My question is probably naive to the experts, but I am curious about the answer. In Weinberg's "Quantum theory of fields", vol. 1, QFT is motivated first by some general principles, such as Lorentz invariance and the cluster decomposition principle.

My question is this. If distant experiments are not (or barely) correlated, why doesn't that contradict the famous thought-experiment of having 2 correlated electrons which are then separated very far from each other, and yet remain correlated (I guess the right technical word is entangled).

So in short, why doesn't entanglement and the cluster decomposition principle contradict each other? I am, most certainly, misunderstanding something, so I hope someone may take the time to clear my confusion.

  • $\begingroup$ @Countto10 Thank you. I think I should have looked more carefully in Weinberg's book, vol. 1, for the answer, because it is indeed there, as Luc J. Bourhis kindly pointed out. That being said, it is almost a 600-page book, which I have in hard copy (not electronic), so it is not easy to search for something that specific. $\endgroup$
    – Malkoun
    Aug 3 '17 at 13:15
  • $\begingroup$ You have these already I would say, but free online is Mark Srednicki : web.physics.ucsb.edu/~mark/qft.html, David Tong : damtp.cam.ac.uk/user/tong/qft.html and Susskind on YouTube ( I think). These are not at Weinberg's level, (and I personally am not any level compared to anyone here), but just on the off chance they cover the same ground (or SW is assuming you are nearly as good as he is, so he is unintentionally obscure). Also, search for lecture notes, apologies if none of this is new to you. Best of luck with it. $\endgroup$
    – user163104
    Aug 3 '17 at 13:55
  • $\begingroup$ @Countto10, I know what people say about Weinberg's book. My advice though, after some exposure to the subject, definitely read Weinberg! It is full with some statements that make what it is you are doing much clearer, some little gems. A simple example of that: why are C, P and T transformations important? Well a study of Lorentz group can be reduced to a study of invariance under the time-preserving, spatial orientation preserving Lorentz subgroup, and P and T. Mathematically simple as a statement, but it opens one's eye as to why P and T transformations are important. Read Weinberg. $\endgroup$
    – Malkoun
    Aug 3 '17 at 14:13
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    $\begingroup$ As another interesting "gem" from Weinberg's book, which I have yet to understand, if you write a Hamiltonian using creation and annihilation operators, then cluster decomposition becomes satisfied. According to Weinberg, this is the deep reason why they are so useful in QFT. This is rarely covered this way in other presentations. $\endgroup$
    – Malkoun
    Aug 3 '17 at 16:01

Weinberg actually answers your question in the discussion around eq. (4.3.9): it is required that the connected part of the S-matrix in momentum space contains a single momentum-conservation delta function. An entangle state has additional delta functions, either on momentum in the original EPR thought experiment, or on spins for experimentally realised ones. Thus, they indeed violate the cluster composition principle. But that's all right: that principle is not required to apply to any ensemble of particles. I mean, you can surely see the difference between the fact that the production of a muon produced in Fermilab and one produced at CERN satisfies the cluster decomposition principle: no history in common. On the contrary, entangled photons, ions or electrons have been at the same location at some points (ignoring the like of entanglement swapping to keep this answer tight).

  • $\begingroup$ No worries, and don't get scared by the downvote bot. $\endgroup$
    – user154997
    Aug 3 '17 at 11:56

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