Do nonlocal entities fulfill assumptions of Bell theorem?

While dynamics of (classical) field theories is defined by (local) PDEs like wave equation (finite propagation speed), some fields allow for stable localized configurations: solitons.
For example the simplest: sine-Gordon model, which can be realized by pendula on a rod which are connected by spring. While gravity prefers that pendula are "down", increasing angle by 2pi also means "down" - if these two different stable configurations (minima of potential) meet each other, there is required a soliton (called kink) corresponding to 2pi rotation, like here (the right one in traveling - is Lorentz contracted):

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Kinks are narrow, but there are also soltions filling the entire universe, like 2D vector field with (|v|^2-1)^2 potential - a hedgehog configuration is a soliton: all vectors point outside - these solitons are highly nonlocal entities.
A similar example of nonlocal entities in "local" field theory are Couder's walking droplets: corpuscle coupled with a (nonlocal) wave - getting quantum-like effects: interference, tunneling, orbit quantization (thread http://www.sciforums.com/threads/ho...e-duality-of-couders-walking-droplets.113148/ ).
The field depends on the entire history and affects the behavior of soliton or droplet.
For example Noether theorem says that the entire field guards (among others) the angular momentum conservation - in EPR experiment the momentum conservation is kind of encoded in the entire field - in a very nonlocal way.

So can we see real particles this way?
The only counter-argument I have heard is the Bell theorem (?)
But while soliton happen in local field theories (information propagates with finite speed), these models of particles: solitons/droplets are extremaly nonlocal entities.

In contrast, Bell theorem assumes local entities - so does it apply to solitons?

 

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Classical field theories are, with or without soliton-like solutions, usually local field theories. A complete solution of such local field equations has to be, of course, defined everywhere, and is in this sense a "global" solution, but this does not mean that it is somehow related with global effects.

The violation of Bell's inequalities is a quantum effect. And it shows that there exists a preferred frame - with the only alternative to give up realism and causality completely, or to distort them in such a way that the "weakened" notions of realism and causality have nothing to do with common sense ideas about reality and causality. In particular, one would have to reject Reichenbach's principle of common cause, which requires that any nontrivial correlation we observe needs a causal explanation, or by one thing causing the other, or by some common cause.

This does not mean that one really has to give up such common sense principles. If one prefers to believe them, all one has to accept is that there exist faster than light causal influences. This is what is assumed in completely classical, non-relativistic theories like Newtonian gravity too, thus, in no way counter-intuitive.

 

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Apart from the evident implication of a dual-ether (speed limit c in 'ordinary' ether, speed limit ? in 'other' ether), there are other issues with superluminal signalling.
Cut and paste from what I wrote in another forum:

"The usual argument goes that QM demands giving up either locality or realism or both. Without getting into the statistics side of things which has been thrashed to death here, one real puzzle for me at least is the implicit notion of FTL 'signalling' as possible explanation for correlations if one retains realism but gives up non-locality. But how could any signalling between members of an 'entangled pair' plausibly work - even for time-like separation scenarios where causality is taken to hold?

Any physically real signal must surely embody thus require expenditure of energy-momentum by the signalling particle(s). Yet afaik there is never any talk of a concomitant back reaction on signalling particles that would alter said particle dynamics accordingly. Why is that? It seems especially troublesome to postulate 'signals' between photons - given photons are themselves normally the signalling medium.

Further, how could any signalling not, at least statistically, reasonably be allowed to propagate other than as a 'dumb' spherical wavefront? With the attendant rapid attenuation in intensity (at least as 1/r) surely rendering such useless as causative agent of QM correlations except over small spatial separations.

Apparently then one need implicitly assume a 'smart' signalling strategy where somehow each 'signal' knows exactly where to track the other particle. And further, since distance cannot, as per last point, be a factor, such 'signals' must not diminish with distance. Which implies propagation via some truly weird 1-D 'transmission-line' linking the entangled pair to any arbitrary separation. And presumably, additional to any signal(s) themselves, such trailed-out 'transmission-line' requires zero expenditure of energy-momentum resources to create! Or does one postulate a preexisting, super dense network of 1-D 'transmission-lines' that somehow have no physical effect other than to efficiently and on demand transmit 'signals' between entangled particles?

Am I being just too naive? Missing something obvious? Can someone point to where any of these evident issues have been sensibly addressed in the literature? If not, why is 'signalling' ever admitted as possibility at all?"

 

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I don't think you're missing anything here, and if you are, it's certainly not obvious. I've seen very little on the actual mechanics of quantum signalling, and what I have seen has been pretty underwhelming. (One that particularly sticks in my mind was from another thread - I think you may have posted the link, actually - in which the author concocted a detailed explanation for the double slit experiment, then addressed the issue of Bell's inequality by saying "Eh, we can always postulate superdeterminism.") I think the main reason that people still consider superluminal signalling is that giving up realism is just too radical an option for some. I mean, depending on one's perspective, one might say all of science is based on the notion that the universe does its thing and our goal is to figure out how and why it does what it does. On some level, giving up realism means abandoning that framework. I can definitely understand why some people would rather appeal to a weirdly-behaved, ill-defined "signalling" mechanism.

 

Sure classical field theories are "local" in sense their dynamics is governed by local PDEs (Euler-Lagrange).
However, their solitons, which would represent particles, are very nonlocal entities ... but all information propagates with finite speed.
Like in the photo above - if soliton gets speed, it becomes Lorentz contracted - up to completely flat (and having infinite energy) while approaching the propagation speed of the field.
So while a single hedgehog soliton in 2D vector field with (|v|^2-1)^2 potential is present in the entire Universe - exactly like electric of a charge (E ~ 1/r^2), the only way to add charge somewhere is by creation of charge-anticharge pair, information about which propagates with finite velocity.

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And Couder's droplets, which are kind of solitons in classical field theory, undergo interference in double slit experiment, tunneling, orbit quantization


Returning to Bell theorem, it assumes particles as local entities - each one has a local spin and that's all ...
In contrast, here we additionally have the entire field for example guarding conservation of angular momentum in EPR due to Noether theorem - this information kind of fills the Universe with speed of light.
How to prove Bell theorem for solitons?

Sadly it is very difficult to test Bell inequalities for solitons as EPR rather requires 3D for directions of spin (?)
We would need for example created pair of vortices of opposite direction (due to angular momentum conservation) and perform measurement through some kind of Stern-Gerlach ...
Have you maybe any concept how one could experimentally test Bell inequalities for solitons?

 

Looks like my posts got lost in the hard drive crash, so I'll repost the short version. When people try to find ways around Bell's theorem, they usually focus on trying to recover quantization, spin, double-slit interference, and other traditionally "quantum" phenomena from a classical model. But this is not what Bell's theorem is about. Bell's theorem says that if Alice and Bob take two measurements in different parts of the universe, there is an upper bound to how correlated the results can be unless Alice's choice of measurement is allowed to affect Bob's outcome. It's not really even an argument about physics, so much as an argument about statistics. Since we measure correlations that violate Bell's inequality, we conclude that Alice's choice of measurement can in fact affect Bob's outcome, and any working model of the universe has to account for such. This excludes all local, realistic theories. Solitons are no exception, because no matter how spatially extensive their angular momentum is, no measurement on one part of a soliton can alter the outcome of another measurement on a distant part of the soliton.

 

Nice post Thanks.

 

NOTE: Following 6 posts are in-sequence restorations following crash, and originally were numbered #6-#11

Q-reeus, Nov 4, 9:03PM

Thanks for those observations Fednis48. Every now and then I toy with the frightening idea Nick Bostrom might be right - we are hapless participants in a computer simulation!

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Schmelzer, Nov 5, 2:27AM

First, the choice is simply between giving up to search for any causal realistic explanations or not giving up. If you give up to search for explanations, you have no problem at all with any violations of BI. Correlations schmorrelations, who cares about them. If you don't give up, you have no choice but to accept a causal explanation which contains FTL influences. Such is live.

Then, the problem you mention is not that hopeless, because there is such a strange thing as "quantum monogamy" http://www.quantiki.org/wiki/monogamy-entanglement which suggests that some explanation based on some 1-D 'transmission-lines' may be not completely off.

Anyway, the problems how to explain the correlations are secondary. Once one has accepted that they have to be explained, a first conclusion is for cheap: We need a preferred frame.

 

Jarek Duda, Nov 5, 4:13AM
So how far can Couder-like (soliton) models be in agreement with the microscopic physics?

In soliton particle models, the sine-Gordon (d_tt - d_xx = - sin(phi) ) itself has:
- multiple quantized particles/charges (alternative to QFT way to handle varying number of particles),
- they have mass due to Higgs-like potential (with topologically nontrivial minimum),
- this mass is released while annihilation of opposite charges: https://en.wikipedia.org/wiki/Topological_defect#Images
- there is Lorentz contraction and time dilation (for breathers): https://en.wikipedia.org/wiki/Breather
- mass/energy grows like gamma * m_0,
Sine-Gordon doesn't have long-range interaction, but it appears for 2/3D solitons - with electric charge modeled as topological charge (Gauss-Bonnet theorem is topological analogue of the Gauss law, but allows only integer charges), with regularized singularity (energy of electric field of point particle would be infinite).
slides: https://dl.dropboxusercontent.com/u/12405967/soliton.pdf

Does Bell theorem exclude soliton particle model for modelling the microscopic world?

 

Fednis48, Nov 5, 7:36AM

Yes, it does. Sorry. When trying to get around Bell's theorem, a lot of people tend to focus on how they can recover quantization, spin, double-slit interference, and other traditionally "quantum" phenomena from a classical theory. But that's not what Bell's theorem is about. Bell's theorem doesn't care what your model of the universe is based on naive hidden variables, extensive solitons, or the quaternionic spins that Joy Christian keeps advocating. It simply assumes that every part of the universe is in some state, and Alice and Bob take measurements that depend only on the local state of their part of the universe. In particular, the crucial part is that Bob's measurement outcome can't depend on which measurement Alice chose to make. This is certainly true for solitons: even if the soliton itself is spatially extensive, the part Bob sees does not change depending on how Alice chooses to measure a different, distant part. This assumption is usually described in shorthand as "locality", and as long as you follow it, Bell's theorem puts upper bounds on the correlations between Alice's and Bob's measurements. It's not really even an argument about physics so much as an argument about statistics. So when we do experiments that violate those upper bounds, we can be sure that no clever model will save us from having to abandon either realism or locality.

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http://www.smbc-comics.com/?id=2535

 

brucep, Nov 5, 12:06PM

Folks keep looking for that clever model. Very informative. Thanks.

 

Jarek Duda, Nov 5, 2:25PM

The Bell theorem I have seen is not that general.
Your "parts of universe" are very specific: they describe a local intrinsic property of a particle (spin), which finally undergoes the measurement process.
The measurement process is something extremely sophisticated: it takes any state and "projects" it (destroys information) into a discrete set of possibilities.

The best idealization of the measurement process I know is the Stern-Gerlach experiment, which can be seen classically:
Imagine particle as a tiny magnet (magnetic dipole moment we know they have) and make it travel in strong magnetic field.
If it is not aligned in parallel or anti-parallel way, there is enforced precession - oscillation of EM field which produces EM waves carrying energy out of the system.
Finally any random initial direction of spin is "projected" into two possibilities: parallel or anti-parallel alignment, which finally lead to different trajectories due to gradient.

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In Bell theorem it is crucial that the measured properties are local.
What is not true for solitons of a classical field theory - their intrinsic properties influence the entire field.
See Noether theorem for field theories: similarly to Gauss law guarding charge inside a surface, the entire field kind of magically encodes the guarded properties like spin conservation.
https://en.wikipedia.org/wiki/Noether's_theorem#Field_theory_version

Can one, as you require, get rid from Bell theorem of the assumption that the local properties are finally measured?

 

Wow, Q-reeus, thanks for salvaging the last few days of posts! I'll try to pick up where Jarek Duda left off:

If you're talking about how something "can be seen classically," you've already misunderstood Bell's theorem. Bell's theorem is, in fact, completely classical. It treats measurements not as projective operators, but simply as black boxes that take in the local state of the system and output a number. The Stern-Gerlach experiment is a great way to think about quantum measurements, which Bell's theorem never invokes. (All treatments of Bell's theorem I've seen do bring up quantum measurements, just to show that they lead to Bell inequality violations, but the theorem itself is perfectly complete without them.)

If you're not performing quantum projective measurements, it doesn't really mean anything to talk about whether intrinsic properties "influence the entire field" or not. A measurement on one part of the soliton cannot have any impact on the outcome of another, distant measurement performed at the same time, and that's all the locality that Bell's theorem needs. Of course, if you do allow quantum projection, a measurement on one part of the soliton can influence a distant part by "snapping" the whole wavefunction into the measured state; this is why quantum mechanics violates Bell's inequality.

 

Bell/CHSH inequalities are restrictions on measurement of entangled spins, assuming these spins, which correspond to angular momentum, are local properties.
However, like single charge is affecting the entire universe, the spin of particle in (local) field theory is also highly nonlocal property - due to Noether theorem (angular momentum conservation), this information is propagating with speed of light over the entire universe.

Could you please explain how do you conclude that Bell inequality should be fulfilled for soliton models - for example if there would be created a pair of opposite swirls (conserved total angular momentum of these constructs of field) and we would measure their angular momentum in some kind of Stern-Gerlach experiment, the obtained correlations would have to fulfill the Bell inequalities?

 

The short answer is that the properties in Bell's theorem only need to be "local" in the sense that changing the properties in one part of the universe can't instantaneously alter them in distant parts of the universe. Nothing in the theorem postulates that the properties have to be pointlike, or otherwise "local" in the sense of how far their influence extends.

Leaving out the math, the structure of Bell's theorem is as follows:
1. A two-particle state is produced, characterized by some random variable X. X can be anything - a scalar, a quaternion, a fancy field state, etc. - it just has to somehow capture the particles' properties.
2. Alice and Bob perform two-outcome measurements, Ma and Mb respectively, on their spins. In the canonical Bell experiment, Ma and Mb are spin measurements along some spatial axis, but the theorem isn't that specific.
3. Each experiment's outcome depends on the state of the particles and the choice of measurement performed, but not on the other experimentalist's choice of measurement. In the language of expected values, <Ma>=f(X,Ma), and likewise for Mb.
4. Using basic rules of conditional probability, we can put bounds on the correlations between measurements, <Ma*Mb>, assuming the measurements are independently and randomly chosen from shot to shot.
5. Experiments violate these bounds, so one of our premises must be wrong.
So which one is it? Any state can be described by some variable as long as it's general enough, and Alice and Bob demonstrably do perform two-outcome measurements, so (1) and (2) cannot be wrong. Various contrived loopholes like superdeterminism could mean that the measurements are not actually independently/randomly chosen, and other contrived loopholes like the detection loophole could mean that our experiments don't actually violate Bell's inequality, so (4) or (5) might be wrong. However, these options are all basically bending over backwards and assuming the laws of physics are conspiring against us, so they're more important to keep the theory rigorous than as serious possibilities. That leaves (3) being wrong: Alice's and Bob's measurement outcomes depend not only on their own choice of measurement, but on each others. This can only be true if one measurement sends a faster-than-light signal to modify the other (non-locality) or if the state is in a superposition that can be collapsed by the first measurement (non-realism). Treating the spins as solitons, by contrast, basically just amounts to modifying the nature of X, which doesn't affect the theorem at all.

If you want to look through the math, I'd suggest this page; it's a modernized version of Bell's theorem that smooths out some of the more difficult elements by using three particles instead of two.

 

And thanks for that thanks - a welcome change from what I've copped elsewhere.

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Thanks, Q-reeus.

Fednis, I still don't understand why solitons should fulfill Bell inequalities.

Instead of instant communication, in soliton picture the information is delocalized - fills the entire universe.
While e.g. creation of two opposite swirls, the information about their directions is not only in the centers of swirls, but due to Noether's conservation of angular momentum - this information propagates to infinity from the moment of pair creation - assuming lack of viscosity (superfluidity like in most field theories), this pair creation will finally cause rotation of the liquid/field in the entire space.
Spin is far non-local property here - finally the entire filed encodes the spin information.
Like in Gauss law: the field on a boundary encodes charge inside a region- making charge a delocalized property.

Regarding (super)determinism, I am talking about solitons in field theories governed by Lagrangian formalism (like from QFT to GRT) - which is completely deterministic for example by the Euler-Lagrange equation: determining the future (and past) by fixing values (of field) and velocities in some point in time.
An equivalent formulation is due to action optimization: with fixed only values but in two points in time - for example we can imagine the history of our universe as the action optimizing result of fixing it in both Big Bang in the past and (hypothetical) Big Crunch in the future.
It is like living in 4D jello (Einstein's block universe) - both past and future affect the present - which is kind of in tension minimizing equilibrium between both directions.
So there is no true randomness while measurement in Lagrangian mechanics - its result can be seen as determined by both past and future local situations, like in CPT conservation suggesting symmetry between past and future.

I personally think that this living in 4D spacetime is the reason for correlations of QM: which are non-intuitive for us, as we think from the perspective of evolving 3D.
Specifically, for my physics PhD I was working on Maximal Entropy Random Walk (thesis, slides) - it shows that the standard choice of diffusion models (GRW) often (in non-homogeneous space) only approximates the basic requirement for statistical physics models: maximization of entropy (or minimization of free energy).
Doing it right, MERW leads to the stationary probability distribution exactly like for thermal equilibrium predicted by QM: squares of coordinates of the dominant eigenvector of Hamiltonian.
For example standard random walk (GRW) would expect nearly uniform probability distribution for electrons in defected lattice of semi-conductor, making that a small potential difference would lead to flow of electrons - making it conductor.
In contrast, all three: nature (experiment), QM and MERW predict that electrons are (Anderson) localized, preventing from conductance.
Here is an example of comparison of their evolution for a lattice with defects (small squares denote missing node):

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So how MERW gets the Born rules (which lead to violation of Bell inequalities): probabilities as squares of amplitudes...
MERW can be equivalently formulated as uniform (or Boltzmann) distribution among paths - infinite toward both past and future (it is very similar to euclidean path integrals).
Amplitudes correspond to probability distribution on the end of ensemble of half-paths: from now toward the past (or toward the future).
Now while measurement, to randomly get a given value, we need to get it from both ensembles of half-paths: toward past and future, so the probability of getting a value is multiplication of both amplitudes, which are usually equal - getting the Born rules:

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Adding a slightly different angle to what Fednis48 has covered: Treated as a classical charged spinning top, field angular momentum L of say an electron owes to integrating over all space for r × (E × B) and one easily finds the contribution from a shell of thickness dr centered about the spin axis, goes down as 1/r^3. That's net over the entire shell. How can it be reasonably expected that for large particle separations, such feeble transverse momentum densities could allow appreciable influence of one particle on the other? And that's leaving out any question of propagation delay.

And how about experiments showing Bell violations involving neutrons, where a largely non-EM field? angular momentum must be tightly confined to a very small effective volume. An example of single-neutron experiment: http://physics.oregonstate.edu/~ostroveo/COURSES/ph651/Supplements_Phys651/Nature2003_EPR.pdf

Taking a very different approach, Bryan Sanctuary claims to be able to reconcile locality and realism with Bell by modelling particle spin (or rather 2D spins) differently to the standard way:
http://quantummechanics.mchmultimedia.com/2011/quantum-mechanics/009-disproof-of-bells-theorem/
Given Fednis48 is evidently the best informed here re Bell inequalities, it might be interesting to get his take on that.

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Cool stuff about random walks! Statistical mechanics always go a little over my head, but it's pretty cool that you can recover Anderson localization and the Born rule by just maximizing entropy over classical paths. Also, the plots are pretty, which is always one of my primary criteria for evaluating papers outside my field.

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Back to the topic at hand, though, I think I may finally see your confusion. I certainly agree that solitons have information about both spins spread throughout all of space, but that's not enough. In fact, let's go a step further and consider the most powerful classical measurement possible: observe data that tells you the exact, pre-measurement state of both particles, then apply some function to that data that outputs one of two outcomes. Such a measurement is still not enough to violate Bell's inequality, because each observer only knows the pre-measurement state of the particles. To reach quantum-mechanical levels of correlations, each measurement outcome needs to depend not only on the states of the particles, but also on what measurement the other observer chose to make. Solitons provide a clear mechanism for the former, but no mechanism for the latter, as far as I can tell.

Beyond that, I'm running up against the limits of how precisely I can explain why Bell's inequality applies to solitons. Bell's inequality (or at least, the two-particle version of it) requires measurements of spin at 45-degree angles to one another, so for me to get any more specific, you'll have to provide some model of solitons that supports more than two directions of spin. Otherwise, all I can say is that your arguments seem to stem from defining "locality" differently from Bell, and if you actually go through any proof of Bell's theorem, you'll never find any steps that stop being correct for solitons.

I didn't know they'd done Bell inequality violations with neutrons - nifty!

As for that link, Bryan Sanctuary seems to be a big fan of Joy Christian's work. In reading his blog, I think I've actually come to a better understanding of where Christian's primary error is. The math is obtuse enough that I can't say for sure, but I think that both authors are looking at direct correlations between spins, rather than correlations between measurements of spins. By applying fancy models of the spins, they come up with more complicated formulas for spin-spin correlations, which can exceed Bell inequality bounds. But in doing so, they sacrifice the connection between their formulas and experimental results. In Bell inequality experiments, some physics leads to Alice's and Bob's binary measurement outcomes, which they write down in their lab notebooks. That's where the physics stops. To look for Bell inequality violations, they punch the numbers into Matlab and calculate a well-defined correlations function. Does that function really capture the correlations between the spins themselves? Maybe not, but Bell's theorem doesn't care, because it puts bounds on the correlation function as traditionally defined. That's what I meant before when I said that Bell's theorem is really about statistics, rather than physics.

(Addendum: I think the most convincing argument that Sanctuary is making this mistake comes near the top of post 009b, where he says Bell's original equation 1 is mistaken. He says that because spin can point in any direction, the functions A and B should not be restricted to having values of plus/minus one. But A and B are measurement outcomes, so they can only be plus/minus one by definition, independent of any physics.)