One of the differences of opinion, at least in the middle of the last century, among quantum mechanics, was the “Hidden Variable” theory.
When a previously unresolved quantum state was resolved, it could either be due to true randomness, decided at the time the resolution takes place, or it could be due to “hidden variables” inside the entities involved. In other words, they “know” what they will do in every situation, it just appears random from outside because we do not have access to the hidden variables.
Now, I have seen a proof that the HV theory is incorrect. However, the mathematics of it was completely incomprehensible to me. Well, perhaps not completely, I seem to recall a summation in there, but other than that it used notation and or variables I had no idea what they meant.
Could someone point me to a place that explains it in basic calculus or algebra? Or could you do so yourself? I dont really see how one can disprove this, it seems to be more of a philosophical issue than a scientific one.
Bell’s inequality in a nutshell: the essential feature of standard quantum mechanics is that there exists a concept called “superpositions” of states: a physical state can be in a state that is neither A nor B, but an odd combination of both. If we measured the state, though, we would measure the state to have property A (with some probability) or property B (with some probability); if we prepared the same state again, from scratch, we wouldn’t necessarily get the same result (A or B) we got the first time. (Schrödinger’s cat is the best-known “example” of this.)
Some people (most notably Einstein) found this philosphically abhorrent, and believed that such states were “really” A or B, but that there some information about the state was hidden to us which made it seem like the state was “choosing” between A and B. In this picture, in other words, a superposition had “really” been A or B all along, we just didn’t know about it.
The problem with this picture, though, is that if you allow the existence of superposition states you can get quantitatively different physical behaviour. This is what Bell proved in 1964, and is often known as “Bell’s theorem”. A nice article on it can be found in the Wikipedia.
You know, you can do this with little more than arithmetic, though the specific equations (or, rather, inequations) I don’t know off the top of my head.
Basically, one place HV could come in is in the EPR experiment. Einstein, Podolsky, and Rosen thought that they could disprove quantum mechanics showing what a ludicrous result it predicts. Specifically, there’s a property called “spin”, which for an electron is either “up” or “down” (people who know more details, hush). If you take two electrons and put them into a superposition of “electron A up, electron B down” and “electron A down, electron B up” (as QM says you can), then separate the two and measure their spins at close enough to the same time that any signal from one measurement time and place to the other would have to travel faster than light. QM predicts that the measurements will still be correlated, but this would violate various philosophical principles such as causality and locality.
The problem is that when people actually did this experiment, the results were correlated. Einstein (and others) snapped back that there must be “hidden variables” in the initial state. That is: neither measurement “caused” the other to turn out in a correlated way, but both measurements had a common cause when the electrons were together.
Now, enter John Bell. He worked out how the experiment would run in theories with hidden variables and derived certain inequalities (“Bell’s Inequalities”) which predict certain statistical properties of the data to fall above or below a fixed number. There is such an inequality for any theory with hidden variables as long as the theory respects locality. Now, the experiment is run, the data analyzed, and the inequality shown false.
So, in short, hidden variables do allow correlations to occur in the data, but only to a certain extent. The real world is better modeled not by a hidden-variable theory, but by a nonlocal one.
Oh, btw: I learned of Bell’s Inequalities in Interpreting the Quantum World by Jeff Bub. I believe that was the book I read that had a very easy derivation of the inequalities, but it’s in my office at the moment. Still, a worthwhile read.
I think it is merely orthodoxy that says hidden variables don’t exist. Here’s something on Bohmian Mechanics with quotes from Bell himself. Of note,
I have no idea how accurate this site is. It is a philosophy site, not a science site. But I have yet to find them in error on such matters. I would greatly appreciate anyone in the know commenting on this. the Wikipedia link mentions this: you have to give up locality, or you have to give up hidden variables. But AFAIK nothing compells us to choose which to give up.
Remeber Bell’s theorum disproves local hidden variables theories only (or more correctly it disproves that any local hidden variable theory can reproduce all the predictions of QM), Bohmian mechanics is in essence a non-local hidden variables theory and fits in fine.
Well, the most satisfying explanation I’ve seen is that once the electrons in the EPR state are tangled, they really have no independant existance. The language in which QM is formulated makes one think that there really are “two electrons”, when it’s better to think in terms of having one EPR pair. I think QFT supports this by extending to the notion of a single “electron field” with various excitation states we call “electrons”.
Really, I’m not sure how bad throwing out locality is. Remember, locality is not the same as causality. Just because the measurements are correlated doesn’t mean that any information has moved faster than light, especially since the results are randomized.
Let’s say you had two decks of cards that, when completely shuffled, would always turn over the same card, even if on opposite sides of the universe. Their results would be perfectly correlated, but you couldn’t use this to send a signal of any sort, since they have to be randomized for the trick to work.
I think the first half of the century was about working through the old philosophical fetishes about locality and determinism. There are working, fantastically accurate theories which just happen to be nondeterministic. Big deal.
Yes, quantum mechanics is essentially non-local, but there’s an obvious problem with a hidden variable theory that is non-local and special relativity.
How so? I just pointed out what Jeff Bub, a student of Bohm’s, explained to me is the essential distinction between locality and causality. You can have a nonlocal theory without violating causality. I’d be glad to hear a reason nonlocality (not just noncausality) violates SR.
In Bohmian mechanics, changes in the potnetial can propagte instaneously which is clearly against the the idea that information can only be transmitted in fields slower than the speed of light.
So that particular nonlocal theory violates SR. Is NL+HV+SR (nonlocal, special relativistic theory with hidden variables) inherently contradictory? Is NL+SR?
As another question, what does EPR look like in QED?
Just like it looks like in non-relativistic QM. The literature on the philosophical implications of quantum field theory is very small and nobody has seriously managed to argue that it makes any significant difference to the arguments about the interpretation of quantum mechanics.
Really non-locality and relativity are contradictory, but it’s not a problem in the standrad QM interepretaions as there is no tachyonic communication. Violation of SR is a recurring feature of theories which attempt to explain QM in a deterministic manner.
Actually, I don’t think tachyon fields violate SM. It can easily be shown from the relevant field equation (d’Alembertian of psi equals mass-squared times psi) that though tachyonic fields have field velocities faster than light, the packet velocity is still below lightspeed. i.e. individual particles travel faster, but signals travel slower than lights.
The experiments were not done while Einstein was alive, so far as I know. Einstein’s analysis was based upon thought experiments alone, and his objections seem to have been on the money–he derived results from thought experiments that were eventually born out in the real experiments.
Some would say that it is the other way around–in fact, that is more or less Einstein’s objection.
If you have a non-local hidden variable theory, then there does not have to be superluminal communication between the two separated systems–they each carry their “hidden variable information” with them. The whole idea of local hidden variables is more or less a red herring–I don’t think anyone seriously considered such a thing, and the fact that local hidden variables have been shown to be impossible does not mean a lot.
If I was mistaken in my memory about Einstein being part of the response, I apologize.
Still, the EPR paper basically says that Locality + Realism + Quantum Mechanics is impossible. One of them has got to go. QM had already been justified by experiment and Locality was thought to be obvious, so Realism was put on the block, saying that though QM gave a good predictive scheme, the objects of the theory did not correspond to properties of the “real world”; that is, it isn’t the whole story and there’s some deeper theory which will behave in a “more orderly” fashion than QM does and will also be “Realistic”. Various means have been proposed to find this “real theory”, and Hidden Variables theories were significant suggestions. Lo and behold, Bell comes along and it turns out that Locality wasn’t so obvious after all.
As an aside, note that only one of HV and Locality had to go. This is why the mentions of “non-local hidden variables theories” (though they exist) sounded so strange to me in this context: it’s like throwing the baby out with the bathwater once you’ve found that (against what you’d thought at first) the bathwater can stay and only the baby needs to go. Also, it should be noted that Einstein, Podolsky, and Rosen were not disproven. They simply were found to have drawn the wrong corrollary to their main result at the end of their paper.
I love the smell of quantum mechanics in the morning. It smells like math with probability the square of the length of its projection onto the math eigenstate.
The way I remember it, no one was a part of the response. The notion of hidden variables to satisfy objections like EPR preceded the actual experiments. But I too apologize for my memory.
Did anyone who ever presented a hidden variables theory suggest it was only local?
Yes, I sometimes wonder why Bell’s inequality was ever even involved.
No I don’t see why that should be. In Bohmian mechanics you have a six dimensional configuration space which ‘holds’ the ‘hidden variable information’ but the ‘real’ potenial is defined and is subject to non-local changes. Without hidden variables the potential is in a suppoistion of states effectively holds the information in it so there is no need for tachyonic communication.
I’m not sure what exactly you mean. To me, “tachyonic information transfer” means sending a signal by means of tachyons. As I’ve just said, if you try to send a signal by particles which travel faster than light, the signal still travels slower than light.
Now if you mean “superluminal” information transfer…