Making Waves
If you look on the Wikipedia page, it gleefully tells you that the symmetries required by quantum mechanics (QM) can’t be duplicated in a discretized system. That’s wrong. Over the next few posts, I’m going to show you how to do it, and even get QM effects happening on your laptop if you want.
What I’m going to show you isn’t a predictive physical theory. I don’t have that and I’m not trying for it. It’s simply a demonstration that anybody who tells you that QM effects require continuum mathematics (eg: Roger Penrose, Rouse Ball Professor of Mathematics at Oxford University) hasn’t done their homework.
The place I’m going to start is with waves. When people try to reformulate QM in a discrete way, they usually do it in terms of waves rather than particles. And when they duplicate waves in a discrete system, they often think of something like a cellular automaton (CA) running a discrete approximation to the wave equation. Here’s an example of what I mean. Indeed, when many physicists used to using more traditional tools hear digital physics proponents describe their ideas, this is what they often to assume is being proposed.
This approach is nice for a demo, but hopeless as an approximation to nature. This is because:
1: It’s anisotropic. In other words, the waves are ever so slightly square-shaped. This is bad, because the system is treating some directions as preferable to others, which we don’t see in nature.
2: It relies on a large but finite number of possible states for each cell. This means that once waves get a certain distance out from their center of propagation, they stop behaving like waves altogether. In contrast, light waves that can travel from one side of the visible universe to the other without breaking down. Hence, anything that relies on a set of amplitude states in cells is in trouble, unless the number of possible states is so high as to make the discreteness of the approach irrelevant.
3: It’s not obviously compatible with Relativity. Relativity shows that spacetime is deformed by gravity. A cellular automaton requires a regular grid to function. As soon as you change the shape of the lattice, the automaton stops working properly. Some CA enthusiasts will tell you that this doesn’t matter because space isn’t really bending. However, I’ve never seen a working model of this and doubt there is one.
4: This approach is at odds with the idea of a Big Bang. CA rules require that the universe is running on a grid of fixed size. However, there is a huge pile of evidence that about fourteen billion years ago, the universe occupied zero volume. In other words, the Big Bang didn’t rush outward from a single point into a void of empty space. It was the space that was doing the rushing outwards. I have yet to encounter a CA model that can comfortably handle this.
Net: Any attempt to model quantum waves that uses this approach looks broadly inconsistent with what we know about how the universe works, and so it’s useful really only as a model.
So if this approach doesn’t help us get a deeper understanding of nature, what will? I’m going to show you the answer in stages so that it makes sense. Otherwise it’s going to look weird.
First, let’s talk about problem (1). If using a grid is a poor approximation to space, what do we use instead? I’ve touched on this in previous posts. The answer is an irregular, locally-connected network. The easiest way to build something like this is to throw a bunch of random points onto a flat surface and connect each point to the ones near it. That’s not the only way to do it, but it’s good enough for now. So that’s it. You’re done. You now have a network that is effectively the same in all directions, so long as you measure at a scale that’s significantly bigger than the mean distance between any two connected points.
Now, let’s talk about problem (2). How do we make waves that will never degrade, regardless of how much they spread out. We solve this by using the kinds of waves that discrete systems like to produce. These are called ‘activation waves’, or ‘excitable media waves’. When you see them in cellular automata, they look like this:
These kind of waves show up in nature in forest fires, heart tissue, and the famous BZ reaction:
These kinds of waves will run forever across a discrete structure without ever stopping or losing coherence. In fact, your heart depends on this fact to keep beating. Excitation waves are keeping you alive right now.
We can get waves like this running on our irregular network using an easy rule. It goes like this:
- Make two sets of points, A and B. Put a single point in set A, and keep B empty (don’t worry, we’ll fill it later).
- Look at all the neighbors to the elements in sets A and B. Score them such that each point’s score goes up by one for every member of A that it’s next to. Make the score go down for every member of B that it’s next to. (So, a point that’s next to 5 elements of A, and 2 elements of B gets a score of 3.)
- Rank the neighbors and take the top half of the set. Call this the new set A. Take the points that used to be in A, and call them the new set B.
- Go back to step 2 and iterate.
When you run this rule on the network we built, it looks like this:
Looks pretty similar, doesn’t it? However, there are some important differences. First, the algorithm for our new wave works by updating sets. That means it’s controlled in a single place rather than being distributed across the network.
Second, the algorithm only needs two sets. Equivalent algorithms that work in CAs tend to require that cells have a lot of different states.
Third, sets of nodes are the same thing as sets of neighbors. This point is subtle, so let’s think about it for a moment. Each point in the network has a set of other points that it’s connected to. Similarly, our wave has sets of points that it’s occupying at any point in time. What this means is that the wave is just like a point in the network we just built, except that it has two sets of neighbors instead of one, and we have a rule for changing those neighbors. Also, just as each node in our network represents a single point, so does our wave. In other words, our wave is a lot like a particle, or even just a piece of space.
You can look at a wave of this sort running here.
While this kind of wave interesting to think about, this wave is still a long way from looking like the ones we see in nature. For a start, it can’t interfere with itself, and it doesn’t have a clearly defined wavelength. However, I’ve probably covered enough ground for this time. The next time we tackle waves, we’ll make them more realistic.
How to Build A Universe 
It certainly seems that CA’s, in the form of a regular lattice where each cell’s state depends only upon its immediate neighbours, appear futile for various reasons. But I think a lot of care is needed to dismiss it completely. Digital physics in general appears hopelessly implausible until you spend time thinking things through and gradually it begins to seem more viable.
Concerning its anisotropic nature, there has been discussion about CA expanding waves that form a perfect circle in the limit although I haven’t personally read the literature to know if they’re saying what I think they’re saying, eg at
http://vm-jn.wspc.com.sg/ijprai/15/1507/S0218001401001301.html Also, in a chaotic network we don’t have local isotropy, so a wave front will be “buffeted” as it moves along, as I’m sure you’re fully aware. Somehow this needs to not accumulate, so that light coming from a distant star does not appear dispersed. I notice you say that the Jellyfish algorithm shows “potential” compatibility with Lorentz invariance.
Let me stress though that I absolutely agree that a locally random network is a better approach.
I note with interest that you mention Roger Penrose as an advocate of continuous physics. Ironically some comments of his gave me the confidence I needed to keep pursuing an interest in digital physics. At the time I had read some of Ed Fredkins’ work and found it immediately compelling. However, being an amateur like yourself, I was constantly troubled that I had stumbled into a pseudoscience that only people who don’t really know anything could make the error of taking seriously. And then I was reading Penrose’s The Road To Reality and came upon a statement in section 32.6 on page 947 where he describes his early years:
“My own particular goal had been to try to describe physics in terms of discrete combinatorial quantities, since I had, at the time, been rather strongly of the view that physics and spacetime structure should be based, at root, on discreteness, rather than continuity.”
I immediately felt then that even if it is wrong, it is not silly, since a highly respected scientist could give it serious thought. I have since become aware of other mainstream scientists who treat it seriously, and apparently a lot of others that are quietly uneasy about the implications of continuity.
Anyway, I love your blog and I hope you keep it going.
Hi Ray,
Thank you loads for your kind words about the blog. I completely agree with you that the CA approach shouldn’t be disregarded, but I think it has a lot of ground to cover before it can achieve the things that network-based approaches seem capable of. I should also say that Ed Fredkin has been a source of inspiration for me too. He is an extraordinary guy, and his quiet influence on really important figures in physics like Feynman and ‘t Hooft is to his huge credit. There are several topics on which I disagree with him, but that doesn’t stop him from being a fully awesome individual.
As for Penrose, he both fascinates and frustrates me. He was instrumental my decision to take digital physics seriously. (A similar response to your own, by the sounds of it.) My background was originally in AI research. When I read Emperor’s New Mind, the thing that struck me was that every single one of the phenomena he quoted as being non-computable was actually something I was pretty confident that I could build. My research since that point seems to bear out that initial instinct. I think it’s unfortunate that such a bright guy should have put down the discrete approach before he’d properly explored the ramifications.
You also mention the pseudoscience perception problem. It’s a real issue, I think, and rather sad. I suspect that a lot of the willful disregard stems from the fact that reconciling algorithmic approaches with the continuum tools that physics is used to using is very hard. People don’t want a discrete approach to work because it entails a severe mathematical headache. Ironically, I think biologists do a better job of overcoming this gap than physicists. Evolutionary theorists, for instance, know that continuum approximations are just that–approximations–and calculate accordingly. I sometimes wonder if some of the defining math for 21st century physics will end up being borrowed from biology, or economics, or somewhere even further afield.
Alex
So is your set of points a grid of pixels, with each point having 4 neighbours? And how are you assigning the colours?
Hey Keir,
These are great questions. Sorry if I wasn’t clear.
The approach I mention first, that I think doesn’t work, uses a grid of pixels, with each point having 8 neighbors usually.
My approach, and the focus of this piece is an approach where the number of neighbors to each node in the network isn’t known before you build it. Each node might have two neighbors, or it might have 500. You drop points onto a smooth surface using a Poisson process, and then connect up the ones that are close to each other. In the picture I show, many nodes have more than 50 neighbors. The graph is purposefully densely connected.
The colors in my particle correspond to membership of the different sets. Red indicates membership of set A. Blue indicates membership of set B. Magenta indicates members of both A and B.
Yes I think you’re right about the mathematical headache that discreteness implies. If I were a career physicist I have to admit I doubt I’d be putting a lot of effort into digital physics unless I had a pretty secure tenure bolted down. I have a suspicion, though, that while conceptualising how some properties of QM or relativity can be accommodated by an algorithm may be impossible, that developing working models may turn out a bit easier than thought. This is because I think many of the properties will turn out to be emergent, you just have to hit on the right form of algorithm. I’m using the word emergent in the stricter mathematical sense, that the easiest shortest way to establish a property is to simply run a simulation. That’s why I like your proof of concept approach. Maybe one day I’ll have the time and energy to do the same.