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A two-bin comparison of a declared classical field model and the quantised model: identical normalised laws do not discriminate

Physical scope

A conditional two-model comparison on a declared two-dimensional transverse amplitude space: the normalised two-bin laws assigned by a declared classical field model with dimensionless amplitude coordinates and relative intensity equal to squared amplitude norm, and by the quantised fixed-mode model with the Born law for the same declared bin projections. The result is that the normalised laws coincide for every declared amplitude, so such a pattern does not discriminate the two declared models; the thought experiment records the cross-term (interference) structure common to both and asserts no experiment, no counts, no detector response and no empirical conclusion.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2 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, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), 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 with polarisation space $V$ and orthonormal basis $(e_{1}, e_{2})$ . 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). Declare the following two-bin comparison:

an orthonormal pair $(u_{+}, u_{-})$ of $V$ , read as the pair of orthogonal fringe bin directions of a declared two-bin detector, with projections $Q_{\pm} x : = ⟨ x, u_{\pm} ⟩ u_{\pm}$ ;
a declared amplitude datum $a \in V ∖ {0}$ whose complex coordinates are dimensionless ratios relative to one fixed reference amplitude $A_{r e f}$ ; $a$ need not have unit norm;
the declared classical field model, whose state datum is the amplitude coordinate $a$ relative to $A_{r e f}$ , whose dimensionless relative bin intensities are $I_{\pm} (a) : = ∥ Q_{\pm} a ∥^{2}$ , and whose normalised two-bin law is $I_{\pm} (a) / (I_{+} (a) + I_{-} (a))$ ;
the quantised model reading of the same datum: the pure preparation of the ray $[a]$ of the polarisation space, measured by the declared ideal sharp measurement with outcome projections $(Q_{+}, Q_{-})$ declared directly in the sense of Declared ideal measurements and instruments of the framework clause (a), with the Born law of P2. This arbitrary-bin sharp measurement need not belong to the real-angle analyser family of the original frame.

Then:

(a) $Q_{+} + Q_{-} = I$ and the normalised stated laws of the two declared models coincide for every datum $a \neq 0$ : $\frac{I_{\pm} (a)}{I_{+} (a) + I_{-} (a)} = p_{\pm} ([a]) = \frac{∥ Q_{\pm} a ∥^{2}}{∥ a ∥^{2}} = \frac{∣ ⟨ u_{\pm}, a ⟩ ∣^{2}}{∥ a ∥^{2}};$

(b) writing $a = a_{1} + a_{2}$ for any declared decomposition of the datum into two path components, both declared laws contain the interference cross-term $\frac{2 Re (⟨ u_{\pm}, a_{1} ⟩ \overline{⟨ u_{\pm}, a_{2} ⟩})}{∥ a ∥^{2}},$ so the presence of a fringe pattern of this two-bin form does not distinguish the two declared models;

(c) the two declared models are nevertheless different physical models: in the classical model the datum is a field amplitude with a deterministic intensity, while in the quantised model it is a state ray with a Born law; a discriminator requires a prediction on which two fully specified models differ, together with a declared detector response, preparation and sampling model in the sense of P7. Count statistics on a declared number-state preparation are a candidate for a further comparison only after a classical field-and-detector count law and the corresponding preparations in both models are declared. The deterministic classical intensities do not supply that count law, so no count-statistics discriminator follows from the present declarations;

(d) this item constructs no such discriminating comparison and reports no data; it asserts no measurement, no count, no frequency and no empirical conclusion, and it does not claim that the classical model agrees with the quantised model beyond the declared two-bin law.

Facts & Assumptions

Given: The declared polarisation space $V$ with orthonormal basis $(e_{1}, e_{2})$ , the orthonormal bin pair $(u_{+}, u_{-})$ , the projections $Q_{\pm}$ , the datum $a \in V ∖ {0}$ , and the two declared models.

[F1]

Projections and resolution. For every orthonormal pair $(u_{+}, u_{-})$ of the two-dimensional space $V$ , the maps $Q_{\pm}$ are orthogonal projections with ranges $C u_{\pm}$ , $Q_{+} Q_{-} = 0$ and $Q_{+} + Q_{-} = I$ (Analyzer projections of a two-dimensional polarisation space and their overlaps clauses (a),(d), applied with $(e_{1}, e_{2}) : = (u_{+}, u_{-})$ ).

[F2]

Parseval. For every $x \in V$ one has $∥ x ∥^{2} = ∥ Q_{+} x ∥^{2} + ∥ Q_{-} x ∥^{2}$ (Parseval equivalences for an orthonormal family, [F1]).

[F3]

Born law. For the declared ideal sharp measurement with outcome projections $(Q_{+}, Q_{-})$ and the pure preparation of a unit vector $ψ$ , the outcome probabilities are $∥ Q_{\pm} ψ ∥^{2}$ ; for a nonzero datum $a$ the unit vector is $ψ = a / ∥ a ∥$ and the probabilities are $∥ Q_{\pm} a ∥^{2} / ∥ a ∥^{2}$ (Declared ideal measurements and instruments of the framework clauses (a),(b), Assigned sharp observables and the outcome law of the framework clause (b), The state space: density operators, pure and mixed states, and purity).

[F4]

Inner product. The inner product is linear in the first argument and conjugate-linear in the second; consequently $∥ Q_{\pm} a ∥^{2} = ∣ ⟨ u_{\pm}, a ⟩ ∣^{2}$ , and for $a = a_{1} + a_{2}$ , $∣ ⟨ u_{\pm}, a ⟩ ∣^{2} = ∣ ⟨ u_{\pm}, a_{1} ⟩ ∣^{2} + ∣ ⟨ u_{\pm}, a_{2} ⟩ ∣^{2} + 2 Re (⟨ u_{\pm}, a_{1} ⟩ \overline{⟨ u_{\pm}, a_{2} ⟩})$ (Analyzer projections of a two-dimensional polarisation space and their overlaps Fact [F1], The state space: density operators, pure and mixed states, and purity).

[F5]

Identification discipline. The claim that a laboratory source, interferometer or detector realizes one of the declared models is a tested identification with uncertainty in the sense of P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

[F6]

Quantum count law, conditional on the separate radiation counting model. For a unit mode $φ \in h$ , the $n$ -quanta preparation is the ray of the unit vector $U_{φ} e_{n} = φ^{\otimes n}$ in the $n$ -photon sector, with $n \in N = {0, 1, \dots}$ . Under the radiation extension and the declared ideal total-number detector, its count probabilities are $p_{m} = δ_{m n}$ , where $δ_{m n} = 1$ for $m = n$ and $0$ otherwise (Number and energy statistics of a declared mode: number states, coherent states and the continuum thermal limitation clause (a), The declared ideal counting, energy-window and analyser detection models of the radiation sector clause (b)). This is a quantum prediction under those additional declarations, not a count law for the classical amplitude datum $a$ .

Proof

1.1F1F2F4algebra

[Resolution and normalised classical law] By [F1] the bin projections resolve the identity, $Q_{+} + Q_{-} = I$ . For $a \neq 0$ put $N (a) : = I_{+} (a) + I_{-} (a)$ ; by [F2] applied to $a$ one has $N (a) = ∥ Q_{+} a ∥^{2} + ∥ Q_{-} a ∥^{2} = ∥ a ∥^{2} > 0$ , so the normalised classical law is well defined and equals $I_{\pm} (a) / ∥ a ∥^{2} = ∣ ⟨ u_{\pm}, a ⟩ ∣^{2} / ∥ a ∥^{2}$ by [F4].

2.1F3F4step 1.1algebra

[Quantised law] The ray $[a]$ is a pure preparation; by [F3] the Born probabilities of the declared measurement with projections $(Q_{+}, Q_{-})$ are $p_{\pm} = ∥ Q_{\pm} a ∥^{2} / ∥ a ∥^{2} = ∣ ⟨ u_{\pm}, a ⟩ ∣^{2} / ∥ a ∥^{2}$ . Together with step 1.1 this proves (a).

3.1F4step 1.1step 2.1algebra

[Common cross-term] Let $a = a_{1} + a_{2}$ be any decomposition. Steps 1.1 and 2.1 identify both normalized laws as $∣ ⟨ u_{\pm}, a ⟩ ∣^{2} / ∥ a ∥^{2}$ . Dividing the expansion of [F4] by the positive number $∥ a ∥^{2}$ gives, in each law, the cross-term $2 Re (⟨ u_{\pm}, a_{1} ⟩ \overline{⟨ u_{\pm}, a_{2} ⟩}) / ∥ a ∥^{2}$ . This denominator is the same for both bins and both models at the fixed datum $a$ . The unnormalized classical intensity has the unscaled cross-term of [F4]; the displayed normalized term is common to both declared laws, proving (b).

4.1givenstep 1.1step 2.1F5F6∎

[Limits of the comparison] Steps 1.1 and 2.1 show equality of the declared normalised two-bin laws for every datum, so this prediction does not distinguish the two models. Their declared objects nevertheless differ: a classical field amplitude with deterministic intensities and a quantum state ray with Born probabilities. For counting, the classical declarations impose no condition on a sequence of count probabilities $q_{m} (a) \geq 0$ with $\sum_{m \in N} q_{m} (a) = 1$ . For example, for any fixed $n \in N$ the assignment $q_{m} (a) = δ_{m n}$ would leave every declared intensity and normalised bin fraction unchanged and would match the quantum number-state law of [F6]. Thus the quantum count prediction alone does not imply a differing classical prediction. A proposed count comparison must first specify a classical field-and-detector count law, preparations in both models and their detector responses; comparison with repeated data also requires the sampling assumptions of [F5]. No such extension or statistical comparison is adopted here, and no data are reported. This proves (c) and (d).

Remarks

Classical waves interfere. The cross-term of clause (b) is present in the declared classical intensity as well as in the Born law, so this two-bin interference prediction alone does not distinguish the two declared models or refute the declared classical model; conversely, the declared quantised model predicts the pattern too, so the absence of a visible pattern in finitely many detections would not by itself falsify it. Statistical comparison requires the sampling model of P7 and reported data.
What "classical" means here. The classical model of this item is the elementary declared transverse-amplitude model with intensity equal to squared amplitude norm; it is not Maxwell's theory, and the item imports no Maxwell theorem. A test of classical electromagnetism would require that framework's own postulates and field solutions.

A two-bin comparison of a declared classical field model and the quantised model: identical normalised laws do not discriminate

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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