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.
This work develops a deterministic, field-based reformulation of quantum mechanics in which entanglement arises naturally from correlated hidden variables, directly realizing Einstein’s prediction. Fermions are modeled as coupled real fields whose internal oscillatory phases provide a physical ontology for spin, interference, and charge. The model yields the standard −cosθ correlation law in the uniform-phase limit, while predicting a small deterministic correction C( θ )=−cosθ+ε sin 2 θ with ε≈0.06 . This deviation is benchmarked against historic and modern Bell tests, showing it is not excluded but lies within the sensitivity of angle-resolved experiments. Thus, the work provides a realist, experimentally testable account of entanglement, aligning with Einstein’s vision of hidden variables. Experimental deviation between the quantum mechanics model to our real coupled-fields model is expected to be of order ε=0.06 . The model provides unique experimental predictions—a finite offset at symmetry points (e.g. Δ = 45˚) not explained by noise or reduced visibility. This work examines the violation of Bell’s inequality through the lens of the Coupled-Fields model. While standard quantum mechanics attributes these results to nonlocal entanglement, the Coupled-Fields framework reproduces the same correlations through deterministic, local interactions between the internal variables of the coupled strings. Each fermion pair shares an intrinsic phase relationship established at creation, which governs measurement outcomes without requiring instantaneous influence. Statistical violation of Bell’s inequality thus reflects the geometry of internal field orientations rather than true nonlocality. This analysis reconciles experimental data with physical realism, demonstrating that local hidden variables—when properly defined within the coupled-fields structure—can reproduce the full quantum correlation pattern. Our conclusion is that Einstein (EPR) [1] was right. Internal phase is the hidden variable.
This work demystifies quantum paradoxes by showing that the phenomena often labeled as “spooky action” arise from unfactorizable classical local processes that lie outside the scope of Bell’s tests, indicating that the universe remains locally real and that physical states exist independently of measurement. It presents a deterministic hidden‑variable framework, relativistically compatible with quantum entanglement, based on Intrinsic Spin Momentum (ISM). In this model, spin is treated as a real internal momentum circulation with definable vector structure, phase, curvature, and acceleration. Phase‑created correlations between intrinsic spin momenta reproduce CHSH‑type values without superluminal signaling or Einstein’s so‑called “spooky action at a distance.” The Bloch sphere emerges as the geometric representation of internal spin structure, and phase alignment on this sphere provides the mechanism for entanglement correlations, demonstrating that classical unfactorizable processes can reproduce Bell‑inequality violations without breaking any known laws of physics. This stands in stark contrast to the widely publicized interpretation of the 2022 Nobel Prize in Physics, which claimed that Bell‑test violations imply that the universe is “not locally real.” The model generalizes naturally to higher‑spin systems, linking intrinsic spin geometry to gravitational curvature. Gravitational waves dynamically modulate intrinsic spin acceleration, offering a physical mechanism for quantum gravity that diverges worldlines through localized curvature rather than as a purely coordinate effect of spacetime geometry. This unified picture replaces “spooky action” with correlated spin geometry embedded in a dynamic relativistic spacetime. Contact: ellis.ammons1@gmail.com Keywords: Intrinsic Spin Momentum, quantum gravity, entanglement, Bloch sphere, CHSH correlations, relativity, hidden variables, gravitational waves, phase geometry, spin structure.
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.
This letter challenges the standard interpretation of Bell test results, which has been used to argue for the abandonment of local realism. We demonstrate that the quantum-mechanical correlation Pc (θ) = cos2 (θ) and the classical Malus' Law result Pc (θ) = 1/2 cos2 (θ) + 1/4 are not contradictory but are complementary descriptions of two distinct source preparations. By modeling light as a real transverse electromagnetic wave, we identify the initial phase orientation of the wavefront as a local hidden variable. Stochastic (randomly prepared) photon sources produce four equally likely phase combinations, yielding the full Malus' Law distribution. Correlated sources, however, are physically constrained to produce only the synchronized subset of these combinations, yielding the cos2 (θ) correlation. We argue that standard Bell test analysis conflates these distinct populations and violates the Fair Sampling Assumption, rendering Bell's inequality inapplicable to the aggregated data. The observed "quantum" correlation is therefore the signature of a correlated source producing pre-synchronized wavefronts, not evidence of non-local communication. We conclude that local realism remains a viable, classical framework for describing these phenomena.
We introduce a local phase-field framework for describing spin entanglement in which measurement correlations arise from an internal scalar phase associated with each fermion. The phase is defined by two underlying real fields and evolves according to local relativistic dynamics. When particle pairs are produced at a common spacetime event, a phase-locking constraint is established at creation, after which the internal phases evolve independently without any nonlocal interaction.Spin measurements performed by Stern–Gerlach analyzers are modeled as local filtering operations that depend only on the internal phase and the analyzer orientation. Using this deterministic local response, we derive the exact quantum correlation function; when inserted into the CHSH expression, it attains the standard Tsirelson bound.The framework preserves locality, parameter independence, and no-signaling, while providing a concrete physical ontology for spin correlations based on internal phase structure. We compare the model with earlier phase-based approaches and outline experimental configurations—such as time-resolved and multi-stage Stern–Gerlach measurements—that could probe the dynamical evolution of the internal phase. The results demonstrate that exact quantum entanglement correlations can emerge from a strictly local phase-field description.Bell’s theorem constrains models in which measurement outcomes are functions of pre-assigned discrete values. The present framework instead employs a continuous internal phase field as the relevant physical variable. Measurement outcomes are deterministic functions of the local phase and analyzer orientation. The model preserves locality, parameter independence, and no-signaling, while allowing outcome dependence, which is permitted within Bell’s framework.
Unlike our basic theories of space and time, quantum mechanics is not a locally causal theory. Moreover, it is widely believed that any hopes of restoring local causality within a realistic theory have been undermined by Bell's theorem and its supporting experiments. By contrast, we provide a strictly local, deterministic, and realistic explanation for the correlations observed in two such supporting experiments, performed independently at Orsay and Innsbruck. To this end, a pair of local variables is constructed to simulate detections of photon polarizations at various angles, chosen freely by Alice and Bob. These generate purely random outcomes, A = +1 or -1 and B = +1 or -1, occurring within a parallelized 3-sphere. These outcomes do not depend on the contexts of measurements, but are determined entirely by the randomly chosen initial orientation of the 3-sphere. When they are compared, however, the correlation between them turn out to be exactly equal to -cos2(alpha - beta), with the corresponding CHSH inequality violated for the polarization angles alpha, alpha', beta, and beta' in precisely the manner predicted by quantum mechanics. The key ingredient in our explanation is the topology of the 3-sphere, which remains closed under multiplication, thus preserving the locality condition of Bell. It allows us to model the physical space as a 3-sphere, and reveals that the illusion of quantum nonlocality in the present case stems from a twist in the Hopf fibration of the 3-sphere.