Why is Quantum Field Theory Local?

[A. Neumaier]

There are many such theorems. Which one are you referring to?

The classic is by Currie, Jordan and Sudarshan.

 

[A. Neumaier]

So by assuming non-existence of wave properties, you find that there is no interference. It's logically correct, but in my opinion too trivial to be interesting.

? Existence or nonexistence does not figure in my proof.

Like Bell I assume some natural properties and prove that they imply things contradicting quantum mechanics.

Just as Bell in his work; his local hidden variable objects cannot interfere either. Hence in your opinion, his work should be equally uninteresting.

Interference is the reason why a Bell inequality can be violated.

Bohmian theory is not classical in that sense, so your analysis does not exclude the Bohmian interpretation.

Why do you mention this triviality?

Bohmian theory is also not excluded by Bell's work.

 

[vanhees71]

The classic is by Currie, Jordan and Sudarshan.

Yes, and the no-go theorem holds for any finite number of degrees of freedom:

https://link.springer.com/article/10.1007/BF02749856

So the "natural" relativistic dynamics is field-theoretical rather than point-particle like.

 

[A. Neumaier]

Actually, this is more or less done in the book by Mandel and Wolf cited in post #154.

However, their exposition is not complete enough to enable me to gain full understanding of what is happening, and why. The reason is that there is no connection between the QFT discussion of coincidence counts using 2-point correlations in Chapter 14 and the nonrelativistic QM discussion of Bell's inequality in Chapter 12.14. A good exposition should connect the two. In particular, what is missing is a discussion of how the two dichotomic observables $A(a)$ and $B(b)$ introduced in Section 12.4.2 are realized in QFT. They are informally postulated but nowhere shown to exist in terms of the QFT machinery introduced.

I am convinced that any construction of these would reveal that they are horribly nonlocal expressions in the quantum fields. This would constitute the natural explanation why Bell nonlocality is experimentally seen.

 

[A. Neumaier]

Yes, and the no-go theorem holds for any finite number of degrees of freedom:

https://link.springer.com/article/10.1007/BF02749856

So the "natural" relativistic dynamics is field-theoretical rather than point-particle like.

Yes, the only well-defined one.

 

[Demystifier]

Just as Bell in his work; his local hidden variable objects cannot interfere either.

I don't think it's true. As far as I can see, his proof does not have an analog of the first line in your 3-line equation.

Let me also add that the experiment you describe can be explained by a (Bell) local hidden-variable theory. Namely, 1-particle Bohmian mechanics is a local theory.

 

[vanhees71]

However, their exposition is not complete enough to enable me to gain full understanding of what is happening, and why. The reason is that there is no connection between the QFT discussion of coincidence counts using 2-point correlations in Chapter 14 and the nonrelativistic QM discussion of Bell's inequality in Chapter 12.14. A good exposition should connect the two. In particular, what is missing is a discussion of how the two dichotomic observables $A(a)$ and $B(b)$ introduced in Section 12.4.2 are realized in QFT. They are informally postulated but nowhere shown to exist in terms of the QFT machinery introduced.

I am convinced that any construction of these would reveal that they are horribly nonlocal expressions in the quantum fields. This would constitute the natural explanation why Bell nonlocality is experimentally seen.

But isn't it enough that the correlators describe the experiments correctly, which violate Bell's inequality? Also, if the however constructed observables are nonlocal, then they are not in accordance with Bell's class of "local realistic" HV theories.

Unfortunately I don't have Mandel and Wolf's book at hand. I've to check the details tomorrow.

 

[Demystifier]

Yes, the only well-defined one.

String theory can also describe interactions in a Lorentz covariant manner.

 

[A. Neumaier]

Bell's argument is applicable to any local beables, ...

In Bell's words (Speakable and unspeakable..., p.8/9): 'That so much follows from such apparently innocent assumptions leads us to question their innocence. Are the requirements imposed, which are satisfied by quantum mechanical states, reasonable requirements on the dispersion free states? Indeed they are not. [...] The danger in fact was not in the explicit but in the implicit assumptions. It was tacitly assumed that [...]''

Indeed, Bell tacitly assumes a lot - whenever he talks about beable he assumes that all beables are local!

Since any local hidden variable theory contains lots of nonlocal beables, the experimental violation of Bell's inequalities does not rule out local hidden variable theories. It only proves that the beables measured are not local.

Indeed, my paper exhibited an explicit case of a local hidden variable field theory that violates Bell inequalities - by the same arguments used in the paper to show the violation of (2).

... local beables, namely variables defined on spacetime positions. This includes both pointlike particles and fields. (But it excludes multi-local beables that appear in your thermal interpretation.)

Bell's argument proves that bounded local beables in local hidden variable theories satisfy a Bell inequality. This includes both pointlike particles and fields. (But it excludes multi-local beables that appear in my thermal interpretation.)

However, local beables are not characterized by being variables defined on spacetime positions. There are many possible theories with nonlocal fields defined on spacetime positions, the prime example being Newtonian gravity.

Instead, local beables are characterized by Bell's locality definition - that they are influenced only by their past light cone. This is a much more restrictive condition!

 

[A. Neumaier]

I don't think it's true. As far as I can see, his proof does not have an analog of the first line in your 3-line equation.

He adds up probabilites in (6) p.37 of 'Speakable...' This is not permitted if there is interference.

Let me also add that the experiment you describe can be explained by a (Bell) local hidden-variable theory. Namely, 1-particle Bohmian mechanics is a local theory.

The free Maxwell field discussed in my paper predated quantum mechanics, is also a (Bell) local hidden-variable theory, and also explains the experiment.

 

[A. Neumaier]

String theory can also describe interactions in a Lorentz covariant manner.

The no-go theorem is about classical theories!

 

[A. Neumaier]

But isn't it enough that the correlators describe the experiments correctly, which violate Bell's inequality?

Whether it is enough depends on the ambitions.

What I want to understand is not that Bell-nonlocality is predicted and observed but how this is related to causal locality as postulated by the commutation relations.

Also, if the however constructed observables are nonlocal, then they are not in accordance with Bell's class of "local realistic" HV theories.

No. They are in accordance with Bell's class of "local realistic" HV theories but not in accordance with his tacit assumption that all measurable beables are local. See my post #149.

 

[Demystifier]

He adds up probabilites in (6) p.37 of 'Speakable...' This is not permitted if there is interference.

That's different. He integrates over $\lambda$, which, by definition, is averaging over all hidden variables. That's just an application of Kolmogorov probability axioms and has nothing to do with absence of interference. You, on the other hand, integrate over all hidden variables $\lambda$ (which is OK), but in addition sum over $k$, $k=1,2$. It is this summation over $k$, not present in the Bell case, that is related to absence of interference.

What's the difference between summation over $\lambda$ and summation over $k$? In any run of an experiment, $\lambda$ has only one value, but you don't know which one, so to make statistical predictions you average over all possible values. On the other hand, in your experiment both arms of the apparatus are present at ones, so $k$ takes both values. Hence summation over $k$ is not merely a statistical averaging. Instead, it corresponds to a physical assumption that the overall effect of two arms of the apparatus is a sum of their individual effects. It's a reasonable assumption as you say, but it's not an assumption that can be derived from Kolmogorov probability axioms. Hence this assumption is much less innocent and much more questionable.

 

[A. Neumaier]

That's different. He integrates over $\lambda$, which, by definition, is averaging over all hidden variables. That's just an application of Kolmogorov probability axioms and has nothing to do with absence of interference. You, on the other hand, integrate over all hidden variables $\lambda$ (which is OK), but in addition sum over $k$, $k=1,2$. It is this summation over $k$, not present in the Bell case, that is related to absence of interference.

That's not different. The hidden variables determine which k is used by the particle, and the probability for the other k is simply zero. The summation over k just simplifies writing this down.

Independent of that, what do you mean by interference of particles in a hidden variable model? I have never seen anything like that, so you should explain your terminology.

 

[Demystifier]

what do you mean by interference of particles in a hidden variable model?

I never talked of interference of particles. In Bohmian mechanics, for instance, it is waves that interfere.

 

[Demystifier]

That's not different. The hidden variables determine which k is used by the particle, and the probability for the other k is simply zero. The summation over k just simplifies writing this down.

OK, but then you assume that your hidden variables are "classical", in a sense in which Bell doesn't assume.

 

[A. Neumaier]

OK, but then you assume that your hidden variables are "classical", in a sense in which Bell doesn't assume.

Classical = satisfy the rules of probability theory,and that the hidden variables determine every physically relevant fact. Just as Bell does.

You can substitute Bell's argumens and results for mine, and they still apply in my setting.

 

[A. Neumaier]

I never talked of interference of particles. In Bohmian mechanics, for instance, it is waves that interfere.

So it is also in my Maxwell explanation - it corresponds to Bohmian theory in Bell's papers.

But this is irrelevant in Bell's derivation, which does not apply to the Bohmian theory, as you well know.

 

[Demystifier]

Classical = satisfy the rules of probability theory,and that the hidden variables determine every physically relevant fact. Just as Bell does.

No, you have an additional assumption of classicality. You assume that your hidden variables involve only particles and not waves. Bell's local hidden variables are much more general than that, in particular they allow a possibility that each particle is guided by its own wave (without entanglement), in a local manner.

 

[Demystifier]

But this is irrelevant in Bell's derivation, which does not apply to the Bohmian theory, as you well know.

Not quite true. The Bell's assumption that hidden variables are local does not exclude the possibility of single-particle Bohmian mechanics, or many-particle Bohmian mechanics without entanglement.

 

[vanhees71]

But how can it then be that Bohmian mechanics makes precisely the same predictions as QT (as far as observable entitities are concerned, while the Bohmian trajectories are not observable) including the violation of Bell's inequality? The answer is that Bohmian mechanics is not local!

 

[Demystifier]

@A. Neumaier perhaps the crucial observation on your experiment is this. In your experiment, there are no correlations between spatially separated measurement outcomes. So, no matter how one interprets your experiment in terms of hidden variables, the experiment itself is not an evidence of nonlocality. Hence the fact that you ruled out one local theory but explained it with another local theory is not an indication that all experiments (in particular those that do involve correlations between spatially separated measurement outcomes) can be explained by a local theory.

 

[A. Neumaier]

@A. Neumaier perhaps the crucial observation on your experiment is this. In your experiment, there are no correlations between spatially separated measurement outcomes. So, no matter how one interprets your experiment in terms of hidden variables, the experiment itself is not an evidence of nonlocality.

Thus it disproves (as claimed in the abstract) the hidden variable particle concept even without assuming locality.

Because of that, polarization is in Bohmian mechanics not a beable, as you observed in this post:
''In the Bohmian interpretation it means that electron, as a pointlike particle, always has a position and never has a spin. When we measure spin, we don't really measure a property of the electron alone, but a property that can be attributed to the electron and the apparatus together.''

Point particles are simply a defective, idealized notion, as also seen in the many instances discussed in

• J.C. Baez, Struggles with the Continuum, arXiv:1609:01421.

 

[A. Neumaier]

You assume that your hidden variables involve only particles and not waves.

Yes, that's the whole point of my paper.

 

[AndreasC]

I like how every thread eventually descends into a Demystifier vs A. Neumaier debate sooner or later lol

 

[A. Neumaier]

every thread eventually descends into a Demystifier vs A. Neumaier debate sooner or later

only with a very diluted notion of 'every' ... (count the number of threads in Quantum Physics, and work out the ratio!)

 

[vanhees71]

Thus it disproves (as claimed in the abstract) the hidden variable particle concept even without assuming locality.

Because of that, polarization is in Bohmian mechanics not a beable, as you observed in this post:
''In the Bohmian interpretation it means that electron, as a pointlike particle, always has a position and never has a spin. When we measure spin, we don't really measure a property of the electron alone, but a property that can be attributed to the electron and the apparatus together.''

Point particles are simply a defective, idealized notion, as also seen in the many instances discussed in

• J.C. Baez, Struggles with the Continuum, arXiv:1609:01421

But this is utter nonsense, because the polarization of electrons and other elementary or composed particles are observables (I don't care about strange philosophical buzz words like "beables"; for me there are observables, and they are defined by a quantity that can be measured), and as any observable it's defined by (an equivalence class of) measurement procedures (e.g., the just now very much discussed (g-2) measurement on anti-muons at Fermilab and hopefully soon also at Jefferson lab. Another fascinating example are polarization measurements on $\Lambda$s in semi-central heavy-ion collisions hinting at an enormous vorticity of the created strongly interacting medium.

Of course, from the most fundamental physical theory, which is local relativistic QFT, it's true that a naive interacting-point-particle description is impossible. That's not surprising, because it's already impossible within classical relativistic physics! Within QFT a particle (or rather particle-like) interpretation of certain states of the quantized fields as "particles" that are to some limited extent localizable are asymptotic free one-particle Fock states. However, polarization (helicity for massless and spin components for massive) is anyway a quantitity that makes much more physical sense as a concept within field theory than within point-particle theory.

 

[A. Neumaier]

But this is utter nonsense, because the polarization of electrons and other elementary or composed particles are observables

I agree that Bohmian mechanics (and every interpretation of quantum mechanics or classical relativistic mechanics) that features point particles) is utter nonsense, and gives only a make-believe interpretation. The unreal spin admitted by @Demystifier and the lack of Lorentz covariance of Bohmian theories are vivid example of this.

 

[AndreasC]

only with a very diluted notion of 'every' ... (count the number of threads in Quantum Physics, and work out the ration!)

I was being hyperbolic. It's interesting though.

 

[AndreasC]

It should also be noted that I said "sooner or later". So perhaps the rest of the threads just haven't had enough time yet!

 

[Demystifier]

Point particles are simply a defective, idealized notion

For Bohmian mechanics it's not important that the particles are exactly pointlike. If you like they can be balls of the Planck size, it doesn't change anything important.

 

[Demystifier]

Yes, that's the whole point of my paper.

So you rule out something that nobody believed in the first place.

 

[Demystifier]

I like how every thread eventually descends into a Demystifier vs A. Neumaier debate sooner or later lol

There are many threads, for instance, where @A. Neumaier and me were together against @vanhees71 .

 

[vanhees71]

For Bohmian mechanics it's not important that the particles are exactly pointlike. If you like they can be balls of the Planck size, it doesn't change anything important.

I'd indeed say that Bohmian mechanics in its version for non-relativistic QM provides not so much a point-particle but rather a hydrodynamical picture. The Bohmian trajectories are contructed after all from the quantum-mechanical probability current $\vec{j} = -\frac{\mathrm{i}}{2m} (\psi^* \vec{\nabla} \psi - \psi \vec{\nabla} \psi^*)$. That's a continuum-mechanical rather than a point-particle mechanical idea. That underslines the fact that the single-particle Bohmian trajectories are not observable but the hydro-like flow pattern from averaging over many single-particle trajectories. This is equivalent to the standard statistical interpretation. So at the end the Bohmian trajectories are nothing that needs to be even calculated to confront QT with experiment and thus are simply a superfluous addition from a physics point of view.

It's not so much the Planck scale that's important here but rather the de Broglie wave lengths of the involved particles since within relativistic QT the uncertainty principle for position and momentum as well as energy and time as defined via the momentum of the particles (which in general makes sense for massive fields only of course) leads to the conclusion that you can localize a particle only within a volume at a length scale of the de Broglie wavelength $h/(mc)$.

Another independent argument, working also within non-relativistic QM, is to consider the Wigner function, which is the closest thing to what's in classical physics is a phase-space-distribution function, but is not positive semidefinite. To get a true phase-space distribution function in the sense of a classical approximation you have to smear the Wigner function out ("coarse graining") over phase-space dimensions given by $\hbar$, which makes sense, because you can only determine the phase-space position of a point particle in non-relativistic QT within a box with volume $\gtrsim \hbar^3$ (given the uncertainty relation $\Delta x \Delta p_x \geq \hbar/2$.

 

[AndreasC]

There are many threads, for instance, where @A. Neumaier and me were together against @vanhees71 .

I call that "anomalous vanhees effect".