Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2108.07869

Geometric model for the electron spin correlation

Ana Marı́a Cetto

Published Aug 17, 2021• 0 citations
arXivHTMLPDF

Abstract

The quantum formula for the spin correlation of the bipartite singlet spin state, $C_{Q}(\boldsymbol{a},\boldsymbol{b})$, is derived on the basis of a probability distribution $ρ(ϕ)$ that is generic, i. e., independent of $(\boldsymbol{a},\boldsymbol{b})$. In line with a previous result obtained within the framework of the quantum formalism, the probability space is partitioned according to the sign of the product $A=αβ$ of the individual spin projections $α$ and $β$ onto $\boldsymbol{a}$ and $\boldsymbol{b}$. A specific partitioning and a corresponding set of realizations {$ ϕ$} are associated with every measurement setting $(\boldsymbol{a},\boldsymbol{b})$; this precludes the transfer of $α$ or $β$ from $C_{Q}(\boldsymbol{a},\boldsymbol{b})$ to $C_{Q}(\boldsymbol{a},\boldsymbol{b'})$, for $\boldsymbol{b'}\neq\boldsymbol{b}.$ A geometric model that reproduces the spin correlation serves to validate our approach, giving a concrete meaning to the quantum result in terms of a (local random variable) probability distribution.

DOI
10.48550/arxiv.2108.07869

Related papers

Physical model for the electron spin correlation

Ana Marı́a Cetto

The quantum formula for the spin correlation of the bipartite singlet spin state, $C_{Q}(\boldsymbol{a},\boldsymbol{b})$, is derived on the basis of a probability distribution $\rho(\phi)$ that is generic, i. e., independent of $(\boldsymbol{a},\boldsymbol{b})$. In line with a previous result obtained within the framework of the quantum formalism, the probability space is partitioned according to the sign of the product $A=\alpha\beta$ of the individual spin projections $\alpha$ and $\beta$ onto $\boldsymbol{a}$ and $\boldsymbol{b}$; this precludes the transfer of $\alpha$ or $\beta$ from $C_{Q}(\boldsymbol{a},\boldsymbol{b})$ to $C_{Q}(\boldsymbol{a},\boldsymbol{b'})$, for $\boldsymbol{b'}\neq\boldsymbol{b}.$ A specific physical model that reproduces the quantum spin correlation serves to validate our approach.

•Aug 17, 2021•0 citations

Electron Spin Correlations: Probabilistic Description and Geometric Representation

Ana Marı́a Cetto

The electron spin correlation is shown to be expressible in terms of a bona fide probability distribution function with an associated geometric representation. With this aim, an analysis is presented of the probabilistic features of the spin correlation within the quantum formalism, which helps clarify the concepts of contextuality and measurement dependence. The dependence of the spin correlation on conditional probabilities allows for a clear separation between system state and measurement context; the latter determines how the probability space should be partitioned in calculating the correlation. A probability distribution function ρ(ϕ) is then proposed, which reproduces the quantum correlation for a pair of single-particle spin projections and is amenable to a simple geometric representation that gives meaning to the variable ϕ. The same procedure is shown to be applicable to the bipartite system in the singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical picture of the electron spin, as discussed at the end of the paper.

Entropy•Oct 9, 2022•9 citations

Electron spin correlations: a geometric representation

Ana Marı́a Cetto

An analysis is made within the quantum formalism of the probabilistic features of the electron spin correlation, with the purpose of clarifying the concepts of contextuality and measurement dependence. The quantum formulas for the spin correlation are then derived on the basis of a probability distribution function and its associated geometric representation, both for a pair of projections of a single-particle spin and for the bipartite system in singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical image of the electron spin, as discussed at the end of the paper.

arXiv:2207.02352•Jul 5, 2022•0 citations

Beyond Quantum Mechanics: Local QM Space‑Time Algebra Prediction of the Singlet State — Computational Notebooks and Simulations Package

Carl F. Diether

I derive the quantum mechanical prediction $-\mathbf{a}\cdot\mathbf{b}$ for the singlet spin state using local measurement functions within the Space-Time Algebra (STA) framework. I establish a compact and computationally tractable STA representation of the two-particle singlet state, $\Psi = \tfrac{1}{2}(I\sigma_2^{(1)} - I\sigma_2^{(2)})$, which is simpler than previously published forms but is not required for the correlation calculation. The analysis shows that STA naturally generates both scalar dot-product terms ($\mathbf{a}\cdot\mathbf{s}_1$ and $\mathbf{s}_2\cdot\mathbf{b}$) and bivector wedge-product terms ($\mathbf{a}\wedge\mathbf{s}_1$ and $\mathbf{s}_2\wedge\mathbf{b}$) from local spin–detector interactions, whereas standard quantum mechanics retains only the scalar contributions through expectation-value projection. Because the standard quantum formalism represents correlations via operator expectation values, the antisymmetric (cross‑product) contributions are averaged out at the level of observables. In contrast, the STA formulation keeps these bivector terms explicit in the intermediate geometric products, raising the question of whether potentially meaningful geometric structure is being hidden by the quantum averaging procedure. I verify the analytical derivation using computational simulations in Mathematica with a Clifford algebra package, confirming that local measurement functions reproduce the standard quantum correlation $-\cos\theta$, where $\theta$ is the angle between measurement directions $\mathbf{a}$ and $\mathbf{b}$. The correlation $-{\bf a}\cdot {\bf b}$ is the scalar coordinate of 3-sphere rotors.

•Feb 25, 2026•1 citation

ISING MODEL IN A MAGNETIC FIELD: THE QUANTUM FIELD THEORY ApPROACH TO THE CALCULATION OF THE SPIN-SPIN CORRELATION FUNCTION *

Gesualdo Delfino A, Giuseppe Mussardo

•Jan 1, 1997•0 citations

A spinor-geometric representation of quantum correlations with an explicit measurement mapping

Engel Franciscus Wichmann

Next research.•Aug 22, 2026•0 citations
The quantum formula for the spin correlation of the bipartite singlet spin state, CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b), is derived on the basis of a probability distribution ρ(ϕ)ρ(ϕ)ρ(ϕ) that is generic, i. e., independent of (a,b)(\boldsymbol{a},\boldsymbol{b})(a,b). In line with a previous result obtained within the framework of the quantum formalism, the probability space is partitioned according to the sign of the product A=αβA=αβA=αβ of the individual spin projections ααα and βββ onto a\boldsymbol{a}a and b\boldsymbol{b}b. A specific partitioning and a corresponding set of realizations {ϕ ϕϕ} are associated with every measurement setting (a,b)(\boldsymbol{a},\boldsymbol{b})(a,b); this precludes the transfer of ααα or βββ from CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b) to CQ(a,b′)C_{Q}(\boldsymbol{a},\boldsymbol{b'})CQ​(a,b′), for b′≠b.\boldsymbol{b'}\neq\boldsymbol{b}.b′=b. A geometric model that reproduces the spin correlation serves to validate our approach, giving a concrete meaning to the quantum result in terms of a (local random variable) probability distribution.
The quantum formula for the spin correlation of the bipartite singlet spin state, CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b), is derived on the basis of a probability distribution ρ(ϕ)\rho(\phi)ρ(ϕ) that is generic, i. e., independent of (a,b)(\boldsymbol{a},\boldsymbol{b})(a,b). In line with a previous result obtained within the framework of the quantum formalism, the probability space is partitioned according to the sign of the product A=αβA=\alpha\betaA=αβ of the individual spin projections α\alphaα and β\betaβ onto a\boldsymbol{a}a and b\boldsymbol{b}b; this precludes the transfer of α\alphaα or β\betaβ from CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b) to CQ(a,b′)C_{Q}(\boldsymbol{a},\boldsymbol{b'})CQ​(a,b′), for b′≠b.\boldsymbol{b'}\neq\boldsymbol{b}.b′=b. A specific physical model that reproduces the quantum spin correlation serves to validate our approach.
The electron spin correlation is shown to be expressible in terms of a bona fide probability distribution function with an associated geometric representation. With this aim, an analysis is presented of the probabilistic features of the spin correlation within the quantum formalism, which helps clarify the concepts of contextuality and measurement dependence. The dependence of the spin correlation on conditional probabilities allows for a clear separation between system state and measurement context; the latter determines how the probability space should be partitioned in calculating the correlation. A probability distribution function ρ(ϕ) is then proposed, which reproduces the quantum correlation for a pair of single-particle spin projections and is amenable to a simple geometric representation that gives meaning to the variable ϕ. The same procedure is shown to be applicable to the bipartite system in the singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical picture of the electron spin, as discussed at the end of the paper.
An analysis is made within the quantum formalism of the probabilistic features of the electron spin correlation, with the purpose of clarifying the concepts of contextuality and measurement dependence. The quantum formulas for the spin correlation are then derived on the basis of a probability distribution function and its associated geometric representation, both for a pair of projections of a single-particle spin and for the bipartite system in singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical image of the electron spin, as discussed at the end of the paper.
I derive the quantum mechanical prediction −a⋅b-\mathbf{a}\cdot\mathbf{b}−a⋅b for the singlet spin state using local measurement functions within the Space-Time Algebra (STA) framework. I establish a compact and computationally tractable STA representation of the two-particle singlet state, Ψ=12(Iσ2(1)−Iσ2(2))\Psi = \tfrac{1}{2}(I\sigma_2^{(1)} - I\sigma_2^{(2)})Ψ=21​(Iσ2(1)​−Iσ2(2)​), which is simpler than previously published forms but is not required for the correlation calculation. The analysis shows that STA naturally generates both scalar dot-product terms (a⋅s1\mathbf{a}\cdot\mathbf{s}_1a⋅s1​ and s2⋅b\mathbf{s}_2\cdot\mathbf{b}s2​⋅b) and bivector wedge-product terms (a∧s1\mathbf{a}\wedge\mathbf{s}_1a∧s1​ and s2∧b\mathbf{s}_2\wedge\mathbf{b}s2​∧b) from local spin–detector interactions, whereas standard quantum mechanics retains only the scalar contributions through expectation-value projection. Because the standard quantum formalism represents correlations via operator expectation values, the antisymmetric (cross‑product) contributions are averaged out at the level of observables. In contrast, the STA formulation keeps these bivector terms explicit in the intermediate geometric products, raising the question of whether potentially meaningful geometric structure is being hidden by the quantum averaging procedure. I verify the analytical derivation using computational simulations in Mathematica with a Clifford algebra package, confirming that local measurement functions reproduce the standard quantum correlation −cos⁡θ-\cos\theta−cosθ, where θ\thetaθ is the angle between measurement directions a\mathbf{a}a and b\mathbf{b}b. The correlation −a⋅b-{\bf a}\cdot {\bf b}−a⋅b is the scalar coordinate of 3-sphere rotors.