Using real fields instead of complex ones, it was recently claimed, that all fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, related to Planck’s constant. Quantum mechanics is described with real fields and real operators. Schrodinger and Dirac equations then are solved. The solution to Dirac equation gives four, real, 2-vectors solutions ψ 1 =( U 1 D 1 ) ψ 2 =( U 2 D 2 ) ψ 3 =( U 3 D 3 ) ψ 4 =( U 4 D 4 ) where ( ψ 1 , ψ 4 ) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ψ 2 , ψ 3 ) are coupled via linear combinations to represent spin-up and spin-down anti-fermions. For an incoming entangled pair of fermions, the combined solution is Ψ in = c 1 ψ 1 + c 4 ψ 4 where c 1 and c 4 are some hidden variables. By applying a magnetic field in +Z and +x the theoretical results of a triple Stern-Gerlach experiment are predicted correctly. Then, by repeating Bell’s and Mermin Gedanken experiment with three magnetic filters σ θ , at three different inclination angles θ , the violation of Bell’s inequality is proven. It is shown that all fermions are in a mixed state of spins and the ratio between spin-up to spin-down depends on the hidden variables.
Using real fields instead of complex ones, it was recently claimed, that all fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, related to Planck’s constant. Quantum mechanics is described with real fields and real operators. Schrodinger and Dirac equations then are solved. The solution to Dirac equation gives four, real, 2-vectors solutions ψ 1 =( U 1 D 1 ) ψ 2 =( U 2 D 2 ) ψ 3 =( U 3 D 3 ) ψ 4 =( U 4 D 4 ) where ( ψ 1 , ψ 4 ) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ψ 2 , ψ 3 ) are coupled via linear combinations to represent spin-up and spin-down anti-fermions. For an incoming entangled pair of fermions, the combined solution is Ψ in = c 1 ψ 1 + c 4 ψ 4 where c 1 and c 4 are some hidden variables. By applying a magnetic field in +Z and +x the theoretical results of a triple Stern-Gerlach experiment are predicted correctly. Then, by repeating Bell’s and Mermin Gedanken experiment with three magnetic filters σ θ , at three different inclination angles θ , the violation of Bell’s inequality is proven. It is shown that all fermions are in a mixed state of spins and the ratio between spin-up to spin-down depends on the hidden variables.
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 book proposes a fundamentally new framework for quantum theory rooted in real physical fields defined in Minkowski space, instead of the conventional Hilbert space formalism. It introduces a physically realistic model of fermions, built from coupled strings that explain spin, interference, entanglement, and the Planck constant, all without imaginary numbers or wavefunctions. The approach resolves longstanding paradoxes like Bell’s inequality violation and Einstein-Podolsky-Rosen correlations through a deterministic model with hidden variables. The book guides the reader from quantum basics to experimental implications.
Entanglement is studied in the framework of Dyson's S-matrix theory in relativistic quantum field theory, which leads to a natural definition of entangled states of a particle-antiparticle pair and the spin operator from a Noether current. As an explicit example, the decay of a massive pseudo-scalar particle into a pair of electron and positron is analyzed. Two spin operators are extracted from the Noether current. The Wigner spin operator characterizes spin states at the rest frame of each fermion and, although not measurable in the laboratory, gives rise to a straightforward generalization of low-energy analysis of entanglement to the ultrarelativistic domain. In contrast, if one adopts a (modified) Dirac spin operator, the entanglement measured by spin correlation becomes maximal near the threshold of the decay, while the entanglement is replaced by the classical correlation for the ultrarelativistic electron-positron pair by analogy to the case of neutrinos, for which a hidden-variables type of description is possible. Chiral symmetry differentiates the spin angular momentum and the magnetic moment. The use of weak interaction that can measure helicity is suggested in the analysis of entanglement at high energies instead of a Stern-Gerlach apparatus, which is useless for the electron. A difference between the electron spin at high energies and the photon linear polarization is also noted. The Standard Model can describe all of the observable properties of leptons.
This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A, B, and C, with a fixed angle θ between A and B and a range of angles θ' between B and C. For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser-Horne-Shimony-Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M. We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M, whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ'.
The Complete Disenchantment of Quantum Entanglement—Failure of Measurement Independence and a Field-Theoretic Explanation of Electron Internal Structure This paper provides a physical explanation of quantum entanglement—the most central puzzle in quantum mechanics—without relying on "nonlocality." The violation of Bell's inequality is widely interpreted by mainstream physics as "experimental falsification of local realism," from which the conclusion "the world is nonlocal" is drawn. This paper points out that this chain of reasoning depends on a premise that has never been independently tested: the measurement process does not change the hidden variable that determines the measurement outcome. This theory divides all matter in the universe into Nie-matter entities (massive, Staticity) and Ji-matter entities (massless, Dynamicity). The electron is not a point particle, but a Nie-matter entity composed of Zhonghua Micro-Particles, possessing an internal Ji-matter field configuration (the electron self-field). When a measurement device operates, it inevitably releases a Ji-matter field; this field undergoes local coupling with the electron self-field, changing the electron's effective configuration from the original configuration λ₀ at production to the coupled effective configuration λ_eff(a) = C[λ₀, F_device(a)]. The measurement outcome is a function of λ_eff(a), not a function of λ₀ alone—therefore, the measurement-independence premise of Bell's theorem does not hold physically. This paper further argues: the electron's having internal structure is precisely the physical origin of the angle-dependent correlations. If the electron were an unstructured point particle, measurements at different angles should give the same result—but experiments observe angle-dependent correlations, which in itself already proves that the electron is not a point particle. Bell-inequality experiments have never been a "victory of nonlocality"—they are ironclad evidence that the electron has internal structure. This paper gives four core arguments: P6-4C (failure of the premises of Bell's inequality derivation), P6-4D (change of electron state by the measurement device's Ji-matter field), P6-5 (how a point particle absorbs a photon—the electron must have internal structure to accommodate an absorbed photon), P6-4A (Bell's inequality—ironclad evidence that the electron has internal structure), and P6-4B (local mechanism of Bell experiments and experimental test paths for electron internal structure). These include an order-of-magnitude estimate of the electron internal structure correction to g-2 (~10⁻¹⁸), a comparison with current experimental precision (~10⁻¹²), and future test paths. The full text does not claim that Bell's inequality is "overturned"—it is a mathematically correct theorem. What this paper claims is: the measurement-independence premise on which Bell's inequality depends does not hold under the physical picture in which measurement is local field coupling. Violating Bell's inequality is precisely evidence that the electron has an internal Ji-matter field configuration, not evidence for nonlocality.
The Stern–Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern–Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles θ between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2±μB. The values ±μB thus do not correspond to the continuum of values −μ·B Einstein and Ehrenfest had conjectured. The energy term V=−μ·B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.