mapsto ℏ c ∣ k ∣$ (Projection valued measure, The unbounded PVM integral is densely defined, closed and normal);

(d) for every unit $f \in D (M_{m}^{h})$ one has ${Var}_{f} (M_{m}^{h}) : = ∥ M_{m}^{h} f ∥^{2} - ⟨ M_{m}^{h} f, f ⟩^{2} > 0;$ consequently no unit vector of $h$ has a sharp $m$ -value.

The identifications of $h$ and of these multipliers with physical transverse photons, momenta or energies are not asserted here; they are a separate model assignment outside this lemma.

Facts & Assumptions

Given: the Axioms of Countable Choice and of Choice, the constants $c, ℏ > 0$ , the transverse subspace $h = ran M_{Π}$ , the Borel sets $B \subseteq R$ , the functions $m \in {ℏ k_{j}, ℏ c ∣ k ∣}$ and the operators $M_{m}^{h}$ .

[F1]

Full-space multiplier. For a real measurable finite-valued $a : R^{3} \to R$ the componentwise multiplier $M_{a} u : = a u$ on $D (M_{a}) = {u \in L^{2} (R^{3}; C^{3}) : a u \in L^{2}}$ is self-adjoint: the scalar statement is Real multiplication operators and free plus bounded-potential Schrödinger operators clause (a), applied to each of the three components of $u$ .

[F2]

Spectral measure of a multiplier. Let $(X, Σ, μ)$ be $σ$ -finite with $μ (X) > 0$ , $m : X \to R$ measurable and finite a.e., $H = L^{2} (X, μ; C)$ and $M_{m}$ as in [F1]. Then $E (B) : = M_{1_{m^{- 1} (B)}}$ is the spectral projection-valued measure of $M_{m}$ , and for a unit vector $ψ \in H$ one has $∥ E (B) ψ ∥^{2} = \int_{m^{- 1} (B)} ∣ ψ ∣^{2} d μ$ (Spectral measure of a real L2 multiplication operator clauses (b), (c)).

[F3]

PVM integrals. Let $E$ be a projection-valued measure on a Hilbert space $K$ and $f$ real-valued measurable. Then $f (E) = \int f d E$ is self-adjoint with domain $D (f (E)) = {x : \int f^{2} d E_{x} < \infty}$ , where $d E_{x}$ is the scalar measure of $E$ at $x$ , and $∥ f (E) x ∥^{2} = \int f^{2} d E_{x}$ ; moreover bounded measurable functions with $∣ g_{n} ∣ \leq C ∣ f ∣$ and $g_{n} \to f$ pointwise satisfy $g_{n} (E) x \to f (E) x$ for every $x \in D (f (E))$ (Integral of a measurable function against a projection-valued measure, The unbounded PVM integral is densely defined, closed and normal).

[F4]

Hilbertian projections. $h = ran M_{Π}$ is a closed linear subspace of $L^{2} (R^{3}; C^{3})$ with the restricted inner product, and $M_{Π}$ is the orthogonal projection onto it; multiplication operators by functions of $k$ commute with $M_{Π}$ (The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive, Real multiplication operators and free plus bounded-potential Schrödinger operators).

[F5]

Null level sets of the momentum components. For $j \in {1, 2, 3}$ and $λ \in R$ the set ${k \in R^{3} : ℏ k_{j} = λ}$ is the graph of a constant, hence Lebesgue measurable, function over $R^{2}$ and is Lebesgue null (The graph of a measurable function R^n to R is Lebesgue null).

[F6]

Null spheres. For $λ \in R$ the set ${k \in R^{3} : ℏ c ∣ k ∣ = λ}$ is empty when $λ < 0$ . When $λ \geq 0$ , put $r_{0} : = λ / (ℏ c)$ . If $r_{0} = 0$ , the level set is the singleton ${0}$ and is Lebesgue null by the countable-subset proposition. If $r_{0} > 0$ , the polar-coordinates formula applied to $g = 1_{{∣ k ∣ = r_{0}}}$ gives zero measure: its radial integrand vanishes except at $r = r_{0}$ , and the singleton ${r_{0}} \subset (0, \infty)$ is Lebesgue null. Thus every level set is null (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ).

Proof

technique · direct; restrict the full-space multiplier to the invariant closed subspace, identify the restricted spectral measure, and use the nullity of the level sets to exclude eigenvectors

Given: The data of the statement and [F1]-[F6].

1.1F2F4algebra

[Restriction of the spectral measure] Let $m$ be one of the two displayed functions; both are real, measurable and finite-valued. By [F4] the multiplier $M_{m}$ commutes with $M_{Π}$ , so $E (B) = M_{1_{m^{- 1} (B)}}$ (componentwise) commutes with $M_{Π}$ for every Borel set $B$ and therefore maps $h$ into itself: for $f \in h$ one has $E (B) f = 1_{m^{- 1} (B)} f \in h$ because transversality holds pointwise. Thus $E_{h} (B) : = E (B) ∣_{h}$ is a family of projections on $h$ with $E_{h} (B)^{2} = E_{h} (B) = E_{h} (B)^{*}$ , $E_{h} (\emptyset) = 0$ , $E_{h} (R) = {i d}_{h}$ , and $σ$ -additivity and strong convergence inherited from $E$ ; hence $E_{h}$ is a projection-valued measure on $h$ .

1.2F5F6algebra

[No eigenvalues] Let $λ \in R$ and let $f \in D (M_{m}^{h})$ satisfy $M_{m}^{h} f = λ f$ , that is $m (k) f (k) = λ f (k)$ for almost every $k$ . Then $f = 0$ almost everywhere on ${k : m (k) \neq λ}$ , so $f$ is supported in $N_{λ} : = {k : m (k) = λ}$ up to a null set. For $m = ℏ k_{j}$ the set $N_{λ}$ is null by [F5]; for $m = ℏ c ∣ k ∣$ it is null by [F6]. In both cases $f = 0$ almost everywhere, hence $f = 0$ in $L^{2}$ . This proves (c).

2.1F2F3F4step 1.1algebra

[The restricted integral is the restricted multiplier] By [F3] applied to $E_{h}$ and the real function $λ \mapsto λ$ the operator $A : = \int λ d E_{h} (λ)$ is self-adjoint with domain ${f \in h : \int λ^{2} d E_{h, f} (λ) < \infty}$ . For every $g \in h$ and Borel $B$ we have $⟨ E_{h} (B) g, g ⟩ = \int_{m^{- 1} (B)} ∥ g (k) ∥^{2} d^{3} k$ : the scalar statements of [F2] applied to the three components of $g$ combine by sesquilinearity, and the unit case extends by scaling in $g$ , so the scalar measure $E_{h, g}$ is the push-forward of $∥ g ∥^{2} d^{3} k$ under $k \mapsto m (k)$ ; hence $\int λ^{2} d E_{h, f} (λ) = \int m (k)^{2} ∥ f (k) ∥^{2} d^{3} k$ by the change-of-variables formula for a push-forward measure against the Borel function $λ \mapsto λ^{2}$ , the two sides agreeing on indicators of Borel sets and extending by linearity and monotone convergence. Therefore $D (A) = {f \in h : \int m^{2} ∥ f ∥^{2} d^{3} k < \infty} = D (M_{m}^{h})$ .

3.1F1F2F3step 1.1step 2.1algebra

[Identification on the domain] Let $f \in D (A)$ and put $m_{n} : = m 1_{{∣ m ∣ \leq n}}$ for $n \geq 1$ ; then $∣ m_{n} ∣ \leq ∣ m ∣$ and $m_{n} \to m$ pointwise, so [F3] gives $m_{n} (E_{h}) f \to A f$ . For $g \in h$ the bounded operator $m_{n} (E_{h})$ satisfies $⟨ m_{n} (E_{h}) g, g ⟩ = \int m_{n} d E_{h, g} = \int m_{n} (k) ∥ g (k) ∥^{2} d^{3} k = ⟨ m_{n} g, g ⟩$ by the push-forward description of step 2.1; both operators are bounded and self-adjoint and their quadratic forms agree, so $m_{n} (E_{h}) = M_{m_{n}} ∣_{h}$ : the bounded functional calculus of the restricted multiplier is multiplication by $m_{n}$ . The convergence $m_{n} f \to m f$ in $L^{2} (R^{3}; C^{3})$ holds by dominated convergence, since $∥ m_{n} f - m f ∥^{2} \leq 4 m^{2} ∥ f ∥^{2}$ is integrable under $f \in D (A)$ ; its limit $m f$ lies in $h$ because transversality is preserved pointwise, with no assertion that $m f$ belongs to $D (A)$ ; that stronger assertion would require $m^{2} f \in L^{2}$ . Hence $A f = m f = M_{m}^{h} f$ for every $f \in D (A)$ ; conversely if $f \in D (M_{m}^{h})$ then $\int m^{2} ∥ f ∥^{2} = ∥ m f ∥^{2} < \infty$ , so $f \in D (A)$ . Therefore $A = M_{m}^{h}$ with the displayed domain, and (a) and (b) follow.

4.1step 1.2algebra∎

[Positive variance] Let $f \in h$ be a unit vector in $D (M_{m}^{h})$ and put $μ : = ⟨ M_{m}^{h} f, f ⟩$ , a real number because $M_{m}^{h}$ is self-adjoint. Then $∥ M_{m}^{h} f ∥^{2} - ⟨ M_{m}^{h} f, f ⟩^{2} = ∥ (M_{m}^{h} - μ I) f ∥^{2} \geq 0,$ and equality holds exactly when $(M_{m}^{h} - μ I) f = 0$ , that is when $f$ is an eigenvector of $M_{m}^{h}$ with eigenvalue $μ$ . By step 1.2 no nonzero eigenvector exists, so the left-hand side is strictly positive. This proves (d) and completes the proof.

Remarks

Choice hypotheses. The real-multiplier and multiplication spectral-measure suppliers assume Countable Choice and the Axiom of Choice; the unbounded-PVM- integral supplier assumes Countable Choice. The item carries Countable Choice and Choice. The cited suppliers do not require Dependent Choice, and the nullity computations use Countable Choice.
What fails for sharp modes. Step 1.2 says that the sharp-mode idealisation of the radiation model is not a state vector of the sector: a normalisable one-photon vector never has an exact frequency or an exact momentum component. The plane-wave labels $k$ are parameters of the spectral representation, not normalisable states.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

The fixed-mode isometry of Fock space and the transfer of number and ladder operators

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). By AC implies DC implies countable choice, Choice supplies the Dependent Choice hypothesis, as defined in The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, of the cited transverse free-field Fock result. Let $H$ be a complex Hilbert space (Hilbert space, Real and complex inner-product spaces and their induced length) with completed tensor powers $H^{\otimes n}$ , symmetric projections $P_{+}^{(n)}$ , Fock space $F (H)$ , sector projections $Π_{n}$ , number operator $N$ and maximal creation and annihilation operators $A_{\pm} (f)$ as in Hilbert direct-sum construction, Fock sector projections, Number operator domain and Closed adjoint sector domains. Fix a unit vector $φ \in H$ . Write $φ^{\otimes 0} : = 1 \in C$ and $φ^{\otimes n} : = φ \otimes \dots \otimes φ \in H^{\otimes n}$ for $n \geq 1$ , and define $U_{φ} : F (C) \to F (H), U_{φ} e_{n} : = (0, \dots, 0, φ^{\otimes n}, 0, \dots)$ with the entry in coordinate $n$ , extended linearly. Then:

(a) each $φ^{\otimes n}$ is a unit vector of $ran P_{+}^{(n)}$ ; the family $(φ^{\otimes n})_{n \in N}$ is orthonormal in $F (H)$ , and $U_{φ}$ is a linear isometry of $F (C)$ onto the closed subspace $\overset{}{span} {φ^{\otimes n} : n \in N}$ (Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family, A bounded linear operator between normed spaces);

(b) for every $ψ = (ψ_{n})_{n \in N} \in F (C)$ and every $n$ , $Π_{n} U_{φ} ψ = U_{φ} Π_{n}^{'} ψ, ∥ U_{φ} ψ ∥^{2} = \sum_{n \in N} ∣ ψ_{n} ∣^{2} = ∥ ψ ∥^{2},$ where $Π_{n}^{'}$ is the Fock sector projection of $F (C)$ ; and for every $ψ \in D (N^{'})$ one has $U_{φ} ψ \in D (N)$ with $N U_{φ} ψ = U_{φ} N^{'} ψ;$

(c) for every $ψ \in D (A_{\pm} (1))$ one has $U_{φ} ψ \in D (A_{\pm} (φ))$ and $A_{\pm} (φ) U_{φ} ψ = U_{φ} A_{\pm} (1) ψ;$ in particular for every $z \in C$ the coherent vector $ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} e_{n}$ of Fixed-mode number law satisfies $U_{φ} ψ_{z} \in D (A_{-} (φ))$ and $A_{-} (φ) U_{φ} ψ_{z} = z U_{φ} ψ_{z}$ ;

(d) for the transverse case $H = h$ of Transverse continuum free Fock Hamiltonian with $ε (k) = ℏ c ∣ k ∣$ , $H_{f} = ⨁_{n} T_{n}$ , and a unit $φ \in h \cap D (ε_{h})$ (so that $ε φ \in h$ ), one has $⟨ H_{f} U_{φ} e_{n}, U_{φ} e_{n} ⟩ = n ⟨ ε φ, φ ⟩ (n \in N),$ and for every $ψ \in F (C)$ with $\sum_{n} n^{2} ∣ ψ_{n} ∣^{2} < \infty$ $⟨ H_{f} U_{φ} ψ, U_{φ} ψ ⟩ = (\sum_{n \in N} n ∣ ψ_{n} ∣^{2}) ⟨ ε φ, φ ⟩ = ⟨ N^{'} ψ, ψ ⟩ ⟨ ε φ, φ ⟩;$ Already under $φ \in D (ε_{h})$ , writing $\overset{ˉ}{ε} = ⟨ ε φ, φ ⟩$ , the finite spectral second moment is $∥ T_{n} φ^{\otimes n} ∥^{2} = n ∥ ε φ ∥^{2} + n (n - 1) {\overset{ˉ}{ε}}^{2} .$ If moreover $φ \in D (ε_{h}^{2})$ , the vector lies in $D (T_{n}^{2})$ and this norm square equals the literal operator pairing below; then $⟨ T_{n} φ^{\otimes n}, φ^{\otimes n} ⟩ = n ⟨ ε φ, φ ⟩, ⟨ T_{n}^{2} φ^{\otimes n}, φ^{\otimes n} ⟩ = n ⟨ ε^{2} φ, φ ⟩ + n (n - 1) ⟨ ε φ, φ ⟩^{2} .$

No physical interpretation of the isometry, of the vectors $U_{φ} e_{n}$ , or of the transferred operators is asserted.

Facts & Assumptions

Given: the complex Hilbert space $H$ , its Fock space $F (H)$ with sectors $ran P_{+}^{(n)}$ , projections $Π_{n}$ , number operator $N$ and maximal operators $A_{\pm} (f)$ , the unit vector $φ \in H$ , the map $U_{φ}$ , a vector $ψ = (ψ_{n}) \in F (C)$ , and in part (d) the transverse space $h$ , the energy multiplier $ε$ and the operator $H_{f} = ⨁_{n} T_{n}$ .

[F1]

Sectors and coordinate embeddings. For $n \geq 1$ the vector $ι_{n} (u) : = (0, \dots, 0, u, 0, \dots)$ with $u \in ran P_{+}^{(n)}$ lies in $F (H)$ , $∥ ι_{n} (u) ∥ = ∥ u ∥$ , and the ranges of distinct $ι_{n}$ are orthogonal; the same formula with $n = 0$ and $ran P_{+}^{(0)} = C$ gives the vacuum embedding, so the embedding is defined for every $n \in N$ ; for $x, y \in F (H)$ one has $⟨ x, y ⟩ = \sum_{n} ⟨ x_{n}, y_{n} ⟩$ and $∥ x ∥^{2} = \sum_{n} ∥ x_{n} ∥^{2}$ (Hilbert direct-sum construction clauses (a),(c),(d), Fock sector projections clauses (a),(b)).

[F2]

Symmetric tensors. $P_{+}^{(n)}$ is the orthogonal projection onto $ran P_{+}^{(n)}$ ; for $u \in H$ the elementary tensor $u^{\otimes n} = u \otimes \dots \otimes u$ satisfies $P_{+}^{(n)} u^{\otimes n} = u^{\otimes n}$ , $∥ u^{\otimes n} ∥ = ∥ u ∥^{n}$ , and $⟨ u^{\otimes n}, v^{\otimes n} ⟩ = ⟨ u, v ⟩^{n}$ (Permutation sector projections on the n-fold tensor power clause (c), Hilbert tensor construction: inner product, completion and product bases clause (b)).

[F3]

Number operator. $N$ is self-adjoint with $D (N) = {ψ : \sum_{n} n^{2} ∥ ψ_{n} ∥^{2} < \infty}$ and $N Π_{n} = n Π_{n}$ on the sector, i.e. $(N x)_{n} = n x_{n}$ ; the Fock space is the orthogonal direct sum of the sector ranges (Number operator domain clauses (a),(b), Fock sector projections clause (c)).

[F4]

Maximal creation and annihilation operators. $A_{+} (f)$ and $A_{-} (f)$ are closed linear operators in the state vector; creation is linear and annihilation conjugate-linear in the mode parameter, with domains $D (A_{+} (f)) = {ψ : \sum_{n} (n + 1) ∥ P_{+}^{(n + 1)} (f \otimes ψ_{n}) ∥^{2} < \infty}$ and $D (A_{-} (f)) = {ψ : \sum_{n} (n + 1) ∥ Ψ_{f}^{(n)} (ψ_{n + 1}) ∥^{2} < \infty}$ , and on them $(A_{+} (f) ψ)_{n} = \sqrt{n} P_{+}^{(n)} (f \otimes ψ_{n - 1})$ , $(A_{-} (f) ψ)_{n} = \sqrt{n + 1} Ψ_{f}^{(n)} (ψ_{n + 1})$ ; the contraction maps satisfy $Ψ_{f}^{(n)} (u_{1} \otimes \dots \otimes u_{n + 1}) = ⟨ u_{1}, f ⟩ u_{2} \otimes \dots \otimes u_{n + 1}$ (Closed adjoint sector domains, Creation contraction bounds clauses (a),(b)).

[F5]

Fixed-mode oscillator coefficients. For $F (C)$ with orthonormal basis $(e_{n})$ and the operators $A_{\pm} (1)$ one has $A_{+} (1) e_{n} = \sqrt{n + 1} e_{n + 1}$ , $A_{-} (1) e_{n} = \sqrt{n} e_{n - 1}$ for $n \geq 1$ and $A_{-} (1) e_{0} = 0$ ; for every $z \in C$ the vector $ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} e_{n}$ is unit and satisfies $A_{-} (1) ψ_{z} = z ψ_{z}$ (Fixed-mode number law clauses (a),(b)).

[F6]

Transverse Fock data. For the transverse space $h$ and $ε = ℏ c ∣ k ∣$ , the operators $T_{n}$ on $ran P_{+}^{(n)}$ are self-adjoint with $T_{n} ψ = ε_{n} ψ$ , $ε_{n} (k_{1}, \dots, k_{n}) = \sum_{i = 1}^{n} ε (k_{i})$ , and $H_{f} = ⨁_{n} T_{n}$ is self-adjoint with domain ${ψ : \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty}$ ; under the Euclidean identification of $h^{\otimes n}$ with a subspace of $L^{2} ((R^{3})^{n}; (C^{3})^{\otimes n})$ , the multiplier $ε_{n}$ is symmetric and commutes with all permutation unitaries, so $ε_{n}$ preserves $ran P_{+}^{(n)}$ (Transverse continuum free Fock Hamiltonian clauses (a),(b),(c), Permutation sector projections on the n-fold tensor power clause (c), Hilbert tensor construction: inner product, completion and product bases clause (b)).

Proof

technique · direct; identify $F(\mathbb C)$ inside $F(H)$ along the coefficient expansion in the single mode $\varphi$, check the isometry and the sectorially diagonal operators coefficientwise, and evaluate the energy expectation on the mode sector by symmetry of the tensor integrals

Given: The data of the statement and [F1]-[F6].

1.1F1F2algebra

[The powers of $φ$ are orthonormal sector vectors] For $n \geq 0$ put $Φ_{n} : = φ^{\otimes n}$ , with $Φ_{0} : = 1$ . By [F2] with $u = φ$ and $∥ φ ∥ = 1$ one has $∥ Φ_{n} ∥ = 1$ for every $n$ , $P_{+}^{(n)} Φ_{n} = Φ_{n}$ , and $⟨ Φ_{n}, Φ_{n} ⟩ = \prod_{i = 1}^{n} ⟨ φ, φ ⟩ = 1$ ; for $m \neq n$ the pairing $⟨ Φ_{m}, Φ_{n} ⟩$ vanishes because $Φ_{m}$ is supported in the direct-sum coordinate $m$ and $Φ_{n}$ in coordinate $n$ , and the inner product of $F (H)$ is the sum of the coordinate pairings by [F1]. Hence $(Φ_{n})_{n \in N}$ is an orthonormal family with $Φ_{n} \in ran P_{+}^{(n)}$ for every $n$ .

2.1F1F2algebra

[The isometry] Define $U_{φ}$ on the finite linear span of $(e_{n})$ by $U_{φ} e_{n} = Φ_{n}$ and extend by linearity; by step 1.1, $∥ U_{φ} ψ ∥^{2} = \sum_{n} ∣ ψ_{n} ∣^{2} = ∥ ψ ∥^{2}$ for every finite combination, so $U_{φ}$ is isometric on the dense finite-particle subspace $F_{0} (C)$ of Hilbert direct-sum construction clause (c) and extends uniquely to an isometry $U_{φ} : F (C) \to F (H)$ with the same norm identity for all $ψ \in F (C)$ (A bounded linear operator between normed spaces, Parseval equivalences for an orthonormal family). Its range is the closed linear span of $(Φ_{n})$ , since an isometry maps the closed span of an orthonormal family onto the closed span of its image and the image family is orthonormal; this proves (a).

2.2F2F4F5step 1.1algebra

[Ladder transfer on the exact domains] Write $ψ = \sum_{n} ψ_{n} e_{n}$ . By [F4] and [F5], $ψ \in D (A_{+} (1))$ exactly when $\sum_{n} (n + 1) ∣ ψ_{n} ∣^{2} < \infty$ , while $ψ \in D (A_{-} (1))$ exactly when $\sum_{n} (n + 1) ∣ ψ_{n + 1} ∣^{2} = \sum_{m \geq 1} m ∣ ψ_{m} ∣^{2} < \infty$ . For the unit mode $φ$ , [F2] gives $P_{+}^{(n + 1)} (φ \otimes ψ_{n} Φ_{n}) = ψ_{n} Φ_{n + 1}$ and [F4] gives $Ψ_{φ}^{(n)} (ψ_{n + 1} Φ_{n + 1}) = ψ_{n + 1} Φ_{n}$ . Therefore the two exact domain sums for $U_{φ} ψ$ equal the respective scalar sums, without any extra weighted output assumption. On these domains, $(A_{+} (φ) U_{φ} ψ)_{n} = \sqrt{n} ψ_{n - 1} Φ_{n} = (U_{φ} A_{+} (1) ψ)_{n}$ for $n \geq 1$ , with zero vacuum coordinate, and $(A_{-} (φ) U_{φ} ψ)_{n} = \sqrt{n + 1} ψ_{n + 1} Φ_{n} = (U_{φ} A_{-} (1) ψ)_{n}$ . This proves (c), including its coherent-vector specialization by [F5].

2.3F3F6step 1.1algebra

[Energy expectation] Assume now $H = h$ and $φ$ unit with $ε φ \in h$ ; write $\overset{ˉ}{ε} : = ⟨ ε φ, φ ⟩$ . For fixed $n$ , $Φ_{n} = φ^{\otimes n} \in ran P_{+}^{(n)}$ and, by [F6], $T_{n} Φ_{n} = ε_{n} Φ_{n}$ ; using the Euclidean identification and the symmetry of $ε_{n}$ under the permutation action, the iterated $L^{2}$ integral gives $⟨ T_{n} Φ_{n}, Φ_{n} ⟩ = \int_{(R^{3})^{n}} (\sum_{i = 1}^{n} ε (k_{i})) \prod_{i = 1}^{n} ∥ φ (k_{i}) ∥_{C^{3}}^{2} d^{3} k_{1} \dots d^{3} k_{n} = n \overset{ˉ}{ε} ∥ φ ∥^{2 (n - 1)} = n \overset{ˉ}{ε},$ each of the $n$ summands contributing the same product $\overset{ˉ}{ε} ∥ φ ∥^{2 (n - 1)}$ by Tonelli's theorem, the product measure being $σ$ -finite and the integrand nonnegative; this is finite because $ε φ \in L^{2}$ and $φ \in L^{2}$ give $\int ε ∥ φ ∥^{2} \leq ∥ ε φ ∥ ∥ φ ∥ < \infty$ . The actual domain is also justified: by the triangle inequality on the product space, $∥ T_{n} Φ_{n} ∥ \leq n ∥ ε φ ∥$ , since each factor multiplier has that norm and all other factors have norm one. Hence $\sum_{n} ∣ ψ_{n} ∣^{2} ∥ T_{n} Φ_{n} ∥^{2} \leq ∥ ε φ ∥^{2} \sum_{n} n^{2} ∣ ψ_{n} ∣^{2} < \infty$ , so $U_{φ} ψ \in D (H_{f})$ and its finite-sector truncations converge in the graph norm: the squared graph-tail norms are the tails of these two convergent nonnegative series. Therefore for a finite combination the sector-diagonal form of $H_{f}$ (clause (c) of [F6]) gives $⟨ H_{f} U_{φ} ψ, U_{φ} ψ ⟩ = \sum_{n} ∣ ψ_{n} ∣^{2} n \overset{ˉ}{ε}$ ; the general case $\sum_{n} n^{2} ∣ ψ_{n} ∣^{2} < \infty$ follows by approximation with truncations in the graph norm of $H_{f}$ (both sides are continuous under this approximation, the left because $H_{f}$ is closed and self-adjoint, the right by dominated convergence). This proves the first two displays of (d).

3.1F1F3step 2.1algebra

[Sector and number transfer] For $ψ = \sum_{m} ψ_{m} e_{m}$ in the dense span and fixed $n$ , $Π_{n} U_{φ} ψ = ψ_{n} Φ_{n} = U_{φ} (ψ_{n} e_{n}) = U_{φ} Π_{n}^{'} ψ$ ; since both sides are bounded in $ψ$ and the span is dense, the identity extends to all $ψ \in F (C)$ , and the norm identity is step 2.1. For $ψ \in F (C)$ the coefficient vector of $U_{φ} ψ$ is again $(ψ_{n})$ , so $\sum_{n} n^{2} ∥ (U_{φ} ψ)_{n} ∥^{2} = \sum_{n} n^{2} ∣ ψ_{n} ∣^{2}$ : $U_{φ} ψ \in D (N)$ exactly when $ψ \in D (N^{'})$ , and then $(N U_{φ} ψ)_{n} = n ψ_{n} Φ_{n} = (U_{φ} N^{'} ψ)_{n}$ for every $n$ by [F3]. This proves (b).

4.1F6step 1.1algebra∎

[Spectral second moment and literal square domain] For $φ \in D (ε)$ , expand $ε_{n}^{2} = \sum_{i} ε (k_{i})^{2} + \sum_{i \neq j} ε (k_{i}) ε (k_{j})$ and integrate against $\prod_{i} ∥ φ (k_{i}) ∥^{2}$ . Tonelli gives $∥ T_{n} Φ_{n} ∥^{2} = n ∥ ε φ ∥^{2} + n (n - 1) {\overset{ˉ}{ε}}^{2}$ , finite by the stated domain and Cauchy-Schwarz. This is the spectral second moment and requires no $T_{n}^{2} Φ_{n}$ . If $φ \in D (ε^{2})$ , the inequality $(\sum_{i = 1}^{n} ε (k_{i}))^{4} \leq n^{3} \sum_{i} ε (k_{i})^{4}$ gives $∥ T_{n}^{2} Φ_{n} ∥^{2} \leq n^{4} ∥ ε^{2} φ ∥^{2} < \infty$ , so $Φ_{n} \in D (T_{n}^{2})$ ; self-adjointness then gives $⟨ T_{n}^{2} Φ_{n}, Φ_{n} ⟩ = ∥ T_{n} Φ_{n} ∥^{2}$ , exactly the original operator-square display. The vacuum has energy zero and all its moments vanish, handled separately when $n = 0$ .

Remarks

Scope. The lemma is a coefficientwise statement about an orthonormal family in a Fock space; it does not assert that a fixed mode is an energy eigenstate. In fact the energy transfer is only an expectation statement: $H_{f}$ does not preserve the span of the $φ^{\otimes n}$ unless $ε φ$ is a multiple of $φ$ , which for the transverse continuum fails for every nonzero $φ$ by Restricted momentum and energy multipliers of the transverse one-photon space clause (c).
Choice hypotheses. The transverse free-field Fock supplier assumes Dependent Choice; the stated Axiom of Choice supplies it by AC implies DC implies countable choice. The result also carries the cited Countable Choice hypothesis; no other choice is introduced.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Total momentum multipliers on transverse symmetric Fock sectors

Statement

Assume Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), carrying the actual hypotheses of the multiplier and Fock suppliers. Fix a positive real parameter $ℏ$ and let $G = L^{2} (R^{3}; C^{3}), h = {f \in G : k \cdot f (k) = 0 for a.e. k}, Π (k) = I - ∣ k ∣^{- 2} k k^{T} (k \neq 0), Π (0) = 0.$ Use the Hilbert tensor powers of Hilbert tensor construction: inner product, completion and product bases, the symmetric projections $P_{+}^{(n)}$ of Permutation sector projections on the n-fold tensor power, and the Fock space of Hilbert direct-sum construction, with sectors $H_{n} = ran P_{+}^{(n)} \subset h^{\otimes n}$ for $n \geq 1$ and $H_{0} = C$ .

For $n \geq 1$ there is a canonical isometric embedding $I_{n} : H_{n} ⟶ L_{n} : = L^{2} (R^{3 n}; (C^{3})^{\otimes n}),$ obtained by inclusion into $G^{\otimes n}$ followed by the iterated finite-component Euclidean product unitary of Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}). On elementary tensors before symmetrization that unitary sends $f_{1} \otimes \dots \otimes f_{n}$ to $(k_{1}, \dots, k_{n}) \mapsto f_{1} (k_{1}) \otimes \dots \otimes f_{n} (k_{n})$ . Write $K_{n} = I_{n} H_{n}$ , a closed subspace. For $j \in {1, 2, 3}$ put $p_{j, n} (k_{1}, \dots, k_{n}) = ℏ \sum_{i = 1}^{n} (k_{i})_{j}, D (P_{j, n}) = {u \in H_{n} : p_{j, n} I_{n} u \in L_{n}}, P_{j, n} u = I_{n}^{- 1} (p_{j, n} I_{n} u) .$ The displayed product always lies in $K_{n}$ whenever it lies in $L_{n}$ , so this action is well-defined. Set $P_{j, 0} = 0$ on $C$ . Then:

(a) Sector self-adjointness and spectral measure. Every $P_{j, n}$ is densely defined and self-adjoint on its sector. For Borel $B \subset R$ , its regular spectral PVM is $E_{j, n} (B) u = I_{n}^{- 1} (1_{p_{j, n}^{- 1} (B)} I_{n} u) (n \geq 1), E_{j, 0} (B) = 1_{B} (0) I_{C} .$ In particular $D (P_{j, n})$ is exactly the square-integrability domain of the signed summed multiplier, without requiring the individual summands to be square integrable separately.

(b) Total momentum. On $F (h) = ⨁_{n \geq 0} H_{n}$ define $D (P_{j}) = {ψ : ψ_{n} \in D (P_{j, n}) for every n, \sum_{n \geq 0} ∥ P_{j, n} ψ_{n} ∥^{2} < \infty}, (P_{j} ψ)_{n} = P_{j, n} ψ_{n} .$ Then $P_{j}$ is densely defined and self-adjoint on precisely this domain. Equivalently its domain is given by $\sum_{n \geq 1} \int_{R^{3 n}} ∣ p_{j, n} (k) ∣^{2} ∥ I_{n} ψ_{n} (k) ∥^{2} d^{3 n} k < \infty .$ Its regular spectral PVM is the sectorwise direct sum $E_{j} (B) = ⨁_{n \geq 0} E_{j, n} (B), P_{j} = \int_{R} t d E_{j} (t) .$ For every $ψ \in F (h)$ , without a domain assumption, its scalar spectral measure is $⟨ E_{j} (B) ψ, ψ ⟩ = 1_{B} (0) ∣ ψ_{0} ∣^{2} + \sum_{n \geq 1} \int_{p_{j, n}^{- 1} (B)} ∥ I_{n} ψ_{n} (k) ∥^{2} d^{3 n} k .$ Thus it is the sum of the sectorwise push-forward measures and the vacuum atom. The vacuum belongs to $D (P_{j})$ and $P_{j} Ω = 0$ . On the one-particle sector the operator is the restricted multiplier $f \mapsto ℏ k_{j} f$ on its maximal square-integrability domain.

These are mathematical operator/domain statements. They adopt no photon assignment, probability law, dispersion relation, dynamical law or relation between momentum, energy and number operators.

Facts & Assumptions

Slot convention. For $ℓ \geq 1$ , $S_{ℓ}$ acts on ${0, \dots, ℓ - 1}$ as in Permutation sector projections on the n-fold tensor power. On one-based tuple positions use $\hat{π} (k) : = 1 + π (k - 1)$ for $1 \leq k \leq ℓ$ . Conjugation by $k \mapsto k - 1$ gives $\hat{π σ} = \hat{π} \circ \hat{σ}$ and $\hat{π^{- 1}} = {\hat{π}}^{- 1}$ ; thus the supplier's tensor action becomes $U_{π} (v_{1} \otimes \dots \otimes v_{ℓ}) = v_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes v_{{\hat{π}}^{- 1} (ℓ)}$ . The parameters of $U_{π}$ and $sgn (π)$ remain canonical zero-based permutations.

Given: CC and AC; $ℏ > 0$ ; $G, h, Π$ , the tensor powers, sectors and $F (h)$ as above; $j \in {1, 2, 3}$ .

[F1]

The completed tensor construction gives density of elementary tensors and the product inner product; bounded tensor maps have their product/adjoint identities. The finite-component Euclidean product unitary has exactly the elementary product formula and is onto (Hilbert tensor construction: inner product, completion and product bases, Bounded tensor maps: extension, norm estimate, adjoint and commutation clauses (a)–(c), Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Statement and proof 1.1–3.1).

[F2]

The permutation unitary acts by $U_{π} (f_{1} \otimes \dots \otimes f_{n}) = f_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes f_{{\hat{π}}^{- 1} (n)}$ ; its average is the orthogonal projection on invariant tensors (Permutation sector projections on the n-fold tensor power clauses (a)–(c)).

[F3]

A real finite-valued Euclidean scalar multiplier is self-adjoint on its maximal multiplication domain (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (a)). Its regular spectral PVM is multiplication by the inverse-image indicators (Spectral measure of a real L2 multiplication operator clauses (b),(c)). Finite vector components follow by finite orthogonal sums; a restriction to a closed transverse/symmetric subspace will be proved below, not inferred from an unstated restriction clause of the real-multiplier supplier.

[F4]

Bounded PVM integrals are linear and satisfy the norm estimate. The integral of a real Borel function is self-adjoint on exactly its square-integrability domain and is the limit of bounded spectral truncations (Bounded borel pvm integral, Integral of a measurable function against a projection-valued measure, The unbounded PVM integral is densely defined, closed and normal, Statement). A self-adjoint operator has a unique regular representing PVM (CC spectral calculus on an arbitrary complex Hilbert space by compatible vector-local Cayley subspaces, Statement and proof 6.2).

[F5]

$F (h)$ is the Hilbert direct sum of the symmetric sectors, with norm square $\sum_{n} ∥ ψ_{n} ∥^{2}$ and mutually orthogonal sector projections whose strong sum is $I$ (Hilbert direct-sum construction clauses (a)-(d), Fock sector projections).

Proof

Given: The data of the statement and [F1]–[F5].

1.1givenF1constructalgebra

[Explicit transverse tensor embedding] The measurable matrices $Π (k)$ are orthogonal projections with norm at most one. Hence $M_{Π}$ is a bounded orthogonal projection on $G$ , with range exactly $h$ , because $Π (k) f (k) = f (k)$ is equivalent to $k \cdot f (k) = 0$ off the null point $0$ . In particular $h$ is closed and Hilbert. The inclusion $h \to G$ induces an isometric inclusion of every finite tensor power: its elementary-tensor inner products agree by [F1], and density extends that isometry to the completion. Its closed range is the range of $(M_{Π})^{\otimes n}$ , since the latter projection maps the dense elementary span onto elementary transverse tensors. Iterating [F1] with dimensions $3, 3 (n - 1)$ and finite components gives a unitary $Θ_{n} : G^{\otimes n} \to L_{n}$ with the stated product formula. Under it $(M_{Π})^{\otimes n}$ becomes multiplication by $Q_{n} (k) = Π (k_{1}) \otimes \dots \otimes Π (k_{n})$ : both bounded maps agree on all elementary tensors, hence on their dense span. No undocumented product-measure or tensor-subspace identification is used.

2.1step 1.1F1F2algebra

[Symmetry includes polarization permutations] Define the unitary $τ_{π}$ on $(C^{3})^{\otimes n}$ by the same tensor-factor permutation as in [F2]. The conjugated ambient permutation is $(C_{π} u) (k_{1}, \dots, k_{n}) = τ_{π} u (k_{\hat{π} (1)}, \dots, k_{\hat{π} (n)})$ : substitution into the elementary product formula verifies this identity, and the bounded maps agree everywhere by density. Its average $S_{n} = (n!)^{- 1} \sum_{π} C_{π}$ is the orthogonal symmetry projection. The tensor projection $(M_{Π})^{\otimes n}$ commutes with every ambient permutation, as is verified on elementary tensors, so $M_{Q_{n}}$ commutes with $S_{n}$ . Their product $R_{n} = M_{Q_{n}} S_{n}$ is an orthogonal projection, whose range is exactly the image of the symmetric transverse tensor sector. Thus $K_{n} = ran R_{n}$ and $I_{n} = Θ_{n}$ restricted to that sector is an isometry onto $K_{n}$ . Every bounded scalar multiplier $b (p_{j, n})$ commutes with $M_{Q_{n}}$ pointwise and with $C_{π}$ , since $p_{j, n}$ is invariant under permutations; therefore it commutes with $R_{n}$ and preserves $K_{n}$ .

3.1step 2.1F3algebra

[Sector PVM and regularity] The scalar function $p = p_{j, n}$ is real, measurable and finite on $R^{3 n}$ . By [F3], multiplication by $1_{p^{- 1} (B)}$ is its spectral projection in each scalar component of $L_{n}$ ; the finite component sum is a regular PVM there. It commutes with $R_{n}$ by step 2.1. Its restriction to $K_{n}$ , transported by $I_{n}$ , is therefore an orthogonal projection for every $B$ , with total projection $I$ , product law and strong countable additivity inherited from the ambient PVM. Its scalar measure at $u$ is exactly $\int_{p^{- 1} (B)} ∥ I_{n} u ∥^{2}$ , the sum of its finite component measures, so it is regular and is the push-forward of $∥ I_{n} u ∥^{2} d^{3 n} k$ . This defines the sector PVM $E_{j, n}$ . For $n = 0$ , the indicated atom at zero is a regular PVM on $C$ .

4.1step 2.1step 3.1F4algebra

[Exact signed sector domain and self-adjointness] For the PVM of step 3.1, push-forward integration gives $\int t^{2} d E_{j, n, u} (t) = \int ∣ p (k) ∣^{2} ∥ I_{n} u (k) ∥^{2} d^{3 n} k$ : it holds first for indicators and simple nonnegative functions and then for increasing nonnegative simple approximations. By [F4], $\int t d E_{j, n} (t)$ is densely defined and self-adjoint on exactly the displayed multiplier domain. For bounded Borel $b$ , indicator/simple-function approximation and the norm estimate of [F4] show that $b (E_{j, n})$ is the restriction of multiplication by $b (p)$ , transported by $I_{n}$ . In particular the defining truncations for the identity function act as $p 1_{∣ p ∣ \leq L} I_{n} u$ . If $p I_{n} u \in L_{n}$ , these converge to $p I_{n} u$ in norm, because the integrable squared tail $\int_{∣ p ∣ > L} ∣ p ∣^{2} ∥ I_{n} u ∥^{2}$ tends to zero. Every truncation lies in the closed $K_{n}$ , so its limit lies there also. Thus the PVM integral is exactly $P_{j, n}$ with the stated action and domain. Spectral uniqueness [F4] identifies $E_{j, n}$ as its spectral PVM; $P_{j, 0} = 0$ is the corresponding vacuum operator. The sign of $p$ played no role in this squared-integrability argument.

5.1step 3.1step 4.1F3F5algebra

[The Fock PVM and push-forward sum] For each Borel $B$ , define $E_{j} (B) = ⨁_{n \geq 0} E_{j, n} (B)$ . It is a bounded orthogonal projection with the product law and total projection $I$ . Strong countable additivity follows sectorwise and then on the full sum: first retain finitely many sectors, where convergence is already known, and bound the omitted squared tail by $\sum_{n > N} ∥ ψ_{n} ∥^{2}$ , which tends to zero uniformly in the partial union, using [F5]. Its scalar measure at $ψ$ is [step 3.1, step 4.1, F3, F5, algebra] $E_{j, ψ} (B) = 1_{B} (0) ∣ ψ_{0} ∣^{2} + \sum_{n \geq 1} \int_{p_{j, n}^{- 1} (B)} ∥ I_{n} ψ_{n} (k) ∥^{2} d^{3 n} k .$ This is a finite Borel measure of mass $∥ ψ ∥^{2}$ . It is regular: given $ϵ > 0$ , choose finitely many sector measures leaving total tail mass below $ϵ / 2$ ; approximate the set from inside by compact sets for those finitely many regular measures and take their finite union, or from outside by open sets and take their finite intersection, allocating the remaining error among those measures. The vacuum measure is supported on the compact singleton ${0}$ and is regular. Thus $E_{j}$ is a regular PVM and the scalar push-forward formula in (b) holds.

6.1step 3.1step 4.1step 5.1F4F5algebra

[Total operator, exact domain and self-adjointness] For the scalar measure of step 5.1, nonnegative simple approximation and monotone convergence give [step 3.1, step 4.1, step 5.1, F4, F5, algebra] $\int_{R} t^{2} d E_{j, ψ} (t) = \sum_{n \geq 1} \int_{R^{3 n}} ∣ p_{j, n} (k) ∣^{2} ∥ I_{n} ψ_{n} (k) ∥^{2} d^{3 n} k = \sum_{n \geq 1} ∥ P_{j, n} ψ_{n} ∥^{2},$ with the vacuum contribution zero; the monotone convergence step uses Monotone convergence for the integral. By [F4], $A_{j} : = \int t d E_{j} (t)$ is self-adjoint on the domain where this second moment is finite, exactly the domain $D (P_{j})$ in (b). For bounded Borel functions, the PVM integral acts sectorwise: this holds for indicators by the definition of $E_{j}$ , for simple functions by linearity, and for bounded Borel functions by bounded approximation and the norm estimate in [F4]. In particular the truncations $t 1_{{∣ t ∣ \leq L}} (E_{j})$ act on sector $n$ as the corresponding truncations of $P_{j, n}$ . On $D (P_{j})$ these converge to $P_{j, n} ψ_{n}$ in each sector, and their squared total error is bounded by $\sum_{n} ∥ P_{j, n} ψ_{n} ∥^{2}$ ; truncating the sector tail and then taking $L \to \infty$ in the finitely many retained sectors proves convergence in $F (h)$ . Hence $A_{j} ψ = (P_{j, n} ψ_{n})_{n} = P_{j} ψ$ on precisely $D (P_{j})$ . This proves self-adjointness of the displayed total operator directly, with the vacuum sector $P_{j, 0} = 0$ included, and identifies $E_{j}$ as its spectral PVM by uniqueness.

7.1step 1.1step 2.1step 4.1step 5.1step 6.1algebra∎

[Vacuum, one-particle restriction and scope] The vacuum has only its zeroth component, where $P_{j, 0} = 0$ , so it belongs to $D (P_{j})$ and $P_{j} Ω = 0$ . For $n = 1$ , $Θ_{1} = I_{G}$ and the symmetric projection is the identity, so the sector construction gives exactly the maximal restricted multiplier $f \mapsto ℏ k_{j} f$ on $h$ . Steps 1.1–6.1 establish all tensor embeddings, sector domains, self-adjointness and PVM formulas without importing any physical radiation postulate. CC and AC are the declared hypotheses of the actual mathematical suppliers; no positivity, dispersion relation, commutation between distinct unbounded observables or physical law was used.

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Radiation extension of the framework: transverse mode and photon assignments

Physical scope

The adopted radiation extension of the non-relativistic quantum framework: a fixed propagation-mode one-photon polarisation model with a two-dimensional transverse internal space, and a quantised free transverse radiation sector with bosonic Fock space and the mode assignments omega=c|k|, photon energy E=hbaromega and photon momentum p=hbark in a declared inertial laboratory chart. The extension is an explicitly adopted model assumption; it is not derived from classical Maxwell theory, from special relativity, or from the massive-particle model by a mass-to-zero limit, and no Lorentz transformation law for the mode labels and no Galilean covariance of the radiation sector are claimed. Detection, coupling to matter and spontaneous emission are not supplied by this postulate.

Postulate

Adopt, as an additional model assumption of the framework on this page and its consumers, the following radiation extension. It is adopted together with the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5, the identical-particle and spin postulate P6 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, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), in the space vocabulary of The state space of the framework: preparations, pure rays and density operators, the observable vocabulary of Assigned sharp observables and the outcome law of the framework, the species and sector vocabulary of Declared species sectors and their preparations: the framework's reading of P6, and the units and dimensions of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). By AC implies DC implies countable choice, the stated Axiom of Choice entails Dependent Choice, as defined in The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, supplying the DC hypothesis of the cited transverse free-field Fock theorem without adopting an additional choice axiom.

(i) Fixed propagation-mode one-photon polarisation model. For a declared propagation mode, described by a nonzero wavevector label $k_{0} \in R^{3} ∖ {0}$ read in $m^{- 1}$ in a declared inertial chart (Inertial frames of the framework and the class of frame changes), the model assigns a two-dimensional complex internal space, the transverse polarisation space of the mode: the complex span of an orthonormal pair $(v_{1}, v_{2})$ of vectors of $C^{3}$ satisfying $k_{0} \cdot v_{1} = k_{0} \cdot v_{2} = 0$ , with the standard Hermitian form. Unit rays of this space are the model's one-photon polarisation preparations of the mode. This assignment is an idealised manifestation of the abstract Born/state framework on a finite-dimensional internal factor; it is not a spatial wave function of a massive particle, not a model of a photon position, and not the massive one-particle Hamiltonian of P5 with mass set to zero. For the massive free family, The massive free-particle family has no strong zero-mass limit on Schwartz states refutes strong action convergence to a self-adjoint operator on every Schwartz state and shows frequency divergence at each fixed nonzero wave number as $m ↓ 0$ . This extension uses its independently declared massless model assignments, with no such limiting construction.

(ii) Quantised transverse radiation sector. The model assigns to the free radiation a bosonic species in the sense of P6, whose one-particle space is the transverse space $h : = {f \in L^{2} (R^{3}; C^{3}) : k \cdot f (k) = 0 for almost every k}$ with the $L^{2}$ inner product; its $n$ -particle sector is $H_{n}^{+} = ran P_{+}^{(n)}$ and its radiation state space is the bosonic Fock space $F (h) = ⨁_{n \geq 0} H_{n}^{+}$ (Declared species sectors and their preparations: the framework's reading of P6, Hilbert direct-sum construction). The choice of a bosonic species is the additional model assignment of P6 clause (iii). It is adopted here, not derived from the preceding assumptions, and this item records no experimental observation that establishes it.

(iii) Mode assignments of frequency, energy and momentum. In the declared inertial chart, each free mode label $k \in R^{3} ∖ {0}$ is assigned the angular frequency $ω (k) : = c ∣ k ∣$ in $s^{- 1}$ , the photon energy $E (k) : = ℏ ω (k) = ℏ c ∣ k ∣$ in joules and the photon momentum $p (k) : = ℏ k$ in $k g m s^{- 1}$ , where $c = 2,99792458 \times 10^{8} m s^{- 1}$ and $ℏ = h / (2 π) > 0$ in $J s$ have the SI values of Dimensions, SI unit conventions and quantity types of the framework clause (c). Ordinary frequency is $ν (k) = ω (k) / (2 π)$ in hertz. These are separate quantum-radiation model assignments: they are neither consequences of classical Maxwell theory nor consequences of the special-relativistic framework, and no Lorentz transformation law for the mode label $k$ , for $ω (k)$ or for the polarisation pair is adopted or claimed; the extension does not assert Galilean covariance of the radiation sector, and it does not identify the massless Lorentz representations of the photon with the representations of the spatial rotation group used in P6.

Littlejohn's Section 7 derives the normal-ordered free-field momentum from the field momentum density and obtains $P = \sum_{λ} ℏ k a_{λ}^{†} a_{λ}$ , then reads an occupation-number state as carrying the corresponding sum of photon momenta (equations (21)-(33)). That derivation uses Gaussian units, Coulomb gauge and box normalization; it supports this assignment but does not derive this framework's other radiation assumptions or its covariance claims.

(iv) Free-field observables. The model assigns the total free-field energy observable $H_{f} = ⨁_{n \geq 0} T_{n}$ with $ε (k) = ℏ c ∣ k ∣$ and $T_{n} ψ = ε_{n} ψ$ , read in joules, the total number observable $N$ read in the dimensionless unit $1$ of counted quanta, and the total momentum components $⨁_{n} (\sum_{i = 1}^{n} ℏ k_{j, i})$ , read in $k g m s^{- 1}$ ; their operator domains and self-adjointness are the mathematics of Transverse continuum free Fock Hamiltonian and Number operator domain for energy and number, and Total momentum multipliers on transverse symmetric Fock sectors for total momentum, read physically in The radiation sector of the framework: modes and the number, energy and momentum assignments. The total momentum components use the maximal signed summed-multiplier sector domains and the squared-norm summability Fock domain of that lemma; the vacuum has total momentum zero.

(v) Counting, detection and light-matter coupling are not supplied. Photon number and energy statistics require a declared detector model in the sense of P2 and P7, which this postulate does not supply; a prescribed classical electromagnetic field does not by itself yield photon counting or vacuum spontaneous emission. No light-matter Hamiltonian, no dipole coupling, no emission or absorption process and no recoil model is asserted here; the absorption/stimulated-emission branch with a finite-level dipole coupling and the quantised-mode emission construction are separate model assumptions for the later pair on light-matter and spectral transitions. The relation of a laboratory field, polariser or detector to this model is a tested identification with uncertainty under P7, not a consequence of the postulate.

(vi) Scope of the framework. The radiation extension is deliberately non-relativistic in its matter sector: it adds external free radiation modes to the fixed-particle-number massive matter core without changing P5. Relativistic matter, antiparticles, pair creation, covariant local field dynamics and full quantum electrodynamics are not contained in this extension and are not disguised as solved core prerequisites; they require a named field-theory destination. No position wave function of the photon satisfying the massive Schrödinger equation is introduced.

(vii) Status. Clauses (i)-(vi) are adopted assumptions of the model. They are motivated by the sources cited above -- the transverse two-state description of polarisation with transmitted fraction $\cos^{2} θ$ , the single-mode oscillator quantisation with photon energy increment $ℏ ω$ , and the field-operator expansion of free modes -- but motivation is not deduction. No empirical claim, no measured value and no claim that any laboratory field, polariser or detector is exactly described by the model is made by this item.

Status and scope

Adopted, not derived. Neither the two-dimensional transverse polarisation space, nor the bosonic Fock assignment, nor the relations $ω = c ∣ k ∣$ , $E = ℏ ω$ and $p = ℏ k$ follow from P1-P7. They are the additional model content of this page, adopted only by the items that cite this postulate.
Relation to the massive model. The quantum-radiation assignments are declared directly, without a limit construction from the massive free Schrödinger model. The counterexample The massive free-particle family has no strong zero-mass limit on Schwartz states refutes strong action convergence on every Schwartz state and a finite frequency limit at each fixed nonzero wave number; it does not decide other operator topologies or exclude independently specified massless models. The polarisation internal space is a finite-state model, not a position representation of a photon.
Empirical qualifications. The statistics choice, the transverse polarisation count, and the frequency-energy-momentum assignments are empirical model choices with uncertainties carried by their laboratory identifications. No experiment, count, efficiency or calibration is recorded or assumed by this postulate; comparisons with data belong to the preparation-and-apparatus discipline of P7.

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The radiation sector of the framework: modes and the number, energy and momentum assignments

Physical scope

The reading of the radiation extension postulate in the framework's vocabulary: the transverse one-particle space, the bosonic Fock sector assignment, the vocabulary of modes and mode families, normalized one-photon wave-packet coordinates f(k) in m^(3/2) under d^3k and dimensionless coherent-state labels z, the assigned total-number observable in the dimensionless unit 1, the assigned free-field energy observable in joules with epsilon(k)=hbarc|k|, and the assigned total momentum components and their one-photon restrictions in kg m s^-1 with the mode assignments omega=c|k|, E=hbaromega and p=hbark. The readings record operator domains and self-adjointness from the cited mathematics; they assert no detector response, no light-matter coupling, no dynamics, no Lorentz transformation law and no empirical result.

Definition

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2 and the identical-particle and spin postulate P6 (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, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations) and the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments in its clauses (i)-(iv), read in the state vocabulary of The state space of the framework: preparations, pure rays and density operators, the observable vocabulary of Assigned sharp observables and the outcome law of the framework, the sector vocabulary of Declared species sectors and their preparations: the framework's reading of P6 and the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axioms of Countable Choice, Dependent Choice and Choice, the choice hypotheses of the cited Fock space mathematics (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice).

(b) Radiation one-particle space and sectors. The transverse one-particle space of the radiation sector is $h : = {f \in L^{2} (R^{3}; C^{3}) : k \cdot f (k) = 0 for almost every k}$ with the restricted $L^{2}$ inner product, a separable complex Hilbert space (Hilbert space); its elements are read as one-photon wave-packet vectors. The wave-vector coordinate is $k$ in $m^{- 1}$ and the normalization is against $d^{3} k$ . With normalized state vectors dimensionless, a unit wave-packet vector $f$ therefore has coordinate representatives $f (k)$ with units $m^{3 / 2}$ , so that $\int ∥ f (k) ∥^{2} d^{3} k$ is dimensionless. A unit mode vector has the same coordinate units. For $n \geq 1$ the $n$ -photon sector is $H_{n}^{+} = ran P_{+}^{(n)} \subseteq h^{\otimes n}$ and the radiation state space is the bosonic Fock space $F (h) = ⨁_{n \geq 0} H_{n}^{+}$ , whose sector projections are $Π_{n}$ and whose vacuum is $Ω = ι_{0} (1)$ (Hilbert direct-sum construction, Fock sector projections). A declared radiation preparation is a state $ρ \in D (F (h))$ in the sense of The state space of the framework: preparations, pure rays and density operators; a declared one-photon preparation is a preparation whose state is supported in the one-photon sector, and for a unit $f \in h$ the ray of $ι_{1} (f)$ is the one-photon preparation of wave-packet vector $f$ .

(c) Modes. A mode of the radiation sector is a ray $[φ] = {z φ : z \in C ∖ {0}}$ of a nonzero class $φ \in h$ , equivalently the rank-one orthogonal projection onto $C φ$ . Its unit representatives differ only by a complex factor of modulus one. A declared mode family is a finite or countable orthonormal family $(φ_{j})$ of $h$ (Orthonormal families, complete orthonormal systems and Hilbert bases). For a unit mode vector $φ$ and a dimensionless label $z \in C$ the vector $U_{φ} ψ_{z}, ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n \in N} \frac{z^{n}}{\sqrt{n!}} e_{n},$ is the mode-coherent preparation vector of label $z$ , and $U_{φ} e_{n}$ is the $n$ -quanta mode vector; these are the readings of the transferred vectors of The fixed-mode isometry of Fock space and the transfer of number and ladder operators. A mode vector is a normalisable wave packet; the sharp-mode label $k$ of the propagation-mode idealisation is a spectral parameter and is not a state of $h$ (Restricted momentum and energy multipliers of the transverse one-photon space clause (c)).

(d) Assigned observables. The framework assigns to the radiation sector:

the total number observable $(N, 1)$ , the self-adjoint number operator of Number operator domain, with domain $D (N) = {ψ : \sum_{n} n^{2} ∥ ψ_{n} ∥^{2} < \infty}$ and read in the dimensionless unit $1$ of counted quanta;
the free-field energy observable $(H_{f}, J)$ , the operator $H_{f} = ⨁_{n \geq 0} T_{n}$ with $ε (k) = ℏ c ∣ k ∣$ and $T_{n} ψ = ε_{n} ψ$ , of Transverse continuum free Fock Hamiltonian, read in joules; its one-particle restriction is the energy multiplier $M_{ℏ c ∣ k ∣}^{h}$ of Restricted momentum and energy multipliers of the transverse one-photon space;
for $j \in {1, 2, 3}$ the total momentum component $(P_{j}, k g m s^{- 1})$ is the self-adjoint direct sum of the signed summed multipliers $P_{j, n}$ of Total momentum multipliers on transverse symmetric Fock sectors, with $P_{j, 0} = 0$ . In the lemma's exact Euclidean embedding $I_{n}$ of each symmetric transverse sector, $P_{j, n}$ multiplies by $p_{j, n} (k) = ℏ \sum_{i} (k_{i})_{j}$ on the maximal domain ${u : p_{j, n} I_{n} u \in L^{2}}$ . Its Fock domain is $D (P_{j}) = {ψ : ψ_{n} \in D (P_{j, n}) for every n, \sum_{n} ∥ P_{j, n} ψ_{n} ∥^{2} < \infty} .$ This is the domain of the sum, without an extra requirement that each individual momentum summand be square integrable. Its spectral PVM is the sectorwise direct sum of the inverse-image indicator multipliers, including the atom at zero on the vacuum. The one-photon momentum component is its restriction to sector one, identified with $M_{ℏ k_{j}}^{h}$ of Restricted momentum and energy multipliers of the transverse one-photon space. The postulate assigns these mathematical operators their physical momentum meaning and units.

(e) Reading of the mode assignments. In the declared inertial chart the assignments $ω (k) = c ∣ k ∣$ in $s^{- 1}$ , $E (k) = ℏ ω (k)$ in $J$ and $p (k) = ℏ k$ in $k g m s^{- 1}$ of clause (iii) of the radiation postulate are read through the spectral measures of clause (d): for a one-photon preparation of unit wave-packet vector $f \in h$ , the assigned frequency, energy and momentum component laws are the push-forwards of the measure $∥ f (k) ∥_{C^{3}}^{2} d^{3} k$ under $k \mapsto c ∣ k ∣$ , $k \mapsto ℏ c ∣ k ∣$ and $k \mapsto ℏ k_{j}$ ; these laws, and their finite means under the corresponding first-moment conditions, are proved on this page. Sharp frequency, sharp energy and sharp momentum-component states are not vectors of $h$ ; every unit one-photon vector in the corresponding multiplier domain has finite strictly positive variance in that quantity (clause (d) of Restricted momentum and energy multipliers of the transverse one-photon space). The push-forward laws themselves exist for every unit $f \in h$ , without a moment-domain assumption; all their singleton probabilities vanish by the null level sets in the cited lemma, so they are non-Dirac laws even when finite means or variances do not exist. Thus "photon of energy $ℏ ω$ " names a spectral assignment of the model, not a normalisable state with an exact energy.

(f) Scope. Clauses (b)-(e) fix the sector assignment and the units. They assert no detector model, no sampling model, no dynamics, no interaction with matter, no spontaneous emission and no empirical result; the counting and detection vocabulary is fixed separately on this page. The identification of a laboratory radiation field with a preparation of $F (h)$ is a tested identification with uncertainty under P7, and no Lorentz transformation law for modes or their labels is adopted or claimed.

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The fixed-mode polarisation model of the framework

Physical scope

The declared fixed propagation-mode polarisation model of the framework: a nonzero wavevector label, a declared real orthonormal pair spanning the real transverse plane with an optional common phase, the two-dimensional complex polarisation space with its unit rays as idealised one-photon polarisation preparations, and the two-outcome polarisation analyser family with the Born outcome law of P2 and the Lueders reading of P3 where used. The model is an adopted idealisation of a transverse mode with a definite propagation label; it is not a spatial wave function, has no position or momentum observable, is not the massless limit of the massive model, and asserts no localisation, no detector response, no loss, no efficiency and no empirical result.

Definition

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the ideal conditional-measurement postulate P3, 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) and the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments in its clause (i). Read the state vocabulary in The state space of the framework: preparations, pure rays and density operators, the observable vocabulary in Assigned sharp observables and the outcome law of the framework, the measurement vocabulary in Declared ideal measurements and instruments of the framework, and the units of angles in Dimensions, SI unit conventions and quantity types of the framework: angles and phases are dimensionless under the radian convention. Assume Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice), including the hypotheses inherited from the adopted radiation postulate and its Fock-space suppliers.

(b) Declared fixed-mode data. A declared fixed-mode polarisation model consists of:

a nonzero propagation label $k_{0} \in R^{3} ∖ {0}$ , read in $m^{- 1}$ in a declared inertial chart, which is a parameter of the model and not a state;
a declared transverse frame, that is a pair $v_{j} = ζ u_{j}$ for $j \in {1, 2}$ , where $(u_{1}, u_{2})$ is a real orthonormal basis of $k_{0}^{⊥} : = {u \in R^{3} : k_{0} \cdot u = 0}$ and $ζ \in C$ is one common phase factor, $∣ ζ ∣ = 1$ . Here $k_{0} \cdot u$ is the real Euclidean dot product, and the Hermitian form on $C^{3}$ is $⟨ x, y ⟩ = \sum_{ℓ = 1}^{3} x_{ℓ} \overline{y_{ℓ}}$ , linear in its first argument; thus $⟨ v_{i}, v_{j} ⟩ = δ_{i j}$ and $k_{0} \cdot v_{j} : = \sum_{ℓ = 1}^{3} (k_{0})_{ℓ} (v_{j})_{ℓ} = 0$ ;
the polarisation space $V : = C v_{1} \oplus C v_{2}$ with the restricted Hermitian inner product and the orthonormal basis $(e_{1}, e_{2}) : = (v_{1}, v_{2})$ .

The model's state space is $D (V)$ , the set of density operators of $V$ ; the pure preparations of the model are the unit rays $[ψ]$ of $V$ , that is the preparations assigned the pure state $\hat{ρ} (ψ)$ of The state space: density operators, pure and mixed states, and purity, and a mixed preparation is a statistical mixture of preparations in the sense of P1. A pure ray is linearly polarised when it has a nonzero real representative in $k_{0}^{⊥}$ . The map $θ \mapsto ψ_{θ} : = \cos θ e_{1} + \sin θ e_{2}$ assigns to each angle $θ \in R$ a distinguished pure preparation, the linear polarisation preparation at angle $θ$ in the declared transverse frame. In particular, $ψ_{θ} = ζ (\cos θ u_{1} + \sin θ u_{2})$ has that ray, whose real direction is the unoriented line $R (\cos θ u_{1} + \sin θ u_{2})$ ; hence the physical angle label is modulo $π$ . General complex unit vectors of $V$ remain allowed as pure preparations and need not be linearly polarised.

(c) Declared polarisation analyser. For each angle $θ \in R$ the declared polarisation analyser at angle $θ$ is the declared ideal sharp measurement of clause (a) of Declared ideal measurements and instruments of the framework with two outcomes labelled $+ 1$ and $- 1$ and outcome structure $(P_{θ}, P_{θ}^{⊥}), P_{θ} x = ⟨ x, e_{θ} ⟩ e_{θ}, e_{θ} : = \cos θ e_{1} + \sin θ e_{2}, P_{θ}^{⊥} : = I - P_{θ},$ whose projections $P_{θ}, P_{θ}^{⊥}$ are those of Analyzer projections of a two-dimensional polarisation space and their overlaps. The outcome law of the analyser in a preparation $ρ$ is the reading of P2 for the projections of the outcome structure, and the labels are dimensionless with the reading " $+ 1$ : transmitted", " $- 1$ : blocked". Where a conditional post-measurement state is used, it is the Lueders update of clause (b) of Declared ideal measurements and instruments of the framework; the framework does not assert that a laboratory absorber leaves an outgoing one-photon polarisation state, and the absorption outcome is used only through its discarded-outcome law unless a separate instrument is declared (clause (v) of P3).

(d) Frame covariance and angle labels. For another complex orthonormal transverse pair $(e_{1}^{'}, e_{2}^{'})$ (not necessarily a transverse frame of clause (b)), let $U : V \to V$ be the unitary map with $U e_{i} = e_{i}^{'}$ , explicitly $U x = \sum_{i} ⟨ x, e_{i} ⟩ e_{i}^{'}$ . Its transported sharp-measurement family is $P_{θ}^{'} = U P_{θ} U^{*}$ , because $U e_{θ} = \cos θ e_{1}^{'} + \sin θ e_{2}^{'}$ . Under the simultaneous transport of a preparation to $ρ^{'} = U ρ U^{*}$ the Born probabilities agree: $tr (ρ^{'} P_{θ}^{'}) = tr (U ρ P_{θ} U^{*}) = tr (ρ P_{θ})$ . The last equality follows by evaluating the finite trace in a transported orthonormal basis; the same argument preserves trace one, and conjugation preserves positivity and self-adjointness of $ρ$ . Thus covariance concerns the transported states and projectors together. A general complex basis change need not leave the original linear-analyser family fixed: replacing $e_{2}$ by $i e_{2}$ gives rays of $\cos θ e_{1} + i \sin θ e_{2}$ , generally absent from the original family, and does not satisfy clause (b). For two transverse frames of clause (b), the bases differ by a common phase times a real orthogonal two-by-two matrix. Write its first column as $(\cos χ, \sin χ)$ ; its second column is $(- \sin χ, \cos χ)$ or $(\sin χ, - \cos χ)$ according to whether its determinant is $+ 1$ or $- 1$ . The families are equal with the angle reparametrisation $θ \mapsto χ + θ$ or $χ - θ$ , respectively, by the sine and cosine addition formulas. Arbitrary complex basis transport gives a valid sharp measurement but need not give a linear polariser. The value of $∣ k_{0} ∣$ enters no polarisation probability: it enters the separate mode frequency and energy assignments of the radiation postulate.

(e) Scope. The model is an idealisation of a transverse propagation mode with a definite mode label. It is a finite-dimensional internal-factor model in the sense of P1 and P6: its state space is $D (V)$ , its pure preparations are rays, and its analyser is a declared ideal sharp measurement. It contains no position observable, no momentum observable, no spatial wave function and no localisation claim, and it is not the massive one-particle model of P5 with $m = 0$ and is not obtained from that model by a mass-to-zero limit; no such limit is used or claimed. The mode label $k_{0}$ is not a normalisable state of any sector of the framework, and a plane wave is not introduced as a state. Whether a laboratory polariser and source realize this declared model, with what efficiency, loss and background, is a tested identification with uncertainty under P7, not a consequence of this definition.

(f) Status. Clauses (a)-(e) fix the model data, its outcome vocabulary and its scope; they add no assumption beyond the cited postulates and the cited finite-dimensional mathematics. The mathematical properties of the projections are the content of Analyzer projections of a two-dimensional polarisation space and their overlaps; the conditional outcome law of the declared analyser is proved on this page. No empirical claim is made here.

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The analyser outcome law of the fixed-mode polarisation model: Born probabilities and the Malus ratio

Physical scope

The conditional outcome law of the declared polarisation analyser of the fixed propagation-mode model: the analyser is a declared ideal sharp measurement with projections P_theta and P_theta-perp, its Born law in a pure linear-polarisation preparation is cos^2(theta-phi) and sin^2(theta-phi), the law is affine over statistical mixtures and invariant under the phases of the preparation, the conditional post-passage state is the analysed polarisation ray, and a second ideal analyser gives the relative cos^2 law. The result is conditional on P1, P2, P3 and the adopted model; no laboratory polariser is identified with the declared ideal analyser and no empirical result is asserted.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2 and the ideal conditional-measurement postulate P3 (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), the radiation extension postulate clause (i) (Radiation extension of the framework: transverse mode and photon assignments) and the declared fixed-mode polarisation model The fixed-mode polarisation model of the framework, in the state vocabulary of The state space of the framework: preparations, pure rays and density operators and the observable and measurement vocabularies of Assigned sharp observables and the outcome law of the framework and Declared ideal measurements and instruments of the framework. Assume Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice), including the hypotheses inherited from the radiation postulate and the cited spectral and measurement mathematics. Fix angles $φ, θ, θ^{'} \in R$ and write $ψ_{φ} = \cos φ e_{1} + \sin φ e_{2}$ and $P_{θ}$ for the analyser projections of the model. Then:

(a) the declared polarisation analyser at angle $θ$ is a declared ideal sharp measurement with outcome structure $(P_{θ}, P_{θ}^{⊥})$ , the outcome labels $+ 1$ and $- 1$ naming the first and second projection, in the sense of clause (a) of Declared ideal measurements and instruments of the framework: $P_{θ}$ and $P_{θ}^{⊥}$ are bounded orthogonal projections with $P_{θ} P_{θ}^{⊥} = 0$ and $P_{θ} + P_{θ}^{⊥} = I$ ;

(b) the spectral projection-valued measure $E$ of the bounded self-adjoint operator $P_{θ}$ (assigned with unit $1$ ; its values on the two outcomes are $1$ and $0$ , while the analyser labels are $+ 1$ and $- 1$ ) satisfies $E ({1}) = P_{θ}, E ({0}) = P_{θ}^{⊥}, E (R ∖ {0, 1}) = 0,$ consequently the outcome law of the assigned sharp quantity $P_{θ}$ in a preparation $ρ \in D (V)$ is $p_{+ 1} (ρ) = tr (ρ P_{θ}), p_{- 1} (ρ) = tr (ρ P_{θ}^{⊥}) = 1 - p_{+ 1} (ρ),$ which is exactly the law of the declared analyser;

(c) for the linear-polarisation preparation $ψ_{φ}$ and a general pure preparation with unit vector $ψ = α e_{1} + β e_{2}$ , $p_{+ 1} (ψ_{φ}) = \cos^{2} (θ - φ), p_{- 1} (ψ_{φ}) = \sin^{2} (θ - φ), p_{+ 1} (ψ) = ∣ α \cos θ + β \sin θ ∣^{2};$ for a statistical mixture $ρ = \sum_{n} w_{n} ρ_{n}$ of preparations the law is affine, $p_{+ 1} (ρ) = \sum_{n} w_{n} p_{+ 1} (ρ_{n})$ ;

(d) the law is invariant under the phase of the preparation and under the phase of the analysed direction: $p_{+ 1}$ depends only on the rays $[ψ_{φ}]$ and $[e_{θ}]$ , is $2 π$ -periodic in each argument, and satisfies $p_{+ 1} (θ, φ) = p_{+ 1} (θ - φ, 0)$ ;

(e) if $p_{+ 1} (ρ) > 0$ , the conditional state after the outcome $+ 1$ of the analyser is $ρ_{+ 1} = P_{θ} ρ P_{θ} / p_{+ 1} (ρ)$ ; for the pure preparation $[ψ_{φ}]$ with $\cos (θ - φ) \neq 0$ it is the pure state of the analysed ray $[e_{θ}]$ , and a second analyser at angle $θ^{'}$ then gives the conditional probability $\cos^{2} (θ^{'} - θ)$ for the outcome $+ 1$ . If the outcome is discarded, the state is $Φ (ρ) = P_{θ} ρ P_{θ} + P_{θ}^{⊥} ρ P_{θ}^{⊥}$ ;

(f) the assertions (a)-(e) are conditional consequences of P1, P2, P3 and the adopted model. They assert no dynamics, no laboratory polariser, no detector efficiency, no loss, no sampling model and no empirical result; in particular the fraction $\cos^{2} (θ - φ)$ is a model probability of the declared ideal analyser, not a measured count.

Facts & Assumptions

Given: the postulates P1, P2, P3 and the declared fixed-mode model with polarisation space $V$ , orthonormal basis $(e_{1}, e_{2})$ , rays $[ψ_{φ}]$ and analyser projections $P_{θ}, P_{θ}^{⊥}$ , together with the cited spectral and measurement mathematics.

[F1]

Projection pair. $P_{θ}$ and $P_{θ}^{⊥}$ are orthogonal projections on $V$ with $P_{θ} P_{θ} = P_{θ} = P_{θ}^{*}$ , $P_{θ} + P_{θ}^{⊥} = I$ , $P_{θ} P_{θ}^{⊥} = P_{θ}^{⊥} P_{θ} = 0$ , $∥ P_{θ} ∥ = 1$ , and their ranges are $C e_{θ}$ and $C e_{θ}^{⊥}$ : $\ker (P_{θ} - I) = C e_{θ}$ , $\ker P_{θ} = C e_{θ}^{⊥}$ (Analyzer projections of a two-dimensional polarisation space and their overlaps clauses (a),(d), The Hilbert orthogonal projection onto a closed subspace).

[F2]

Analyser law. For the declared analyser the outcome law of clause (ii) of P2 is $ρ \mapsto tr (ρ E_{θ} (\cdot))$ for the spectral measure of the assigned observable; a declared ideal sharp measurement has mutually orthogonal projections with strong sum $I$ , outcome probabilities $tr (ρ P_{a})$ and Lueders update of clause (b) of the measurement definition (Declared ideal measurements and instruments of the framework clauses (a),(b), Assigned sharp observables and the outcome law of the framework clause (b)).

[F3]

Spectral atoms. For a self-adjoint operator $A$ with spectral measure $E$ , the projection $E ({λ})$ is the orthogonal projection onto $\ker (A - λ)$ ; hence at an eigenvalue $λ$ with eigenspace $C u$ , $u$ unit, one has $E ({λ}) = T_{u, u}$ and $tr (ρ E ({λ})) = ⟨ u, ρ u ⟩$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clause (a)).

[F4]

Overlaps. For all real $φ, θ$ : $⟨ e_{θ}, ψ_{φ} ⟩ = \cos (θ - φ)$ and $∥ P_{θ} ψ_{φ} ∥^{2} = ⟨ ψ_{φ}, P_{θ} ψ_{φ} ⟩ = \cos^{2} (θ - φ)$ ; moreover $P_{θ} (z ψ) = z P_{θ} ψ$ for all $z \in C$ (Analyzer projections of a two-dimensional polarisation space and their overlaps clauses (b),(c)).

[F5]

Affinity and updates. The outcome law of an assigned sharp observable is affine in the preparation: $μ_{\sum_{n} w_{n} ρ_{n}}^{A} = \sum_{n} w_{n} μ_{ρ_{n}}^{A}$ ; the selective Lueders update of a declared ideal sharp measurement with outcome projection $P_{a}$ and $p_{a} = tr (ρ P_{a}) > 0$ is $ρ_{a} = P_{a} ρ P_{a} / p_{a}$ , and the discarded-outcome state is $Φ (ρ) = \sum_{a} P_{a} ρ P_{a}$ (Affinity and trace-norm continuity of the scalar spectral law, The selective Lueders update by an orthogonal projection, The non-selective Lueders map and the discarded-outcome ensemble, The state space: density operators, pure and mixed states, and purity).

Proof

1.1F1F2algebra

[The analyser is a declared ideal sharp measurement] By [F1] the two projections of the analyser are bounded, orthogonal and mutually orthogonal with $P_{θ} + P_{θ}^{⊥} = I$ ; therefore the pair $(P_{θ}, P_{θ}^{⊥})$ is a declared ideal sharp measurement in the sense of clause (a) of the measurement definition, with outcome labels $+ 1$ and $- 1$ . This is (a).

1.2F1F2F3algebra

[Spectral law] The operator $P_{θ}$ is bounded and self-adjoint by [F1], so it has a spectral projection-valued measure $E$ (Assigned sharp observables and the outcome law of the framework clause (b)). Its eigenvalues are $1$ and $0$ , with eigenspaces $C e_{θ}$ and $C e_{θ}^{⊥}$ by [F1]; by [F3], $E ({1})$ is the orthogonal projection onto $C e_{θ}$ , that is $P_{θ}$ , and $E ({0})$ is the orthogonal projection onto $C e_{θ}^{⊥}$ , that is $P_{θ}^{⊥} = I - P_{θ}$ . Since $P_{θ}$ is a projection one has $P_{θ}^{2} = P_{θ}$ , so for $λ \notin {0, 1}$ the bounded operator $\frac{1}{1 - λ} P_{θ} - \frac{1}{λ} (I - P_{θ})$ is a two-sided inverse of $P_{θ} - λ I$ (multiplying out gives $P_{θ} + (I - P_{θ}) = I$ ), hence the spectrum is contained in ${0, 1}$ (Resolvent and spectrum of an unbounded operator, The Hilbert orthogonal projection onto a closed subspace); therefore $E (R ∖ {0, 1}) = E (R ∖ σ (P_{θ})) = 0$ by Support of the spectral measure, the spectrum and the values of the scalar law clause (a). For any preparation $ρ$ the outcome law of the assigned quantity $P_{θ}$ at the value $1$ is $tr (ρ E ({1})) = tr (ρ P_{θ})$ by [F2]; the analyser label $- 1$ corresponds to the quantity's spectral value $0$ , whose projection is $E ({0}) = P_{θ}^{⊥}$ , and the two weights sum to $tr (ρ I) = 1$ . This is (b).

1.3F1F3F4algebra

[Pure preparations] Let $ψ = α e_{1} + β e_{2}$ be a unit vector. Since $E ({1}) = P_{θ}$ , the probability is $∥ P_{θ} ψ ∥^{2} = ⟨ ψ, P_{θ} ψ ⟩ = ∣ ⟨ ψ, e_{θ} ⟩ ∣^{2} = ∣ α \cos θ + β \sin θ ∣^{2}$ ; for $ψ = ψ_{φ}$ this equals $\cos^{2} (θ - φ)$ by [F4], and then $p_{- 1} = 1 - p_{+ 1} = \sin^{2} (θ - φ)$ .

1.4F4F5algebra

[Mixtures and phases] If $ρ = \sum_{n} w_{n} ρ_{n}$ is a statistical mixture, [F5] gives $p_{+ 1} (ρ) = \sum_{n} w_{n} tr (ρ_{n} P_{θ}) = \sum_{n} w_{n} p_{+ 1} (ρ_{n})$ . The number $∣ ⟨ z ψ, e_{θ} ⟩ ∣^{2} / ∥ z ψ ∥^{2} = ∣ ⟨ ψ, e_{θ} ⟩ ∣^{2}$ for $z \neq 0$ is unchanged by scaling the representative, so the law depends only on the rays; the identities $P_{θ + 2 π} = P_{θ}$ and $P_{θ + π} = P_{θ}$ of [F4] give the stated periodicities, and $p_{+ 1} (θ, φ) = \cos^{2} (θ - φ) = p_{+ 1} (θ - φ, 0)$ . This proves (c) and (d).

2.1F4F5step 1.3algebra

[Conditional and discarded states] Let $ρ$ be a preparation with $p_{+ 1} (ρ) > 0$ . By [F5] the conditional state after the outcome $+ 1$ is $ρ_{+ 1} = P_{θ} ρ P_{θ} / p_{+ 1} (ρ)$ . For the pure preparation of $ψ_{φ}$ with $\cos (θ - φ) \neq 0$ one has $P_{θ} ψ_{φ} = ⟨ ψ_{φ}, e_{θ} ⟩ e_{θ} = \cos (θ - φ) e_{θ}$ by [F4]; hence $P_{θ} ρ P_{θ} = ∣ \cos (θ - φ) ∣^{2} T_{e_{θ}, e_{θ}}$ and $ρ_{+ 1} = T_{e_{θ}, e_{θ}}$ , the pure state of the ray $[e_{θ}]$ (The state space: density operators, pure and mixed states, and purity). Applying the pure-preparation formula of step 1.3 to the preparation $[e_{θ}]$ at analyser angle $θ^{'}$ gives conditional probability $∣ ⟨ e_{θ}, e_{θ^{'}} ⟩ ∣^{2} = \cos^{2} (θ^{'} - θ)$ by [F4]. The discarded-outcome state is $Φ (ρ) = P_{θ} ρ P_{θ} + P_{θ}^{⊥} ρ P_{θ}^{⊥}$ by [F5]. This proves (e).

3.1step 1.1step 1.2step 1.3step 1.4step 2.1∎

[Assembly and scope] Steps 1.1 and 1.2 prove (a) and (b), steps 1.3 and 1.4 prove (c) and (d), and step 2.1 proves (e). All assertions are conditional on P1, P2, P3 and the adopted model; no laboratory apparatus, no sampling model and no measured number occurs in the argument, which is (f).

Remarks

The same number classically. The number $\cos^{2} (θ - φ)$ is also the intensity ratio of the declared classical transverse-field model of Malus and Jones ratios of the declared analyser: the same cos^2 from the Born law and from a classical transverse amplitude ↗; the comparison shows that this ratio alone does not distinguish the two declared models.
Absorption. For a laboratory absorber the "blocked" outcome of the declared analyser is used only through its discarded-outcome law: clause (v) of P3 asserts no post-absorption one-photon state, and no such state is used here.

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Number, energy and momentum laws of a one-photon preparation

Physical scope

The outcome laws of the assigned radiation observables in a declared one-photon preparation of the model: the number observable has the sharp value one; the frequency and free-field energy laws are the push-forwards of the transverse momentum density under k -> c*|k| and k -> hbarc|k|, with means whenever the first moment is finite; and the one-photon momentum-component readouts are the push-forwards under k -> hbar*k_j, with means whenever the corresponding absolute first moment is finite. The frequency, energy and momentum-component variances are finite and strictly positive on their multiplier domains. The result is conditional on P1, P2 and the adopted radiation model, records no detector, no sampling model and no measured number, and shows that no normalisable one-photon vector has sharp frequency, energy or momentum.

Statement

Adopt the state and composition postulate P1 and the observable and probability postulate P2 (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), 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 state vocabulary of The state space of the framework: preparations, pure rays and density operators and the observable vocabulary of Assigned sharp observables and the outcome law of the framework. Assume Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice), including the hypotheses inherited from the radiation postulate and the cited operator and spectral items. Let $f \in h$ be a unit one-photon wave-packet vector. Under the $d^{3} k$ normalization of clause (b) of The radiation sector of the framework: modes and the number, energy and momentum assignments, its coordinate representatives $f (k)$ have units $m^{3 / 2}$ . Put $ε (k) = ℏ c ∣ k ∣$ , and let $ρ_{f}$ be the one-photon preparation of the ray of $ι_{1} (f)$ . Then:

(a) number law: the outcome law of the assigned total-number observable $(N, 1)$ in $ρ_{f}$ is $μ_{ρ_{f}}^{N} (B) = 1_{B} (1)$ for every Borel set $B \subseteq R$ : the number readout of the one-photon preparation is one with probability one;

(b) frequency and energy laws: the assigned frequency law $μ_{ρ_{f}}^{ω}$ and the outcome law of the assigned free-field energy observable $(H_{f}, J)$ in $ρ_{f}$ are the push-forwards of $∥ f (k) ∥_{C^{3}}^{2} d^{3} k$ under $k \mapsto c ∣ k ∣$ and $k \mapsto ℏ c ∣ k ∣$ , respectively: $μ_{ρ_{f}}^{ω} (B) = \int_{{k : c ∣ k ∣ \in B}} ∥ f (k) ∥_{C^{3}}^{2} d^{3} k, μ_{ρ_{f}}^{H_{f}} (B) = \int_{ε^{- 1} (B)} ∥ f (k) ∥_{C^{3}}^{2} d^{3} k (B \subseteq R Borel),$ and if $\int_{R^{3}} ∣ k ∣ ∥ f (k) ∥_{C^{3}}^{2} d^{3} k < \infty$ , their spectral-law means are $\int_{R} ω d μ_{ρ_{f}}^{ω} (ω) = c \int_{R^{3}} ∣ k ∣ ∥ f (k) ∥_{C^{3}}^{2} d^{3} k, \int_{R} E d μ_{ρ_{f}}^{H_{f}} (E) = ℏ c \int_{R^{3}} ∣ k ∣ ∥ f (k) ∥_{C^{3}}^{2} d^{3} k;$

(c) momentum readouts: for $j \in {1, 2, 3}$ the one-photon momentum-component readout of clause (d)(3) of The radiation sector of the framework: modes and the number, energy and momentum assignments has, in the identification of the one-photon sector with $h$ by $ι_{1}$ , the outcome law $μ_{ρ_{f}}^{P_{j}} (B) = \int_{{k : ℏ k_{j} \in B}} ∥ f (k) ∥_{C^{3}}^{2} d^{3} k (B \subseteq R Borel),$ the push-forward of $∥ f ∥^{2} d^{3} k$ under $k \mapsto ℏ k_{j}$ , with mean $ℏ \int k_{j} ∥ f (k) ∥^{2} d^{3} k$ whenever $\int ∣ k_{j} ∣ ∥ f (k) ∥^{2} d^{3} k < \infty$ ;

(d) no sharp frequency, energy or momentum: for every unit vector $f \in D (ε_{h})$ , the variances of the frequency and energy laws of (b) are strictly positive, and for every unit vector in $D (M_{ℏ k_{j}}^{h})$ the momentum-component variance of (c) is strictly positive; consequently the mode assignment "photon energy $E = ℏ ω$ , $ω = c ∣ k ∣$ " is a spectral assignment of the model and never an exact value of a normalisable one-photon state;

(e) all assertions are conditional on P1, P2 and the adopted radiation model; they record no detector model, no sampling model, no measured count and no empirical claim.

Facts & Assumptions

Given: the postulates P1, P2 and the adopted radiation model with transverse space $h$ , sectors $H_{n}^{+}$ , number operator $N$ , free-field energy $H_{f}$ with $ε (k) = ℏ c ∣ k ∣$ , one-photon momentum readouts $M_{ℏ k_{j}}^{h}$ , and a unit $f \in h$ .

[F1]

Sector structure. $Π_{n}$ is the orthogonal projection of $F (h)$ onto the closed subspace $ι_{n} (H_{n}^{+})$ , the ranges are pairwise orthogonal, $\sum_{n} Π_{n} = I$ in the strong operator topology, and $∥ ψ ∥^{2} = \sum_{n} ∥ Π_{n} ψ ∥^{2}$ (Fock sector projections clauses (a)-(c)).

[F2]

Number operator. $N$ is self-adjoint with $D (N) = {ψ : \sum_{n} n^{2} ∥ ψ_{n} ∥^{2} < \infty}$ , $N ι_{n} (u) = n ι_{n} (u)$ for every $u \in H_{n}^{+}$ , and $σ (N) = N$ (Number operator domain clauses (a),(b)).

[F3]

Spectral atoms. For a self-adjoint operator $A$ with spectral measure $E_{A}$ , $E_{A} ({λ})$ is the orthogonal projection onto the eigenspace $\ker (A - λ)$ ; for a unit eigenvector $u$ at $λ$ and a pure state $ρ = T_{u, u}$ one has $μ_{ρ}^{A} ({λ}) = 1$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a),(b)).

[F4]

Free-field energy. $H_{f} = ⨁_{n \geq 0} T_{n}$ with $T_{n}$ self-adjoint on $ran P_{+}^{(n)}$ , $T_{n} ψ = ε_{n} ψ$ , is self-adjoint and positive with $D (H_{f}) = {ψ : ψ_{n} \in D (T_{n}) \forall n, \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty}$ ; its one-particle restriction is $ε_{h}$ (Transverse continuum free Fock Hamiltonian clauses (a)-(c)).

[F5]

The exact free-field direct-sum spectral measure. For the actual sector operators $T_{n}$ of [F4], including $T_{0} = 0$ on $C$ , write $E_{n}$ for their regular spectral PVMs. Then $E_{H_{f}} (B) = ⨁_{n \geq 0} E_{n} (B), E_{0} (B) = 1_{B} (0) I_{C} .$ This is proved locally in Transverse continuum free Fock Hamiltonian, proof 5.1: strong countable additivity follows by finite-sector truncation and a squared-norm tail bound; scalar regularity follows by finite compact/open approximations after discarding a tail of measure; nonnegative integration gives $\int t^{2} d E_{ψ} = \sum_{n} \int t^{2} d (E_{n})_{ψ_{n}}$ . Thus the identity integral has exactly the domain ${ψ : ψ_{n} \in D (T_{n}) \forall n, \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty}$ , and bounded spectral truncations converge there to $(T_{n} ψ_{n})_{n}$ . Uniqueness of the regular representing PVM identifies the result with $E_{H_{f}}$ . The same supplier proves that $E_{1} (B)$ is the restriction of the inverse-image indicator multiplier for $ε (k)$ to $h$ . These exact proved energy statements, rather than an unsupported generic PVM-sum assertion, are what step 1.2 uses.

[F6]

Restricted multipliers. For $m \in {ℏ c ∣ k ∣, ℏ k_{j}}$ the operator $M_{m}^{h}$ is self-adjoint on $h$ with spectral measure $E_{m} (B) = M_{1_{m^{- 1} (B)}} ∣_{h}$ , the law of a unit $g \in h$ is the push-forward of $∥ g ∥^{2} d^{3} k$ under $k \mapsto m (k)$ , and $M_{m}^{h}$ has no eigenvalues; for unit $g \in D (M_{m}^{h})$ the variance is strictly positive (Restricted momentum and energy multipliers of the transverse one-photon space clauses (a)-(d)).

[F7]

Means and variances for pure states. For a self-adjoint $A$ and a unit $u \in D (A)$ the mean of the outcome law in the pure state $T_{u, u}$ is $⟨ A u, u ⟩$ ; the variance is $∥ A u ∥^{2} - ⟨ A u, u ⟩^{2}$ when $u \in D (A)$ (Assigned sharp observables and the outcome law of the framework clause (e)).

[F8]

Monotone convergence. For nonnegative measurable functions $g_{n} ↑ g$ , their integrals against a measure increase to the integral of $g$ (Monotone convergence for the integral).

Proof

1.1F1F2F3algebra

[Number law] For $n \in N$ the eigenspace of $N$ at $n$ is ${ψ \in D (N) : N ψ = n ψ}$ ; comparing coordinates with the definition $(N ψ)_{m} = m ψ_{m}$ , this eigenspace equals $ι_{n} (H_{n}^{+})$ , the closed subspace of vectors supported in coordinate $n$ . Hence by [F3] the spectral projection is $E_{N} ({n}) = Π_{n}$ , and for the one-photon state $ρ_{f} = T_{ι_{1} f, ι_{1} f}$ we get $μ_{ρ_{f}}^{N} ({n}) = tr (ρ_{f} Π_{n}) = ⟨ Π_{n} ι_{1} f, ι_{1} f ⟩ = δ_{n, 1}$ by [F1]; since the atoms cover $N$ and the law is a measure, $μ_{ρ_{f}}^{N} (B) = 1_{B} (1)$ for every Borel set $B$ . This proves (a).

1.2F4F5F6algebra

[Frequency and energy laws] By [F5] the spectral measure of $H_{f}$ is $E (B) = ⨁_{n} E_{T_{n}} (B)$ , and on the one-photon sector $ι_{1} H_{1}^{+}$ it acts as $ι_{1} E_{T_{1}} (B) ι_{1}^{- 1}$ ; since $T_{1} = ε_{h}$ by [F4], the restriction is the spectral measure $E_{ε} (B) = M_{1_{ε^{- 1} (B)}} ∣_{h}$ of [F6]. Therefore $μ_{ρ_{f}}^{H_{f}} (B) = ⟨ E (B) ι_{1} f, ι_{1} f ⟩ = ⟨ E_{ε} (B) f, f ⟩ = \int_{ε^{- 1} (B)} ∥ f (k) ∥^{2} d^{3} k .$ The assigned frequency law is the push-forward under $k \mapsto c ∣ k ∣$ by clause (e) of The radiation sector of the framework: modes and the number, energy and momentum assignments. These are the laws in (b), and they exist for every unit $f \in h$ .

2.1F6F7F8step 1.2algebra

[Push-forward integration and frequency and energy first moments] Let $ν_{f} (A) : = \int_{A} ∥ f (k) ∥^{2} d^{3} k$ , a probability measure, and, for any real measurable $m$ , let $μ_{m} (B) : = ν_{f} (m^{- 1} (B))$ . For an indicator $g = 1_{C}$ , the defining identity gives $\int g d μ_{m} = \int g (m (k)) d ν_{f} (k)$ ; linearity gives it for nonnegative simple $g$ , and increasing simple approximation with [F8] gives it for every nonnegative Borel $g$ . Applying this identity to $g (t) = t$ for the nonnegative frequency and energy maps gives the means in (b), finite under the stated first-moment condition. If also $f \in D (ε_{h})$ , the energy mean agrees with $⟨ ε_{h} f, f ⟩$ by [F7]; the first-moment condition alone need not put $f$ in that operator domain.

3.1F6F7step 2.1algebra

[Momentum readouts and means] The same sector restriction gives $μ_{ρ_{f}}^{P_{j}} (B) = ⟨ E_{m} (B) f, f ⟩ = \int_{{m \in B}} ∥ f ∥^{2} d^{3} k$ for $m (k) = ℏ k_{j}$ , the push-forward under $k \mapsto ℏ k_{j}$ . For the signed identity function, apply the push-forward integral identity proved in step 2.1 separately to its positive and negative parts; the assumed finite absolute first moment makes both integrals finite, so their difference is the mean stated in (c). If also $f \in D (M_{m}^{h})$ , this mean agrees with $⟨ M_{m}^{h} f, f ⟩$ by [F7] and the isometry of $ι_{1}$ . This proves (c).

4.1F6step 1.2step 3.1algebra

[Strict positivity of the variances] By [F6] the multiplier $M_{ℏ c ∣ k ∣}^{h}$ has no eigenvalues and every unit $f \in D (ε_{h})$ has strictly positive finite energy variance; together with step 1.2 this gives the energy variance claim. The frequency law is the positive constant rescaling $ω = E / ℏ$ , so on the same domain its variance is $ℏ^{- 2}$ times the energy variance and is strictly positive. The same clause of [F6] applied to $m = ℏ k_{j}$ gives strictly positive finite momentum-component variance whenever $f \in D (M_{ℏ k_{j}}^{h})$ . Hence no unit vector of $h$ has a sharp value of frequency, energy or any momentum component, and the assignments of the radiation postulate are spectral assignments. This proves (d).

5.1step 1.1step 1.2step 2.1step 3.1step 4.1∎

[Assembly and scope] Step 1.1 proves (a), steps 1.2 and 2.1 prove the frequency and energy claims in (b), step 3.1 proves (c), and step 4.1 proves (d). No step introduces a detector, a sampling model, a measured count or an empirical premise, which is (e).

Remarks

Reading the mean. The energy mean $ℏ c \int ∣ k ∣ ∥ f ∥^{2}$ is the first moment of the transverse momentum density weighted by $ℏ c$ ; it is a preparation mean of the model, not a measured energy. It exists whenever $\int ∣ k ∣ ∥ f ∥^{2} d^{3} k < \infty$ , which is weaker than $f \in D (ε_{h})$ ; on that operator domain it also equals $⟨ ε_{h} f, f ⟩$ .
Why "photon energy $ℏ ω$ " remains meaningful. For a model preparation concentrated near a wavevector $k_{0}$ , the energy law is concentrated near $ℏ c ∣ k_{0} ∣$ in the sense of the push-forward measure; the item claims no limit statement and no localisation, and the exact value is never attained by a normalisable vector.

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

Physical scope

The declared ideal detection models of the radiation sector: the ideal total-number counting model with the Fock sector projections as outcome structure and the Born law of P2, the ideal energy-window model with the spectral projections of the free-field energy, and the polarisation analyser of the fixed-mode model. The declaration fixes vocabulary only; it asserts no laboratory counter, no absorption instrument, no efficiency, no dark counts, no dead time, no background, no calibration and no sampling model, and repeated trials require a separately declared statistical model.

Definition

(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 ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -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 $N$ , outcome projection $P_{n} : = Π_{n} (n \in N),$ the sector projections of the Fock space $F (h)$ ; the family $(Π_{n})$ 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 $ρ$ is the reading of clause (ii) of P2, $p_{n} (ρ) = tr (ρ Π_{n}), \sum_{n \in N} p_{n} (ρ) = 1,$ 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 $B \subseteq [0, \infty)$ 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 $(H_{f}, J)$ ; its window law in the preparation $ρ$ is $p_{B} (ρ) = tr (ρ E_{H_{f}} (B))$ and its complement probability is $tr (ρ (I - E_{H_{f}} (B)))$ , 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 $[0, \infty)$ , with outcome projections $(E_{H_{f}} (B_{a}))_{a \in A}$ . Countable additivity of the spectral PVM and $E_{H_{f}} ([0, \infty)) = I$ 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 ${P_{θ}, P_{θ}^{⊥}}$ 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.

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Number and energy statistics of a declared mode: number states, coherent states and the continuum thermal limitation

Physical scope

The conditional number and energy statistics of a declared normalisable mode of the adopted radiation sector: the n-quanta mode state has sharp count n; the mode-coherent state has the Poisson count law of dimensionless parameter |z|^2 with mean and variance |z|^2; the energy means and variances of these preparations are computed from the mode profile, are strictly positive in variance for n>=1 and z!=0 (the vacuum cases have zero variance), and the full-sector Gibbs operator e^(-beta H_f) is not trace class for beta>0 in J^-1. This excludes only its trace-normalized Gibbs density form; other preparations are outside the theorem. The result is conditional on P1, P2 and the adopted radiation model, uses the ideal counting model only, reports no measured count and asserts no thermalisation or temperature identification.

Statement

Adopt the state and composition postulate P1 and the observable and probability postulate P2 (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), the radiation extension postulate (Radiation extension of the framework: transverse mode and photon assignments), the radiation sector definition The radiation sector of the framework: modes and the number, energy and momentum assignments and the declared ideal counting model The declared ideal counting, energy-window and analyser detection models of the radiation sector. Assume the Axioms of Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice). Fix a unit mode vector $φ \in h$ , a dimensionless label $z \in C$ , and write $U : = U_{φ}$ for the fixed-mode isometry of The fixed-mode isometry of Fock space and the transfer of number and ladder operators. Then:

(a) for every $n \in N$ the $n$ -quanta mode vector $U e_{n} = φ^{\otimes n}$ is a unit vector of the $n$ -photon sector; the count law of the declared ideal total-number detector in the preparation of its ray is $p_{m} = δ_{m n}$ , and the number readout is $n$ with probability one;

(b) the count law of the mode-coherent preparation $U ψ_{z}$ of label $z$ is the Poisson law $p_{n} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!} (n \in N),$ with mean and variance $\sum_{n \in N} n p_{n} = ∣ z ∣^{2}, \sum_{n \in N} n^{2} p_{n} - (\sum_{n \in N} n p_{n})^{2} = ∣ z ∣^{2};$ in particular the mode-coherent preparation is not a number state unless $z = 0$ ;

(c) if $φ \in D (ε_{h})$ , with unit-mode coordinates $φ (k)$ in $m^{3 / 2}$ under $d^{3} k$ as in clause (b) of The radiation sector of the framework: modes and the number, energy and momentum assignments, and writing $\overset{ˉ}{ε} : = ⟨ ε_{h} φ, φ ⟩ = ℏ c \int ∣ k ∣ ∥ φ (k) ∥^{2} d^{3} k$ , the energy means of the assigned free-field energy observable are $⟨ H_{f} ⟩_{U e_{n}} = n \overset{ˉ}{ε}, ⟨ H_{f} ⟩_{U ψ_{z}} = ∣ z ∣^{2} \overset{ˉ}{ε};$

(d) under the same condition $φ \in D (ε_{h})$ , writing $\overline{ε^{2}} : = ∥ ε_{h} φ ∥^{2}$ , the energy variances are ${Var}_{U e_{n}} (H_{f}) = n (\overline{ε^{2}} - {\overset{ˉ}{ε}}^{2}), {Var}_{U ψ_{z}} (H_{f}) = ∣ z ∣^{2} \overline{ε^{2}};$ the first is strictly positive for every $n \geq 1$ and the second is strictly positive for every $z \neq 0$ . Thus $U e_{n}$ has non-sharp energy for $n \geq 1$ and $U ψ_{z}$ has non-sharp energy for $z \neq 0$ . At $n = 0$ or $z = 0$ the corresponding preparation is the vacuum $Ω$ , with $H_{f} Ω = 0$ and energy law $δ_{0}$ ;

(e) full-sector Gibbs limitation: let $β > 0$ have SI unit $J^{- 1}$ , so that $β H_{f}$ is dimensionless. The bounded operator $e^{- β H_{f}}$ is not compact and hence not trace class; therefore it does not define a density operator by trace normalization on the full transverse continuum sector. This conclusion rules out only that trace-normalized full-sector Gibbs form. Other mode-selected, regularized or otherwise separately defined preparations are outside this theorem; no identification of $β$ with an inverse thermodynamic temperature and no thermal interpretation of any state is asserted here;

(f) all assertions are conditional on P1, P2 and the adopted radiation model and use only the declared ideal counting model. No measured count, frequency, efficiency or temperature is asserted, and no sampling model is introduced.

Facts & Assumptions

Given: the postulates P1, P2 and the adopted radiation model with transverse space $h$ , Fock space $F (h)$ , sectors $H_{n}^{+}$ , projections $Π_{n}$ , number operator $N$ , energy $H_{f}$ with $ε (k) = ℏ c ∣ k ∣$ , a unit mode vector $φ$ with coordinate units $m^{3 / 2}$ under $d^{3} k$ , a dimensionless label $z \in C$ and the isometry $U = U_{φ}$ .

[F1]

Fixed-mode isometry. $U e_{n} = φ^{\otimes n} \in ran P_{+}^{(n)}$ , $(U e_{m}, U e_{n})$ is orthonormal, $∥ U ψ ∥ = ∥ ψ ∥$ , $N U ψ = U N^{'} ψ$ for $ψ \in D (N^{'})$ , $A_{-} (φ) U ψ = U A_{-} (1) ψ$ for $ψ \in D (A_{-} (1))$ , and for the transverse case with $φ \in D (ε_{h})$ , $⟨ H_{f} U e_{n}, U e_{n} ⟩ = n ⟨ ε_{h} φ, φ ⟩$ and $⟨ H_{f} U ψ, U ψ ⟩ = (\sum_{n} n ∣ ψ_{n} ∣^{2}) ⟨ ε_{h} φ, φ ⟩$ for $\sum_{n} n^{2} ∣ ψ_{n} ∣^{2} < \infty$ (The fixed-mode isometry of Fock space and the transfer of number and ladder operators clauses (a),(b),(d)).

[F2]

Fock sectors and number atoms. $Π_{n}$ is the orthogonal projection onto the coordinate- $n$ subspace, the ranges are pairwise orthogonal with strong sum $I$ , and for the number operator $N$ one has $N ι_{n} (u) = n ι_{n} (u)$ for every $u \in H_{n}^{+}$ ; the count law of the declared ideal total-number detector is $p_{n} (ρ) = tr (ρ Π_{n})$ (Fock sector projections, Number operator domain clause (b), The declared ideal counting, energy-window and analyser detection models of the radiation sector clause (b)).

[F3]

Coherent vector coefficients. $ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} e_{n}$ is unit with $⟨ e_{n}, ψ_{z} ⟩ = e^{- ∣ z ∣^{2} / 2} \frac{{\overset{ˉ}{z}}^{n}}{\sqrt{n!}}$ and $∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!}$ (Fixed-mode number law clauses (b),(c)).

[F4]

Energy moments on mode sectors. For a unit $φ \in D (ε_{h})$ , put $\overset{ˉ}{ε} = ⟨ ε φ, φ ⟩$ and $\overline{ε^{2}} = ∥ ε φ ∥^{2}$ . Then $Φ_{n} = φ^{\otimes n} \in D (T_{n})$ , $⟨ T_{n} Φ_{n}, Φ_{n} ⟩ = n \overset{ˉ}{ε}$ and $∥ T_{n} Φ_{n} ∥^{2} = n \overline{ε^{2}} + n (n - 1) {\overset{ˉ}{ε}}^{2}$ . The last is the spectral second moment, not an assertion of membership in $D (T_{n}^{2})$ (The fixed-mode isometry of Fock space and the transfer of number and ladder operators clause (d)).

[F5]

Exponential series. For every $x \geq 0$ , $\sum_{n} \frac{x^{n}}{n!} = e^{x} < \infty$ , and the series converges absolutely (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument).

[F6]

Non-compactness of the free Gibbs operator. For every $β > 0$ read with SI unit $J^{- 1}$ , the operator $e^{- β H_{f}}$ is not compact and in particular not trace class (Transverse continuum free Fock Hamiltonian clause (d)).

Proof

1.1F1F2algebra

[Number states] By [F1] each $U e_{n} = φ^{\otimes n}$ is a unit vector of the sector $ran P_{+}^{(n)}$ ; by [F2] $N$ acts on it as $n$ times itself, so the number readout is $n$ with probability one by the atom law of P2 (Atoms of the spectral measure and eigenvalues of a self-adjoint operator), and the count law of the declared detector is $p_{m} = tr (T_{U e_{n}, U e_{n}} Π_{m}) = ⟨ Π_{m} U e_{n}, U e_{n} ⟩ = δ_{m n}$ . This proves (a).

1.2F1F2F3algebra

[Poisson law] For the mode-coherent preparation $U ψ_{z}$ the count law is $p_{n} = ⟨ Π_{n} U ψ_{z}, U ψ_{z} ⟩ = ⟨ U Π_{n}^{'} ψ_{z}, U ψ_{z} ⟩ = ⟨ Π_{n}^{'} ψ_{z}, ψ_{z} ⟩ = ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!},$ using [F1] for the sector transfer and the isometry, and [F3] for the coefficient.

2.1F3F5step 1.2algebra

[Poisson moments] Put $x : = ∣ z ∣^{2} \geq 0$ . By [F5], $\sum_{n \geq 1} n \frac{x^{n}}{n!} = x \sum_{m \geq 0} \frac{x^{m}}{m!} = x e^{x}$ and $\sum_{n \geq 2} n (n - 1) \frac{x^{n}}{n!} = x^{2} e^{x}$ , so the mean is $e^{- x} x e^{x} = x$ and the second moment is $e^{- x} (x^{2} + x) e^{x} = x^{2} + x$ ; hence the variance is $x$ . For $z \neq 0$ the law is supported on more than one value, so the preparation is not a number state. This proves (b).

3.1F1F2F4step 2.1algebra

[Energy means] The sector diagonality of $H_{f}$ and orthogonality of the sectors give, for the coherent preparation, $⟨ H_{f} U ψ_{z}, U ψ_{z} ⟩ = \sum_{n \in N} ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} ⟨ T_{n} φ^{\otimes n}, φ^{\otimes n} ⟩ = (\sum_{n} ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} n) \overset{ˉ}{ε} = ∣ z ∣^{2} \overset{ˉ}{ε},$ where the middle identity is [F1] and [F4] and the last is the Poisson mean of step 2.1; the number-state value is the $n$ -th instance of [F1]. The Poisson second number moment is finite by step 2.1, so [F1] places $U ψ_{z}$ in the actual domain of $H_{f}$ and justifies the graph-truncation passage. The mean exists because both sides are finite: $\overset{ˉ}{ε} < \infty$ by $φ \in D (ε_{h})$ and Cauchy-Schwarz, and $\sum_{n} n ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = x < \infty$ . This proves (c).

4.1F1F4step 2.1step 3.1algebra

[Energy variances] For the number state, $∥ H_{f} U e_{n} ∥^{2} = ∥ T_{n} φ^{\otimes n} ∥^{2} = n \overline{ε^{2}} + n (n - 1) {\overset{ˉ}{ε}}^{2}$ by [F4], so the variance is $n (\overline{ε^{2}} - {\overset{ˉ}{ε}}^{2})$ , which is strictly positive for $n \geq 1$ by clause (d) of Restricted momentum and energy multipliers of the transverse one-photon space (the energy multiplier has no eigenvalues, so $\overline{ε^{2}} - {\overset{ˉ}{ε}}^{2} = {Var}_{φ} (ε) > 0$ ). For the coherent preparation, its Poisson second moment and [F1] give $U ψ_{z} \in D (H_{f})$ ; the orthogonal-sector norm expansion therefore gives $∥ H_{f} U ψ_{z} ∥^{2} = \sum_{n} ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} (n \overline{ε^{2}} + n (n - 1) {\overset{ˉ}{ε}}^{2}) = x \overline{ε^{2}} + x^{2} {\overset{ˉ}{ε}}^{2},$ using the Poisson first and second factorial moments of step 2.1; subtracting the square of the mean $x \overset{ˉ}{ε}$ from step 3.1 gives $x \overline{ε^{2}}$ , which is strictly positive for $x = ∣ z ∣^{2} > 0$ because $\overline{ε^{2}} = ∥ ε_{h} φ ∥^{2} > 0$ : $ε_{h} φ$ vanishes only where $φ$ vanishes off the null set ${k = 0}$ , so a unit $φ$ cannot satisfy $ε_{h} φ = 0$ in $h$ . For $n = 0$ and $z = 0$ , respectively, $U e_{0} = U ψ_{0} = Ω$ ; the vacuum sector has $T_{0} = 0$ , hence $H_{f} Ω = 0$ , and the atom/eigenvalue law of P2 gives the sharp energy law $δ_{0}$ . This proves (d), including its vacuum exceptions.

5.1F6step 1.1step 1.2step 2.1step 3.1step 4.1∎

[Full-sector Gibbs limitation and assembly] A trace-normalized density operator proportional to $e^{- β H_{f}}$ would require that operator to be trace class, which [F6] excludes for every stated $β$ . This proves only the full-sector Gibbs limitation in (e). Steps 1.1 and 1.2 prove (a) and (b), steps 3.1 and 4.1 prove (c) and (d), and no step introduces a detector response, a sampling model or an empirical premise, which is (f).

Remarks

Poisson versus geometric. The mode-coherent count law is Poisson, with variance equal to its mean; the declared single-mode Gibbs-type preparation of Number statistics of a single declared mode: sharp, Poisson and geometric preparations ↗ has a geometric count law with larger variance at the same mean. Both are conditional model computations; distinguishing them in a laboratory requires a declared detector, preparation and sampling model and reported data.
No temperature identification. The parameter $β$ in (e) has unit $J^{- 1}$ so that $β H_{f}$ is dimensionless. Identifying it with $(k_{B} T)^{- 1}$ is an additional physical assumption not made here.

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Scope and empirical status of the photons and radiation modes page

Physical scope

The scope and empirical status of the photons and radiation modes page: the adopted radiation extension, the conditional results proved on the page with their units, and the physical comparisons, laboratory identifications and extensions that the page does not claim.

Scope of the 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 $ω = c ∣ k ∣$ , $E = ℏ ω$ and $p = ℏ k$ 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 $\cos^{2} (θ - φ)$ , $\sin^{2} (θ - φ)$ 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 $∥ f (k) ∥^{2} d^{3} k$ under $c ∣ k ∣$ and $ℏ c ∣ k ∣$ (a unit packet coordinate $f (k)$ has units $m^{3 / 2}$ 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 $e^{- β H_{f}}$ is not trace class for the full continuum sector when $β > 0$ has unit $J^{- 1}$ . 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 $\cos^{2} (θ - φ)$ 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 ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain and The Axiom of Choice record these assumptions.

5 · Examples, experiments and counterexamples

None yet.

Qm Photons And Radiation Modes

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Analyzer projections of a two-dimensional polarisation space and their overlaps

Statement

Facts & Assumptions

Proof

Remarks

Restricted momentum and energy multipliers of the transverse one-photon space

Statement

Facts & Assumptions

Proof

Remarks

The fixed-mode isometry of Fock space and the transfer of number and ladder operators

Statement

Facts & Assumptions

Proof

Remarks

Total momentum multipliers on transverse symmetric Fock sectors

Statement

Facts & Assumptions

Proof

Radiation extension of the framework: transverse mode and photon assignments

Postulate

Status and scope

The radiation sector of the framework: modes and the number, energy and momentum assignments

Definition

The fixed-mode polarisation model of the framework

Definition

The analyser outcome law of the fixed-mode polarisation model: Born probabilities and the Malus ratio

Statement

Facts & Assumptions

Proof

Remarks

Number, energy and momentum laws of a one-photon preparation

Statement

Facts & Assumptions

Proof

Remarks

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

Definition

Number and energy statistics of a declared mode: number states, coherent states and the continuum thermal limitation

Statement

Facts & Assumptions

Proof

Remarks

Scope and empirical status of the photons and radiation modes page

Scope of the page

5 · Examples, experiments and counterexamples

Sources