Resolving the Paradox: Combining Quantum Mechanics and Special Relativity

[Helena Wells]

According to Bell's theorem quantum mechanics is not local.How can we combine it with Special Relativity which is local and gives us another successful theory?

 

[PeroK]

According to Bell's theorem quantum mechanics is not local.How can we combine it with Special Relativity which is local and gives us another successful theory?

There's a good answer here:

https://physics.stackexchange.com/questions/76036/how-does-qft-help-with-entanglement

 

[Helena Wells]

There's a good answer here:

https://physics.stackexchange.com/questions/76036/how-does-qft-help-with-entanglement

*__*

 

[Vanadium 50]

According to Bell's theorem quantum mechanics is not local.

That's not what it says.

 

[PeterDonis]

According to Bell's theorem quantum mechanics is not local.

It depends on what you mean by "local". Bell gave a specific condition for "locality", namely that the joint probability function factorizes; all quantum models, including QFT, violate that condition.

However, quantum field theory, which combines QM with special relativity, defines "locality" a different way; as shown in the Stack Exchange answer linked to, the QFT definition of "locality" is that measurements at spacelike separated events commute--the results don't depend on the order in which they're made (which makes sense since the order of spacelike separated events is frame-dependent anyway). This kind of "locality" is perfectly consistent with violations of the Bell inequalities, so it is "nonlocal" in Bell's sense. So there is no contradiction anywhere, just different meanings given to the term "local".

 

[DrChinese]

According to Bell's theorem quantum mechanics is not local.How can we combine it with Special Relativity which is local and gives us another successful theory?

A more general description of the meaning of Bell's Theorem:

No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

You will quickly find that although QM appears to have nonlocal elements (sometimes called "quantum nonlocality"), there is no one specific element of relativity that is in opposition to QM. For example, there is no FTL signaling. There are no particles moving FTL. Etc.

For those elements that appear to violate the spirit of locality: there are quantum interpretations that address this. Check out the Quantum Interpretations and Foundations subforum to learn more about those.

 

[atyy]

According to Bell's theorem quantum mechanics is not local.How can we combine it with Special Relativity which is local and gives us another successful theory?

Classical special relativity has (at least) two notions of locality: relativistic causality and local causality. Bell's theorem says that quantum phenomena are incompatible with local causality, but does not rule out compatibility with relativistic causality (terminology varies, I follow https://arxiv.org/abs/1503.06413). QFT may be considered nonlocal in the sense that it lacks local causality, but it is still local in the sense of preserving relativistic causality. Roughly, although one may imagine that nonlocal things do happen in QFT, these nonlocal things do not allow faster-than-light communication.

 

[PeterDonis]

Bell's theorem says that quantum phenomena are incompatible with local causality, but does not rule out compatibility with relativistic causality (terminology varies, I follow https://arxiv.org/abs/1503.06413).

Looking at the terminology in the paper, the key "locality" criterion (the one I described in post #5 and the one that implies the Bell inequalities, and which is violated by QM) is the one the paper calls "Bell-local" (Definition 6, p. 9). As for "relativistic causality", that condition, as the paper defines it (Postulate 2, p. 12), is weaker than the "QFT locality" condition I gave in post #5--it only says that spacelike separated events cannot be the cause of each other, not that they must commute.

 

[Demystifier]

According to Bell's theorem quantum mechanics is not local.How can we combine it with Special Relativity which is local and gives us another successful theory?

As others told you, there are different notions of "locality". QFT is local in one sense but nonlocal in another.

One way to put it is as follows. The QFT Hamiltonian is local, so the time evolution governed by the Hamiltonian is local (and deterministic!). However, there is more about QFT then deterministic Hamiltonian evolution of the state. There are also measurement outcomes that seem nondeterministic. How exactly those measurement outcomes appear is not entirely clear, but nonlocality is associated with those appearances of the measurement outcomes.

 

[atyy]

Looking at the terminology in the paper, the key "locality" criterion (the one I described in post #5 and the one that implies the Bell inequalities, and which is violated by QM) is the one the paper calls "Bell-local" (Definition 6, p. 9). As for "relativistic causality", that condition, as the paper defines it (Postulate 2, p. 12), is weaker than the "QFT locality" condition I gave in post #5--it only says that spacelike separated events cannot be the cause of each other, not that they must commute.

I guess the commutation of spacelike-separated events would satisfy both "locality" (Definition 3) and "signal locality" (Definition 4). And yes, relativistic locality is weaker than locality - looking at their final figure, it looks like agent causation must be added to relativistic causality to obtain locality.

 

[Vanadium 50]

This thread is going down the interpretations rabbit hole. (And why not? There are dead horses to be beaten!) However it is important to read Bell carefully. He doesn't specify what is - he specifies what isn't: i.e. it's a discussion of what is not possible.