# Locality in QFT vs "non-local" in QM

In quantum mechanics (QM), teacher always emphasizes on the "weird" parts, like EPR paradox, Bell inequality and so on. The Bell inequality tells us that QM is either nonlocal or non-realistic or both.

However in quantum field theory (QFT), teacher says that physics requires locality and causality, and never mentions that the "nonlocality" or "nonrealisitc". While QFT is also a quantum theory, is there some contradiction of locality requirement of QFT with nonlocal in QM? Or does it mean that locality of QFT just implies that QM is local and nonrealistic?

The locality of a QFT refers to the operator algebra. The (non-) locality of Bell's theorem refers to the states (rays) of the Hilbert space. These are different notions of locality, and they coexist peacefully.

To quote, Fredenhagen1

Apart from these problems, there is a deeper reason why it is fortunate to separate the construction of observables from the construction of states. This is the apparent conflict between the principle of locality, which in particular governs classical field theory, and the existence of nonclassical correlations (entanglement) in quantum systems, often referred to as non-locality of quantum physics. As a matter of fact it turns out that the algebra of observables is completely compatible with the locality principle whereas the states typically exhibit nonlocal correlations.

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1 An Introduction to Algebraic Quantum Field Theory, from the book Advances in Algebraic Quantum Field Theory.

• Note to self: remove the comma after "quote". Nov 6, 2017 at 22:09
• So why must we require that operator algebra must be local but the state can be nonlocal? Nov 7, 2017 at 4:03
• @AccidentalFourierTransform With all due respect, saying that "that is an axiom" is like saying nothing at all. Of course, a mathematician would never ask why an axiom. But from a physicist's point of view, why do we assume that the operators cannot have spacelike correlations whereas states can? Why characterize causality using operators rather than states? Mar 13, 2018 at 13:48
• @AccidentalFourierTransform Yes, motivation! Vague is fine, so long as it helps understanding. I understand that QFT is built on a local algebra of operators, and it works. But I would like to know what made people think oh, let's forbid non-locality in operators and see what happens, although the states are non-local: let them be. Surely, the decision was not ad-hoc and the founders of QFT had reasons to believe that operators could be local, whereas states not. Axioms come after intuition, not the other way around, at least for physicists I would think. Mar 13, 2018 at 14:07
• @AccidentalFourierTransform Let me see if I got this right. So, locality of operators is another name for relativistic causality, whereas non-locality of states has nothing to do with causality. That makes more sense. Yes, I agree that at some point, even physicists have to categorically agree on some basic axioms to build more complicated theories upon that foundation. But some axioms can be unclear and a little explanation can go a long way. Thanks a lot. :) Mar 13, 2018 at 14:20

Yes, QFT is a subset of QM theories – quantum field theories are theories whose observables are naturally constructed from field operators – so everything that holds for all QM theories holds for QFTs, too. In particular, all QFTs are "non-realist" because all QM theories are "non-realist".

QM theories may be in principle non-local but the QM theories relevant for the description of our Universe are local. They're either QFTs or their generalizations like string theory. The locality means that an event or a measurement can never impact or change the probabilities of events that are spacelike-separated. Mathematically in QFTs, this is derived from the vanishing (graded) commutator of fields at spacelike separations.

Everyone who says that there is non-locality in our Universe in any sense is just plain wrong – violates elementary insights of the special theory of relativity of 1905. There is no non-locality in Nature. One may consider hypothetical theories in QM which are non-local but those aren't relevant for our world. In particular, the non-locality in these theories has nothing whatever to do with the right explanation of entanglement experiments.

• How do you see the non local "nature" of field quanta in the sense that they are said to appear or disappear in space instantaneously in this respect? Jan 6, 2018 at 12:57
• There is nothing nonlocal about field quanta in our Universe - or any other object in the Universe. They are not appearing or disappearing instantaneously. Instead, what happens when one entangled field quantum is observed is that the observer learns something about it - and because of the known entanglement (quantum-described correlation), he learns something about the other, too. The other quantum wasn't "absent" or "present" before the measurement - so it didn't appear or disappear. Instead, its properties were unknown to the observer, and they became known. Jan 10, 2018 at 15:35
• The instantaneous change is only the change of the observer's knowledge, and it's a change that takes place inside his brain or mind, which is a localized object. For the information to be fully absorbed etc., the electric pulses have to travel by the speed of light or slower. But the observation of particle A in an entangled pair doesn't affect B - it doesn't change any probabilities of B itself in the absence of the knowledge of the result of measurement of A. The knowledge of the A result changes predictions for B but that's not action at a distance, it's just a proof of a correlation. Jan 10, 2018 at 15:37
• @LubošMotl The viewpoint you take above is usually called "$\psi$-epistemic," and this type of interpretation has been essentially ruled out by the PBR theorem: en.wikipedia.org/wiki/PBR_theorem Apr 22, 2021 at 21:48
• Out of curiosity, do you still hold by this interpretation, where collapsing the wavefunction only updates an observer's knowledge and not something objective about nature? If so, I'm interested in how you reconcile it with PBR. Apr 22, 2021 at 21:50