This is basically the text of an email I sent to a couple of friends the summer before I came to UNC. If you've read anything on this by N. David Mermin you'll probably find the exposition familiar; he's who I got it from. I recommend his book Boojums All the Way Through. He draws different conclusions from Bell's theorem than I do. Of course, you should believe me rather than him, because I am a CS grad student, and he is merely an eminent physicist.
I think I've internalized the whole EPR/Bell's Inequality issue, and I want to check myself by trying to explain it to somebody, so I'm trying you all. I warn you, it's rather long, but I hope you'll read it anyway.
The key quantitative fact to know about this particular gedankenexperiment is that a photon which passes through one polarized filter has a 1/4 chance of passing through a filter oriented at an angle of 60 degrees to the first. By rotating in either of two directions, you get three possible orientations for a filter, any two of which are separated by 60 degrees (or by 120 degrees, which amounts to the same thing). If the detectors are 90 degrees apart, no photons get through both of them.
A key qualitative fact to know is that you can excite certain atoms in such a way that they will give off pairs of photons in opposite directions that are guaranteed by a conservation law to exhibit the same polarization statistics as if they were a single photon being measured by two filters in sequence. Actually, that's a bit poorly stated, but it helps me remember what's going on. More precisely: If the two photons are measured by identically oriented filters, they either both go through or both don't. Both possibilities are equally likely. If the filters differ by 60 degrees then the photons agree 1/4 of the time, both going through and being blocked with equal frequency.
A natural interpretation for the first fact is that making it through the first (say vertically oriented) filter puts the photon in a vertically polarized state, and that a vertically polarized photon has a 1/4 chance of making it through a filter oriented 60 degrees to the vertical. If it does, then it will have now be in a 60-degree-angle-polarized state. The thing that is supposed to encode this state is the wave function, and the process of making it (or not) through the filter is the collapse of the wave function.
This interpretation encourages one to think that a photon, at any time, has a polarization state that is well-defined with respect to at most one particular pair of orthogonal axes. If it makes it through one (ideal) filter, then it's sure to make it through another oriented in the same way. Its behavior with respect to filters with a different, non-orthogonal orientation is indeterminate. This is a key premise of the Copenhagen interpretation of QM.
The key qualitative fact I mentioned is puzzling under this interpretation. Say we put the emitting atom in the center, one detector on the left, and another on the right. Each detector has three possible orientations, separated by 60 degrees. Let's label them 1, 2, and 3, with 2 being vertical for both the left and the right detector.
Say that both detectors have setting 1. We know that all photons accepted on the left will be accepted on the right, and vice versa. But they can't all leave the emitter with their polarizations well-defined with respect only to orientation 1, because we could have just as easily done the experiment with the detectors set at 2 or 3. And any pairs that leave the emitter with their polarizations not well-defined with respect to orientation 1 are going to have trouble ensuring that they agree when they hit their respective detectors.
So the obvious solution is that photons must have a well-defined polarization with respect to any angle, subject to the constraint that two directions 90 degrees apart must have contrary polarization states. Now, this violates the conventional metaphysics of quantum mechanics, but we seem driven to it by the predictions of the theory itself (the conservation law requiring pairs of photons to exhibit the same polarization is one such prediction). This was reassuring to Einstein, Podolsky, and Rosen (who originated the gedankenexperiment in a different, less plausible, form using position and momentum rather than polarization at different angles).
However, the weirdness of QM seems to have a way of defending itself. Assuming that photons do have well-defined polarization states with respect to all angles, we should be able to list a given photon's polarization states with respect to each of the three orientations 1, 2, and 3 (of course, we could never determine all three simultaneously, but I've just argued that the photon must somehow know what they are). There are 8 possibilities:
where 1 means ``makes it through the filter'' and 0 means ``doesn't'' We have to assume each photon carries the same ``instruction set'' as its partner, to make sure that they give the same answer if the pass through identically oriented disks. But remember that, if the filter settings are different, then the photons behave the same way only 1/4 of the time. In fact, they're both rejected 1/8 of the time, and accepted 1/8 of the time. Since each of the instruction sets 2-7 has 2 readings the same, no one of them can occur with a frequency greater than 1/8, or their misbehavior would be detected by an appropriate choice of filter settings. But instruction sets 1 and 8 only make things worse - for every appearence of 1, there must over the long haul be one fewer appearence of 2, 3, and 4. And similarly for 8. So the sum of the possible frequencies for all the instruction sets must be less than 1, which is impossible. This inequality between the totals of the possible frequencies and 1 is a variant of Bell's inequality.
So we are forced to the conclusion that the photons do not have ``instruction sets'', otherwise known as hidden variables. In other words, the polarization state in a given direction cannot in general have a specific value before being measured. This notion had always been a premise of classical QM, where the wave function is collapsed by the process of observation. Before Bell's inequality, skeptics had thought that the predictions of QM might be consistent with a more philosophically satisfying theory. Bell's result showed otherwise.
Unfortunately, the notion that observation collapses the wave function has problems of its own. The observer and observed together just make a larger quantum mechanical system, and when is its wave function collapsed?
Perhaps more damning (as it is less philosophical), consider the two detectors in the above experiment, perhaps separated by a couple of light minutes. Detection at one of them is supposed to collapse the wave function for the whole two-photon system instantaneously. Otherwise how does the other photon know how to behave consistently? But there is no such thing as an instantaneous event occurring over an extended region of space. Observers moving at different velocities will differ as to which photon is detected first, so we cannont meaningfully say that one or another of the detectors caused the wave function to collapse.
It is this state of affairs that tends to support the many-worlds approach. When the detector receives the photon, it is thrown into a superposed state, one fraction rejecting it and another accepting it. (Photons can be detected as they pass through a detector without scrambling their polarization.) When you interact with the detector, you, like Schroedinger's cat, are thrown into a superposed state. What this amounts to is splitting into two copies of you. For each copy, the other may as well not exist.
I'm not sure this simplifies things much, but think of the universe as a function
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It may be better still to think of the universe as a function g from the space of all possible histories of the entire universe, to the open interval from 0 to 1. The function g, applied to a particular history, gives the probability density for that particular history actually occurring. To use the function as a predictive tool, we look at the set of all histories that match what we have observed so far, and consider the fraction of those that exhibit particular properties in the future. That fraction is the probability of the given property being exhibited when we actually observe the future. To the best of my understanding, this is sort of how Feynman's approach to QED works.
These last two paragraphs aren't entirely clear, I realize. Let me conclude with a couple of somewhat more concrete comments.