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Quantum mechanics is non-local in that long distance correlations are present, though there is no signalling possible. But QFT is Lorentz invariant and contains quantum mechanics as a special case. I assume this is not a paradox as paradoxes do not exist but I do not understand the details. Can anyone supply a reference or satisfactory explanation?

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Correlations of results of measurement procedures of entangled system in QM (and thus also in QFT) are fixed at "the moment" of the observation and not previously as instead it happens for long-range correlated systems of statistical mechanics. In this sense, because the two measured parts of the system can stay arbitrarily far form each other (so that no physical signal can propagate from one part to the other with speed $<c$ during the measurement procedures), non-locality is manifest in quantum theories. Even in QFT in spite of the fact that fields obey covariant and local equations. It is because non locality is due to entangled quantum states and not to field equations.

This is just what the experimental failure of Bell's inequalities proves: (1) these correlations show up and (2) they were not fixed before performing measurement on the system (as it would be if there were local hidden variables, more fundamental than the quantum description of the system).

It is worth stressing that these correlation do not imply any transfer of energy or momentum or other physical quantities from one part of the system to the other, and there is no violation of causality with them (also because the time order of the pair of distant observations may depend on the used reference frame, since the involved pair of events are spacelike separated). Moreover, since the outcome of measurements is stochastic one cannot transfer information through these correlations.

The situation is similar to this one where the entangled system is replaced by a pair of magical quantum dice. I have a die and you have another one. It happens that, no matter the distance between us, once you get a number from your die, I get the same number from mine.

In principle we could communicate through these correlations, in practice we cannot, because as the outcome is stochastic I cannot impose to my die to produce the outcome I want.

There is another possibility to communicate through our magic quantum dice: I could communicate you something simply by throwing my die. You should see your die to reproduce my numbers and you would know, this way, that I am throwing my die.

Conversely, with correlations of QM even this possibility is forbidden: It is possible to prove that the outcome of your measurement procedures on your part of the system have the same statistics, independently from the fact that I perform measurement on my part of system or not, though each pair of outcomes (on both sides of the system) appear to be correlated. So you cannot know whether or not I am "observing" my part of system.

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  • $\begingroup$ thanks everyone for the answers ... it is a little more clear. I also got Mermins physics today description and that helps. But the dice are a real simple way to think about that paper! $\endgroup$ Jan 11, 2014 at 5:29
  • $\begingroup$ While I never downvote, I must say that some of your assertions are false. Quantum mechanics respect causality/locality, and correlations exist before (joint) measurement. The expression "non-local correlations" is a nonsense. Causlaity/Locality has to do with the fact that an information or energy is not transfered instantaneously (or quicker than the speed of light) from one space-time point to an other space-time point. Correlations are what they are. $\endgroup$
    – Trimok
    Jan 11, 2014 at 11:17
  • $\begingroup$ Excellent answer, cross referenced here $\endgroup$
    – Xlsx2020
    Sep 10, 2017 at 15:49
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Quantum Mechanics is non-local because you are working with particles and interactions all occur instantaneously. This doesn't allow locality. Quantum Field Theory is a framework that instead of working with particles, works with fields. A field has values at each point in space and allows local interactions. However, it doesn't enforce locality. It is only local because we choose equations of motion for our fields that are relativistic. If you wish you can write a non-relativistic Schroedinger equation in QFT and do non-relativistic calculations which will no longer obey locality.

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Pseudo- apparent "Long-distances" correlations exist in statistical classical physics too.

In fact, if you consider a system, which at some time, for some reason, in which two subsystems (of this system) are being locally correlated, and supposing, that, in the future, these two sub-systems are spatially separated , the correlations between these sub-systems continue to exist (these correlations are completely independent of the distance between the sub-systems), while no information or energy can be sent instantaneously from one sub-system to the other.

So, correlations have nothing to do with non-locality, in statistical classical physics, in Quantum mechanics, or in Quantum Field Theory. And Quantum mechanics and QFT respect causality/locality

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