Tuesday, December 10, 2013

The resurgence of Hidden Variables.

No hidden variables, EPR. 
I studied quantum mechanics (QM) under the tutoring of A.Aspect.  In the 70's, while at CERN, Bell had shown that so called 'hidden variable' models of QM (the one championed by Einstein) and proper QM models gave different predictions.  Aspect's experiment in the 80's had ruled in favor of QM. Hidden variables models were a no-go. 30 years later the debate still isn't settled. 

30 years later a hidden variable analog emerges.
 The 2006 Walkers of Couder (ENS Paris) were profiled in 'through the wormhole'. They have recently been re-filmed and precisely mathematically modeled by John Bush at MIT.  See the accompanying video. I have started modeling these 'walkers' in java software code. 

This is a silicon bath (viscous) with damping standing waves excited at the frequency of Faraday. The combination of standing waves in confined spaces gives a phase where the particle and the wave are in sync. The particle creates the wave, the wave guides the particle. The resulting system is unstable and starts walking in specific phases close to the Faraday instability.  The particles are literally "walking on water".

The deBroglie promise, 1927
This also has been attracting attention because it is an analog of a deBroglie-Bohm system.   Taking the wave-particle duality idea completely literally, deBroglie thought about the problem by talking about a wave at the deBroglie wavelength moving at the same frequency "as if a clock". Why this singularity is there in the first place is mentioned by deBroglie, it is 'the matter', today we think of them as 'solitons'. DeBroglie however studies the wave that englobes the particle.  It guides the particle. 

DeBroglie (pronounced 'duh-bro-eee') presented the basic idea in 1927 at the Solvay conference.  There he got shut down by Bohr and Heisenberg, who were developing their own interpretation of Schrodinger and the Copenhagen interpretation won the day.  Compared to deBroglie, the axiomatic presentation was by far the simplest. It postulated the randomness, instead of trying to recreate it, and it postulated a vectorial space which allowed for comparatively simple calculations. 

In retrospect deBroglie's proposal is complex as it involves non-linear math.  They had no computers to explore these.  Towards the end of his life, deBroglie admitted that randomness needed to be re-introduced manually in his system anyway.  In 1957 deBroglie put out an analytical study of 'la double solution' and 'theorie de la mesure'. They are fantastic reads.  


God does play dice. 
Bell showed in the 70's that a  'hidden variables' systems would obey his famous inequalities. However the walker system falls under the 'stochastic (hidden) variable" category, not the same hidden variables Bell uses.  Couder when talking at Perimeter Institute, in the Q&A session, vaguely invokes path memory to answer the question of Bell.   

It prompted me to go to the source and read Bell's "Speakable and Unspeakable in Quantum Mechanics".  Bell was a convincing deBroglie-Bohm evangelist while at CERN.  I reread the proof he came up with and the newer variants.  I have tried to understand this idea of the path memory.  I approach the problem computationally in java. The variables one uses here have chaotic patches. 

The computational image,  
The picture that comes into focus from the computational angle is one of a stochastic dynamic system. Like a pachinko, the path a give particle is taking is in fact random within a bigger order. Chance and randomness emerge from the dynamics. Just like QM models  and unlike "deterministic hidden variables".  Those are 'random hidden variables'. 

The path is everything.
Seen as a computational process, at each step the bouncing droplet is accelerated by the slope it is at, that is the 2D picture. It already creates complexity.  The wave at each point (you can generalize this to 3D) is the sum of all waves emitted earlier from a collection of points and reaching a concrete point.  Think reverb for musicians.  All the influences from the past sum up at each point of the field. And that is the local information that accelerates the particle.  If it oscillates "like a clock" (in the words of deBroglie) this feedback gives rise to very complex dynamics even in 2D.  It includes "echo-location" information about all the sources, including those bouncing on a surface. Path memory is an encoding of the geometry. The greater geometry reappears in the resonant modes of the cavity, something that drives the expectations via standing waves.  How standing waves appear, a central construct of deBroglie, is here seen as emergent and due to geometrical constraints.

Elements of stochasticity. 
Computed wavefield in java simulation
Standing waves and uniform velocity.

 Here pictured is a single "water walker" (based on the Couder/Fort formula) with constant speed. This is the equivalent of a "HelloWorld" for this class of problems and took me about a week worth of work so the data may just be all garbage.  But most areas are predictable and in the middle randomness occurs. These are standing waves interfering resulting from a periodic walk on a surface.  

Bell's Houdini: Stochasticity
The walkers escape Bell's construct. Bell assumes a distribution of the hidden variable that is set at the time of preparation of the entangled pair and doesn't vary after.  It is not time dependent. The randomness is all in the preparation.  Here the randomness is continuously injected into the system along the path. 

Emergence of random walk.
In this picture, the progress is stochastic, each step like a pachinko where randomness is introduced over the length of the path. When the wavefield gets too chaotic, as in some spots in the middle of the picture above, then randomness is introduced. In the Bush video for regimes close to faraday instability the surfaces washes over these details, since the oscillations are too frequent to have a physical meaning and in Bush you see the emergence of a random walk around the Faraday instability. The frequency of oscillation in the simulations suggest a measure of 'how much random'.
  The path is randomized at the slit in the Couder experiments or those seen here. 

Emergent QM
Each particle will take a different path that will seem random.  But it will be guided by a wavefield that conforms to the geometry of the confining space. In the Bush video during the corral film, the parts where the particle is slowest are the parts where the particle will spend more time.  The probability of finding it there is proportional to the time it spends there. It maps to a probability density in QM.  This also is a prescription on a way to recapture the probability of presence computationally. To the right is a capture of a walker walking back from the slit.     This effect is decisively not QM. Most interesting. Captured by Heligone (dot/wave). 

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