If quantum mechanics leaves something out, what might the missing pieces be? Physicists call them hidden variables: extra properties, not described by the quantum state, that would fix the outcome of each measurement in advance. Hidden means only that the standard theory does not mention them.

The appeal of hidden variables

There is a familiar precedent. In the nineteenth century, the behavior of gases was described by laws of pressure and temperature that involved averages. Behind those averages, physicists came to believe, lay countless molecules each moving in a definite way. The randomness of a gas was ignorance of details, not randomness in nature. Einstein hoped quantum probabilities would turn out the same way, as averages over a deeper, definite reality.

Von Neumann's proof and its critics

In 1932 the Hungarian American mathematician John von Neumann published a celebrated book on the mathematics of quantum mechanics. In it he offered a proof that hidden variables could not exist, and for decades many physicists treated the question as closed. Yet the proof rested on an assumption about how hidden variables must combine that was reasonable for averages but not for individual cases. The German philosopher and mathematician Grete Hermann pointed out this weakness in 1935, but her criticism went largely unread. Bell independently made the same objection in a paper published in 1966.

Bohm's surprise

The clearest refutation of the impossibility claim came in 1952, when David Bohm published a working hidden variable theory. In it, particles always have definite positions, guided by a wave that follows the usual quantum equation. Bohm's theory reproduced every prediction of standard quantum mechanics, which showed that hidden variables were possible after all. Its interpretation is the business of QUAN 202. What matters here is a feature Bell noticed when he studied it: Bohm's theory is explicitly nonlocal. In it, how a particle moves can depend instantly on what is done to its distant partner. Bell asked whether that was an accident of Bohm's design or something any hidden variable theory would need.

Three assumptions, carefully stated

To ask the question properly, it helps to separate the ideas EPR combined. Physicists usually name three.

First, realism in this narrow sense means that measurement results are determined by properties the system has before it is measured. Second, locality means that a measurement in one place cannot affect a result in another place faster than light could travel between them. This fits Einstein's own theory of relativity, which treats the speed of light as a limit on cause and effect. Third, measurement independence means that the settings the experimenters choose are not correlated with the hidden properties of the particles. If nature somehow knew in advance which angle Ana would pick, the whole test would be rigged.

Together the first two are often called local realism; Bell came to prefer the term local causality. A theory with all three features is called a local hidden variable theory. The EPR authors hoped one existed. Bell's achievement was to show that any local hidden variable theory makes predictions that can be checked against quantum mechanics in a real experiment. The question stopped being only philosophy.

Why the question was neglected

For about thirty years after EPR, the issue sat at the margins of physics. Quantum mechanics was producing lasers, transistors and nuclear power, and most physicists were busy using it. The EPR and Bohm cases gave perfect correlations for matching settings, and those, as the glove story shows, could be explained by hidden instructions. No one had found a situation where local hidden variables and quantum mechanics predicted different results, so the disagreement seemed to be a matter of taste rather than fact. Questioning the foundations could also harm a young physicist's career, and some who pursued it, including John Clauser a few years later, have described being discouraged by senior colleagues.

Worked example: designing a local hidden variable model

Suppose you want to explain Bohm's spin pairs with hidden variables. A natural model: at the moment the pair is created, the source writes out an instruction card for each particle. The card for particle A lists an answer, up or down, for every direction in which it might be measured, and the card for B lists the opposite answer for each direction. The cards travel with the particles. When Ana measures along some direction, her particle reads its card and shows that answer. Ben's does the same.

This model is local, because nothing passes between the particles after they part. It is realistic, because each result was fixed at the source. It respects measurement independence if the cards are written without any knowledge of the future settings. And it perfectly reproduces the case EPR described: whenever Ana and Ben measure along the same direction, their results are opposite. If the cards are shuffled at random from pair to pair, each person's own record looks random too.

So far, the model matches quantum mechanics exactly. The trap lies in the cases where the two choose different directions, and to see it we need Bell's argument.