On the Spin Projection Operator and the Probabilistic Meaning of the Bipartite Correlation Function

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 ( a , b ) = σ ^ · a σ ^ · b for the bipartite singlet state, and show it to be amenable to an unequivocal probabilistic reading. In particular, the calculation of C Q ( a , b ) entails a partitioning of the probability space, which is dependent on the directions ( a , 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.