Abstract
Spin is a fundamental and distinctive property of the electron, having far-reaching implications. Yet its purely formal treatment often blurs the physical content and meaning of the spin operator and associated observables. In this work we propose to advance in disclosing the meaning behind the formalism, by first recalling some basic facts about the one-particle spin operator. Consistently informed by and in line with the quantum formalism, we then proceed to analyse in detail the spin projection operator correlation function \(C_{Q}(\varvec{a},\varvec{b})=\left\langle \left( \hat{\varvec{\sigma }}\cdot \varvec{a}\right) \left( \hat{\varvec{\sigma }}\cdot \varvec{b}\right) \right\rangle \) for the bipartite singlet state, and show it to be amenable to an unequivocal probabilistic reading. In particular, the calculation of \(C_{Q}(\varvec{a},\varvec{b})\) entails a partitioning of the probability space, which is dependent on the directions \((\varvec{a},\varvec{b}).\) The derivation of the CHSH- or other Bell-type inequalities, on the other hand, does not consider such partitioning. This observation puts into question the applicability of Bell-type inequalities to the bipartite singlet spin state.
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Notes
In trying to get a more geometrical picture of the effect of the operator \(\hat{{\varvec{\sigma }}}\cdot \varvec{a}\) in 3D space, it may help to look at the problem from the perspective of algebraic geometry. Take \(\varvec{s}\) as the spin vector, and \(\varvec{a}\) the unitary vector as before; the product \(\varvec{s}\varvec{a}\) is then defined as a combination of the scalar (or internal) product and the vector (or external) product [3],
$$\begin{aligned} \varvec{s}\varvec{a}=\varvec{s}\cdot \varvec{a}+\mathbf {\text {i }}\varvec{s}\times \varvec{a}=\cos \vartheta _{a}+i\sin \vartheta _{a}=\exp \left( i\vartheta _{a}\right) , \end{aligned}$$where \(\varvec{s}_{1},\varvec{s}_{2},\varvec{s}_{3}\) make up a right-handed set of orthonormal vectors in Euclidean space such that
$$\begin{aligned} \varvec{s}_{i}^{2}=1,\varvec{s}_{i}\cdot \varvec{s}_{j}=0\quad \text {and} \quad \varvec{s}_{i}\cdot \varvec{s}_{j} =\varvec{-s}_{j}\cdot \varvec{s}_{i}\quad \text { if }i\ne j, \end{aligned}$$and
$$\begin{aligned} i=\varvec{s}_{1}\varvec{s}_{2}\varvec{s}_{3}, \end{aligned}$$whence \(i^{2}=-1;\)\(\vartheta _{a}\) is the angle formed by \(\varvec{s}\) and \(\varvec{a}.\) Note that \(\varvec{s}\) in this language is the vector representing the state of the spin, in contrast to \(\varvec{\hat{\sigma }},\) which acts on the state vector in Hilbert space. The product \(\varvec{s}\varvec{a}\) so defined contains the full information about the geometrical relationship between the vectors \(\varvec{s}\) and \(\varvec{a}.\) In the operator formalism, in its turn, Eqs. (10) and (11) together contain the full information about the geometrical relationship between the Bloch vector \(\varvec{r}\) associated with the spin state, and the direction \(\varvec{a}.\) By looking just at the observable, given by (10), one loses valuable information contained in (11).
Let us briefly look at the spin product for the bipartite singlet state from the perspective of algebraic geometry. If \(\varvec{s}\) and \(\varvec{s}'\) are two antiparallel vectors,
$$\begin{aligned} \varvec{s}\varvec{a}=\exp (i\vartheta _{a}),\quad \varvec{s}'\varvec{b}=-\varvec{s}\varvec{b}=-\exp (i\vartheta _{b}), \end{aligned}$$their product takes the form [3]
$$\begin{aligned} \left( \varvec{s}\varvec{a}\right) ^{*}\left( \varvec{s}'\varvec{b}\right) =-\exp \left( i\theta _{ab}\right) . \end{aligned}$$Products containing \(\varvec{s}\) and \(\varvec{s}'\) depend of course in general on both vectors. However, in this case the result does not depend on \(\varvec{s}\) and \(\varvec{s}'\) because \(\varvec{s}+\varvec{s}'=0,\) as corresponds to the (spherically symmetric) singlet state.
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Acknowledgements
The authors acknowledge support from DGAPA-UNAM through Project PAPIIT IA101918. Valuable comments from two referees are also acknowledged.
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Cetto, A.M., Valdés-Hernández, A. & de la Peña, L. On the Spin Projection Operator and the Probabilistic Meaning of the Bipartite Correlation Function. Found Phys 50, 27–39 (2020). https://doi.org/10.1007/s10701-019-00313-8
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DOI: https://doi.org/10.1007/s10701-019-00313-8