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arXiv:2502.01696 · Journal of Quantum Information Science

Local Quantum Mechanical Prediction of the Singlet State Using Geometric Algebra

Carl F. Diether

Published Jan 1, 2025• 1 citation
arXivHTMLPDF

Abstract

We deduce the quantum mechanical prediction of −a⋅b for the singlet spin state employing local measurement functions following Bell’s approach. This result represents the quantum mechanical expectation value for the joint measurement of spin projections in the singlet state. And is equal to the negative cosine of the angle between vectors a and b . Our derivation is corroborated through a computational simulation conducted in the Mathematica programming environment using geometric algebra.

DOI
10.4236/jqis.2025.154010

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We deduce the quantum mechanical prediction of −a⋅b for the singlet spin state employing local measurement functions following Bell’s approach. This result represents the quantum mechanical expectation value for the joint measurement of spin projections in the singlet state. And is equal to the negative cosine of the angle between vectors a and b . Our derivation is corroborated through a computational simulation conducted in the Mathematica programming environment using geometric algebra.
I derive the quantum mechanical prediction −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= 1 2 ( I σ 2 ( 1 ) −I σ 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 ( a· s 1 ) and ( s 2 ·b ) and bivector wedge-product terms ( a∧ s 1 ) and ( s 2 ∧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 −cosθ , where θ is the angle between measurement directions a and b . The correlation −a·b is the scalar coordinate of 3-sphere rotors.
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.
Abstract. In conformity with our previous work on the measurement process in quantum mechanics in terms of first passage random walks in Hilbert space through interactions with the measuring devices, we here consider the problem of measurement of entangled states at spatially separated measuring devices. We find that Bell’s inequality need not be an obstacle if the set of hidden variables include local ones, and we present a simple model that includes specific probability distributions of the source and local hidden variables, giving the quantum mechanical result for the expectation value of the spin-spin correlation for arbitrary orientations of the frames of measurement, which is known to violate Bell’s relation for chosen angles of orientation. Local measurements and entangled states 2 1.
Abstract A new representation for spin 1/2 in the even 3D subalgebra of the spacetime algebra (STA) combines in a single geometric object the roles of the standard Pauli spin vector and spin state. It is a vector quantity comprising a gauge phase. In the one-particle case the representation (1) is Hermitian; (2) chiral; (3) reproduces all standard expectation values, including the total one-particle spin modulus ; (4) constrains a spinor basis representing opposite spins to preserve handiness (chirality); (5) the gauge phase allows to explicitly formalize irreversibility in spin measurement. In the two-particle case it (1) identifies entangled spin pairs as having opposite handiness and precise gauge phase relations; (2) doubles the dimensionality of the spin space due to variation of handiness; (3) the four maximally entangled states are naturally derived by pairing spins that are reflections (triplets) and inversions (singlet) of each-other. The cross-product terms in the expression for the squared total spin of two particles can be affected by experiment and they yield the standard expectation values after measurement. Here I directly define and transform spin in 3D orientation space, without invoking concepts like abstract Hilbert space and tensor product as in the standard formulation. The STA formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.
Abstract Entangled quantum systems can connect to the environment by means of a Bell state measurement. This is true for instance for teleportation and entanglement swapping. While the results are well understood it is not quite clear if they involve nonlocal action or if they are determined in advance. Models based on the fact that the partners of an entangled pair have the same value of a statistical parameter do not apply here. Therefore, in this work a model is presented which reproduces the quantum mechanical predictions for expectation values with spin measurements, but is not based on shared statistical parameters. The coupling of the entangled particles is instead based on the conservation of the spin angular momentum. The model refutes Bell's theorem and explains teleportation and entanglement swapping in a local manner as well. Multilevel entanglements can also be explained locally by the model. The manuscript is thus a step forward towards a complete theory describing quantum physical reality as thought possible by Einstein, Podolsky and Rosen.
Given a sequence of pairs (pi{p}_{i}pi​, p‾i{\overline{p}}_{i}p​i​) of spin-1/21/21/2 particles in the singlet state, assume that Alice measures the normalized projections ai{a}_{i}ai​ of the spins of the pi{p}_{i}pi​'s along vector $\mathbit{a}$ while Bob measures the normalized projections bi{b}_{i}bi​ of the spins of the p‾i{\overline{p}}_{i}p​i​'s along vector $\mathbit{b}$. Then quantum mechanics (QM) lets one evaluate the correlation ⟨a,b⟩⟨a,b⟩⟨a,b⟩ as $\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{\mathbit{a}}\ensuremath{-}{\ensuremath{\theta}}_{\mathbit{b}})$ where ${\ensuremath{\theta}}_{\mathbit{v}}$ is the angle between the vector $\mathbit{v}$ and a reference vector chosen once and for all and in a fixed plane. Assuming classical microscopic realism (CMR) there exist also normalized projection pairs (${a}_{i}^{\ensuremath{'}}$, ${b}_{i}^{\ensuremath{'}}$) of the spins of the pairs (pi{p}_{i}pi​, p‾i{\overline{p}}_{i}p​i​) along (${\mathbit{a}}^{\ensuremath{'}}$, ${\mathbit{b}}^{\ensuremath{'}}$) so that $⟨{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}⟩=\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{{\mathbit{a}}^{\ensuremath{'}}}\ensuremath{-}{\ensuremath{\theta}}_{{\mathbit{b}}^{\ensuremath{'}}})$. Since all projections are in ${\ensuremath{-}1,1}$, $|⟨c,d⟩+⟨c,e⟩|+|⟨f,d⟩\ensuremath{-}⟨f,e⟩|\ensuremath{\le}2$ for ccc, ddd, eee, and fff in ${a,b,{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}}$. Assuming locality (the impossibility of any effect of an event on another event when said events are spatially separated) beside QM and CMR, Bell's theory lets one deduce various violations of this inequality at some choices of quadruplets $Q\ensuremath{\equiv}(\mathbit{a},\mathbit{b},{\mathbit{a}}^{\ensuremath{'}},{\mathbit{b}}^{\ensuremath{'}})$. Our main result is the existence of such QQQ's where at least one of the above inequalities is violated if one only assumes QM, CMR, and some very mild further hypotheses that only concern the behavior of correlations that appear in these inequalities near special QQQ's.