[N.B. this is from a usenet posting to sci.philosophy.meta from May, 1996. Dan M. is another usenet poster. The exposition of Bell's theorem is paraphrased from Tim Maudlin's _Quantum Nonlocality and Relativity_, ch. 1 (q.v.) --Michael Huemer] Dan M. and I have been arguing, with little headway, about this. I say that the theory of relativity is false, and so is the orthodox (Copenhagen) interpretation of quantum mechanics (hereafter, QM). The following will be a simple explanation that anyone can understand of why. First, I will explain the EPR experiment (originally conceived by Einstein to refute QM). Then I will explain why the results of the experiment refute quantum mechanics. Finally (this is the hard part), I will explain why they also refute the theory of relativity. I. THE EPR EXPERIMENT (If you're already familiar with this, skip to section II.) This was originally a thought experiment, but since was performed as an actual experiment. You have a light-filter called a polarizer. What it does is, when you shoot photons at it, each photon either passes through it, or gets absorbed (with a certain probability of each). It also matters how the polarizer is oriented: for a photon in a given state, its probability of passing through a polarizer varies as you rotate the polarizer. It's possible to have a pair of photons that are in a state such that, quantum mechanics predicts, if they both encounter a polarizer oriented the same way, *either* they will both pass, *or* they will both get absorbed. There is a 100% probability that the two photons will do the same thing. However, whether they will both pass or both get absorbed is also, according to QM, random -- there is a 50% probability that they both pass, and a 50% probability that they both get absorbed. This probability is irreducible -- that is, it is not due in any way to our ignorance of relevant causal factors; the way the world works is inherently random. Let us say the photons 'agree' if they both pass or both get absorbed, and they 'disagree' if one passes and the other doesn't. Then in this case, the photons are guaranteed (100% probability) to agree. One more thing: If you take one of the polarizers and rotate it by 30 degrees (relative to the other), then the probability of 'agreement' goes down to 3/4. If, instead, you rotate one of the polarizers by 60 degrees, then the probability of agreement will be 1/4. Finally, if you rotate one of the polarizers by 90 degrees, then the probability of agreement is 0 (i.e. they always disagree). These statistics have been experimentally verified. Now, Einstein's thought experiment went like this: You shoot the two photons off in opposite directions. When they get a good distance apart, you simultaneously let them both encounter polarizers with the same orientation. At the time they encounter the polarizers, the photons are far enough apart that no signal, travelling at or below the speed of light, could travel from the location of one of the photons to the location of the other before the latter had been measured (i.e., in time to 'tell' the other photon what to do). This kind of separation is called 'space-like separation' -- two events are separated in such a way that a light signal could not connect them. This is significant because, in the theory of relativity, it is impossible for any event to have any influence on a space-like separated event (i.e. no influence can be propagated faster than the speed of light). With that set up, let's turn to how Einstein (correctly, IMO) used this thought experiment to refute QM. II. EPR REFUTES THE COPENHAGEN INTERPRETATION OF QUANTUM MECHANICS The argument is really pretty simple. Many scientists have a prejudice against simple arguments, but if you can overcome that, imagine the following: Suppose you and a friend decide to play a little game. The object of the game is to mimic the behavior of photons (hey, you're smarter than a couple of photons). It goes like this: You and the friend start out in a room together. While you are in the room, you are free to communicate all you like, and to devise any strategy you like for playing the game. After that, you both leave the room through opposite doors. You walk about a mile, and then each of you encounters a questioner. The questioner says to each of you one of three things: "30?", "60?", or "90?" (analog to the orientation of polarizers to the vertical). You are required to answer either "Pass" or "Absorbed". However, you are not permitted, after you have left the room, to communicate with your partner any more. Furthermore, you do not know what question you are going to be asked until you are actually asked, and you do not know what question your friend is asked. And he cannot communicate to you either what question he was asked or what he answered. This game will be played over and over, about a thousand times. Now, your task (in collaboration with your friend) will be to devise a strategy -- any strategy at all, subject to the constraints just indicated -- for choosing how to answer the questions that will reproduce the statistics the photons with the polarizers reproduce. That is: 1. When both of you are asked the same question, your answers must agree 100% of the time. 2. When you are asked questions that differ by 30 (i.e., one of you is asked "30?" and the other is asked "60?", or one of you is asked "60?" and the other is asked "90?"), your answers agree 3/4 of the time. 3. When you are asked questions that differ by 60 (i.e., one of you is asked "30?" and the other is asked "90?"), then your answers agree only 1/4 of the time. All of this is an exact analogy to the EPR experiment, only more picturesque. One more thing: each of you has a coin in his pocket which you may flip if you want to. If you like, you can also have a many-sided die to roll, to help you generate random numbers, if you think this will help you. Now, the orthodox interpretation of quantum mechanics describes the 'strategy' that the photons play. So why not start there? Can you win this game by taking the strategy that (QM says) the photons take? Clearly no. The photons' strategy, allegedly, is the following: While in the room, you both agree that you will answer all questions in the same way; however, you do not decide which way that will be. That is, you both agree that if asked "30?" you will agree in your answers, but you don't know yet whether you will both say "pass" or both say "absorbed," and that won't be decided until the time comes. Then you both leave the room. When you meet your questioner, he asks you "30?" let's say. You flip a coin to determine how to answer. End of story. Obviously, this strategy does not work -- it does not satisfy constraint (1) above. That is, if you play this strategy, then when both of you are asked "30?", your answers will agree only 50% of the time, not 100% of the time as desired. For when you are asked this question, you know that IF your partner was just asked "30?" and HE answered "pass," then you *have to* answer "pass" as well. Likewise, if he was just asked "30?" and he answered "absorbed," then you have to answer "absorbed." If you have no idea what answer he gave, then you have no idea how to answer. The only way you could be guaranteed of giving the same answer as your partner, would be if you knew what answer he gave (or would have given if he was asked this question). According to the current strategy, however, you don't know this -- for (a) you and he did not decide upon this while in the room; and (b) he has not communicated with you in any way since then, so there's no way you could know what answer he decided to give. Given the constraints of the game, the only possible strategy for satisfying condition (1) is that, in each iteration of the game, you decide while in the room how you're both going to answer each question. You cannot decide later and hope to have your answers correspond 100% of the time without communication. This refutes the orthodox interpretation of quantum mechanics -- Einstein was right. The standard interpretation simply cannot explain a basic fact -- it cannot explain the 100% correlations in this experiment. For those of you who are more into QM: When the photons leave the room, they are both in a 'superposition' -- neither of them has a determinate polarization. Let's say photon A is the first to be measured. When photon A reaches its polarizer, it passes, let's say. Its state therefore collapses; it is no longer in a superposition, but in a definite state. Question: what is the state of photon B at this time, just prior to photon B's being measured? If photon B is in a superposition still, then there is only a 50% chance that it will pass; hence, only a 50% chance of agreement -- which is contrary to the 100% prediction. But if photon B is not in a superposition still, then the measurement at the location of A has instantaneously changed its state -- hence, action at a distance. III. EPR REFUTES RELATIVITY Now, this is the startling thing -- John Bell proved that there is no strategy that you can pick that will satisfy conditions 1, 2, and 3, subject to my constraints -- that is, there is *no way* that you and your friend can play this game and win, if you are not allowed to communicate after you leave the room. Okay, here's the mathematical part. As we just established, in order to have any hope of winning, you and your friend must decide, while in the room, how you're going to answer each question. Now, even if you weren't convinced by the above argument, consider this: Let's suppose for the moment that I was wrong about that, and it is possible to win by flipping a coin after you leave the room. Then you might as well just flip the coin while you're IN the room, and tell your partner what result you got. Having more information cannot possibly DEGRADE his performance, since in the worst case, he can just ignore it. So even if there is a strategy that could work by deciding how to answer *after* you leave the room, there must be a corresponding strategy *at least as good* in which you decide while *in* the room. That being established, let's consider the possible ways you and your friend could decide to answer the questions, on any given iteration of the game ("P" stands for "Pass" and "A" for "Absorbed"): 1. PPP 5. AAA (a) 2. PPA 6. AAP (b) 3. PAP 7. APA (c) 4. APP 8. PAA (d) Strategy 1 means you both will answer "Pass" if asked "30?", "Pass" if asked "60?" and "Pass" if asked "90?" Strategy 2 is you answer "pass" is asked "30?", "pass" if asked "60?", and "absorbed" if asked "90?" Etc. (We here ignore any possible strategies in which you don't both answer the same way to all questions.) These are the only logical possibilities. Of course, you will have to use different strategies on different iterations of the game (you cannot, for example, always choose #1). Now let a = the proportion of times you pick either strategy 1 or 5 (i.e., lump #1 and #5 together). Let b = the proportion of times you and your friend use either 2 or 6. Let c = the proportion with which you use 3 or 7. And let d = the proportion with which you use 4 or 8. Now, since you must always use 1,2,3,4,5,6,7, or 8, we know that a + b + c + d = 1. Also, when one of you is asked "30?" and the other is asked "60?", your answers have to have a 3/4 probability of being the same. They will be the same only under strategies 1,2,5, and 6. Therefore, a + b = 3/4. Next, when one of you is asked "60?" and the other is asked "90?", your answers will agree iff you have picked strategy 1,4,5, or 8. Since you are supposed to agree 3/4 of the time if these are the questions, that means you have to pick 1,4,5, or 8 3/4 of the time. Ergo, a + d = 3/4. Finally, when one of you is asked "30?" and the other is asked "90?" your answers must agree 1/4 of the time. You can only achieve this objective by choosing 1,5,3, or 7 1/4 of the time. Hence, a + c = 1/4. This gives you 4 equations in 4 unknowns -- enough to solve and determine what your strategy should be. Solve these equations (left as an exercise for the reader). Surprisingly, (if I've done my calculations right) you will obtain the following values: a = 3/8 b = 3/8 c = -1/8 d = 3/8. But this is impossible. It is not possible to choose strategy 3 or 7 *negative one eighth of the time*! Therefore, you cannot succeed. That is to say, you and your friend cannot reproduce the statistics that quantum mechanics predicts, subject to the constraints we delineated (namely, that you can't communicate after you leave the room, and you can't know beforehand which questions you're going to be asked). What does this prove? It proves that you *have to communicate*. Of course, if you are allowed to communicate with each other after you leave the room -- say you're given a pair of walkie talkies -- then there is absolutely no problem. Then you can just have whichever of you encounters the questioner first call up the other one, tell him what question you were asked and how you answered, and the other can then roll his dice to get the appropriate probability of agreement and answer accordingly. The above is Bell's Theorem. Bell's Theorem is widely misunderstood. I think Dan misunderstands it. It is sometimes thought to be a proof of something different (viz. that the EPR experiment can't be explained with 'hidden variables'). What it proves is that to produce the correlations predicted by QM in the EPR experiment, you have to communicate after you leave the room; hence, that the photons must 'communicate' (i.e. influence each other) faster than the speed of light. This isn't speculation, and it's not a 'philosophical' or 'metaphysical' point that I've personally invented. This is a proven result that anyone can understand, and, what's more, the statistical predictions used in the above (the 1/4, 3/4, etc.) have been experimentally verified by Alain Aspect. So there is no escape from this conclusion: There is faster-than-light causal influence. This refutes the theory of relativity. The theory of relativity entails that the photons could not 'communicate' after they were outside each other's light cones (at space-like separation), because (a) nothing can travel faster than the speed of light; (b) more importantly, according to relativity, there is never any objective fact about which event happened first, when the two events are space-like separated (the temporal order is relative to a reference-frame). Therefore, the theory of relativity is false. Finally, for you QM enthusiasts, here's one more, even simpler argument that QM is an inherently non-local theory ('non-local' means the theory requires instantaneous action at a distance). Here is a formulation of what locality requires for a probabilistic theory: Let A and B be space-like separated events. Let K be a complete description of all the events preceding A (i.e. in A's backward light cone). Then the objective probability of A given K should be identical to the probability of A given K and B: P(A|K) = P(A|K,B) In other words, no event outside A's backward light cone should alter the probability of A occurring. In QM, this condition is obviously violated, where A and B are the measurement-results for the two photons. Hence, QM is, straightforwardly, a non-local theory. _ _ _ Dan, if you're reading this, what is your response to these arguments? So far, I am not aware of any. I do not believe there is any response. To recap, there have been three main arguments, all based on the EPR experiment: 1. The Copenhagen interpretation of QM could not explain the EPR correlations. (section II) 2. Bell's Theorem proves that the correlations require action at a distance (non-locality). (section III) 3. It's obvious anyway that QM requires this, inasmuch as the theory requires that events outside A's absolute past alter the probability of A. 2 & 3 make QM inconsistent with relativity. Hence, I conclude that relativity is false (from 2,3), and the Copenhagen interpretation is false (from 1). Geez, I hope this wasn't too long . . .