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It is well accepted that quantum theory has well adapted itself to the requirements of special relativity. Quantum field theories are perfect examples of this peaceful coexistence. However I sometimes tend to feel little uneasy about some aspects. Consider an EPR pair of particles light years apart. Suppose there are 2 observers moving relative to each other with constant relative velocity. Let us consider, there are spin detection mechanism at both end for each particle. Now suppose one of the observer is at rest w.r.t. the detector for the first particle. As soon as the detection made, the wave function of the 2 particle entangled system will collapse instantaneously and the second particle must realize a definite opposite spin value. Now due to relativity of simultaneity, the second observer may claim that the collapse of the wave function for the two particle system is not simultaneous. He may even claim that the second particle is measured first. In that case a special frame of reference will be privileged, the frame at which the wave function collapsed instantaneously. This will cause a significant strain on the core principle of special relativity.

I am sure the above reasoning is flawed. My question is where?

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Two comments, without time for an answer. (1) You've missed out the state preparation apparatus in your description of the experiment. There are 3 frames of reference. (2) SR ultimately does not play nice with QM for particles, we have to use QFT; it's a field theory, so you can't talk about particles without establishing which localized discrete structures in the field theory you are talking about. In general, it's hard to tie down a robust localized discrete structure. –  Peter Morgan Feb 20 '11 at 17:00
    
I also wonder what happens to a system like this when you turn gravitational radiation on –  Jerry Schirmer Feb 21 '11 at 14:41
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4 Answers

The reasoning is flawed for multiple reasons. One of them is that there is no physical wave-function collapse so in particular there can't be any problem with relativity. More precisely, entanglement is only about correlations between particles. That is, if you carry out experiments and later (after both observers who measured individual particle meet) they will notice that there was a correlation.

And this has to do with another flaw: locality. You are simply not allowed to say anything about stuff you don't measure. And that is only the stuff in your past light-cone. You just can't say anything about the other particle which is space-like separated from you (for all you know, it might no longer exist, having been absorbed by a BH).

He may even claim that the second particle is measured first.

Sure, this is because there is no notion of causality between space-like separated events. This alone should be enough to convince you that wave-function collapse is completely unphysical. It is just a convenient tool of Copenhagen interpretation (which simply ignores the measurement problem) that has its limitations.

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I am well aware of this standard answer and I accept it to be the right answer. Yet there is some nagging feeling that it somehow undermines the spirit of SR. One may think of the wave function as "our knowledge of the system". But the way the different possibilities interfere with each other seems something more physical than that. I believe the issue is little more delicate. That's all. I don't want to attract too much arguments. –  user1355 Feb 20 '11 at 14:41
    
@sb1: I agree that there is more to discuss but I believe this answers your question. Anyway, maybe someone else will provide you with something more to your liking. By the way, what do you mean by "the different possibilities interfere with each other"? Wave-function itself is of course physical (well, depending on interpretation it need to be, but let's not delve too deep into metaphysics). What's unphysical is its collapse. Collapse is just an analogue of e.g. action-at-a-distance forces in classical physics. They make things simpler in certain cases. But they just aren't physical. –  Marek Feb 20 '11 at 14:51
    
"the different possibilities interfere with each other" this statement refers to the fact that in quantum theory the wave function of a system may be the linear superposition of different possible states. –  user1355 Feb 20 '11 at 14:59
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Nice answer, @Marek, +1. Dear @sb1, your uneasy feeling is pretty clearly coming from the fact that you are imagining that the wave function is a classical wave. It's not. It's just an ensemble of complex numbers that may be used to predict the probabilities of any outcomes - any combination of outcomes of the entangled subsystems. The "collapse" is only a simplification of our bookkeeping - instead of the full probabilities $P(A=A_i,B=B_j)$, we may use the conditional probabilities with $A=A_m$ once it's known that tne $m$th result of $A$ was measured. The collapse only occurs in our heads. –  Luboš Motl Feb 20 '11 at 19:24
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There's no privileged reference frame here. Just two observers with consistent measurements and a disagreement on the order of events (as is typical in relativity).

Remember the collapse of the wave function isn't something you can observe. You can observe the spin of one of the particles being measured, and then you can observe the spin of the other particle being measured, and these measurements are guaranteed to be consistent due to the entanglement. Each observer will conclude that the first measurement (as determined in his reference frame) collapsed the wave function.

If your observers disagree on which measurement happened first, then perhaps they'll dispute which measurement collapsed the wave function, or when this collapse happened. But certainly there won't be an observer who thinks the wave function collapse wasn't instantaneous, (unless that observer doesn't understand quantum mechanics).

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Just add a few points to the other answers. There are two immediate topics that are being conflated in the question. For me the ontology of QM has not been resolved and so I can only discuss some points from my current perspective.

Firstly QFT deals with relativistic fields and so is unlikely to be associated with any Special Relativity paradoxes or oddities. By contrast the EPR experiment was about QM, which is expressed in a Newtonian manner with phrases like "instantaneous collapse" which are at odds with Special Relativity - whether "collapse" is physical or not the word "instantaneous" should not also be used. Of course it sometimes is because QM is a Newtonian theory - you need QFT for the relativistic aspects!

As other answers have suggested this might be resolved if the "collapse" is not physical. Related to this is the issue as to whether $\Psi(x)$ is defined over physical space (in a field like manner as presented in elementary QM) or whether it should be taken as a member of an abstract Hilbert space only (and thus not subject to spacelike processes). Alternatively a conclusion could be that "collapse" is physical - but somehow local. GRW (and its variants) is such a physical collapse theory. For such a theory your example then becomes a valid gedanken experiment for that model of collapse.

The second point is that one needs to separate the issue of "collapse" - where/when/if and how - from the correlation issues in this EPR experiment. The story of EPR and Bell's theorem is still ongoing. See Disproof of Bells Theorem if you wish to pursue that aspect further.

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Let me rephrase your question with the simple, but wery illustrative example

Let's call the event of detection of the first spin as $A$ and the event of detection of the second spin as $B$. Suppose that $A$ and $B$ are timelike events. Then you can choose such a reference frame, where $t_A>t_B$. But also there exist a reference frame, where $t_B>t_A$.

The conclusion is inavoidable:

  1. In the first reference frame collapse happened before the detection of first spin.
  2. In the second reference frame collapse happened before the detection of second spin.

In other words: The moment of collapse of the wave function depends on reference frame.

Is there any contradiction? -- No. You cannot devise an experiment that "detects" this collapse. You cannot use this collapse to transmit a signal. e.t.c. If you feel like "there is a condtradiction" -- try to get down to the level of a concrete experiment that demostrates this contradiction. You will see that any attempt to formulate such experimet will fail.

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