# On the spin projection operator and the probabilistic meaning of the bipartite correlation function

## TL;DR

This paper explores the probabilistic interpretation of the bipartite spin correlation function, revealing that Bell inequalities may not be applicable due to the underlying probability space partitioning dependent on measurement directions.

## Contribution

It provides a detailed analysis of the spin projection operator correlation function and challenges the conventional application of Bell inequalities to the singlet state.

## Key findings

- Correlation function admits an unequivocal probabilistic interpretation
- Partitioning of probability space depends on measurement directions
- Questions the applicability of Bell inequalities to the singlet state

## Abstract

Spin is a fundamental and distinctive property of the electron, having far-reaching consequences in wide areas of physics. Yet, further to its association with an angular momentum, the physics underpinning its formal treatment remains obscure. 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 $C_Q(\boldsymbol{a}, \boldsymbol{b}) = \langle(\hat{\sigma}\cdot\boldsymbol{a})(\hat{\sigma}\cdot\boldsymbol{b})\rangle$ for the bipartite singlet state, and show it to be amenable to an unequivocal probabilistic reading. In particular, the calculation of $C_Q(\boldsymbol{a}, \boldsymbol{b})$ entails a partitioning of the probability space, which is dependent on the directions $(\boldsymbol{a}, \boldsymbol{b})$. 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.