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I'm a bit confused. QFT is claimed to incorporate both Quantum Mechanics and Special Relativity. Therefore it should address the problem of non-locality caused by entanglement. However when I search for an answer on the Internet, I found nothing.

I'm not complaining. But it seems that most people only use QFT to do some fancy particle stuff and forgot we should care more about the more fundamental stuff.

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What do you mean you found nothing when you searched the Internet? –  Qmechanic Sep 3 '13 at 0:16
3  
You might want to try the keywords Relativistic Quantum Information. The Nottingham group or the recent workshop they hosted may provide good starting points. –  Emilio Pisanty Sep 3 '13 at 0:17

3 Answers 3

Even in non-relativistic quantum mechanics, entanglement is no symptom of any non-locality. The explanation of the entanglement as a non-local effect is a mystification spread by numerous popular books that ultimately boils down to a misinterpretations of quantum mechanics by Albert Einstein who misleadingly called entanglement "the spooky action at a distance".

Even in non-relativistic quantum mechanics, entanglement is just a correlation – expressed in the most general way that quantum mechanics allows (so it allows different, more general predictions than correlations predicted from a classical theory) – and this correlation isn't the same as causation. Recall: correlation isn't causation. So in reality, there are no signals whatsoever being sent in between the two entangled particles while they are being measured. Instead, the correlation between their measured properties is caused by the common origin of these two particles – by their mutual contact sometime in the past.

When we talk about two entangled faraway electrons or two entangled faraway photons in non-relativistic quantum mechanics, the Hamiltonian is usually assumed to be the sum of the two free Hamiltonians for the two free particles – there are no interaction terms operating in between the two particles at all! Because there are no interactions, there is no influence, and the observed correlations clearly can't have anything to do with any non-local interactions. They all boil down to the initial state that was prepared to be entangled.

Quantum field theory makes it totally manifest and universally valid that there can't be any non-localities. After all, experts sometimes use the term "local quantum field theory" for what may also be called just "quantum field theory". The perfect locality of QFTs is shown by the vanishing of certain propagators in the spacelike-separated region or, more precisely, by the vanishing (anti)commutators of fields at spacelike-separated points.

What Bell's theorem shows is that local realist theories predict correlations that are, in some cases, smaller than the larger correlations predicted by quantum mechanics (and seen experimentally). So local realist theories are excluded. Most of the popular writers about quantum mechanics are deeply confused about this point and they assume that realism "can't possibly fail" so it must be the locality that does fail. But this is a completely wrong, safely excluded possibility.

In 1905, we learned relativity that has guaranteed that locality is a perfectly valid law of physics, a principle that any newer theory obeys (faster-than-light influences are equivalent to clearly forbidden influences changing the past; the equivalence is achieved by the allowed change of the inertial system). The description of entanglement isn't incompatible with locality and quantum field theory actually prohibits non-locality of any kind. Instead, it's "realism" – the classical intuition that the state of the physical system has some "objective properties" even before the observation – that is wrong. The fact that the quantum description of the reality doesn't contain any "objective properties" of physical systems prior to the measurement isn't arts, isn't open to some "personal preferences" or "interpretations". It is one of the basic, fundamental, indisputable, universal postulates that underlie all of quantum physics which pretty much means all of modern physics.

Any "classical model" that would attempt to simulate the predictions of quantum mechanics would have to be non-local to achieve the sometimes high correlations. But Nature isn't described by any classical theory so this is not a problem. There's no need for classical models to be right and they are not right. Nature is described by a quantum mechanical theory that is, by definition, fundamentally non-realist. Quantum field theory is a quantum and therefore non-realist theory that is also perfectly local because it is Lorentz-covariant.

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There is what appears to be a very good lecture that deals with many of the issues. It is a Google Techtalk from Ron Garret entitled "The Quantum Conspiracy:What the Popularizers of QM don't want you to know". A rather tongue-in-cheek title that should not mislead you into thinking it is not a serious and worthwhile talk. www.youtube.com/watch?v=HQIJgheuYNU

He concludes with an information based interpretation of QM (if I've understood correctly), one that personally I don't like. I prefer the Transactional Interpretation, but there are many to choose from, which I suppose illustrates that the big problem with QM comes down to interpretation of the results as something that is meaningful to human beings.

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The answer that I see here is rather cagily worded, it seems to me, at best, and misleading. There is this: 'Even in non-relativistic quantum mechanics, entanglement is no symptom of any non-locality.' There is, then, I suppose, no 'problem of non-locality caused by entanglement'. This contradicts the rumors that entanglement violates Special Relativity as it allows for the seemingly instantaneous transmission of information.

I'm considering this, then: 'So in reality, there are no signals whatsoever being sent in between the two entangled particles while they are being measured. Instead, the correlation between their measured properties is caused by the common origin of these two particles – by their mutual contact sometime in the past.'

I take the key point of this to be an assertion that you needn't flat-out describe entanglement as an effect operating "outside of time and space." There is reference here to 'the common origin of these two particles - by their mutual contact sometime in the past'. But what does this mean? The entangled particles aren't moving at all. The typical scenario that's considered in discussions about entanglement involves two particles with the same velocity. Leaving that aside, as I read this, the idea is that the wave-function could simply describe correlations that are caused locally by other means, in the same way you can synchronize two or more distant clocks, by light-speed communication with a master clock. Which harmonizes w/this: 'quantum field theory actually prohibits non-locality of any kind'. Now, tracking this very closely, then, the next assertion is that: 'Instead, it's "realism" – the classical intuition that the state of the physical system has some "objective properties" even before the observation – that is wrong.'

To sum up, 'realism', is referred to as 'the classical intuition etc.', and there is a reference to 'local realist theories', and '[A]ny "classical model"', 'any classical theory'. There's reportedly 'no need for classical models to be right and they are not right.' I just want to emphasize the point, here, that electrodynamics is, in this parlance, a classical model, and so is general relativity. The blandly dismissive reference to 'any classical theory' and such, might convey the impression that we are merely transcending, maybe, classical mechanics, here, --which might not so urgently strain your credulity. My suspicion is that the term 'local realism' is being thrown around loosely here, one may not remember on the spot that a definition that excludes general relativity and electrodymanics is hardly obviously implied, or even obviously possible, and if you try to formulate one, I'll be able to dismiss it, very likely, as an unreasonably narrow definition.

Now, the point is not to dispute whether Bell experiments can prove (in principle) that Nature cannot be described by any local realistic theory. On the interpretational side, I think there is something of a crisis here, about the term 'the quantum description of the reality', whatever you mean by that. I'll try to play along, though. Apparently, it's the mathematical formalism + the minimal statistical interpretation for non-relativistic quantum mechanics. I note merely in passing, that for other readers it might mean something different.

It is mentioned, here, that 'The perfect locality of QFTs is shown by the etc.' Frankly this strikes me, superficially, as an odd bit of misdirection. I'll try to be more understanding than that, I might learn something, but couldn't we say that unlike QM, the quantum field theory (QFT) bends over backwards to be a "local" theory. Couldn't we, then, also say, that in fact, it is unclear how to write down a self-consistent non-local field theory. What I find so odd about this exposition, is it somehow doesn't come up, that QFT is, then, incomplete, since QM's entanglement has been shown to be in agreement with all experiments. Couldn't we, in fact, actually, say that nonlocality emerges from entanglement (which does not have a classical analog), which is equally present in both QM and QFT.

I've tossed a few points out, I see. I was moved to post, because I've encountered this odd sort of ideological presentation of QM as an unproblematic sort of thing, this is sometimes accompanied by patronizing remarks about Einstein, and very confident interpretation of Bell's theorem & experiments, all looking rather odd (I mean, whoever started these rumors, it's not the traditional presentation of the issue, and I don't know how to pithily convey how ideological it seems to me). QM contradicts Relativity. There is a real issue here, whether you decide to blame QM or Relativity. I suppose that we are committed, then, to simply tossing our Relativity. However, it makes lots of predictions. Are we also tossing those out? And, classical electrodynamics and classical mechanics do not lack on drama either. I see this: 'Nature is described by a quantum mechanical theory that is, by definition, fundamentally non-realist.' If we work with more concrete examples, I think we can recapture the difficulty w/simply asserting that local realism does not accurately describe the real world.

To be crystal clear, and again, I do agree that no local hidden variable theory is consistent with the predictions of standard quantum theory. The more traditional and mainstream conclusion drawn from this, is that standard quantum theory is inconsistent with itself. At the very least, I would want to draw the distinction between what was given in the answer here, and 'traditional/mainstream conclusions'.

Also, I'd like to clarify a simple point. I'm trying to interpret (pun intended) this statement: 'The fact that the quantum description of the reality doesn't contain any "objective properties" of physical systems prior to the measurement isn't arts, isn't open to some "personal preferences" or "interpretations".'

Okay, well, Bell used the term beables (properties of a system that are not observed). There is, by definition, no possibility of disproof of this view. And further, there is no possibility - by definition - that it could ever lead to any advance in science. That is because it is strictly an ad hoc theory.

Can I bring this down to the layman level, my level. This answer here, to which I am responding, is trying hard to convince us all that something, or other, is really not happening. And what is that? I might describe it as a malicious, clever, omnipotent, and invisible fairy that is messing with our measurements. Yet what is offered in its stead? This angry assertion that is not open to some 'personal preferences', or 'interpretation', that it is reasonable to believe that unobserved objects exist. Now, this may be right, but doesn't this bring us right back where we started? With, of course, a growing suspicion that there are people out there who enjoy the hell out of the fact that they are able to "understand" things that make most people's brains hurt.

Let me try to be unambitious, and just sort out what 'locality' means, a little bit. To have a use for 'locality', you have to have a notion of causality. Note this: 'Because there are no interactions, there is no influence, and the observed correlations clearly can't have anything to do with any non-local interactions. They all boil down to the initial state that was prepared to be entangled.'

This is to say, that everything has to be computed from initial values. And this, is to simply have no notion of causality. But thus, it would actually make no sense to talk about locality. But also, meanwhile, we have no fundamental theory that is deterministic, here. So the answer to which I'm replying, is presumably supposed to explain something, but it doesn't. And again, I think we can recapture the difficulty w/simply asserting that local realism does not accurately describe the real world. The alternative would be, that every particle would need to have all the details of all other particles contained locally to work. One might be inclined to debate this, but it has, at the very least, baggage.

Mind, I think that that the wavefunction is a beautiful, healthy baby. I just do not wholeheartedly agree with the idea that QM-as-we-know-it offers more than very little 'insight' into the inner workings of Mother Nature. I might even go so far as to stipulate that in my book, the wavefunction -- as a mathematical equation that allows us to solve for real dynamical standing waves -- is an absolute godsend. However, I decide what is plausible and what is not. And, one example of 'instant something' is the collapse of the wave function in every path of a split beam or all over a spherical surface created by a photon spreading out. When a photon is observed then 'it knows' it's been observed at every possible place the wave function could be - instantly. And, the many-particle wave function is a nonlocal object, why, exactly? Because, you cannot specify it by specifying psi at each point of space. Instead, for 2 particles you need to specify psi for each PAIR of points on space, and similarly for n particles.

So, there is nonlocal - something. Now, a theory with maximum speed of information transfer c is named "local". Somewhat breezily, 'nonlocal' strongly suggests a theory which is completely out of control, where things far away, out of our control, can influence and distort everything. Less breezily, I'd prefer to get away from 'local', or 'causal', and talk about Lorentz-symmetric. Realism or not, that doesn't matter that much -- the problem is that entanglement, for example, gives us evidence that there is a >R(c). There is a logic problem here and it's not just a violation of special relativity sensibilities. Consider 2 particles approaching and interacting in an exchange. As you go up the Time axis, the particles approach, approach and approach until "Suddenly a Miracle occurs!", as the old cartoon went. How did the particles know to exchange at that moment?

This is a question about the signaling mechanism. It is "There are 2 circles of equal radius in the same plane that intersect with the center of one circle being on the circle of the other and (necessarily) vice versa."

I used the term 'Lorentz symmetric'. How does this dovetail w/this quote from the other answer?: 'Quantum field theory is a quantum and therefore non-realist theory that is also perfectly local because it is Lorentz-covariant.'

I'll note that in this whole mouthful, 'Lorentz-covariant' is kind of interesting, I think this might mean something different, if indeed it means anything, than 'Lorentz invariant'. Then, it also might mean something different than 'we have an irreparable violation of Lorentz invariance'. I'm not sure that it is implied, here, either, that the process of wavefunction collapse blatantly violates Lorentz invariance (as in, take the Copenhagen interpretation, with its wavefunction collapse). Which it does.

How deep into this do we need to go --I think the other answer is cagey, on the subject of whether it's actually advocating, at the end of the day, an interpretation without wavefunction collapse (this is no dam interpretation!). Which might matter more, if you see, anyways, how one could merely get rid of wavefunction collapse and have lorentz invariance. What about the nonlocality of quantum mechanics? So I have to assume, that 'Lorentz covariance', an endlessly amusing phrase, to me, has something to do with the guiding equation for the particles being nonlocal and in violation of lorentz invariance. Does 'covariant' mean, well, like if I were to say 'Measurement is inherently not covariant'? I mean, we have quantities which must be covariant here? Which ones? Well, only the measurable quantities..

I'm amused by the phrase Lorentz covariance, in particular, because it is a real term, but this is in reality a key property of spacetime following from the special theory of relativity. It has two meanings. A physical quantity may be said to be Lorentz covariant, and so may an equation. I'm tempted to delve into the customary use of the Lorentz transformation in quantum mechanics, but suffice to say, that I see something fishy in how the term was introduced..I don't want to go all Feynman here, it's probably not helping. My point is then, perhaps only, to flag the other answer -- it's not what it presents itself to be, it's not remedial physics for undergraduates or something, it's, rather, quite odd stuff, and misleading. I'll conclude with another thought experiment.

Assume we create every microsecond an entangled spin pair, for a total of 1000000 pairs. Assume we will measure the entangled pairs when they are separated by 1 lightyear. Assume the measuring device on side A will measure 0.1 microseconds before the measuring device on the other side B. Assume the spin orientation on side A can be chosen freely for each individual measurement.

And here, I will only add that the no-go work on hidden variables is quite extensive. The reply may be coming, that it is not persuasive. But, no amount of experimental effort has been able to uncover even a hint of a hidden variable anywhere. The reply may come in, that this is not conclusive. However, I think it is strongly suggestive. So I am a bit impatient with at least how this other answer is presented, when it's asserting that the hidden variables are literally everywhere.

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Despite all you've written I can't really make sense of it one way or another. –  Brandon Enright Jan 26 at 8:05
    
You refer to 'a mystification..a misinterpretations of quantum mechanics by Albert Einstein'. Well, Einstein couldn't stomach the notion that particles didn't have properties, with real, determinable preexisting values. Is this a misinterpretation? That notion was spelled out in the uncertainty principle. From Heisenberg's relationship, we know that measuring the position of, A will lead to an uncertainty in its momentum. Einstein pointed out that by measuring the position of A, we gain precise knowledge of the position of B; the state of B depends on what we choose to do with A in our lab. –  user38224 Jan 26 at 8:50

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