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An ideal electric-field zero and a declared detector-bin response

Physical scope

Conditional classical-wave example separating an ideal pointwise electric zero from a declared measurable bin response with background. All measurable bins of positive measure and nonnegative measurable backgrounds remain allowed, with a response in $[0, + \infty]$ ; a finite-rate interpretation requires a finite response and the declared apparatus/calibration dictionary. The finite-volume constant-positive-background case provides an explicit finite positive response despite the point zero. No data, sampling law, event probability or classical/quantum discrimination is inferred.

Example

Adopt the two-beam data and conclusions of Superposition intensity and the coherent interference cross term: for $v \in C^{3}$ write $∣ v ∣^{2} : = \sum_{j = 1}^{3} v_{j} \overline{v_{j}} = \sum_{j = 1}^{3} ∣ v_{j} ∣^{2}$ , using the scalar conjugation and modulus of Real and imaginary parts, complex conjugation, and modulus; this agrees with the Euclidean squared norm on real vectors. Use the complex electric amplitude $α (x) = a_{1} e_{1} + a_{2} e_{2} e^{i δ (x)}$ and magnetic amplitude $β (x) = a_{1} b_{1} + a_{2} b_{2} e^{i δ (x)}$ of its clause (a), with $δ (x) = (k_{2} - k_{1}) \cdot x$ , $b_{j} = k_{j} \times e_{j} / ω$ (The cross product in $R^{3}$ ) and the plane-wave solutions $(E_{j}, B_{j})$ of Real plane-wave fields encoded by complex amplitudes; and its clause (b) averaged energy density $⟨ u ⟩ (x) = \frac{ε_{0}}{4} ∣ α (x) ∣^{2} + \frac{1}{4 μ_{0}} ∣ β (x) ∣^{2}$ with the SI constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and assume the Axiom of Countable Choice ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )) exactly as inherited by the Lebesgue-measure and integral items cited in clause (ii), which are stated under that hypothesis; no other choice principle is used. Assume dimensionless $a_{1} = a_{2} = : a > 0$ , $e_{1} = e_{2} = : e$ with real amplitude vector $e$ , and $k_{1} \neq k_{2}$ with $(k_{2} - k_{1}) \times e \neq 0$ ; let $x_{0}$ be a point with $δ (x_{0}) \equiv π (m o d 2 π)$ .

(i) [Ideal prediction and its vanishing] At such $x_{0}$ one has $α (x_{0}) = a e (1 + e^{i δ (x_{0})}) = 0$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ), hence the electric field of the superposition vanishes at $x_{0}$ for every $t$ and the electric term of the averaged energy density vanishes there. The full averaged energy density need not vanish: $β (x_{0}) = a (b_{1} - b_{2}) = a ((k_{1} - k_{2}) \times e) / ω \neq 0$ , so $⟨ u ⟩ (x_{0}) = ∣ a (b_{1} - b_{2}) ∣^{2} / (4 μ_{0}) > 0$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ for the positive-definite norm). Thus a claim that 'the ideal intensity vanishes at $x_{0}$ ' must name the quantity that vanishes.

(ii) [Declared bin response and its finite-rate branch] Adopt a declared detector model with a Lebesgue-measurable bin $W \subseteq R^{3}$ , $x_{0} \in W$ and $λ_{3} (W) > 0$ , a finite real response coefficient $κ \geq 0$ , and a declared nonnegative density $b : W \to [0, \infty)$ whose zero extension $\tilde{b}$ (equal to $b$ on $W$ and to $0$ off $W$ ) is Lebesgue measurable; this measurability of the zero extension is a declared hypothesis of the detector model. The declared response functional uses numerical values in fixed declared units: spatial coordinates are in metres, so $λ_{3}$ carries $m^{3}$ ; $α$ carries $V m^{- 1}$ as in Real plane-wave fields encoded by complex amplitudes. Fix a declared SI unit $U$ of accumulated response (possibly dimensionless); a response rate means response per unit time, in $U s^{- 1}$ . Declare $κ$ in $U s^{- 1} V^{- 2} m^{- 1}$ and $b$ (including any constant $b_{0}$ ) in $U s^{- 1} m^{- 3}$ . Thus $κ ∣ α ∣^{2} + b$ is a response-rate density in $U s^{- 1} m^{- 3}$ , and $R (W)$ carries $U s^{- 1}$ . This is a declared response model, without an empirical identification with a detector output. It is defined by the nonnegative Lebesgue integral $R (W) : = \int_{W} (κ ∣ α (x) ∣^{2} + b (x)) d λ_{3} \in [0, + \infty] .$ Here the continuous real function $∣ α ∣^{2}$ is Borel measurable by Continuous functions on Euclidean spaces are Borel measurable and Lebesgue measurable by Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable; closure under nonnegative scaling and addition is supplied by Closure properties of measurable functions used by the integral, and Integral over a measurable subset defines the extended integral. No boundedness or finite-volume condition on $W$ is imposed. If $\int_{W} b d λ_{3} > 0$ , monotonicity gives $R (W) \geq \int_{W} b d λ_{3} > 0$ even though $∣ α (x_{0}) ∣^{2} = 0$ . For a constant $b = b_{0} > 0$ this lower bound is $b_{0} λ_{3} (W)$ by The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions and Monotonicity and nonnegative homogeneity of the nonnegative integral; it may be $+ \infty$ . The finite predicted rate branch additionally requires $R (W) < \infty$ , in the response-rate units just declared. A concrete sufficient case is $0 < λ_{3} (W) < \infty$ and constant $0 < b_{0} < \infty$ : the bound $∣ α ∣^{2} \leq 4 a^{2} ∣ e ∣^{2}$ gives $0 < b_{0} λ_{3} (W) \leq R (W) \leq (4 κ a^{2} ∣ e ∣^{2} + b_{0}) λ_{3} (W) < \infty .$ For $κ = 0$ the field contribution is the zero function, without forming $0 \cdot (+ \infty)$ . Identifying a finite predicted response with a recorded comparison quantity still requires the apparatus, calibration and analysis declarations of Report-backed operational evidence with uncertainty. Thus a finite positive bin prediction is compatible with the ideal pointwise zero, but an extended infinite response is not a recorded finite rate, and neither response is an event probability.

(iii) [Record versus prediction; statistical qualification] The data of Report-backed operational evidence with uncertainty are a finite tuple of real recorded values or reported observations with declared conditions, calibration and uncertainty. The predictions of (i)-(ii) are deterministic field quantities or a declared extended response functional, with a finite real rate only on its stated finite branch. Here an ideal fringe pattern describes spatial variation of a specified intensity or response; a single detection and a finite data tuple are separate records. Whether a finite record displays a specified visibility or pattern requires a declared criterion, calibration, binning, background and sampling assumptions, none supplied by the field calculation alone. No sampling law, probability of failing a visibility criterion, detection statistics or measured histogram is asserted. This example discriminates no classical or quantum ontology: classical electromagnetic waves interfere and the coherence calculation is a deterministic prediction of the classical model.

Verification

Proof technique: direct substitution, the elementary periodicity of the complex exponential phase, and the declared measure-theoretic interface.

Given: the two-beam data of Superposition intensity and the coherent interference cross term clauses (a)-(b) with $a_{1} = a_{2} = a > 0$ , $e_{1} = e_{2} = e$ , $k_{1} \neq k_{2}$ , $(k_{2} - k_{1}) \times e \neq 0$ and a point $x_{0}$ with $δ (x_{0}) \equiv π (m o d 2 π)$ ; the declared detector model of clause (ii) with measurable bin $W ∋ x_{0}$ , $λ_{3} (W) > 0$ , $κ \geq 0$ and nonnegative density $b$ whose zero extension $\tilde{b}$ is Lebesgue measurable; the Axiom of Countable Choice ${A C}_{ω}$ ; the SI constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

[F1] Clause (a) of Superposition intensity and the coherent interference cross term gives, at each fixed $x$ , $E (x, t) = Re (α (x) e^{i (k_{1} \cdot x - ω t)})$ and $B (x, t) = Re (β (x) e^{i (k_{1} \cdot x - ω t)})$ with $α (x) : = a_{1} e_{1} + a_{2} e_{2} e^{i δ (x)}$ , $β (x) : = a_{1} b_{1} + a_{2} b_{2} e^{i δ (x)}$ , $δ (x) : = (k_{2} - k_{1}) \cdot x$ , $b_{j} = k_{j} \times e_{j} / ω$ and $ω > 0$ .

[F2] Clause (b) of Superposition intensity and the coherent interference cross term gives $⟨ u ⟩ (x) = \frac{ε_{0}}{4} ∣ α (x) ∣^{2} + \frac{1}{4 μ_{0}} ∣ β (x) ∣^{2}$ for every $x$ .

[F3] For $u, v \in R^{3}$ , $u \times v = (u_{y} v_{z} - u_{z} v_{y}, u_{z} v_{x} - u_{x} v_{z}, u_{x} v_{y} - u_{y} v_{x})$ (The cross product in $R^{3}$ ).

[F4] The Euclidean inner product is $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ and the Euclidean norm is $∥ x ∥_{2} = \sqrt{⟨ x, x ⟩}$ ; the form is positive definite: $⟨ x, x ⟩ \geq 0$ , and $⟨ x, x ⟩ = 0$ if and only if $x = 0$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F5] For all $z, w \in C$ , $z \overline{z} = ∣ z ∣^{2}$ , $∣ z ∣ \geq 0$ , $∣ z ∣ = 0 ⟺ z = 0$ , $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ and $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F6] For real $x, y$ one has $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ and $∣ \exp (x + i y) ∣ = e^{x}$ ; in particular $\exp (i θ) = \cos θ + i \sin θ$ for real $θ$ , and $e^{i π} + 1 = 0$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F7] For every real $x$ , $\sin (x + π) = - \sin x$ and $\cos (x + π) = - \cos x$ , and in particular $\sin π = 0$ and $\cos π = - 1$ (Quarter-turn values and shifts by pi/2 and pi).

[F8] The framework's magnetic constant satisfies $μ_{0} > 0$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F9] The Axiom of Countable Choice ${A C}_{ω}$ is the statement that every at most countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F10] A set $E \subseteq R^{n}$ is Lebesgue measurable when $λ_{n}^{*} (A) = λ_{n}^{*} (A \cap E) + λ_{n}^{*} (A ∖ E)$ for every $A \subseteq R^{n}$ , and $λ_{n}$ is the restriction of $λ_{n}^{*}$ to the family $L (R^{n})$ (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ).

[F11] Under ${A C}_{ω}$ , $L (R^{n})$ is a sigma-algebra on $R^{n}$ and $λ_{n}$ is a measure on it (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume, clause 1).

[F12] Under ${A C}_{ω}$ , every continuous map $f : R^{n} \to R^{m}$ is Borel measurable (Continuous functions on Euclidean spaces are Borel measurable).

[F13] Under ${A C}_{ω}$ , $B (R^{n}) \subseteq L (R^{n})$ : every Borel subset of $R^{n}$ is Lebesgue measurable (Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

[F14] For a measure space $(X, A, μ)$ , a measurable $f : X \to [0, + \infty]$ and $E \in A$ , the integral over $E$ is $\int_{E} f d μ : = \int f χ_{E} d μ$ (Integral over a measurable subset).

[F15] If $f, g : X \to \overline{R}$ are measurable and $f + g$ is defined pointwise, then $f + g$ is measurable; if $c \in R$ and $f : X \to R$ is measurable, then $c f$ is measurable (Closure properties of measurable functions used by the integral).

[F16] For measurable $f, g : X \to [0, + \infty]$ and $c \geq 0$ : if $f \leq g$ then $\int f d μ \leq \int g d μ$ ; if $c > 0$ then $\int c f d μ = c \int f d μ$ ; and for $c = 0$ the integral of the zero function is $0$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F17] A nonnegative simple measurable function $s = \sum_{j = 1}^{m} c_{j} χ_{E_{j}}$ has simple integral $\int s d μ : = \sum_{j = 1}^{m} c_{j} μ (E_{j})$ with the convention $0 \cdot (+ \infty) = 0$ (The integral of a nonnegative simple function).

[F18] If $s$ is a nonnegative simple measurable function, then its nonnegative Lebesgue integral equals its simple integral, $\int s d μ = \int_{s i m p l e} s d μ$ (The nonnegative integral agrees with the simple integral on simple functions).

[F19] The complex exponential is entire, and $\exp^{'} (z) = \exp z$ for every $z \in C$ (The complex exponential is entire and its complex derivative is itself).

[F20] If $f : U \to C$ is complex differentiable at $a \in U$ , then $f$ is continuous at $a$ (Complex differentiability at a point implies continuity there).

[F21] For $k \in N$ , finite componentwise sums and products of $C^{k}$ Euclidean maps are $C^{k}$ , and a composite of composable $C^{k}$ Euclidean maps is $C^{k}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

[F22] In an ordered field, $a > 0$ and $b > 0$ imply $a b > 0$ (Sign rules for products and monotonicity of multiplication), and $R$ is a totally ordered field (The reals form a totally ordered field).

[F23] A record of Report-backed operational evidence with uncertainty is a finite tuple of recorded data and reported observations with declared condition annotations, and it contains no probability space, random variable, distribution, sampling law, independence assumption, likelihood or coverage statement (clause (d)).

1.1F6F7givenalgebra

Since $δ (x_{0}) \equiv π (m o d 2 π)$ , there is an integer $n$ with $δ (x_{0}) = π + 2 π n$ ; the cartesian form of [F6] gives $e^{i δ (x_{0})} = \cos δ (x_{0}) + i \sin δ (x_{0})$ , and [F7] gives $\cos (y + 2 π) = - \cos (y + π) = \cos y$ and $\sin (y + 2 π) = - \sin (y + π) = \sin y$ for every real $y$ , so by induction on $∣ n ∣$ one has $\cos (π + 2 π n) = \cos π = - 1$ and $\sin (π + 2 π n) = \sin π = 0$ ; hence $e^{i δ (x_{0})} = - 1$ .

1.2F1F4F5F19F20F21algebra

[Regularity of $α$ ] The phase $x \mapsto δ (x) = (k_{2} - k_{1}) \cdot x$ is an affine map $R^{3} \to R$ , hence of class $C^{\infty}$ ; the exponential is entire [F19], hence complex differentiable everywhere and continuous [F20]; by the algebra-and-composition closure of [F21] the composite $x \mapsto e^{i δ (x)}$ and the finite combinations defining the components of $α = a_{1} e_{1} + a_{2} e_{2} e^{i δ (x)}$ of [F1] are continuous, and the modulus squared $∣ α ∣^{2} = α \cdot \overline{α}$ [F5] is a finite sum of products of continuous real functions, hence continuous on $R^{3}$ .

1.3F9F10F11F14F17F18givenalgebra

[Constant background] For a constant $b = b_{0} > 0$ the zero extension is the nonnegative simple function $\tilde{b} = b_{0} χ_{W}$ with $W \in L (R^{3})$ [F10, F11]; its simple integral is $b_{0} λ_{3} (W)$ by [F17] and equals its Lebesgue integral by [F18], so $\int_{W} b d λ_{3} = b_{0} λ_{3} (W)$ by [F14], a value in $(0, + \infty]$ that may be $+ \infty$ when $λ_{3} (W) = + \infty$ .

1.4F14F16givenalgebra

[Zero response coefficient] For $κ = 0$ the field contribution is the zero function because $∣ α ∣^{2}$ is real-valued, so no product $0 \cdot (+ \infty)$ is formed; then $\int_{W} κ ∣ α ∣^{2} d λ_{3} = 0$ by the $c = 0$ clause of [F16], and $R (W) = \int_{W} b d λ_{3}$ ; in particular the response remains defined in $[0, + \infty]$ on every measurable bin.

2.1F1step 1.1givenalgebra

Substituting the amplitude data of [F1] and step 1.1 gives $α (x_{0}) = a_{1} e_{1} + a_{2} e_{2} e^{i δ (x_{0})} = a e (1 + (- 1)) = 0$ , using $a_{1} = a_{2} = a$ and $e_{1} = e_{2} = e$ .

2.2F1F3givenstep 1.1algebra

For $β (x_{0})$ : substituting $a_{1} = a_{2} = a$ , $e_{1} = e_{2} = e$ and step 1.1 into [F1] gives $β (x_{0}) = a (b_{1} - b_{2})$ , and $b_{1} - b_{2} = k_{1} \times e / ω - k_{2} \times e / ω = ((k_{1} - k_{2}) \times e) / ω$ by the coordinate bilinearity of [F3]; if $β (x_{0}) = 0$ then multiplying by the nonzero scalar $ω / a$ (nonzero because $a > 0$ and $ω > 0$ from [F1]) gives $(k_{1} - k_{2}) \times e = - (k_{2} - k_{1}) \times e = 0$ by [F3], contrary to the hypothesis $(k_{2} - k_{1}) \times e \neq 0$ ; hence $β (x_{0}) \neq 0$ .

2.3F9F10F11F12F13F14F15step 1.2givenalgebra

[Integrand measurability and the response functional] Under ${A C}_{ω}$ the family $L (R^{3})$ is a sigma-algebra and $λ_{3}$ a measure on it [F11], and the bin $W$ is measurable with $λ_{3} (W)$ defined [F10]; $∣ α ∣^{2}$ is continuous by step 1.2, hence Borel measurable [F12] and Lebesgue measurable [F13]; since $κ \in R$ , the scaled function $κ ∣ α ∣^{2}$ is measurable by [F15], and the sum $κ ∣ α ∣^{2} + \tilde{b}$ is measurable by [F15] because $\tilde{b}$ is real-valued and measurable by the declared detector model; consequently [F14] defines $R (W) : = \int_{W} (κ ∣ α ∣^{2} + b) d λ_{3} = \int (κ ∣ α ∣^{2} + \tilde{b}) χ_{W} d λ_{3} \in [0, + \infty]$ .

2.4F1F4F5F6F14F15F16F17F18F22givenstep 1.3algebra

[Finite-rate branch] Suppose $0 < λ_{3} (W) < \infty$ and $b \equiv b_{0}$ with $0 < b_{0} < \infty$ . For every $x$ one has $∣ e^{i δ (x)} ∣ = 1$ by [F6], so $∣ 1 + e^{i δ (x)} ∣ \leq 1 + ∣ e^{i δ (x)} ∣ = 2$ by the triangle inequality of [F5], and $∣ α (x) ∣ = ∣ a ∣ ∣ e ∣ ∣ 1 + e^{i δ (x)} ∣ \leq 2 a ∣ e ∣$ by the multiplicative law of [F5], where $∣ e ∣^{2} = ⟨ e, e ⟩$ is the Euclidean squared norm of the real amplitude [F4]; hence $∣ α ∣^{2} \leq 4 a^{2} ∣ e ∣^{2}$ and, since $κ \geq 0$ , the pointwise bounds $b_{0} χ_{W} \leq (κ ∣ α ∣^{2} + b_{0}) χ_{W} \leq C χ_{W}$ hold with the finite real constant $C : = 4 κ a^{2} ∣ e ∣^{2} + b_{0}$ . Monotonicity and the simple-integral values of step 1.3 and [F17], [F18] give $0 < b_{0} λ_{3} (W) \leq R (W) \leq C λ_{3} (W) < \infty$ , the strict lower bound by the sign rule $b_{0} λ_{3} (W) > 0$ of [F22] applied to $b_{0} > 0$ and $λ_{3} (W) > 0$ ; this proves a finite positive numerical response. By the units declared in clause (ii), $κ ∣ α ∣^{2}$ and $b_{0}$ both carry $U s^{- 1} m^{- 3}$ , while $λ_{3}$ carries $m^{3}$ , so $R (W)$ is a finite positive predicted response rate in $U s^{- 1}$ .

3.1F1F2step 2.1algebra

By clause (a) of [F1], $E (x_{0}, t) = Re (α (x_{0}) e^{i (k_{1} \cdot x_{0} - ω t)}) = Re (0) = 0$ for every real $t$ ; hence the electric field of the superposition vanishes at $x_{0}$ at every time, and with step 2.1 the electric term $\frac{ε_{0}}{4} ∣ α (x_{0}) ∣^{2}$ of the averaged density of [F2] vanishes there.

3.2F2F4F5F8F22step 2.1step 2.2algebra

[F2] and step 2.1 give $⟨ u ⟩ (x_{0}) = \frac{ε_{0}}{4} ∣ α (x_{0}) ∣^{2} + \frac{1}{4 μ_{0}} ∣ β (x_{0}) ∣^{2} = \frac{1}{4 μ_{0}} ∣ β (x_{0}) ∣^{2}$ ; since $β (x_{0}) \neq 0$ by step 2.2, positive definiteness [F4] and the modulus law $∣ z ∣ = 0 ⟺ z = 0$ of [F5] give $∣ β (x_{0}) ∣^{2} > 0$ , and $μ_{0} > 0$ from [F8] gives $1 / (4 μ_{0}) > 0$ , so the sign rules [F22] yield $⟨ u ⟩ (x_{0}) > 0$ ; thus the full averaged energy density does not vanish at $x_{0}$ , only its electric term does.

3.3F14F15F16step 2.1step 2.3algebra

[General background lower bound] The functions $\tilde{b}$ and $κ ∣ α ∣^{2} + \tilde{b}$ are measurable, nonnegative and finite-valued, with $\tilde{b} \leq κ ∣ α ∣^{2} + \tilde{b}$ pointwise because $∣ α ∣^{2} \geq 0$ [F5] and $κ \geq 0$ ; the monotonicity clause of [F16] and $R (W) = \int (κ ∣ α ∣^{2} + \tilde{b}) χ_{W} d λ_{3}$ of step 2.3 give $\int_{W} b d λ_{3} = \int \tilde{b} χ_{W} d λ_{3} \leq R (W)$ , so $\int_{W} b d λ_{3} > 0$ implies $R (W) \geq \int_{W} b d λ_{3} > 0$ even though $∣ α (x_{0}) ∣^{2} = 0$ by step 2.1.

4.1F23F1step 2.1step 3.1step 2.3step 3.3step 2.4given∎

[Record versus prediction] The data of [F23] are finite tuples of recorded values or reported observations with declared conditions, calibration and uncertainty, and contain no sampling law; the predictions of steps 1.1-3.3 are deterministic field quantities or a declared extended response functional with a finite real rate only on the branch of step 2.4; a detection and a finite record are different objects from the ideal intensity, so no sampling law, probability of failing a visibility criterion, detection statistics or measured histogram is asserted here, and the finite positive bin prediction of step 2.4 is compatible with the pointwise ideal zero of steps 2.1 and 3.1 without discriminating the classical and quantum descriptions of light, since classical electromagnetic waves interfere.

An ideal electric-field zero and a declared detector-bin response

Example

Verification

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