Graduate Why is Quantum Field Theory Local?

[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".

 

[A. Neumaier]

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

That's the case for most theorems.

 

[A. Neumaier]

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.

Then they can even be football size, since they are unobservable, and only their center of mass appears in the equations.

But they are point particles in all publications on the matter.

 

[Demystifier]

But they are point particles in all publications on the matter.

So are the planets in Newtonian mechanics.

 

[A. Neumaier]

So are the planets in Newtonian mechanics.

Yes, and Newtonian mechanics has the typical resulting defects: It can be formulated only as nonrelativistic theory, and has problems with collision trajectories (see the paper by Baez). Just like Bohmian mechanics.

For more than a century we are past this state of affairs.

 

[Kolmo]

It shows such theories, as a class, allow for many features of QM, with QM perhaps the simplest

Reading this thread a while later, if you find it interesting quantum theory as a GPT (Generalized Probability Theory) can be characterized in two ways:

(a) As the most general one satisfying the Exclusion principle. Namely that if each of the pairs from a set of observables $A,B,C$ are compatible/co-measurable, then the whole set is co-measurable

(b) The most general one that permits Bayesian updating

"Most general" here means "has the broadest set of possible correlations". So in a CHSH test classical probability gives $2$ as the bound and QM gives $2\sqrt{2}$, then any theory with correlations beyond $2\sqrt{2}$ breaks both (a) and (b).

Regarding (b), all GPTs allow updating but by "Bayesian" we mean there is a unique way to update in late of data. In GPTs going beyond the Tsirelson bound $2\sqrt{2}$ there is an element of arbitrary choice in how one updates in light of data. This is what leads to a recent phrase: it's the most general GPT where one can still learn.