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Do nonlocal entities fulfill assumptions of Bell theorem?
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Do nonlocal entities fulfill assumptions of Bell theorem?
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Jarek Duda
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Do nonlocal entities fulfill assumptions of Bell theorem?
«
on:
01/11/2015 19:24:51 »
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
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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:
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.thenakedscientists.com/forum/index.php?topic=46639.0
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).
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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Last Edit: 01/11/2015 22:07:01 by chris
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jeffreyH
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The graviton sucks
Re: Do nonlocal entities fulfill assumptions of Bell theorem?
«
Reply #1 on:
05/11/2015 23:27:56 »
What if a soliton experienced time dilation? How would that affect the amplitude and wavelength? This has a bearing on conservation of energy. Locally the effect should not be apparent. This also has a bearing upon the concept of length contraction. I have no answer to this BTW.
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Jarek Duda
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Re: Do nonlocal entities fulfill assumptions of Bell theorem?
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Reply #2 on:
06/11/2015 04:37:01 »
You could see Lorentz contraction for moving soliton in the figure above, here you can find more details about all SRT properties:
https://dl.dropboxusercontent.com/u/12405967/soliton.pdf
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Regarding time dilation, in sine-Gordon you can see it for breathers (
https://en.wikipedia.org/wiki/Breather
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): imagine all pendula hanging down - you lift one of them and release - there will start oscillations.
Here is graph of the angle when such breather has zero velocity (left, oscillations toward both directions) and when it is traveling with a constant direction (right) - the number of ticks of such clock has decreased:
Energy (and momentum) of soilton in Lorentz-invariant field theory also behaves exactly like in SRT: E = m_0 * gamma.
If you want to understand SRT, just study sine-Gordon.
Returning to Bell inequality for solitons, the discussion has evolved here:
http://www.sciforums.com/threads/do-nonlocal-entities-fulfill-assumptions-of-bell-theorem.153000/
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Last Edit: 06/11/2015 04:39:23 by Jarek Duda
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Jarek Duda
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Re: Do nonlocal entities fulfill assumptions of Bell theorem?
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Reply #3 on:
12/11/2015 08:39:22 »
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
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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
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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 (*).
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Jarek Duda
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Re: Do nonlocal entities fulfill assumptions of Bell theorem?
«
Reply #4 on:
15/11/2015 11:57:50 »
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
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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.
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Last Edit: 15/11/2015 12:00:06 by Jarek Duda
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Re: Do nonlocal entities fulfill assumptions of Bell theorem?
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Reply #4 on:
15/11/2015 11:57:50 »
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