Wrap a head around Bell's inequality
On Bell’s inequality
Bell’s inequality is a notoriously difficult subject, not because the basic conclusion is impossible to state, but because the conclusion rests on distinctions that are easy to blur. A simplified explanation therefore has to sacrifice some precision. Still, the broad shape of the argument is this:
Classical physics tends to assume a few things about the world. First, locality: things can only be affected by other things close enough to influence them, meaning inside their light cone. Second, realism: measurable properties have definite values whether or not anyone has measured them yet. Third, ordinary causality: later events do not reach backward and cause earlier ones. These assumptions feel so natural that they can seem almost invisible.
Bell’s insight was that they are not just philosophical preferences. Taken together, they imply a mathematical limit on what kinds of correlations experiments can produce. In other words, if locality, realism, and ordinary causality are all true, then certain repeated measurements must obey a specific upper bound. That bound is what Bell’s inequality expresses.
A simple way to picture the experiment is as a game. A source, Alice, creates a pair of entangled particles and sends one to Bob and one to Claire. Bob and Claire each choose a direction along which to measure their particle’s spin. Their goal is to get matching results as often as possible, even though they cannot communicate once the particles are separated.
If the particles merely carried hidden instructions from the start, and if nothing nonlocal is allowed to happen between them, then there is a hard ceiling on how often Bob and Claire can win. In the standard version of the game, the best they can do is 75%. Quantum mechanics predicts something higher: about 85.4%, namely ((2+\sqrt{2})/4). Experiments agree with the quantum prediction.
That is the problem. The world seems to violate the limit that classical assumptions say it should obey. So at least one of those assumptions has to give.
The Copenhagen interpretation gives up realism, at least in the strong hidden-variable sense. It says the particle does not already possess a definite spin value waiting to be discovered. Measurement does not reveal a pre-existing answer; it helps produce one. This avoids the need for hidden instructions, but it creates the infamous measurement problem: what exactly happens when a spread of possibilities becomes a single observed outcome?
Bohmian mechanics takes a different route. It keeps definite particle positions and adds a nonlocal guiding structure, the pilot wave. On this view, the world really does have hidden variables, but they are not local ones. The behavior of a particle can depend on the quantum state of the whole system. That makes Bohmian mechanics satisfyingly realist, at least to me, but it pays for that realism by accepting nonlocality.
Everett’s many-worlds interpretation makes a different move again. It does not say that one result mysteriously wins out over the others. Instead, it denies that there is only one outcome in the first place. All the outcomes occur, but in different branches of the universal wavefunction. I personally find this interpretation less attractive, but it is one of the serious options.
The important thing is that these are interpretations, not rival experimental theories. Copenhagen, Bohmian mechanics, Everett, and the other serious contenders all reproduce the same quantum predictions. They differ not in what number comes out when you crunch the equations, but in what they say reality must be like for those numbers to come out at all.