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.
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.
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.
We derive the Born rule as a consequence of two physical processes: environment-induced decoherence, which selects the pointer-basis decomposition of the density matrix, and a contraction map on projective Hilbert space, which assigns each pointer state to a definite outcome basin. The contraction map partitions projective space into basins of attraction whose boundaries are determined by the measurement projectors. We prove that the decoherence-selected decomposition of the reduced density matrix, combined with the basin assignment, reproduces the Born probabilities p_i = ⟨ψ|Π_i|ψ⟩ exactly for projective measurements in any finite dimension, including dimension 2 where Gleason's theorem does not apply. For generalized measurements (POVMs), the result extends via Naimark dilation. The derivation resolves the "probability problem" of decoherence-based approaches: standard decoherence produces the correct diagonal density matrix but leaves open why diagonal entries should be interpreted as outcome probabilities. Our answer is geometric — each diagonal component corresponds to a pointer state lying deep inside a definite basin of the contraction map, and the mixture weights become outcome probabilities through the basin assignment. The derivation is geometric rather than decision-theoretic, does not assume the probability interpretation of the density matrix, requires no ensemble averaging, and applies to single systems. Perturbations to the contraction map produce bounded corrections to the Born probabilities, providing a falsifiable prediction: any measured deviation from Born statistics constrains the perturbation strength. This work connects quantum measurement theory to the geometry of projective Hilbert space and contraction dynamics, offering a new perspective on the origin of quantum probabilities that complements existing approaches by Gleason, Zurek, and Deutsch-Wallace.
We derive the Born rule as a consequence of two physical processes: environment-induced decoherence, which selects the pointer-basis decomposition of the density matrix, and a contraction map on projective Hilbert space, which assigns each pointer state to a definite outcome basin. The contraction map partitions projective space into basins of attraction whose boundaries are determined by the measurement projectors. We prove that the decoherence-selected decomposition of the reduced density matrix, combined with the basin assignment, reproduces the Born probabilities p_i = ⟨ψ|Π_i|ψ⟩ exactly for projective measurements in any finite dimension, including dimension 2 where Gleason's theorem does not apply. For generalized measurements (POVMs), the result extends via Naimark dilation. The derivation resolves the "probability problem" of decoherence-based approaches: standard decoherence produces the correct diagonal density matrix but leaves open why diagonal entries should be interpreted as outcome probabilities. Our answer is geometric — each diagonal component corresponds to a pointer state lying deep inside a definite basin of the contraction map, and the mixture weights become outcome probabilities through the basin assignment. The derivation is geometric rather than decision-theoretic, does not assume the probability interpretation of the density matrix, requires no ensemble averaging, and applies to single systems. Perturbations to the contraction map produce bounded corrections to the Born probabilities, providing a falsifiable prediction: any measured deviation from Born statistics constrains the perturbation strength. This work connects quantum measurement theory to the geometry of projective Hilbert space and contraction dynamics, offering a new perspective on the origin of quantum probabilities that complements existing approaches by Gleason, Zurek, and Deutsch-Wallace.
We derive the Born rule as a consequence of two physical processes: environment-induced decoherence, which selects the pointer-basis decomposition of the density matrix, and a contraction map on projective Hilbert space, which assigns each pointer state to a definite outcome basin. The contraction map partitions projective space into basins of attraction whose boundaries are determined by the measurement projectors. We prove that the decoherence-selected decomposition of the reduced density matrix, combined with the basin assignment, reproduces the Born probabilities p_i = ⟨ψ|Π_i|ψ⟩ exactly for projective measurements in any finite dimension, including dimension 2 where Gleason's theorem does not apply. For generalized measurements (POVMs), the result extends via Naimark dilation. The derivation resolves the "probability problem" of decoherence-based approaches: standard decoherence produces the correct diagonal density matrix but leaves open why diagonal entries should be interpreted as outcome probabilities. Our answer is geometric — each diagonal component corresponds to a pointer state lying deep inside a definite basin of the contraction map, and the mixture weights become outcome probabilities through the basin assignment. The derivation is geometric rather than decision-theoretic, does not assume the probability interpretation of the density matrix, requires no ensemble averaging, and applies to single systems. Perturbations to the contraction map produce bounded corrections to the Born probabilities, providing a falsifiable prediction: any measured deviation from Born statistics constrains the perturbation strength. This work connects quantum measurement theory to the geometry of projective Hilbert space and contraction dynamics, offering a new perspective on the origin of quantum probabilities that complements existing approaches by Gleason, Zurek, and Deutsch-Wallace.
This paper presents a deterministic formulation of the Born rule and Bell correlations based on conserved analytic flux. The framework combines objective realism, Einstein propagation locality, and emergent measurement within a single dynamical theory. Physical reality is identified with a continuous field and the conserved quantities it transports. All influences propagate locally, while discrete detector outcomes emerge through local accumulation and detection of conserved flux. A bipartite state is represented by a relational coefficient matrix connecting local analytic modes. Independent local analyzer transformations redistribute the conserved relational structure among outgoing pair channels. For the antisymmetric two-mode state, the resulting quadratic pair-channel weights depend only on the relative analyzer angle, while local marginals remain independent of the remote setting. A quadratic pair-flux decomposition theorem establishes that, under lossless local analyzer transformations, orthogonality, and equal current normalization of outgoing channels, the conserved coincidence flux satisfies the quadratic channel weights. A deterministic flux-counting theorem then proves that, when each registered coincidence corresponds to a fixed quantum of conserved flux, the asymptotic coincidence frequencies reproduce the Born rule. The singlet correlation and the Tsirelson bound follow directly from these deterministic statistics. The framework unifies Bell, GHZ, Stern--Gerlach, tunnelling, and interferometric experiments. It eliminates the Heisenberg cut, derives the Pauli exclusion principle from flux absence, and obtains uncertainty relations from Fourier wave localization.
We derive the Bell–CHSH inequality as a conditional theorem inside Charlton Geometry (CG), rather than as an independent axiom about “local realism”. The key CG primitive is non-invertibility, η(ω) = η(−ω), which forces an observational quotient and produces irrecoverable “lost state” carried by polarity and intrinsic phase. In a bipartite experiment, Bell’s bound is recovered exactly when one adds an additional factorization condition asserting that outcomes depend only on local settings and a local lift choice, i.e. that the lost-state fiber decomposes into independent local fibers. Within CG, the natural geometric coupling of two regions through shared lift and holonomy generically violates that factorization while preserving operational no-signaling, so the Bell inequality becomes a theorem about a restricted subclass of quotients rather than a universal constraint. Using the Born brick as an upstream theorem that fixes the unique binary law cos2(θ/2) on the spin-lifted setting space, we recover the standard singlet correlation E(a, b) = −a °§ b and the Tsirelson bound 2√2 as the sharp CG-compatible CHSH maximum. In this framing, Bell violation is not an ontic miracle but the expected signature of a globally shared, non-invertible geometric substate whose reversal cannot be represented