Mathematics

Photon Entanglement, Bell Inequality Violation, and Energy Interpretation of the Born Rule in Maxwell–Schwartz Field Theory

David Carfì

Published Apr 28, 2026• 0 citations

Abstract

In this paper we study photon entanglement in the framework of Maxwell–Schwartz field theory. The ambient state space is the complex Maxwellian distribution space W=S′(M4,C3), whose elements are fields of the form F=E→+icB→. Polarization is realized as a two-dimensional complex subspace of W, generated by suitable linearly polarized Maxwellian solutions associated with opposite propagation directions. This yields canonical polarization sectors PA and PB, each naturally isomorphic to C2. Within this setting, the Bell singlet state is represented by a non-factorizable tensorial Maxwellian field in PA⊗PB⊂W⊗W. By means of the induced rotated polarization bases, the standard joint probabilities of the photon polarization experiment are recovered exactly, and the correlation law E(a,b)=−cos(2(a−b)) is obtained. Consequently, the usual CHSH value 22 is reproduced in the Maxwell–Schwartz framework. To clarify the meaning of this violation, we first formulate the CHSH inequality in a purely measure-theoretic form, as a theorem about four correlators represented on a single probability space by bounded measurable functions. We then show that the correlators produced by the intrinsic Maxwellian Bell state do not admit such a common representation. The obstruction is structural: the ontic state is a global non-product field configuration, and the four correlations arise from different polarization resolutions of the same tensorial Maxwellian state. A second main result concerns the Born rule. For L2 scalar quantum states in the domain of the Maxwellian correspondence, we prove that the squared Hilbert norm, times the constant ε0, coincides with the electromagnetic energy of the associated field. This leads to an energy interpretation of the Born rule: the Born probability density is identified with the normalized electromagnetic energy density up to an interference term depending on the chosen Maxwell–Schwartz isomorphism, which assumes the role of a quantum context. In the context of the Aspect and collaborators’ experiment, we prove that, on the other hand, the polarization probabilities become energy contributions of the corresponding field components. These results show that photon entanglement, Bell inequality violation, and the Born rule admit a coherent interpretation within Maxwell–Schwartz field theory, where the basic ontological objects are electromagnetic-like fields rather than abstract state vectors.

DOI: 10.3390/math14091490

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Mathematics

Photon Entanglement, Bell Inequality Violation, and Energy Interpretation of the Born Rule in Maxwell–Schwartz Field Theory

David Carfì

Published Apr 28, 2026• 0 citations

Publisher

Abstract

DOI: 10.3390/math14091490

Related papers

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Sudeer Punnery

Abstract This paper proposes a local-deterministic completion of quantum entanglement by identifying the instantaneous phase of rotating electric field vectors as the primary hidden variable. We demonstrate that the observed violation of the Bell-CHSH inequality is not a signature of non-locality, but a natural consequence of the Intensity-Phase doubling mechanism inherent in Maxwellian electrodynamics. Through analytical derivation and discrete-time numerical integration, it proves that the correlation of synchronized transverse waves—governed by the quadratic nature of Malus’s Law $\:\left(\varvec{I}\:\propto\:\:{\varvec{E}}^{2}\right)$—generates the exact quantum correlation function $\:E\left(\varDelta\:\theta\:\right)=\:\:Cos(2\varDelta\:\theta\:)$ . We resolve the long-standing discrepancy regarding measurement angles by showing that the maximum violation of the CHSH inequality $\:(S=2\sqrt{2})$ occurs at a physical polarizer setting of $\:22.5^\circ\:$ , which corresponds to a $\:45^\circ\:$ , shift in the internal field phase . By distinguishing between mechanical rotation and field-space geometry, this model saturates the Tsirelson bound while strictly maintaining Einsteinian locality and causality. This framework suggests that the "spooky action" of entanglement is a manifestation of temporal phase-locking between the source and the detector, providing a fully realistic and Maxwellian interpretation of Bell-type statistics.Abstract This paper proposes a local-deterministic completion of quantum entanglement by identifying the instantaneous phase of rotating electric field vectors as the primary hidden variable. We demonstrate that the observed violation of the Bell-CHSH inequality is not a signature of non-locality, but a natural consequence of the Intensity-Phase doubling mechanism inherent in Maxwellian electrodynamics. Through analytical derivation and discrete-time numerical integration, it proves that the correlation of synchronized transverse waves—governed by the quadratic nature of Malus’s Law $\:\left(\varvec{I}\:\propto\:\:{\varvec{E}}^{2}\right)$—generates the exact quantum correlation function $\:E\left(\varDelta\:\theta\:\right)=\:\:Cos(2\varDelta\:\theta\:)$. We resolve the long-standing discrepancy regarding measurement angles by showing that the maximum violation of the CHSH inequality $\:(S=2\sqrt{2})$ occurs at a physical polarizer setting of $\:22.5^\circ\:$, which corresponds to a $\:45^\circ\:$, shift in the internal field phase . By distinguishing between mechanical rotation and field-space geometry, this model saturates the Tsirelson bound while strictly maintaining Einsteinian locality and causality. This framework suggests that the "spooky action" of entanglement is a manifestation of temporal phase-locking between the source and the detector, providing a fully realistic and Maxwellian interpretation of Bell-type statistics.

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Coincidence Interpretation of the Born Rule: The Two-Photon Bound State as the Physical Origin of

|\psi|^2

The Born rule assigns probability density

|\psi|^2 = \psi^* \psi

to a single wavefunction. It is postulated here that this interpretation, while computationally successful, conflates two physically distinct situations. For a bound state,

\psi^*\psi

is not the square of a single wavefunction but the coincidence product of two independent conjugate wavefunctions

\psi_1\psi_2

— one incident, one its counter-propagating conjugate — simultaneously present. Detection of a quantum event requires the coincidence of these two conjugates at the detection point; a single wavefunction alone cannot populate a bound energy state or trigger detection. Stated carefully, the postulate is a bilinear detection law: detection is postulated to depend bilinearly on two conjugate field amplitudes through their product, rather than on an independently squared probability for each field. Under stated hypotheses this law reproduces standard results of quantum mechanics — the double slit interference pattern in the pair-arrival regime, and an invariant composite rest mass for the bound pair — without wavefunction collapse. A complete account of Bell correlations is not claimed: a locally factorizable coincidence model cannot reproduce the quantum correlations for arbitrary analyzer settings, and constructing a CHSH-complete treatment is registered as the central open problem of the interpretation (Section 5). The two-photon electron of Delta-c Mechanics provides the mechanical realization of this postulate: the electron is formed by two conjugate photons in counter-propagating circular orbit, their combined flow density forming a standing wave that is Lorentz invariant by construction. A structural correspondence between the Feynman path-integral hierarchy and this standing-wave construction is proposed in companion work; the exact amplitude mapping remains open. A Thomson scattering experiment capable of distinguishing the coincidence interpretation from standard QM is identified and described in a companion paper.

\:E\left(\varDelta\:\theta\:\right)=\:\:Cos(2\varDelta\:\theta\:)

. We resolve the long-standing discrepancy regarding measurement angles by showing that the maximum violation of the CHSH inequality

\:(S=2\sqrt{2})

occurs at a physical polarizer setting of

\:22.5^\circ\:

, which corresponds to a

\:45^\circ\:

, shift in the internal field phase . By distinguishing between mechanical rotation and field-space geometry, this model saturates the Tsirelson bound while strictly maintaining Einsteinian locality and causality. This framework suggests that the "spooky action" of entanglement is a manifestation of temporal phase-locking between the source and the detector, providing a fully realistic and Maxwellian interpretation of Bell-type statistics.