Many in the crypto land (us at Freeside included) share the opinion that "Asset Backed Tokens" or "Security Token Offerings" are the new black in legal token offerings and that they will bring new levels of liquidity to the crypto exchanges. At the same time "Stable Coins" are all the rage. What we are going to talk about is the link between the two which is deep and both obvious and obfuscated.

The gist of the STO/ABT constructions is that actual assets, be it debt, equity, or what have you, are backing the token. Real (future) cash flows back the token created out of thin air. These can represent REITs, Accounts receivables, debt, VC or really any cash flow coming from the real world.

To make things clear, and in an interesting parallel, we will describe FIAT money as an ABT. This is not a stretch or abuse of language but rather a factual staple of modern monetary theory (MMT) and a illuminating meditation for those that wish to spend some time on it.

All FIAT money, at least in the west:US/EU/JPY, is BIRTHED AS DEBT. What that means is that we create the token representation we call the dollar out of thin air in the banking system. It is a token which is worth ZERO at birth for it came out of nothing. It is however backed by debt, meaning that the new emissions are linked to debt repayment at least for those circulating in the economy (as opposed to the abstract QE for banking which was just patching balance sheets in 2009). A token emitted in the economy needs to be repaid in full (plus interest). The interest does not interest us here, bad pun aside, and it is the principal that we focus on. Once the debt has been repaid, the token actually represent A REAL DOLLAR FROM THE REAL WORLD for it has created it's own value in real life and has created the corresponding cash flow.

In the parlance of the stable coins, the dollar is backed 1:1 via debt repayment.

Discussing poster at emQM17 with R. Brady (Cambridge)

Abstract

We describe results from an implementation of a Monte-Carlo simulation of Bell-CHSH type correlations with hydrodynamic walkers as suggested by [Vervoort2017]. We observe the formation of pairs of walkers strongly anti-correlated in position and velocity under various random initial conditions. With a non-relativistic representation of the walkers, i.e. one where the hydrodynamic waves propagate faster than the walkers, as in real life walkers, we observe numerical S values of CHSH correlations above 2, violating the Bell limit as an explicitly non-local system. We observe Bell violations up to the Tsirelson limit of 2sqrt(2), but not violating it, under fine-tuned observation and post selection conditions. We report various such runs in the 2 < S <= 2.82 range under the non-separable assumptions. However when we numerically enforce programmatic separability of the walkers, a numerically enforced locality, then we lose the violations and recover, as predicted if locality holds, a classical S <= 2.

Conceptual framework

M(x,y)= sum(sigma1*sigma2*P(sigma1,sigma2)).

We do the correlation measure for four vectorial values a,a’,b and b’ following the rules of the standard Bell game and we calculate the scalar value of the CHSH form:

S = M(a,b)+M(a,b’)+M(a’,b)-M(a’,b’)

Figure 1: Anti-correlated walkers in red dots seen with their path as filmed from above. Bath dimensions in our simulation: 1.2m*0.6m. Pin measures at position a and b in blue. Sigma1 and sigma2 as UP and DOWN outcomes of the measure. Source of anticorrelated walkers S from [VERVOORT 2017].

Source of AntiCorrelated Walkers

The original paper [Vervoort2017] called for the creation of two walkers, each one with opposite velocities, and thus perfectly anti-correlated by construction. This construction, in theory, does not require wave mediation for the walkers would get created with perfectly anti-correlated values at birth and propagated by inertia. Here we choose velocities randomly (although narrowly) as a more general starting point for the creation of pairs. Anti-correlation emerges as a rather generic outcome for two field mediated walkers. Walker to walker interaction, via wakes, creates perfect anti-correlation in our simulations. Below you can see pictures of walkers going in opposite directions. Bear in mind that these walkers had random initial conditions. The differences at birth quickly get wave mediated and a high degree of anti-correlation appear even numerically as in figure 3 where we show the velocities as vx and vy (2D). Take a moment to read through the raw data if you can.

Figure 2: anti-correlation, the walkers get initiated with different and random velocities.

Figure 3: anti-correlation, random initial velocities.

PSI =1/sqrt2* (|-+>+|+->)

We characterize our source as a statistical mix of up-down and down-up combinations. The following data shows 4 different runs and P(-+)=P(+-)=1/2. We abuse the language to talk about psi, for we have a statistical ensemble here, and still need to show S-Value >2 to talk about entanglement and superposition.

Figure 4: psi =1/sqrt2* (|-+>+|+->)

Importance of Anechoid Cavity

It should be noted that anticorrelated walkers do not seem to appear generically in real walker systems as they do in the simulation. In private communications, both Couder's team (Fort) and MIT's team (Bush), confirm that they haven't seen them yet for we haven't actively looked for them either. The walkers in the presence of cavity noise integrate it chaotically [ChaosBook] and default to confined walkers such as the 2 walker Oroborous. This rationalization can be tested in software by artificially removing all wall effects (just not coding them will do). Removing all wall effects from the walkers requires more care.

Figure 5: 2 walker Oroborous

Bell-CHSH Methods

Walker particle assumptions

We operate in the so-called stroboscopic approximation where we only consider the horizontal point dynamics of a walker.

Walker field assumptions

Each impact of each walker creates a standing wave centered at said point of impact. We represent these wavelets with standing Bessel profiles. The walkers offer a non-Lorentzian ontology. The waves, both travelling and standing, propagating faster than the walkers. Given the finite nature of the velocities, we can choose to simulate separability in walkers or not, knowing real life walkers by default do not meet ideal separability criteria.

Figure 6: A 2D Bessel function of order 0

Pin wave assumptions

The pins are represented by Static Bessel functions which do not decay in time. We set the wavelength at Faraday and the amplitude on the order of 1mm, in order to roughly mimic the hydrodynamic range and as a first approximation of a mirror image of the walker field on the pins. We mimic separability of measure to measured e.g. the last minute insertion of the pins of [Vervoort2017]. We can turn it on or off and it proves a necessary condition for the observation of the Bell violations. In effect we only allow the pins to interact with walkers that are close to them and without cross-talk in both channels, mimicking the photonic Bell conditions of in-flight setting of the polarizers as done in [AA1982].

Figure 7: newtonian fluid differential equations of motion for particle and self-created waves, non-linear effect in wave form (memory).

Force computation

We use a a-dimensional numerical representation in the hydrodynamic walker domain [Bush2014] acceleration = VISCOUS_force + FIELD_force

field_force comes from the gradient of the wake.

Time Decay

Pin waves do not decay in time (static). Walker waves do decay in time.

Walker Separability (outcome independence), OI

We do not include waves coming from another walker if the walkers are separated by a large distance in a simplified implementation of causal relativism. We code for that separability as a boolean. It should be noted that our Bell results, Bell violation vs no Bell violation, were not sensitive to this OI. While the Bell status did not depend on it, the correlated outcome statistics were.

Parameter separability (parameter independence), PI

When calculating the field force we can choose to not include waves coming from the other PIN and account only for the pin present in the arm. The Bell violations were dependent on the value of this parameter.

Artificial separability and real walker systems.

These two separability conditions OI (Walker-Walker) and PI (Pin-Walker), probably do not exist in the real walker systems simply because the waves vastly outpace the walkers.

Violation of Bell inequality w/ non separability (real walker).

Without Parameters Indepence (PI=false) we let the left pin influence the right walker and vice versa. We obtain violations of Bell

Figure 8: Bell violation (S-VALUE>2) under non-separability (ENFORCE_PI set to false)

Loss of Bell violation with separability (software walker).

With Parameters Indepence (PI=true) we lose the violations and recover Clauser separability.

Figure 9: No-Bell violation

Emergence of the Tsirelson limit in post-selection

When we observe violations and when we narrow the admission window on the back walls to +-50mm (Y_BLOWUP) then we observe various violations at 2.80+/-0.05. In other words we observe Bell violations at the Tsirelson limit 2*sqrt2= 2.82. We have not seen violations above the Tsirelson limit. We have not specifically looked for them either.

Figure 10: Tsirelson limit emergence (2.82)

For anyone interested, which should mean about 2 people, you already have my paper version of this blog/poster entry, if you read this far and want to know more I can send you the 26 pages of it.

In retrospect the previous blog, posted about a year ago, shows how naively I approached the problem. The correct calculation, done around September gives a 1/4 correlation in certain configuration. The paper was written with QuTools, Munich based entanglement experts. The experiment was run by QuTools in March in Germany and gave us some interesting anomalous results which required several rounds of debugging and interpretation.

By inserting only one QWP in one arm we detect phase effects with great sensitivity. A modulation in the signal correlation, otherwise computed to be 1/4 still remains. The updated interpretation puts the blame on the source itself and that the pumps would not be perfect resulting in the amplitude and phase we see and detect. This was double checked with a full state tomography on the source and seems to give results that roughly agree with what we measure in the simpler QWP experiment. We therefore have a way of characterizing the entangled sources in a simple way.

I have received the system after much hassle with the customs. And train myself on it.

A great article from the Simons foundation came out back in June covering the walker research. These objects, a wave/particle association, are attracting attention because they have the capacity to self-excite themselves into quantized orbits and reproduce an increasing catalog of quantum behavior previously only observed in electrons but never in 'classical' systems. For a more in depth look at the walkers, the article does a good job. The article interviews 3 Nobel prize winners.

Here is what t'Hooft had to say (from the article): "Personally, I think it has little to do with quantum mechanics,” said Gerard ’t Hooft, a Nobel Prize-winning particle physicist at Utrecht University in the Netherlands. He believes quantum theory is incomplete but dislikes pilot-wave theory.

Wilczek was just as encouraging: "Many working quantum physicists question the value of rebuilding their highly successful Standard Model from scratch. “I think the experiments are very clever and mind-expanding,” said Frank Wilczek, a professor of physics at MIT and a Nobel laureate, “but they take you only a few steps along what would have to be a very long road, going from a hypothetical classical underlying theory to the successful use of quantum mechanics as we know it.”

And finally: "Anthony Leggett, a professor of physics at the University of Illinois, Urbana-Champaign, and a Nobel laureate. “Whether one thinks this is worth a lot of time and effort is a matter of personal taste,” he added. “Personally, I don’t.” Personally, it's about understanding. The puzzling thing about QM is that it is a very successful theory and we have no idea why. We don't know where it comes from. QM, from a logical standpoint, really is structured as a set of axioms (the postulates of symmetries) and then a lot of recipes on how to make it all work. And it works really well, it is an observed theory complete with observed predictions (Higgs). What justifies the choice of symmetries is their result, they work. Epistemologically, the puzzling bit of it is that we don't know where those symmetries come from, we simply hypothesize them.

As pointed out in the article and comments, certain "working" physicists don't care about the 'why' at all. They don't need to, they don't want to. It is a toxic choice. To them it's philosophy. Personally I only care about the why. Let's not forget that understanding the 'why' is a big motivation in physicists to begin with. That's why we do it.

Lack of Interpretation

I always cringe when people talk about the "Copenhagen interpretation". The 'Copenhagen' is not an interpretation. It is a formalism, with bits of stories and smiling cat interpretations bolted around it. I will always remember Alain Aspect yelling at me as I was a wide eyed undergrad trying to make sense of QM and his 'spooky action at a distance' experiment. I needed to stop "interpreting" the formalism, it would lead me nowhere and that some people were very very happy NEVER interpreting the formalism. In retrospect the reason is rather simple in that that if you try to 'interpret' the formalism with everyday categories you quickly start talking about magical stuff such as superimposed cats (/electrons) that exist in multiple states at the same time. It also leads to this superposition instantly disappearing, in a mysterious "wave packet collapse" that supposedly seems to happen instantly in the EPR class of experiments (see below). The 'interpretation' of the formalism wasn't magical, it was rather simply 'not there'. Objects for interpretation

What the walker framework does for me is equip me with tools, objects, categories and abstractions with which to think about QM. All of the sudden I have a picture in my mind. Not just a math formalism. And this picture helps me think clearly, it guides the intuition. Whether it is right or wrong is almost besides the point at this stage. I can write industrial grade code on multi-core machines at home (in java) and explore what the computational models say.

The mental picture and physical intuition that one derives from the walker experiment is indeed 'mind expanding'. For example it is rather simple to revisit superposition and decoherence in this framework, see here for an entry on the re-interpretation of these notions. In this approach the Schrodinger cat is always dead, superposition is a mind view, it never existed, it is replaced by 'chaotic intermittence' and the notions of coherence and decoherence take on a very definitive form in contrast to the magical wave collapse that is supposed to happen upon measurement and observation. In this picture the wave collapse is a false category. It is replaced by a notion of intermittence between states (the particle can self excite into those states) mediated by the presence of chaotic dynamics in the wave/particle duo. In short, the particle exist in a sort of 'superposition', where there are many states and we just transition between states with probabilities that remind us of "transition probabilities" of QM. The difference with the classical superposition is that our particle are not in those several states *at the same time*. There is no superposition per se but rather intermittence between those states at different times. The walkers will self-quantize their orbits in discrete states, just like an electron does and they will oscillate between those states (the intermittence) just like QM particles are supposed to do albeit with everyone at the same time. Reality really exists.

As one commenter on the Simons article said "As stated by Bell’s theorem one has to abandon either locality or reality and it seems that for most physicists (me included) giving up on reality is the favored choice."

But I am honestly not too concerned about comments like this one. It is just going to take time. Prof Bush whose mathematical models provided the starting point for many simulations including ours put it very well in the article: "The more things we understand and can provide a physical rationale for, the more difficult it will be to defend the ‘quantum mechanics is magic’ perspective.”

The walker framework is already taught by Prof Anderson and Brady with the mental pictures of the walkers to explain things like 'spin'. I wish I had had those classes as a undergrad instead of the "Copenhagen".

As a matter of taste, I am firmly convinced that the "Copenhagen interpretation" will fade eventually as a deeper understanding emerges, possibly informed by the walker picture. It will be seen as not an interpretation at all, but simply a formalism, "The Copenhagen Formalism". But there is no magic. Ontological Reality It seems the objects required to describe the formalism (vectors in a Hilbert space) are just mathematical objects and that's it. They capture the language of statistical transitions in a matrix. They are abstractions. To assign ontological reality to these abstractions, for example to think of superposition as something 'real' is what leads us to magic in the first place. What the walker framework says is that indeed the layer at which classical QM is formulated (the state function) cannot be interpreted. It is a mathematical abstraction. 'Shut up and calculate' was indeed the right frame of mind to have. Today a lot of people think that the 'real' objects exist at a lower layer of reality than the 'standard model'. In this particular walker picture, QM would be a sort of an emergent *statistical* composite model. The QM objects were not the 'real objects' we were looking for. QM as emergent objects Strictly speaking 'the standard model" needs to be emergent from any candidate underlying framework. In all generality whatever model you adopt as your underlying reality has to have the standard model somewhere downstream as an emergent property in its logical consequences. Whatever emerges from our axioms has to conform with QCD and the Standard Model. Period. So the trick is to show that QM is an emergent phenomena. The behavior that emerges from our models and the experiments are QM analogs, but they are not the 'real thing'.

Computational chaos

The article mentioned the latest Couder experiment, the ones with the elastic force, this setup leads to chaotic path that shows intermittence and self-quantization of orbits of the walker. It is memory induced quantization. But it takes either the experiment or computers to 'observe' them. Today we try to characterize *how much chaos* we need to create the proper QM-like behavior. Historical detour The article did a great job with 'the history of physics' approach. I can project myself to this famous conference where the Copenhagen formalism won the day. I can see a young Louis De Broglie waving his arms about, talking about an inventive but unpractical wave/particle model and Bohr simply dropping his magical (and mathematically expedient) matrix formalism with Einstein choking on his bagel in the front row that, "God doesn't play dice". In hindsight, it is normal that this formalism would win the day simply because it was the only one that could predict things correctly. The Copenhagen model was a working model while the particle/wave one was a 'philosophical' model. Bohr et al, chose a more abstract starting point, sacrificed understanding in favor of 'working physics'. It was a faustian bargain (captured somewhere in a play) that I would personally take. Those in the article who lament the choices made are in turn a little naive. Only with today's powerful computers and the clever experiments, which were encountered by chance, can we make predictions and build a more fruitful mental picture.

In any case, it was not only the easy way (mathematically), it was also the only way (no computers) and it turned out to be the right way. What more do you want? An explanation as to why? Of Philosophy and in praise of 'understanding'

So I will not disagree with the quotes in the article and their famous authors that this line of work while "mind-expanding" is clearly a philosophical endeavor. Indeed, personally it is also why I am attracted to it as well. I glamorize it as 'old school' natural philosophy. Someone's got to do it. Those who point out that recreating the standard model is going to take a long time are probably right. Those who point out that it is futile since we already have said standard model are only partially right. To me it is obvious that understanding QM in terms of wave/particle composite objects, probably different than these walkers and going through the rebuilding of the standard model as Leggett advises will yield new and valuable insights. To me clearing up the air around superposition and decoherence was already reward enough. Local causality vs non-local correlations. Einstein-Podolski-Rosen in walkers

Finally, as pointed out in the comment section, it would be significant if the framework had something to say about the statistics of Bell inequalities. It has been one of my starting points on the walker research for the simple reason that the formalism was getting too abstract and I needed *something* to guide my intuition. Spooky action at a distance has been bothering me for about 20 years ever since I studied them under Aspect. The logical flow would then be to take 2 walkers let them interact as they drift apart and see if there are non-local correlations. It should be noted that the correlations build up over time, like in a memory, it is path dependent and it is not the all or nothing picture that was used for the hidden variable in the Bell theorem derivation. The variables are not set at birth but rather developed chaotically over time. Some correlation will build up and show up.

This post is going to be highly technical and will actually speak to about 20 people in the world (if that). So if you are a casual reader of this blog you should skip this one and instead focus on the more general considerations of the walker study as it relates to the interpretation of Quantum Mechanics

We are going to review the physics of the walker problem in the context of computational models. This will deal with the in-silico approaches to the problem and shed some light on what we call 'surfers vs walkers'.

The walker problem
The object of study is the association of a particle and the wave it creates in a media. We focus on the silicon walkers as observed by Couder et al and modeled by Bush et al. The physics is rather straightforward in the Newton (force/acceleration) view. The forces are as such:

Slope force. Aka "field force". Each time the particle bounces it creates a Bessel standing wave. These waves sum (interfere) and create a wavefield. The gradient of the wavefield at the point of impact gives an impulse to the particle when it bounces off the surface (like a ball bouncing off an inclined surface). The important points here are a/ the memory of the field (loosely defined in the literature as the number of waves that contribute to the wavefield). It is the main control parameter in our studies. As the memory increases so does the height of the total wave (you sum many little waves). For example in our simulation, the basic wave is of height 0.02 faraday length and the sum is about 0.1. b/ this is a DISCRETE set of waves.

Viscous force. Every time the particle bounces on the surface it is slowed down. This force is proportional to the speed.

Elastic force. This force was introduced recently by the Couder team in a technical tour de force. They injected the bouncing particle with ferromagnetic material and submitted the particle to a centropotential EM field. While it is described as a EM force, it is akin to an elastic potential and the force is modeled as -kx with the exact form of a elastic force. The author prefers the 'elastic force' description for various reasons not having to do directly with the QM interpretation. It is in this elastic potential that the walker system exhibits intermittence and provides the most intriguing insights into coherence/decoherence and the superposition principle.

These 3 forces are the ones identified in the problem so far. It should be stressed that the 'historical' walker study focused on 1 and 2. For example the initial Couder/Fort paper balanced the field force with the viscous force to calculate the speed of walkers. The Bush papers dealt with 1 and 2 as well. The elastic force is a recent addition in experiments.

The Bush approximation: analytical solutions

The mathematical description of the hydrodynamics at play in the walker system has been done by Bush et al. Essentially they make analytical way by approximating the discrete sum of Bessels by a integral. Long story short, instead of a discrete sum, we hypothesize the intermediate steps and calculate with a step that is shorter than the actual bounce. This is not physical, in the sense that these intermediate waves do not exist in reality but they are expedient from a mathematical standpoint. We are able to compute the trajectory with this 'smooth' approximation by inverting matrices.

Surfer vs Walker

In the case of the Bush approximation the particle is always informed by the wave. The mental picture is 'the surfer'. The waves are constantly generated and driving the particle, the particle surfs on its waves.

The silicon droplet system is a 'walker' system. The particle just 'bounces' off the surface and only picks up an acceleration at the time of impact.

The important part here is that surfers assume a continuous set of waves generated along the path, while the walkers deal with a discrete set of waves. This really impacts the 'field force' only.

Integration routines

The main advantage of the 'surfer' approximation is that following Bush et al, we can compute an exact solution to the problem by inverting matrices. This results in a smooth flow and has been implemented by the Dotwaves crowd in Python (by Burak Budanur) and Matlab (by Samuel Bernadet). The disadvantage of this routine from a computational standpoint is that it is heavy and therefore slow. "Slow" is a relative term in computing and a lot of information is coming from these simulations, they also recover many of the features of the 'real' walkers. The important part from a model standpoint is that all forces (including field force) are continuously integrated and smoothly changing.

The discrete routine has been implemented in java (by Marc Fleury) and it's main advantage is that it is very fast (by an order of 10000) so allows for quick visual inspection of the behaviors. The disadvantage is that the integration routine, if done naively, introduces a lot of noise. To address this noise, and after a lot of prodding, the author has implemented a hybrid walker integration. The integration routine has been refined as such:

Field force is still discrete (there is a discrete set of waves in the real problem) and the integration done in one step. In the future we may move the force to a 'Runge Kutta" like average but this is not straightforward and will be arbitrary.

EM force is continuous. The integration routine has been smoothed out to account for the continuous nature of the EM force (it applies when the particle is in flight) and gives us much a much smoother integration.

Viscous force is continuous. This is debatable as the viscous force in the real experiments only apply when the particle is in contact with the liquid. It does however increase the smoothness of the integration routine.

Physical accuracy

In a nut-shell the main features of walkers integration in java is a discrete treatment of waves field with a continuous EM and viscous. The surfer integration is a continuous treatment of all forces including the field force. The discrete nature is closer to the physical picture (as the walkers only create discrete waves, as simple as that). The speed is a secondary point, as we can run both types of simulations over long periods of time.

Noise: discrete vs continuous

The a posteriori justification for the surfer approximation, besides the fact that it is mathematically expedient, is that it seems to replicate the observed facts about walkers at least in the straight walker regime (from the published literature) and gives us good results with the elastic force in the dotwaves efforts.

However we know that discrete sets of waves create a different wavefield than continuous sets of waves, specifically at short distance. Tesselations are important in field theory, continuous and discrete sets giving different results. It should be pointed out that the walkers orbit at about 1 faraday length, meaning are subjected to the short term structure of the field. The presence of a 'lattice' (discrete points) as opposed to a continuous source of waves gives very different wavefields in general.

In other words, the surfer approximation introduces noise of it's own in the computational simulations (mathematical in nature) that is not present in the physical system or the walker implementation.

Surfers vs Walkers: the 3D SURFER case and QM analog

The interest in these systems comes from the fact that they offer a compelling mental picture with which to understand the (obscure) formalism of QM. As detailed in the blog referenced above it is the emergence of intermittences, characteristic of chaotic systems, that gives us the transition probabilities between states and a clear image of what 'superposition' means.

However this walker system is inherently 2D. And because we are 2D we have a notion of "steps" and the walker image only can exist in 2D, we bounce in the vertical direction and create discrete waves on the 2D surface. In 3D, the particle would ALWAYS be in contact with the media and would be generating waves continuously. In 3D we cannot escape to another dimension to only create discrete waves.

For any student of Quantum Mechanics (QM), the interpretation of the QM formalism is at first a puzzling proposal. Simply put it is 'counter intuitive' and most people "shut up and calculate" essentially bowing to the myth that "QM is very weird".

How can things be in 'several states' at the same time. QM matter must be of a different, slightly magical, nature. It is perhaps best exemplified by paradox of the Schrodinger cat that is both dead and alive, supposedly at the same.

In this post we will apply the formalism of walkers, an emergent model of QM dynamics that comes about from the association of a particle and a wave and show how it sheds a new light on the fundamental interpretation of QM and show that in this interpretation, the cat is simply always dead.

We will also use this formalism to shed light on the typically QM notion of coherence and decoherence.

We study the walkers by simulations and our little group operates on facebook:

A brief overview of Walkers.

Walkers are the association of a particle bouncing on a liquid silicon surface and the waves this bouncing creates. Everytime the particle bounces on the surface it creates a standing wave (modeled by a Bessel function in 2D) which in turn drives the particle. The sum of the history of bouncing informs the future bouncing. When the particle bounces it picks up an acceleration due to the gradient of the local surface (like a surfer on a wave). This dynamical system, the association of the particle and its wave exhibits interesting self-quantized dynamics.

A strict interpretation of the deBroglie wave/particle duality.
Louis deBroglie, one of the fathers of QM, postulated wave/particle duality. Observing that sometimes matter behaves as a wave and sometimes as a particle, he postulated a relation between the energy of a particle and it's 'deBroglie' wavelength in his PhD thesis. This earned him the Nobel prize in 1929. The walker are a strict implementation of this idea as we have both a particle and its adjoined wave.

Surfers vs Walkers
This point is a little technical but is worth mentioning. The walkers 'bounce'. Each bounce creates a wave. This is a 'discrete' phenomena as opposed to 'continuous'. We sum a finite number of waves. A continuous phenomena, requiring integration, would be a 'surfer'. It is worth mentioning because some of the current formalism for the walkers (most notably the Bush formalism we use in this research) is really a 'surfer' formalism, assuming a continuous wave creation along the path.The surfer creates the waves it surfs on.

Self-excitation, Self quantization. QM behavior
The dynamics of this system is interesting in it's path. The particle will self-excite and start 'walking' as seen in the video. More importantly to the QM study, some of the behavior observed mimics QM. For example if the particle is submitted to an elastic potential (by way of an EM field) it starts orbiting and showing 'quantized' (discrete sets) of possible orbits. The path cannot be "anything" it is quantized by it's own history, the path creates the path. The history quantizes (by offering a discrete set of possibilities) the future.

Chaotic Dynamics

To understand the solution, pay close attention to the video above. It is a run captured in matlab by Heligone of dotwave.org. The point is that the orbit goes from trefoils to ovals and (not seen here) sometimes circles. These are the quantized orbits. The change between these orbits is called 'intermittence' and is a characteristic signature of chaotic dynamics. The dynamics will randomly change between these stable, discrete, orbits.

Superposition revisited as intermittence
If one observes these objects over long periods of time, one can compute the 'time' the particle spends in each orbit. With this mental picture we can revisit QM. It is a different interpretation than QM. In QM, the particle is in 'several states' AT ONCE, meaning it would be in the oval and the trefoil at the same time. This is difficult to picture for macro objects such as the cat. In this model, the particle oscillates between those states over time, but NOT AT THE SAME TIME. There is no superposition, just intermittence, an oscillation between the states with certain transition probabilities. Weights as probabilities.
If one computes the times, one can arrive at the probability one would observe the particle in a particular state. These are transition probabilities. This formalism demystifies the Hilbert space formalism that says that the particle is in all the states with various probabilities. Here there is a logical and simple interpretation for those 'weights' they are the probabilities to find the particle in the particular quantized states available to the dynamics. It is simply the time it spends in each quantized orbit.

Schrodinger's cat is always dead
Let's revisit the Schrodinger cat paradox in this picture. The idea that a cat would be dead and alive has confused generations of students for the past 100 years and while it makes for good magical mythology it is just false logic in this picture. More importantly, it is the act of observing it that kills the cat. There are several problems with the paradox: a/ we are not in 2 states at the same time. In this model we either are in orbit A or B but not in both at the same time. We oscillate between those states at different times. We are only in ONE state at a time (even if changing rapidly). b/ the category dead/alive are different than intermitting between orbit a and b. Simply put we cannot 'intermit' between dead and alive like we can jump back and forth from orbit A to B and vice versa. But ounce you reach "dead", you stay dead. So essentially the particle would eventually orbit to 'dead' state for the cat, and once it has done it, the cat IS DEAD. End of story.

Observation
In this picture the idea that observation is what killed the cat, which is the usual interpretation, is laughable. You didn't magically kill the cat with your thoughts, the cat died because the particle decayed. End of story.

But Quid of Decoherence and the measurement problem. This is technical but is the heart of the issue at hand. In QM, the act of observation is what creates the 'wave collapse' and 'decoherence'. The system is classical, not QM after the act of observation.

There is a big difference in the walker system: Essentially in the case of walkers, the observation of the particle (here in silicon and silico) DOES NOT DESTROY THE FIELD. This is the important part. If the act of measuring interferes with the field (uses tools that are on the order of magnitude of the deBroglie wavelength), which is the case for 'classical QM" (remember the observation of slits in the Young double slit experiments is enough to destroy the QM result) then one loses the chaotic dynamics induced by the field. Observing a QM dynamic induced by a 'field' destroys the field. So the QM dynamics stop the moment we 'observe' them. The measurement problem is at the heart of our problems with QM.

The notion of 'coherence' and 'decoherence' which is usually magically associated with the existence of 'superposition' and it's disappearance (the mythical wave collapse) is very simply the existence of intermittence due to chaotic dynamics. "Coherence" in the walker/surfer picture is the existence of the association of the particle and the field and the resulting chaotic/intermittent dynamics. Destroying the field destroys that dynamic. the dynamic becomes classical dynamics and ceases to be quantum dynamics. Observing a QM system destroys its QM dynamics if one destroys the field. Decoherence is the destruction of the wavefield self-generated by the particle bouncing about and thus the end of the QM dynamic. QM dynamics is a chaotic dynamics brought about by memory of its past via a wavefield that is easy to destroy.

And that is all there is to it.

A brief historical philosophy consideration

In conclusion, the latest Couder work, simply explains many of the QM behaviors and the magical coherence/decoherence and seems very simple in retrospect. Why wasn't this 'particle/wave' duality taken to it's logical conclusion by the founding fathers of QM? One has to project back to the Solvay conference in 1927, where deBroglie proposed these ideas and was shot down by the Copenhagen school. It is not hard to see why.

To make headway with these models (which we do in the 21st century) one needs the formalism of chaotic dynamics (which was developed in the 80's and 90's) and powerful computers to run the integration as there is little headway to be had analytically. In short, while philosophically satisfying, the wave/particle duality interpretation is difficult from a practical standpoint. In front of this model there was the straightforward (if magical) interpretation of the Copenhagen school: namely a vectorial superposition of states in a abstract 'hilbert space' that could give practical results in calculation. The 'shut up and calculate' mentality was justified as it gave us the atom bomb, most the technological advances of the late 20th century (think lasers, computer spin storage etc) and the CERN Higgs. The practical nature of the formalism was justification enough. The interpretation was secondary.