April 14, 2020

Website:?Wolfram Physics Project

Technical Intro:?A Class of Models with the Potential to Represent Fundamental Physics

How We Got Here:?The Backstory of the Wolfram Physics Project

I Never Expected This

It’s unexpected, surprising—and for me incredibly exciting. To be fair, at some level?I’ve been working towards this for nearly 50 years. But it’s just in the last few months that it’s finally come together. And it’s much more wonderful, and beautiful, than I’d ever imagined.

In many ways it’s the ultimate question in natural science: How does our universe work? Is there a fundamental theory? An incredible amount has been figured out about physics over the past few hundred years. But even with everything that’s been done—and it’s very impressive—we still, after all this time, don’t have a truly fundamental theory of physics.

Back when I?used do theoretical physics for a living, I must admit I didn’t think much about trying to find a fundamental theory; I was more concerned about what we could figure out based on the theories we had. And somehow I think I imagined that if there was a fundamental theory, it would inevitably be very complicated.

But in?the early 1980s, when I started studying?the computational universe of simple programs?I made what was for me a very?surprising and important discovery: that even when the underlying rules for a system are extremely simple, the behavior of the system as a whole can be essentially arbitrarily rich and complex.

And this got me thinking:?Could the universe work this way? Could it in fact be that underneath all of this richness and complexity we see in physics there are just simple rules? I soon realized that if that was going to be the case, we’d in effect have to go underneath space and time and basically everything we know. Our rules would have to operate at some lower level, and all of physics would just have to emerge.

By the early 1990s I had a definite idea about how the rules might work, and by the end of the 1990s I had figured out quite a bit about their implications for?space,?time,?gravity?and?other things in physics—and, basically as an example of what one might be able to do with science based on studying the computational universe, I devoted?nearly 100 pages?to this in my book?A New Kind of Science.

I always wanted to mount a big project to take my ideas further. I?tried to start?around 2004. But pretty soon I got swept up in building?Wolfram|Alpha, and the?Wolfram Language?and everything around it. From time to time I would see physicist friends of mine, and I’d talk about my physics project. There’d be polite interest, but basically the feeling was that finding a fundamental theory of physics was just too hard, and only kooks would attempt it.

It didn’t help that there was something that bothered me about my ideas. The particular way I’d set up my rules seemed a little too inflexible, too contrived. In my life as a?computational language designer?I was constantly thinking about abstract systems of rules. And every so often I’d wonder if they might be relevant for physics. But I never got anywhere. Until, suddenly, in the fall of 2018, I?had a little idea.

It was in some ways simple and obvious, if very abstract. But what was most important about it to me was that it was so elegant and minimal. Finally I had something that felt right to me as a serious possibility for how physics might work. But?wonderful things were happening?with the Wolfram Language, and I was busy thinking about all the?implications of finally having a full-scale computational language.

But then, at our annual?Summer School?in 2019, there were two young physicists (Jonathan Gorard and Max Piskunov) who were like, “You just have to pursue this!”Physics had been my great passion?when I was young, and in August 2019 I had a?big birthday?and realized that, yes, after all these years I really should see if I can make something work.

So—along with the two young physicists who’d encouraged me—I began in earnest in October 2019. It helped that—after a lifetime of developing them—we now had great computational tools. And it wasn’t long before we started finding what I might call “very interesting things”. We reproduced, more elegantly, what I had done in the 1990s. And from tiny, structureless rules out were coming space, time, relativity, gravity and hints of quantum mechanics.

We were doing zillions of computer experiments, building intuition. And gradually things were becoming clearer. We started understanding how quantum mechanics works. Then we realized what energy is. We found an outline derivation of my late friend and mentor?Richard Feynman’s path integral. We started seeing some deep structural connections between relativity and quantum mechanics.?Everything just started falling into place. All those things I’d known about in physics for nearly 50 years—and finally we had a way to see not just what was true, but why.

I hadn’t ever imagined anything like this would happen. I expected that we’d start exploring simple rules and gradually, if we were lucky, we’d get hints here or there about connections to physics. I thought maybe we’d be able to have a possible model for the first 10^-100 seconds of the universe, but we’d spend years trying to see whether it might actually connect to the physics we see today.

In the end, if we’re going to have a complete fundamental theory of physics, we’re going to have to find the specific rule for our universe. And I don’t know how hard that’s going to be. I don’t know if it’s going to take a month, a year, a decade or a century. A few months ago I would also have said that I don’t even know if we’ve got the right framework for finding it.

But I wouldn’t say that anymore. Too much has worked. Too many things have fallen into place. We don’t know if the precise details of how our rules are set up are correct, or how simple or not the final rules may be. But at this point I am certain that the basic framework we have is telling us fundamentally how physics works.

It’s always a test for scientific models to compare how much you put in with how much you get out. And I’ve never seen anything that comes close. What we put in is about as tiny as it could be. But what we’re getting out are huge chunks of the most sophisticated things that are known about physics. And what’s most amazing to me is that at least so far we’ve not run across a single thing where we’ve had to say “oh, to explain that we have to add something to our model”. Sometimes it’s not easy to see how things work, but so far it’s always just been a question of understanding what the model already says, not adding something new.

At the lowest level, the rules we’ve got are about as minimal as anything could be. (Amusingly, their basic structure can be expressed in a?fraction of a line?of symbolic Wolfram Language code.) And in their raw form, they don’t really engage with all the rich ideas and structure that exist, for example, in mathematics. But as soon as we start looking at the consequences of the rules when they’re applied zillions of times, it becomes clear that they’re very elegantly connected to a lot of wonderful recent mathematics.

There’s something similar with physics, too. The basic structure of our models seems alien and bizarrely different from almost everything that’s been done in physics for at least the past century or so. But as we’ve gotten further in investigating our models something amazing has happened: we’ve found that not just one, but many of the popular theoretical frameworks that have been pursued in physics in the past few decades are actually directly relevant to our models.

I was worried this was going to be one of those “you’ve got to throw out the old” advances in science. It’s not. Yes, the underlying structure of our models is different. Yes, the initial approach and methods are different. And, yes, a bunch of new ideas are needed. But to make everything work we’re going to have to build on a lot of what my physicist friends have been working so hard on for the past few decades.

And then there’ll be the physics experiments. If you’d asked me even a couple of months ago when we’d get anything experimentally testable from our models I would have said it was far away. And that it probably wouldn’t happen until we’d pretty much found the final rule. But it looks like I was wrong. And in fact we’ve already got some good hints of bizarre new things that might be out there to look for.

OK, so what do we need to do now? I’m thrilled to say that I think we’ve found a path to the fundamental theory of physics. We’ve built a paradigm and a framework (and, yes, we’ve built lots of?good, practical, computational tools?too). But now we need to finish the job. We need to work through a lot of complicated computation, mathematics and physics. And see if we can finally deliver the answer to how our universe fundamentally works.

It’s an exciting moment, and I want to share it. I’m looking forward to being deeply involved. But this isn’t just a project for me or our small team. This is a project for the world. It’s going to be a great achievement when it’s done. And I’d like to see it shared as widely as possible. Yes, a lot of what has to be done requires top-of-the-line physics and math knowledge. But I want to expose everything as broadly as possible, so everyone can be involved in—and I hope inspired by—what I think is going to be a great and historic intellectual adventure.

Today we’re officially launching our?Physics Project. From here on, we’ll be?livestreaming what we’re doing—sharing whatever we discover in real time with the world. (We’ll also soon be releasing?more than 400 hours of video?that we’ve already accumulated.) I’m posting?all my working materials?going back to the 1990s, and we’re releasing?all our software tools. We’ll be putting out bulletins about progress, and there’ll be?educational programs?around the project.

Oh, yes, and we’re putting up a?Registry of Notable Universes. It’s already populated with nearly a thousand rules. I don’t think any of the ones in there yet are our own universe—though I’m not completely sure. But sometime—I hope soon—there might just be a rule entered in the Registry that has all the right properties, and that we’ll slowly discover that, yes, this is it—our universe finally decoded.

How It Works

OK, so how does it all work? I’ve written a?448-page technical exposition?(yes, I’ve been busy the past few months!). Another member of our team (Jonathan Gorard) has written?two 60-page technical papers. And there’s other material available at the?project website. But here I’m going to give a fairly non-technical summary of some of the high points.

It?all begins?with something very simple and very structureless. We can think of it as a collection of abstract relations between abstract elements. Or we can think of it as a?hypergraph—or, in simple cases, a?graph.

We might have a collection of relations like

{{1, 2}, {2, 3}, {3, 4}, {2, 4}}

that can be represented by a graph like

All we’re specifying here are the relations between elements (like?{2,3}). The order in which we state the relations doesn’t matter (although the order within each relation does matter). And when we draw the graph, all that matters is what’s connected to what; the actual layout on the page is just a choice made for visual presentation. It also doesn’t matter what the elements are called. Here I’ve used numbers, but all that matters is that the elements are distinct.

OK, so what do we do with these collections of relations, or graphs? We just apply a simple rule to them, over and over again. Here’s an example of a possible rule:

{{x,?y}, {x,?z}}?→?{{x,?z}, {x,?w}, {y,?w}, {z,?w}}

What this rule says is to pick up two relations—from anywhere in the collection—and see if the elements in them match the pattern?{{x,y},{x,z}}?(or, in the?Wolfram Language,?{{x_,y_},{x_,z_}}), where the two?x’s can be anything, but both have to be the same, and the?y?and?z?can be anything. If there’s a match, then replace these two relations with the four relations on the right. The?w?that appears there is a new element that’s being created, and the only requirement is that it’s distinct from all other elements.

We can represent the rule as a transformation of graphs:

Now let’s apply the rule once to:

{{1, 2}, {2, 3}, {3, 4}, {2, 4}}

The?{2,3}?and?{2,4}?relations get matched, and the rule replaces them with four new relations, so the result is:

{{1, 2}, {3, 4}, {2, 4}, {2, 5}, {3, 5}, {4, 5}}

We can represent this result as a graph (which happens to be rendered flipped relative to the graph above):

OK, so what happens if we just keep applying the rule over and over? Here’s?the result:

Let’s do it a few more times, and make a bigger picture:

What happened here? We have such a simple rule. Yet applying this rule over and over again produces something that looks really complicated. It’s not what our ordinary intuition tells us should happen. But actually—as I first discovered in the early 1980s—this kind of intrinsic, spontaneous generation of complexity turns out to be?completely ubiquitous?among simple rules and simple programs. And for example my book?A New Kind of Science?is about this whole phenomenon and why it’s so important for science and beyond.

But here what’s important about it is that it’s what’s going to make our universe, and everything in it. Let’s review again what we’ve seen. We started off with a simple rule that just tells us how to transform collections of relations. But what we get out is this complicated-looking object that, among other things, seems to have some definite shape.

We didn’t put in anything about this shape. We just gave a simple rule. And using that simple rule a graph was made. And when we?visualize that graph, it comes out looking like it has a definite shape.

If we ignore all matter in the universe, our universe is basically a big chunk of space. But what?is?that space? We’ve had mathematical idealizations and abstractions of it for two thousand years. But what really is it? Is it made of something, and if so, what?

Well, I think it’s very much like the picture above. A whole bunch of what are essentially abstract points, abstractly connected together. Except that in the picture there are 6704 of these points, whereas in our real universe there might be more like 10400?of them, or even many more.

All Possible Rules

We don’t (yet) know an actual rule that represents our universe—and it’s almost certainly not the one we just talked about. So let’s discuss?what possible rules there are, and what they typically do.

One feature of the rule we used above is that it’s based on collections of “binary relations”, containing pairs of elements (like?{2,3}). But the same setup lets us also consider relations with more elements. For example, here’s a collection of two ternary relations:

{{1, 2, 3}, {3, 4, 5}}

We can’t use an ordinary graph to represent things like this, but we can use a?hypergraph—a construct where we generalize edges in graphs that connect pairs of nodes to “hyperedges” that connect any number of nodes:

(Notice that we’re dealing with directed hypergraphs, where the order in which nodes appear in a hyperedge matters. In the picture, the “membranes” are just indicating which nodes are connected to the same hyperedge.)

We can make rules for hypergraphs too:

{{x,?y,?z}}?→?{{w,?w,?y}, {w,?x,?z}}

And now here’s what happens if we run this rule starting from the simplest possible ternary hypergraph—the ternary self-loop?{{0,0,0}}:

Alright, so what happens if we just start picking simple rules at random? Here are?some of the things they do:

Somehow this looks very zoological (and, yes, these models are definitely relevant for things other than fundamental physics—though probably particularly molecular-scale construction). But basically what we see here is that there are various common forms of behavior, some simple, and some not.

Here are some samples of?the kinds of things we see:

And the big question is: if we were to run rules like these long enough, would they end up making something that reproduces our physical universe? Or, put another way, out in this computational universe of simple rules, can we find our physical universe?

A big question, though, is: How would we know? What we’re seeing here are the results of applying rules a few thousand times; in our actual universe they may have been applied 10500?times so far, or even more. And it’s not easy to bridge that gap. And we have to work it from both sides. First, we have to use the best summary of the operation of our universe that what we’ve learned in physics over the past few centuries has given us. And second, we have to go as far as we can in figuring out what our rules actually do.

And here there’s potentially a fundamental problem: the phenomenon of?computational irreducibility. One of the great achievements of the mathematical sciences, starting about three centuries ago, has been delivering equations and formulas that basically tell you how a system will behave without you having to trace each step in what the system does. But?many years ago?I realized that in the computational universe of possible rules, this very often isn’t possible. Instead, even if you know the exact rule that a system follows, you may still not be able to work out what the system will do except by essentially just tracing every step it takes.

One might imagine that—once we know the rule for some system—then with all our computers and brainpower we’d always be able to “jump ahead” and work out what the system would do. But actually there’s something I call the?Principle of Computational Equivalence, which says that almost any time the behavior of a system isn’t obviously simple, it’s computationally as sophisticated as anything. So we won’t be able to “outcompute” it—and to work out what it does will take an irreducible amount of computational work.

Well, for our models of the universe this is potentially a big problem. Because we won’t be able to get even close to running those models for as long as the universe does. And at the outset it’s not clear that we’ll be able to tell enough from what we can do to see if it matches up with physics.

But the big recent surprise for me is that we seem to be lucking out. We do know that whenever there’s computational irreducibility in a system, there are also an infinite number of pockets of computational reducibility. But it’s completely unclear whether in our case those pockets will line up with things we know from physics. And the surprise is that it seems a bunch of them do.