### Section II: Formulation

In this section, Bell formulates the Einstein-Podolsky-Rosen (EPR) paradox mathematically. He uses the example of two spin-½ particles in a singlet state, moving in opposite directions. According to quantum mechanics, if a measurement of one particle’s spin component along some axis yields , the measurement of the other particle’s spin component along the same axis must yield .

Bell then introduces the hypothesis of *locality*, which means that the result of a measurement on one particle should not be influenced by a distant measurement of the other particle. In other words, if two measurements are made on these separated particles, the outcome of one should not depend on the setting of the other. This leads to the idea that the results of measurements must be predetermined by some additional “hidden” variables denoted by . These hidden variables allow for a more complete specification of the state than quantum mechanics provides.

The results of measurements on the two particles are denoted by and , where and are the directions along which measurements are made, and represents the hidden variables. Bell then writes the product of the measurement results as and considers the expectation value of this product, denoted by , which is expressed as an integral over the probability distribution of the hidden variables.

Bell shows that if locality holds, the expectation value should be equal to the quantum mechanical prediction for the singlet state. However, the crux of Bell’s theorem lies in proving that no hidden variable theory that satisfies locality can reproduce the quantum mechanical predictions.

### Section III: Illustration

Before proving the main result, Bell provides a few examples to clarify the mathematical framework. He first shows that a hidden variable theory can easily explain the measurement outcomes of a single spin-½ particle in a pure state. By introducing a hidden variable that determines the outcome of any spin measurement, Bell demonstrates that the expectation value of the spin measurement along a given direction matches the quantum mechanical prediction.

Next, Bell attempts to construct a hidden variable model for two-particle systems that reproduces some of the essential features of quantum mechanics. However, he finds that while such models can approximate quantum mechanical results for some measurement settings, they fail to fully capture the quantum mechanical correlation functions, particularly for the singlet state. Specifically, the models tend to deviate from the quantum mechanical predictions for certain angles between the measurement settings.

In the final example, Bell shows that if the measurement results and are allowed to depend on both measurement settings and , it is possible to reproduce the quantum mechanical correlation functions. However, this comes at the cost of violating locality, which contradicts the key requirement that the measurement results at one location should not depend on the measurement setting at a distant location.

In conclusion, Section III illustrates that reproducing the quantum mechanical predictions is possible only by giving up locality, thus paving the way for Bell’s proof in the next section.

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