We present a local hidden variable model for bipartite correlations where the hidden variable distribution is derived from first principles rather than postulated. The model posits that particles accumulate a geometric phase from proper time fluctuations in a stochastic spacetime. The causal Lorentzian theory (CLT) [26] ensures that for many independent fluctuations, φ is normally distributed; wrapping to the circle [0, 2π) yields a uniform distribution when the phase variance σ_φ ≫ 2π. Two particles sharing the same fluctuating geometry share the same φ, which serves as a common cause for their measurement outcomes. For measurement functions we compute the correlation function exactly. In the uniform limit, we obtain a triangle wave that respects Bell inequalities. For finite σ_φ, the correlation interpolates between deterministic outcomes and the triangle wave, with the interpolation controlled by particle mass, temperature, and proper time. This mass-dependent signature provides a novel experimental test of geometric hidden variable theories using molecular interferometry and cold-atom experiments. The framework offers a physical origin for hidden variable distributions and connects quantum foundations to stochastic gravity.
We present a local hidden variable model for bipartite correlations where the hidden variable distribution is derived from first principles rather than postulated. The model posits that particles accumulate a geometric phase from proper time fluctuations in a stochastic spacetime. The causal Lorentzian theory (CLT) [26] ensures that for many independent fluctuations, φ is normally distributed; wrapping to the circle [0, 2π) yields a uniform distribution when the phase variance σ_φ ≫ 2π. Two particles sharing the same fluctuating geometry share the same φ, which serves as a common cause for their measurement outcomes. For measurement functions we compute the correlation function exactly. In the uniform limit, we obtain a triangle wave that respects Bell inequalities. For finite σ_φ, the correlation interpolates between deterministic outcomes and the triangle wave, with the interpolation controlled by particle mass, temperature, and proper time. This mass-dependent signature provides a novel experimental test of geometric hidden variable theories using molecular interferometry and cold-atom experiments. The framework offers a physical origin for hidden variable distributions and connects quantum foundations to stochastic gravity.
Quantum entanglement is widely regarded as the most puzzling feature of quantum theory, apparently violating local realism through nonlocal correlations between spacelike-separated particles. Although violations of Bell inequalities are experimentally confirmed, their standard theoretical explanation relies on wavefunction collapse or nonlocal hidden variables. In this work, we present an alternative explanation based on Trembling Spacetime Relativity Theory (TSRT). In this framework, intrinsic metric fluctuations generate proper-time-synchronized geodesic ensembles whose deterministic trajectories give rise to statistical correlations across spacetime. Measurement outcomes are reinterpreted as geometric responses to these fluctuating backgrounds, without invoking quantum collapse or nonlocal influence. We show that the Clauser–Horne–Shimony–Holt (CHSH) inequality can be violated deterministically within a locally causal geometric structure. These violations emerge as macroscopic imprints of shared spacetime curvature, not quantum superposition. The resulting correlation function reproduces quantum predictions at short ranges but decays with separation—yielding testable deviations in long-baseline and gravitationally modulated experiments. TSRT thus offers a realist, Lorentz-invariant, and falsifiable account of entanglement grounded in the causal structure of spacetime.
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.
We develop a complete geometric framework in which each quantum particle possesses its own private spacetime—a world-block—constructed from Fermi–Walker coordinates. The intrinsic spatial metric on each proper-time slice is a dynamical field governed by an action with a universal stiffness constant A0. A single world-block exhibits kinematic non-locality: the metric perturbation at a point depends on the entire wavefunction through an integral relation, while maintaining dynamic locality via causal wave equations. This duality captures the essential non-locality of quantum mechanics without violating relativistic causality. When two particles interact, their world-blocks are stitched along a common boundary, forming a single compound world-block in ordinary four-dimensional spacetime. The stitching imposes local matching conditions that yield a non-separable strain field, providing a geometric account of quantum entanglement. Measurements correspond to fixing boundary conditions on one part of the compound block; the correlated outcome on the distant part is automatically determined by the shared global geometry, not by any superluminal signal. Thus, the apparent non-locality of EPR correlations is explained as a manifestation of geometric connectivity within a single 4D manifold, consistent with Bell’s theorem because the geometry itself is non-local. In the continuum limit of many overlapping blocks, coarse-graining restores an effective local description, and Newton’s law of universal gravitation emerges exactly, with Newton’s constant given by G = 3c4/(8πA0). The model offers a unified, deterministic, and fully relativistic foundation for quantum mechanics and gravity, without invoking extra dimensions or stochastic elements. Experimental signatures in ultrafast interferometry and possible connections to dark energy are discussed. The framework aligns with recent developments in emergent gravity and provides a concrete geometric realization of spacetime from quantum entanglement.
There are two issues concerning hidden variables in quantum mechanics: 1.Whether the outcomes of quantum measurements are hidden variables has long been debated. Experiments violating Bell's inequalities have essentially refuted local hidden variable theories. 2.Quantum entanglement provides nonlocal effects that alter wave functions. However, these changes cannot transmit signals, making the exact time of such nonlocal effects a hidden variable. Theories about hidden variables are difficult to verify experimentally due to their unobservability. This paper connects these two hidden variable theories in quantum mechanics, proposing the spacetime relationship that nonlocal effects from entanglement must satisfy to prevent quantum measurement outcomes from being local hidden variables. It further outlines the spacetime relationship that any causal effects should adhere to. Based on this, the paper demonstrates the universal temporal order between relativistic spacelike events. On this foundation, it integrates the locally deterministic relativistic spacetime structure with the nonlocal phenomena of probabilistic quantum mechanics, thereby establishing the universal ordered spacetime structure. key words: ordered spacetime structure ; local hidden variable ; nonlocal effects; uncertainty principle;
There are two issues concerning hidden variables in quantum mechanics: 1.Whether the outcomes of quantum measurements are hidden variables has long been debated. Experiments violating Bell's inequalities have essentially refuted local hidden variable theories. 2.Quantum entanglement provides nonlocal effects that alter wave functions. However, these changes cannot transmit signals, making the exact time of such nonlocal effects a hidden variable. Theories about hidden variables are difficult to verify experimentally due to their unobservability. This paper connects these two hidden variable theories in quantum mechanics, proposing the spacetime relationship that nonlocal effects from entanglement must satisfy to prevent quantum measurement outcomes from being local hidden variables. It further outlines the spacetime relationship that any causal effects should adhere to. Based on this, the paper demonstrates the universal temporal order between relativistic spacelike events. On this foundation, it integrates the locally deterministic relativistic spacetime structure with the nonlocal phenomena of probabilistic quantum mechanics, thereby establishing the universal ordered spacetime structure.
Bell inequalities have long served as the standard criterion to verify local hidden variable theories. Experimental violations of these inequalities are commonly interpreted as fundamental quantum nonlocality at the microscopic scale. This paper proves rigorously that Bell inequalities hold only under three prerequisites: countable discrete variables, countably additive probability measures, and classical absolute simultaneity. The state space of entangled quantum systems forms an uncountable infinite continuum. By the property of Vitali non-measurable sets in measure theory, the integral deduction of the CHSH inequality collapses naturally over continuous state spaces. Furthermore, special relativity excludes universal global time synchronization, so the absolute time embedded in Bell’s theory is an artificial assumption inconsistent with physical reality. Inspired by the framework dependence of probability revealed in Bertrand’s paradox, we demonstrate that the excess correlation observed in experiments is an apparent effect caused by the mismatch between classical measurement frameworks and the intrinsic continuum property of quantum systems, rather than evidence against local hidden variables. Statistical correlation between entangled particles originates from their shared evolutionary sequence defined at pair generation, with no superluminal interaction, fully complying with relativistic causality. This paper does not explore the ultimate origin of entanglement, but treats it as a given primitive physical law. Two falsifiable predictions are proposed to distinguish the continuum model from standard quantum mechanics. The conclusion is that hidden variable theories have never been falsified by experiments; they only need to be extended from discrete forms to continuum models compatible with relativity.