We revisit the Bell--CHSH scenario for two spin-$\tfrac{1}{2}$ particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-$\tfrac{1}{2}$ is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-$\tfrac{1}{2}$ particles, with the singlet state yielding the usual correlation $E(a,b) = -\,a\cdot b$ and Tsirelson's bound $2\sqrt{2}$. On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as $\{\pm1\}$-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra $\mathcal{C}(Λ)$. We show that there is no representation of the spinor algebra of spin-$\tfrac{1}{2}$ (with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the $\{\pm1\}$ spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C$^*$-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.

quant-pharXiv:2512.11476Dec 15, 20250 citations

An Einstein manuscript on the EPR paradox for spin observables

Tilman Sauer

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsApr 30, 200762 citations

arXiv:1504.04227 · Foundations of Physics

Replacing the Singlet Spinor of the EPR-B Experiment in the Configuration Space with Two Single-Particle Spinors in Physical Space

Michel Gondran, Alexandre Gondran

Published Apr 27, 2016• 4 citations

arXiv HTML PDF

DOI: 10.1007/s10701-016-0011-1

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The time-dependent quantum Monte Carlo method represents a many-electron state by an ensemble of replicas, in which each electron is described by a cloud of walkers which samples its density in physical space, one walker and one guide wave per replica, and it replaces the Hartree potential by a conditional interaction built from the walker positions. The sampling is in physical space rather than in configuration space, which keeps the cost polynomial. Here we derive that conditional interaction. Treating the walker as a localization of its electron to a finite resolution, we show that Bayes' theorem and a single empirical substitution yield the Nadaraya-Watson form which the method has used heuristically. The nonlocality length then becomes the width of the conditioning, and the pair and mean-field limits follow from one construction. Because the conditioning removes the gauge freedom which eliminates the orthonormality multipliers in Hartree-Fock, we enforce orthonormality by a term derived from the constraint. We generalize the formulation to spinors, where the Pauli suppression at coincidence becomes graded by the local spin alignment, and it reduces through the collinear and mean-field limits to the known two-particle spin equations. For two electrons in a double quantum dot in an inhomogeneous magnetic field, compared with the exact solution, the spinor equations reproduce the non-collinear spin geometry which collinear calculations cannot represent and recover the larger part of the entanglement missing from a single determinant. In the crossover between the exchange and the Zeeman regimes, where the singlet-triplet coherence sets the spin geometry, the replicas must carry that coherence themselves, as in the spin-coupled pair of valence-bond theory.The time-dependent quantum Monte Carlo method represents a many-electron state by an ensemble of replicas, in which each electron is described by a cloud of walkers which samples its density in physical space, one walker and one guide wave per replica, and it replaces the Hartree potential by a conditional interaction built from the walker positions. The sampling is in physical space rather than in configuration space, which keeps the cost polynomial. Here we derive that conditional interaction. Treating the walker as a localization of its electron to a finite resolution, we show that Bayes' theorem and a single empirical substitution yield the Nadaraya-Watson form which the method has used heuristically. The nonlocality length then becomes the width of the conditioning, and the pair and mean-field limits follow from one construction. Because the conditioning removes the gauge freedom which eliminates the orthonormality multipliers in Hartree-Fock, we enforce orthonormality by a term derived from the constraint. We generalize the formulation to spinors, where the Pauli suppression at coincidence becomes graded by the local spin alignment, and it reduces through the collinear and mean-field limits to the known two-particle spin equations. For two electrons in a double quantum dot in an inhomogeneous magnetic field, compared with the exact solution, the spinor equations reproduce the non-collinear spin geometry which collinear calculations cannot represent and recover the larger part of the entanglement missing from a single determinant. In the crossover between the exchange and the Zeeman regimes, where the singlet-triplet coherence sets the spin geometry, the replicas must carry that coherence themselves, as in the spin-coupled pair of valence-bond theory.

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Séparer l’inséparable

Michel Gondran, Alexandre Gondran

Nous montrons qu'il est possible de remplacer la fonction d'onde de deux particules intriquées de l'expérience EPR-B, la version de Bohm de l'expérience d'Einstein-Podolsky-Rosen, par deux fonctions d'ondes de particule unique. Pour réaliser cette décomposition, nous utilisons une solution explicite de la fonction d'onde de l'expérience EPR-B, correspondant à une intégration complète en espace et en temps de l'équation de Pauli à deux corps. La nouvelle interprétation causale de l'expérience EPR-B que nous proposons [1] corrige les défauts de l'interprétation précédente [2] et permet la décomposition.[1] A. et M. Gondran: A new causal interpretation of the EPR-B Experiment, AIP Conf. Proc. 1508, 370 (2012), quant-ph/0806.4577v2.A. et M. Gondran: Replacing the singlet spinor of the EPR-B experiment in the configuration space with two single-particle spinors in physical space, à paraitre dans Foundations of Physics, 2016.[2] C. Dewdey, P.R. Holland and A. Kyprianidis : A causal account of non-local Einstein-Podolsky-Rosen spin correlations, J.Phys. A: Math.Gen.20, 4712-32 (1987).Nous montrons qu'il est possible de remplacer la fonction d'onde de deux particules intriquées de l'expérience EPR-B, la version de Bohm de l'expérience d'Einstein-Podolsky-Rosen, par deux fonctions d'ondes de particule unique. Pour réaliser cette décomposition, nous utilisons une solution explicite de la fonction d'onde de l'expérience EPR-B, correspondant à une intégration complète en espace et en temps de l'équation de Pauli à deux corps. La nouvelle interprétation causale de l'expérience EPR-B que nous proposons [1] corrige les défauts de l'interprétation précédente [2] et permet la décomposition.[1] A. et M. Gondran: A new causal interpretation of the EPR-B Experiment, AIP Conf. Proc. 1508, 370 (2012), quant-ph/0806.4577v2.A. et M. Gondran: Replacing the singlet spinor of the EPR-B experiment in the configuration space with two single-particle spinors in physical space, à paraitre dans Foundations of Physics, 2016.[2] C. Dewdey, P.R. Holland and A. Kyprianidis : A causal account of non-local Einstein-Podolsky-Rosen spin correlations, J.Phys. A: Math.Gen.20, 4712-32 (1987).

Feb 11, 20160 citations

Spin Hamiltonian for two singlet states

J. Shaham-Bronstein

Physics Letters AOct 1, 19690 citations

Bell, Spinors, and the Impossibility of a Classical Spin-Vector Model

G. A. Koroteev

We revisit the Bell--CHSH scenario for two spin- $\tfrac{1}{2}$ particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin- $\tfrac{1}{2}$ is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin- $\tfrac{1}{2}$ particles, with the singlet state yielding the usual correlation $E(a,b) = -\,a\cdot b$ and Tsirelson's bound $2\sqrt{2}$ . On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as $\{\pm1\}$ -valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra $\mathcal{C}(Λ)$ . We show that there is no representation of the spinor algebra of spin- $\tfrac{1}{2}$ (with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the $\{\pm1\}$ spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C $^*$ -algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.We revisit the Bell--CHSH scenario for two spin-$\tfrac{1}{2}$ particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-$\tfrac{1}{2}$ is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-$\tfrac{1}{2}$ particles, with the singlet state yielding the usual correlation $E(a,b) = -\,a\cdot b$ and Tsirelson's bound $2\sqrt{2}$. On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as $\{\pm1\}$-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra $\mathcal{C}(Λ)$. We show that there is no representation of the spinor algebra of spin-$\tfrac{1}{2}$ (with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the $\{\pm1\}$ spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C$^*$-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.

quant-pharXiv:2512.11476Dec 15, 20250 citations

An Einstein manuscript on the EPR paradox for spin observables

Tilman Sauer

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsApr 30, 200762 citations

We revisit the Bell--CHSH scenario for two spin-

\tfrac{1}{2}

particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-

\tfrac{1}{2}

is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-

\tfrac{1}{2}

particles, with the singlet state yielding the usual correlation

E(a,b) = -\,a\cdot b

and Tsirelson's bound

2\sqrt{2}

. On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as

\{\pm1\}

-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra

\mathcal{C}(Λ)

. We show that there is no representation of the spinor algebra of spin-

\tfrac{1}{2}

(with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the

\{\pm1\}

spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C

^*

-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.