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The Bell Curve as the Interference-Only Component of the Dipole-Wave Joint Detection Probability

James Arneberg

Published Aug 31, 2026• 0 citations

Abstract

Abstract The dipole-wave framework proposes that all fundamental particles are point-like entities with oscillating dipole moments that radiate real guiding waves into three-dimensional space. Previous work has shown that this framework reduces to the Schrödinger equation for single massive particles and provides a deterministic account of single-photon interference. In this paper, we extend the dipole-wave framework to two-photon Bell-type correlations. We derive the joint detection probability for two photons with orthogonal dipoles and show that the quantum mechanical Bell curve—P = ½sin²(Δ)—is the interference-only, background-subtracted, normalized component of the dipole-wave prediction. The dipole-wave theory yields a raw coincidence rate proportional to 1 + 2sin²(Δ), which contains the Bell curve after background subtraction and normalization. The π/2 shift that relates this to the aligned-dipole case is derived from the orthogonality of the dipoles produced in Type-II parametric down-conversion. We further show that the four outcome probabilities follow from the geometry of the orthogonal dipoles and lead to the standard Bell correlation function E(a,b) = −cos(2Δ) and the CHSH violation |S| = 2√2. We also compare the raw prediction with published raw coincidence data and find that neither the simple raw expression nor the raw Bell curve matches the observed raw rates. This is expected, because the Bell curve is a derived quantity that emerges only after background subtraction and normalization. The central result remains: the dipole-wave framework reproduces the normalized Bell curve, the full correlation function, and the CHSH violation.

DOI: 10.5281/zenodo.22213535

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We extend the electromagnetic mode competition framework to entangled photon pairs. The framework derives the Born rule from classical electromagnetism plus a single empirical input: each emission produces exactly one detection (single-event exclusivity). Here we show that the same postulates, applied to the joint mode structure of spontaneous parametric down-conversion, reproduce the quantum predictions for polarisation-entangled pairs: the sin²(θ₁ − θ₂) coincidence probability and the maximal CHSH violation S = 2√2. We also derive GHZ correlations and the Hong–Ou–Mandel dip. The framework dissolves the mystery of Bell correlations: there are not two photons coordinating across space, but one extended electromagnetic mode undergoing one detection event. Alice and Bob are endpoints of a single structure; their correlation is built in from the moment of emission and enforced by energy conservation. No action at a distance occurs—one detector absorbs the energy, and the shared pool is depleted. Bell's inequality is violated not because of nonlocality, but because Bell's factorisation assumption does not apply to measurements on a single extended object.

An interference experiment in which signal and idler photons from a parametric oscillator (paramp) were mixed by a beam splitter and the coincidence count rate was measured has been outlined by Ou Mandel , [ Phys. Rev. Lett.61, 50 ( 1988)]. The theory of the joint two-photon detection probability has been generalized to include nondegenerate photons and time delays between the photons at the beam splitter. Previously, two-photon Fock states were employed in this analysis. We consider the squeezed vacuum output of a paramp in a cavity. The results derived here reduce to the special case derived by Ou and Mandel. Furthermore, a cascade emission process is treated by using a rate equation approach.

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An experiment has been carried out to test a prediction of the deBroglie guided wave theory for photons. According to this theory, there exist electromagnetic waves as well as photons, with the former acting as guide for the latter. The wave not only determines the probability for the detection of a photon, but it also has an independent physical reality which can result in photodetection, As a result the guided wave theory predicts an interference effect when two photons pass through an interferometer with a particular geometry and one measures the rate of simultaneous detections by two photodetectors in coincidence as a function of optical path difference, By contrast, quantum mechanics predicts no interference under the same conditions. The experimental results clearly violate the prediction of the guided wave theory, but are in agreement with quantum mechanics.

We decompose the Bell-CHSH correlator for the Bell state into a diagonal (classical mixture) contribution and a coherent (off-diagonal) contribution using the density matrix formulation.The diagonal part produces correlations satisfying |S|≤2; the entire enhancement to |S|=2√2 originates from the coherent cross term Re(A₁*A₂), which generates a setting-dependent redistribution of probability among measurement outcomes. This cross term sums to zero over all outcomes (preserving normalisation) but modifies individual coincidence probabilities in a manner that depends on the measurement basis. The coherent contribution is precisely the component that takes Born-rule probabilities outside the class of factorisable probabilities assumed in the derivation of Bell's inequality. We show that this structure is formally parallel to the variance normalisation producing apparent Bell violations in classical coherent-detection systems: in both cases, a setting-dependent operation on coherent amplitudes transforms correlations from |S|≤2 to |S|=2√2.

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