Do nonlocal entities like soltions fulfill assumptions of Bell theorem?

[Duda Jarek]

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 is moving - Lorentz contracted):



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.scienceforums.net/topic/65504-how-quantum-is-wave-particle-duality-of-couders-walking-droplets/ ).
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?

Edited November 2, 2015 by Duda Jarek

[studiot]

Can you apply the inverse scattering transform to the correlation (probability) version of Bell's theorem, in the same way that the fourier transform applies to ordinary correlation (probability) version?

 

https://en.wikipedia.org/wiki/Inverse_scattering_transform

Edited November 2, 2015 by studiot

[Duda Jarek]

I don't know

 

The discussion about Bell inequalities for solitons has evolved a bit here:

http://www.sciforums.com/threads/do-nonlocal-entities-fulfill-assumptions-of-bell-theorem.153000/

[Duda Jarek]

Regarding Bell - we know that nature violates his inequalities, so we need to find an erroneous assumption in his way of thinking.
Let's look at a simple proof from http://www.johnboccio.com/research/quantum/notes/paper.pdf
So let us assume that there are 3 binary hidden variables describing our system: A, B, C.
We can assume that the total probability of being in one of these 8 possibilities is 1:
Pr(000)+Pr(001)+Pr(010)+Pr(011)+Pr(100)+Pr(101)+Pr(110)+Pr(111)=1
Denote by Pe as probability that given two variables have equal values:
Pe(A,B) = Pr(000) + Pr (001) + Pr(110) + Pr(111)
Pe(A,C) = Pr(000) + Pr(010) + Pr(101) + Pr(111)
Pe(B,C) = Pr(000) + Pr(100) + Pr(011) + Pr(111)
summing these 3 we get Bell inequalities:
Pe(A,B) + Pe(A,C) + Pe(B,C) = 1 + 2Pr(000) + 2 Pr(111) >= 1

Now denote ABC as outcomes of measurement in 3 directions (differing by 120 deg) - taking two identical (entangled) particles and asking about frequencies of their ABC outcomes, we can get
Pe(A,B) + Pe(A,C) + Pe(B,C) < 1 what agrees with experiment ... so something is wrong with the above line of thinking ...

The problem is that we cannot think of particles as having fixed ABC binary values describing direction of spin.
We can ask about these values independently by using measurements - which are extremely complex phenomena like Stern-Gerlach.
Such measurement doesn't just return a fixed internal variable.
Instead, in every measurement this variable is chosen at random - and this process changes the state of the system.

Here is a schematic picture of the Bell's misconception:



The squares leading to violation of Bell inequalities come e.g. from completely classical Malus law: the polarizer reduces electric field like cos(theta), light intensity is E^2: cos^2(theta).
http://www.physics.utoronto.ca/~phy225h/experiments/polarization-of-light/polar.pdf

To summarize, as I have sketched a proof, the following statement is true:
(*): "Assuming the system have some 3 fixed binary descriptors (ABC), then frequencies of their occurrences fulfill
Pe(A,B) + Pe(A,C) + Pe(B,C) >= 1
(Bell) inequality"

Bell's misconception was applying it to situation with spins: assuming that the internal state uniquely defines a few applied binary values.
In contrast, this is a probabilistic translation (measurement) and it changes the system.
Beside probabilistic nature, while asking about all 3, their values would depend on the order of questioning - ABC are definitely not fixed in the initial system, what is required to apply (*).

[Duda Jarek]

There is a recent article in a good journal (Optics July 2015) showing violation of Bell inequalities for classical fields:
"Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields"
https://www.osapublishing.org/optica/abstract.cfm?URI=optica-2-7-611

 

Hence, while Bell inequalities are fulfilled in classical mechanics, they are violated not only in QM, but also classical field theories - asking for field configurations of particles (soliton particle models) makes sense.
It is obtained by superposition/entanglement of electric field in two directions ... analogously we can see a crystal through classical oscillations, or equivalently through superposition of their normal modes: phonos, described by quantum mechanics, violating Bell inequalities.

Edited November 15, 2015 by Duda Jarek