The interpretation of quantum mechanics has been a subject of rich debate since the theory’s infancy. Among the proposed interpretations are (and this is certainly non-exhaustive) the so-called *Copenhagen* interpretation and the *hidden variables* interpretation.

The former interpretation is based on the probabilistic understanding of quantum mechanics. What puts it at odds with our usual intuition is that it is not deterministic. Take the position of a particle (this can be an electron, proton, even an atom, anything sufficiently small) as an example. This is what is known as an *observable*, a physical quantity which can be measured through experiment. According to the Copenhagen interpretation, the most we can know about a particle before a measurement is made is the probability of measuring its position in a certain region of space. This is a clear contradiction with classical determinism, which dictates that we should have some means of predicting where the particle will be when we make a measurement.

The hidden variables theory championed most notably by Einstein, suggests that quantum mechanics is incomplete. An example of this argument is the *Einstein-Podolsky-Rosen (EPR) paradox*, introduced by the three authors in their 1935 paper.

They make the claim that quantum mechanics can only be complete if a physical observable has no well-defined value before it is measured, since if it had one, a complete theory would allow us to predict what this would be. The usual example given which leads to the paradox is the physical observable known as *spin*. The original EPR argument does not deal with spin, but rather is general, it can however be applied to spin. I will avoid a detailed description of what spin is and focus on the fundamental fact that if the spin of an electron is measured in some direction, only two outcomes are possible. These two possibilities are referred to as *up* and *down* respectively. Furthermore, spin is a conserved quantity, so if we have two electrons, and we measure the spin in some direction (say the z direction, for simplicity) of electron 1 to be up, the spin in the z direction of electron 2 must be down. This is where the paradox is contained. In this example we could deduce what the spin of the second electron should be without measuring it. The claim made by Einstein, Rosen, and Podolsky is thus that quantum mechanics cannot be a complete theory and there should be some hidden variable. The physicist John Bell killed this idea in his famous 1964 paper.

To summarise Bell’s argument, consider the two electron example from before. It is necessary to introduce the notion of an *expectation value* here. It is essentially a mean. More precisely, it is the mean result one gets upon performing a measurement of some observable on a collection of identical systems, provided a sufficient large number of systems are used. Bell investigates whether the usual quantum mechanical calculation of the expectation value of the product of the spins in two different directions is compatible with the equivalent calculation in the hidden variable model. The way in which the expectation value is incorporated into the hidden variable model is by assuming there exists a probability distribution associated with the hidden variable. Bell then compares the mean computed this way to the way it is computed in quantum mechanics. Indeed, the two formulations lead to different expectation values, and therefore, different physics. He shows this by deriving *Bell’s inequalities*, a necessary condition for compatibility of the quantum mechanical and hidden variable results which are not always satisfied.

Bell’s result leads us to a conclusion which is either frustrating or fascinating depending on your perspective. In essence, we must discard our classical notion of determinism, in favour of the stranger, more wonderful ideas that quantum mechanics provides us.

**References:**

- A. Einstein, B. Podolsky, N. Rosen, Phys. Rev.
**47**, 777 (1935). - J.S. Bell, Physics.
**1**, 195 (1964).