See the announcement: Finally We May Have a Path to the Fundamental Theory of Physics… and It’s Beautiful (April 14, 2020)
(This post was originally published on the Wolfram Blog.)
I don’t have much time for hobbies these days, but occasionally I get to indulge a bit. A few days ago I did a videoconference talking about one of my favorite hobbies: hunting for the fundamental laws of physics.
Physics was my first field (in fact, I became a card-carrying physicist when I was a teenager). And as it happens, the talk I just gave (for the European Network on Random Geometry) was organized by one of my old physics collaborators.
Physicists often like to think that they’re dealing with the most fundamental kinds of questions in science. But actually, what I realized back in 1981 or so is that there’s a whole layer underneath.
There’s not just our own physical universe to think about, but the whole universe of possible universes.
If one’s going to do theoretical science, one had better be dealing with some kind of definite rules. But the question is: what rules?
Nowadays we have a great way to parametrize possible rules: as possible computer programs. And I’ve built a whole science out of studying the universe of possible programs—and have discovered that even very simple ones can generate all sorts of rich and complex behavior.
Well, that’s turned out to be relevant in modeling all sorts of systems in the physical and biological and social sciences, and in discovering interesting technology, and so on.
But here’s my big hobby question: what about our physical universe? Could it be operating according to one of these simple rules?
If the rules are simple enough, one might be able to do something that seems quite outrageous: just search the universe of all possible rules, and find our own physical universe.
It’s certainly not obvious that our universe has simple rules at all. In fact, looking at all the complex stuff that goes on in the universe, one might think that the rules couldn’t be terribly simple.
Of course, as early theologians pointed out, the universe clearly has some order, some “design”. It could be that every particle in the universe has its own separate rule, but in reality things are much simpler than that.
But just how simple? A thousand lines of Mathematica code? A million lines? Or, say, three lines?
If it’s small enough, we really should be able to find it just by searching. And I think it’d be embarrassing if our universe is out there, findable by today’s technology, and we didn’t even try.
Of course, that’s not at all how most of today’s physicists like to think. They like to imagine that by pure thought they can somehow construct the laws for the universe—like universe engineers.
The physicists at my recent videoconference are a little closer to my point of view, though the methodology and technicalities of what I’m doing are still pretty alien to them.
But OK, so if there’s a simple rule for the universe, what might it actually be like? I’ve done a lot of work on this, and written quite a lot about it.
One important thing to realize is that if the rule is simple, it almost inevitably won’t explicitly show anything familiar from ordinary everyday physics. Because in a really small rule, there just isn’t room to fit an explicit “3” for the effective dimension of space, or the explicit masses of one’s favorite particles.
In fact, there almost certainly isn’t even room to fit an explicit notion of space, or of time.
So in a sense we have to go below space and time—to more fundamental primitives. So what might these be?
There are undoubtedly many ways to formulate them. But I think most of the promising possibilities are ultimately equivalent to networks like this:
There’s no “space” here—just a bunch of points, connected in a certain way. But I think it’s a little like, say, a liquid: even though at the lowest level there are just a bunch of molecules bouncing around, on a large enough scale a continuum structure emerges.
Normally in physics one thinks of space as some kind of background, in which matter and particles and so on separately exist.
But I suspect it’s really more integrated: that everything is “just space”, with the particles being something like special little lumps of connectivity in the network corresponding to space.
In his later years, Albert Einstein actually tried hard to construct models for physics a bit like this, in which everything emerged from space. But he had to use continuum equations as his “primitives,” and he could never make it work.
Many years later, there are a certain number of physicists (many of whom were at my videoconference) who think about networks that might represent space. They haven’t quite reached the level of abstractness that I’m at. They still tend to imagine that the points in the network have actual defined positions in some background space—or at least that there’s some topology of faces defined.
I’m operating at a more abstract level: all that’s defined is the combinatorics of connections. Of course, one can always make a picture using GraphPlot or GraphPlot3D.
But the details of that picture are quite arbitrary.
What’s interesting, though, is that when a network gets big enough, its combinatorics alone can in effect define a correspondence with ordinary space.
It doesn’t always work. In fact, most networks (like the last two below) don’t correspond to manifolds like 3D space. But some do. And I suspect our universe is one of them.
But, OK, having space isn’t really enough. There’s also time.
Current physics tends to say that time is just like space—just another dimension. That’s of course very different from the way it works in programs. In programs, moving in space might correspond to looking at another part of the data, but moving in time requires executing the program.
For networks, pretty much the most general kind of program is one that takes a piece of network with one structure, and replaces it with another.
Often there’ll be many different ways to apply rules like that to a particular network. And in general each possible sequence of rule applications might correspond to a “different branch of time”.
But it turns out that if one thinks about an entity inside the network (like us in the universe), then the only aspect of applying the rules that we can ever perceive is their “causal network”: the network that says what “updating event” influences what other one.
Well, here’s an important thing: there exist rules which have the property that whatever order they’re applied in, they always give the same causal network.
And now there’s a big fact: these causal invariant rules not only imply that there’s just a single perceived thread of time in the universe; they also imply the particular relation of space and time that is special relativity.
Actually, there’s even more than that. If the microscopic updatings of the underlying network end up being random enough, then it turns out that if the network succeeds in corresponding in the limit to a finite dimensional space, then this space must satisfy Einstein’s equations of general relativity.
It’s again a little like what happens with fluids. If the microscopic interactions between molecules are random enough, but satisfy number and momentum conservation, then it follows that the overall continuum fluid must satisfy the standard Navier–Stokes equations.
But now we’re deriving something like that for the universe: we’re saying that these networks with almost nothing “built in” somehow generate behavior that corresponds to gravitation in physics.
This is all spelled out in the NKS book. And many physicists have certainly read that part of the book. But somehow every time I actually describe this (as I did a few days ago), there’s a certain amazement.
Special and general relativity are things that physicists normally assume are built into theories right from the beginning, almost as axioms (or at least, in the case of string theory, as consistency conditions). The idea that they could emerge from something more fundamental is pretty alien.
The alien feeling doesn’t stop there. Another thing that seems alien is the idea that our whole universe and its complete history could be generated just by starting with some particular small network, then applying definite rules.
For the past 75+ years, quantum mechanics has been the pride of physics, and it seems to suggest that this kind of deterministic thinking just can’t be correct.
It’s a slightly long story (often still misunderstood by physicists), but between the arbitrariness of updating orders that produce a given causal network, and the fact that in a network one doesn’t just have something like local 3D space, it looks as if one automatically starts to get a lot of the core phenomena of quantum mechanics—even from what’s in effect a deterministic underlying model.
OK, but what is the rule for our universe? I don’t know yet.
Searching for it isn’t easy. One tries a sequence of different possibilities. Then one runs each one.
Then the question is: has one found our universe?
Well, sometimes it’s easy to tell. Sometimes one’s candidate universe disappears after a tiny amount of time. Or has some bizarre exponential version of space in which nothing can ever interact with anything else. Or some other pathology.
But the difficult cases are when what happens is more complicated. One starts one’s candidate universe off. And it grows to millions or billions of nodes. And one can’t see what it’s doing. One uses GraphPlot. And lots of fancy analysis techniques. But all one can tell is that it’s bubbling around, doing something complicated.
Has one caught our universe, or not? Well, here’s the problem: one of the discoveries of NKS is a phenomenon I call computational irreducibility—which says that many systems that appear complex will have behavior that can never be “reduced” in general to a simpler computation.
It’s inevitable that at some level our universe will have this property. But what we have to hope is that a candidate universe that we “catch in our net” will have enough reducibility that we can tell that it really is our universe.
What we’ve been doing for the past few years is to try to build technology for “universe identification.” It’s not at all trivial. In effect what we’re trying to do is to build a system that can automatically recapitulate the whole history of physics—in a millisecond or something.
We need to be able to take what we observe in our candidate universe, and somehow establish what its effective physical laws are, and see whether they correspond to our universe.
Of course, it’s somehow more like mathematics than traditional physics. Because in a sense we have the underlying “axioms”, and we’re trying to see what laws they imply, rather than having to base everything on pure experiment.
There’s an analogy that I find useful. When I was working on the NKS book, I wanted to understand some things about the foundations of mathematics.
In particular, I wanted to know just where the mathematics that we do lies within the universe of all possible mathematics.
So I started enumerating axiom systems, and trying to discover where in the space of possible axiom systems our familiar areas of mathematics show up.
One might think this was crazy—like searching for our universe in the space of possible universes.
But NKS suggests it’s not. Because it suggests that systems with simple rules can have the richness of anything.
And indeed, when I searched, for example, for Boolean algebra (logic), I did indeed find a tiny axiom system for it: it turned out to be about the 50,000th axiom system in the enumeration I used.
Proving that it was correct took all sorts of fancy automated-theorem-proving technology—though I’m happy to say that as of Mathematica 6, FullSimplify can just do it!
I think it’s going to work a bit like this for the universe. It’s going to take a lot of effort—and a little luck—to avoid the long arm of computational irreducibility. But the hope is that we’ll be able to do it.
Physicists at the videoconference were very curious about whether I had candidate universes yet. The answer is yes. But I have no idea yet just how difficult they’ll be to analyze.
A good friend of mine has kept on encouraging me not to throw away any even vaguely plausible universes—even if we can show that they’re not our universe. He thinks that alternate universes have to be good for something.
I certainly think it’ll be an interesting—almost metaphysical—moment if we finally have a simple rule which we can tell is our universe. And we’ll be able to know that our particular universe is number such-and-such in the enumeration of all possible universes.
It’s a sort of Copernican moment: we’ll get to know just how special or not our universe is.
Something I wonder is just how to think about whatever the answer turns out to be. It somehow reminds me of situations from earlier in the history of science. Newton figured out about motion of the planets, but couldn’t imagine anything but a supernatural being first setting them in motion.
Darwin figured out about biological evolution, but couldn’t imagine how the first living cell came to be.
We may have the rule for the universe, but it’s something quite different to understand why it’s that rule and not another.
Universe hunting is a very technology-intensive business. Over the years, I’ve gradually been building up the technology I think is needed—and quite a bit of it is showing up in strange corners of Mathematica.
But I think it’s going to be a while longer before there are more results. And before we can put “Our Universe” as a Demonstration in The Wolfram Demonstrations Project. And before we can take our new ParticleData computable data collection and derive every number in it.
But universe hunting is a good hobby. And it was nice to have a chance to talk about it a few days ago.