Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas - 155 | Stephen Wolfram on Computation, Hypergraphs, and Fundamental Physics

Episode Date: July 12, 2021

It's not easy, figuring out the fundamental laws of physics. It's even harder when your chosen methodology is to essentially start from scratch, positing a simple underlying system and a simple set of rules for it, and hope that everything we know about the world somehow pops out. That's the project being undertaken by Stephen Wolfram and his collaborators, who are working with a kind of discrete system called "hypergraphs." We talk about what the basic ideas are, why one would choose this particular angle of attack on fundamental physics, and how ideas like quantum mechanics and general relativity might emerge from this simple framework. Support Mindscape on Patreon. Stephen Wolfram received his Ph.D. in physics from Caltech. He is the founder and CEO of Wolfram Research, and the creator of Mathematica, Wolfram|Alpha, and the Wolfram Language. Among his awards are a MacArthur Fellowship. Among his books is A New Kind of Science. He recently launched the Wolfram Physics Project. Web site Wolfram Research Talk on Computation and Fundamental Physics Amazon.com author page Wikipedia Twitter

Hey, everyone, it's Cal Penn. I'm inviting you to join the best-sounding book club you've ever heard with my podcast, Earsay, the Audible and I-Heart Audio Book Club. Every episode, I nerd out with amazing guests and dive into the best new audiobooks available on Audible. It's the book club for your ears. Listen to Earsay, the Audible and I-Heart Audio Book Club on the I-Heart Radio app or wherever you get your podcasts. Do you want to find a stress-free way to buy your next car? Start at CarMax and shop your way.

If you want to browse with confidence, get pre-qualified online with no impact on your credit score and shop cars within your budget, from luxury cars to family rides. CarMax has options for almost every price range, including more than 25,000 cars priced under $25,000. So, hey, want to get started? Just head to CarMax.com for details and get pre-qualified today. Want to drive CarMax? Hello, everybody, and welcome to the Mindscape Podcast. I'm your host, Sean Carroll. And this is going to be one of those podcast episodes that people have been clamoring for for a long time. I get a lot of suggestions, and a lot of people suggest the same guest over and over again. Sometimes I ignore the people. You know, I love the people, but I don't always agree with them. Other times, I think, you know what, the people are right. And in this case, I think that they are. So I invited Stephen Wolfram to be on the podcast. Now, Stephen Wolfram was a physics prodigy at a young. young age, one of the youngest winners of the MacArthur Genius Grant, started out in particle

physics before he moved to think about complexity and cellular automata, may be best known right now for inventing the Mathematica programming language and analysis system, as well as the underlying Wolfram Language that was used to write it, and Wolfram Alpha answer engine. I use Mathematica all the time. Many people in my field use Mathematica, but he's also still interested in fundamental physics. A few years ago, some years ago, he wrote a big book called A New Kind of Science, where he suggested that cellular automata might give a clue as to different ways of thinking about all sorts of science, right? Biology, economics, as well as physics. More recently, he has launched what is called the Wolfram Physics Project, which is simply an effort to

start with almost nothing and get the fundamental laws of physics out of them. Now, there's a lot to say, this. And we don't say everything in this podcast, even though it's a long one. I wanted to focus the podcast on giving a sketch for what the main ideas are of the Wolfram Physics Project and how they are not yet fully developed, but might be developed going down the line. There were a lot of criticisms and some kudos about how he developed it sort of outside the mainstream of physics research and so forth, and then, you know, laid on us high. hundreds of pages of work with he and his collaborators at the same time. And, you know, it's not the mainstream of theoretical physics. Most physicists who are doing their own thing did not drop

everything to work on it, but some became interested, and they're inviting people to help out. So you'll learn a lot in this podcast, not about, I'm not going to talk about, you know, how the research was done, or the fact that it was uploaded and shared with the world or whatever. You can draw your own conclusions about that. I care about the substance of the actual ideas. And mostly I'm giving Stephen here the opportunity to explain those ideas, but there is a family resemblance between what he's trying to do and what I and my collaborators are trying to do in quantum mechanics, stuff like that. So let me very briefly tell you how I see that relationship. What we're trying to do is to start with the simplest, most stripped down version of the best

theory we currently understand, namely quantum mechanics, right? To start from almost nothing but quantum mechanics and see how things like space and time and matter and energy emerge from that fundamental description. Stephen Wolfram and his friends are trying to be way more ambitious than that. They're starting with a kind of automata, a kind of discrete system called a hypergraph and generalizations thereof. And basically, you have a guess as to what the rules are for how hypergraphs are updated. And the point that they're trying to make is that some guesses, to things that kind of look like the real world. So the hope is that eventually we find specific guesses

for what those update rules could be that actually get us the specific real world or very, very close to it. My own attitude is, you know, good luck to them. I'm very much in favor of lots of different approaches to fundamental physics. This is one of them. Great, let a thousand flowers bloom, et cetera.

I actually think that our, my and my collaborators approach is more promising, because it leans heavily on something we already understand quite well, which is quantum mechanics. As I see it, if the Wolfram Physics project works, they're first going to have to find quantum mechanics in exactly the right way, and then find all of the features that we have about the real world. So it's kind of a precursor to what I'm trying to do, et cetera.

But, you know, I might be wrong about that. It could be that this particular way of doing things leads to not just quantum mechanics, but a specific implementation of quantum mechanics that gives us other clues to understand the nature of space and time, et cetera. So, as I said, what we're trying to do here in this podcast is help you, the listener, actually understand what the ideas are and where they're going. Hopefully, we succeed in that.

Let me very briefly take an opportunity to, as I occasionally do, remind you that we have resources for the Mindscape Podcast. We have a web page, preposterousuniverse.com slash podcast. So every episode has, you know, links to stuff about the authors and other resources and also full transcripts of the complete episode. So you can search every Mindscape episode for the last three years. We're hitting our three-year anniversary, roughly speaking.

And also we have a Patreon. If you would like to support Mindscape, you can go to patreon.com slash Sean M. Carroll, donate a dollar or two or whatever your local currency is per episode, and you get ad-free versions of the podcast, as well as the ability to ask questions in our monthly Ask Me Anything episodes.

Mostly it's about gratitude. I'm very happy, very pleased that so many people enjoy Minescape very much and therefore choose to donate just a little bit. It's the thought that really does count in this case. So with all that, let's go. Stephen Wolfer, welcome to the Minescape podcast. Hello. So it's been a little over a year, I guess, since you announced your physics project. And I think I finally, by no means, have become an expert, but at least I know enough about it to ask a whole bunch of ignorant questions.

to help us all understand it a little bit better. Maybe to start off, is this, do you consider this current project be a continuation of the cellular automata type of stuff you did in your book a new kind of science, or is it a new branch somehow? Yeah, so I mean, it builds on basically a tower of ideas, technology, intuition, and so on, that actually I'm sort of amazed we've managed to get to the point we've managed to get to, But all those previous pieces seem to be necessary to get to where we are.

And, for example, I studied a lot in New Kind of Science 20 years ago. I studied a lot of cellular automata, which are these very simple programs with arrays of black and white squares and so on. And from studying those things, I developed both a bunch of intuition about how sort of things work in the computational universe. And I managed to figure out some fairly general principles, which we've been able to get more and more evidence for, which are necessary to kind of launch into figuring out things about physics. Now, when I wrote New Kind of Science, you know, it has 12 chapters, and one of those chapters was called Fundamental Physics. And it had two sections.

One was about thermodynamics, and one was about kind of the fundamental underlying theory of physics. And for me at the time, I viewed it as being an incomplete potential application of the general kinds of ideas that I was developing a new kind of science. and I had a bunch of specific ideas about thinking about space as a network, thinking about the way time would operate and so on. I was very pleased with the fact that I had managed to get a fair distance and deriving general relativity from that kind of theory of gravity from that kind of approach. I was not satisfied with where I had got with quantum mechanics. and, you know, that project, so now restarting that project nearly 20 years later with energetic young physicists and so on.

And one kind of initially rather technical new idea that kind of made the model just seem much cleaner to me. It was kind of, it's been a new development. And really, now I've seen a lot of things that I say to myself, I should have been able to figure that out 25 years ago. Welcome to the club. That does not distinguish you from anyone else working on science, right? Yeah, you always look back with that.

No, I think what's, so I view it as being, what's really neat, the formalism that we've now got is something which was sort of, there were precursors of it in a new kind of science, but the full sort of story of it is just emerging now. And it's really neat because it really, it gives one sort of a, to me there's sort of this way of understanding things in terms of how simple programs behave, developed a bunch of intuition and new kind of science. Now there's a whole other level of intuition and results that we're getting that provide in the sort of formalism for this project.

And that's so it's both applicable to fundamental physics and to a bunch of other things. And I'm pretty excited about that. So let's imagine that first, I'm sure this is not true, but let's imagine that there's going to be a listener who only listens to the first few minutes of this podcast. We want to let them have a takeaway for what is going on here. I mean, is it accurate my vague impression to say that the cellular automata had a grid, right? Like you had some checkerboard or something like that and some rules for updating them and some discrete model, which you're going to hope that if you squint and look very far away, it looks like physics.

Whereas here, you still have a discrete set of models, but you're not starting with a fixed grid. You're kind of building things up by rules as you go. Right. Yeah, that's right. I mean, I never thought that that cellular automata would be relevant to fundamental physics. I mean, other people were sort of saying, cellular automator are going to solve fundamental physics. And I was like, no, please don't say that.

That's just not going to work. And, you know, cellular automata are very minimal models once you have a fixed notion of space and time. And they've been extremely fertile models for huge numbers of different kinds of things, from, you know, road traffic flow to chemical catalysis to, you know, leaf growth, all kinds of different things. But they assume a pre-existing notion of space and time. And so in thinking about physics, I had realized ages ago that we really need to go underneath the notions of space and time that we're familiar with

and see how to build those up from something more fundamental. And so the real starting point of our project is to think about just space is made of something. That has not been sort of in the tradition of physics and the tradition of mathematics as well. That's not really been a thing that people think about. Space just is something in which things are placed at certain positions and so on. And so, you know, in this, and it's kind of like what people might have thought about a fluid like water.

It's just water. It's just flows. It does all these kinds of things. But, you know, 150 years ago, we found out, no, those Greeks were actually right. Some of those Greeks were right. It's made of discrete molecules bouncing around. And with space, we haven't yet sort of made that definitive leap. And one of the sort of starting points of this project is to think space is made of something.

What's it made of? It's just made of these disembodied, discrete points. And each point is just an abstract element. It doesn't have a position. There's nothing. It's just this element. And then what we say is all we know about those elements is how they're related to other elements, how they're effectively connected to other elements.

So it's like you think about all these points in space, all they have is a friend network, so to speak. They don't know where they are. They just know who their friends are, so to speak. And so the first big realization is from something as unstructured as that, by looking at that on a large enough scale, it's kind of like how continuous fluids emerge from lots of discrete molecules underneath bouncing around, there's a notion of a kind of continuous-like space that can emerge when you look at that on a large scale.

And that's kind of the, you know, that's one starting point. I mean, another important thing to realize is that in our setup, the only thing in the universe is space. And there's no, you know, we're not saying space is a background and there are things placed in space. We're saying everything in the universe is just kind of made of space. So what I have in mind is a bunch of dots, yite nodes that are connected by lines to form some kind of graph. And do I imagine that the line, that the dots, rather than nodes, are they labeled? Do they come, are there different kinds of dots,

or is there just an er dotness that they all share? It's an er dot. Okay. That's just, they're all identical. The only thing, the only thing about them is they are, two dots are either identical or they're not identical. That is, there's a particular dot or there's another dot. And so the only, the only thing about them is kind of their, you know,

their identity, so to speak. They don't have colors, they don't have, they don't have positions. They just know, I'm this dot and not that other dot. Right. So there's some dots and there's some lines, they make a graph or a hypergraph, I guess, and then there's a rule. There's a rule for updating to go from one collection of dots and lines to the next one, and then we just iterate that forever, I suppose. That's right. And the whole universe comes out of that. That's very nice. That's the idea. So, I mean, you know, there are many. pieces to that setup. For example, one thing is, all there is is these kind of atoms of space

and connections. There's nothing in space. There's nothing. It's not like you then say, we've got space. Now, let's put an electron in space. If you want an electron, you have to make it out of space, so to speak. You have to make it from features of that pattern of connections between the atoms of space. So that's kind of the, that's sort of the base. story of what is the data structure of the universe? What is the universe sort of made of, and that's the idea. It's made of these discrete elements and relations between those elements, which we can think of as being kind of lines joining them.

It's actually convenient, and this was the sort of technical innovation of a couple of years ago, to realize that the best way to think about this is in terms of a so-called hypergraph, where we just have elements and relations. In a graph, you have nodes, and you have nodes. have edges connecting those nodes and each edge connects two nodes. In a hypergraph, you can have any number of nodes involved in a hyperedge. And that sounds like it is really a technical detail, but it makes the whole model much, much easier to deal with, or at least I thought it made it much easier to deal with. In the end, it doesn't make much of a difference. And I could have

figured this out 25 years ago. But that's the way these things work. It seemed to make it easier technically to make forward progress. I think you were then talking about what do these atoms of space do, and you're describing kind of the rules that they use to get updated. And yeah, you're right. I mean, the basic idea is you look at the local structure

of this graph and you say, whenever the local structure of this graph looks like this particular pattern, then it should be updated to be a local piece of graph that looks different. Right. And that's the whole story. Some things are just better at home.

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so they can go look it up because especially, I mean, I should note, you, I don't want to talk too much about like style and procedure here because there's far too much physics to talk about, but you did announce this project with a website and a call to participate, right? If people want to dig into the details, what is the URL they should be going to? Well, fromphysics.org.

There you go. And people can find out what the details are and do their own calculations. So that's it. Yeah, I mean, the other thing we've done kind of in terms of the public, which has been really a big success, is we've, we've live streamed a lot of our internal working meetings. And we've, you know, all the notes from this project are all, you know, posted on the web, basically the day after they're made, typically. And it's been really interesting because a lot of people who are, you know, a lot of professional physicists have gotten involved, but also a lot of people who are, for example, involved in computer kinds of things and understand sometimes more of some of our jargon than the physicists would understand have also gotten involved. And it's really been an interesting process to sort of do science live and in public, so to speak. It's like, you know, and I, the, it's like, it's a good kind of generator of reality because, you know, we say some things that are right, we say some things that are wrong.

It's kind of a place where you can get to see kind of the process really happen. And there is something that is very different about what you're trying to do in terms of fundamental physics. I would argue, and then correct me if I'm wrong, usually, not always, but usually physicists work from the phenomena to try to come up. with an explanation, right? Like there are stars in the sky, there are planets moving in certain orbits, what can we do about that? There's an apple falling from a tree. But in some sense, you're just saying, well, here is the simplest possible thing that physics could be. Let's just see if we can guess the right one. Is that more or less accurate? No, I mean, it is closer to that than the reverse engineering approach. And in fact, when I see people try to sort of understand what we're

doing and sort of say, well, what about doing this and that and the other, the great tendency is to try to reverse engineer. And this was a thing that I kind of, you know, I used to do sort of mainstream standard physics back in the late 1970s, early 1980s. And, you know, that was what one did in those days. And the thing that was really, for me, very liberating was when I started studying, well, I got interested in how do complex phenomena occur in the natural world. And the question was, what are the right models to study that? And my initial thought was, let's use all the fancy stuff from physics, all those differential equations and all these kinds of things, I tried doing that total failure. So I said, let's step back and see, you know,

is there an alternative approach to modeling that we can use? And I had the very fortunate experience that I had been working as a practical matter on building my first computer language. It's a forerunner of mathematical and morphine language. This was back in 1979, 1980. And what one does when one builds a computer language is one saying, let's take all these computations people might want to do, and lets in a kind of natural science-type way drill down and try and find out what are the primitives from which we can build up all these computations, what are the kind of atoms of computation, and then build up from there. And the thing that's really interesting about that process is you're just making up these atoms

of computation. It's not like, you know, it's not really the same kind of, oh, we've also, we've got to make sure we get, you know, Einstein's equations here, we've got to do this and this. You're just making it up and you're hoping that it will be, in that case useful. the people. And so that experience got me into this idea of just figure out the most minimal model and see what it does. And that's what I did with cellular automata and it worked great in that case.

So in a sense with physics, what I was trying to do with this project is find the most structuralist model of structure in the universe. Because that was, and it is not, it is by no means self-evident that that will work. I mean, that's sort of the ultimate Occam's Razor-type idea is, you know, just make it, you know, absolutely minimal. Now, the thing that's really neat, and maybe we'll talk about this a bunch later, is that now that we have this foundation, we can start to see how a lot of ideas that people have had in mathematical physics over the last many decades actually plug into this idea and both inform it and help make those other ideas more, more founded. And it's, I think, so this notion, which basically, you know, this idea of find the simplest

primitives and then see what they do, this is what I get for having spent 40 years being a computational language designer. This is what, that's what that involves. And it's different from what people do traditionally in sort of reductionistic science where you say, start from the phenomena and then say, how do I get this phenomenon? Right. And I do want to get, it's a somewhat different methodology. I do want to definitely get to how we recover all of known physics, or at least, you know, move in the direction of recovering all of known physics. But there also seems to be like a couple of deep principles that you have in mind when you're working on this stuff that maybe it's worth sharing.

I mean, one of them is this idea of computational equivalence, right? I could try to gloss it, but why don't you tell the audience what that means? Yeah, yeah, right. So, you know, you set up some simple program. You run it. You see what it does. You might think if it's a simple program, it will always do simple things. That turns out to not be true.

That's sort of the big thing that I discovered in the early 1980s, that for several years, I kind of resisted that conclusion and eventually realized that is really the way things are. When you look in the sort of computational universe of possible programs, if you just sort of pick a program at random, even though the program may be very simple, its behavior may be very complicated. Programs that we typically build as humans

for some particular known purpose will tend to not have that feature because we want to have them achieve the particular purpose we want them to achieve, which is some purpose that we can describe. And so you avoid this thing that the simple programs producing complicated behavior. That idea of simple programs producing complicated behavior

I think is sort of the essence of the secret that nature uses, to make complicated stuff. And that's why we look around in nature, we see all this complicated stuff, we say, how could it possibly make all this complicated stuff? That's just the formal fact about what happens in the universe of possible programs.

So first thing is the realization, simple program doesn't necessarily have to have simple behavior. And then the next question is, okay, so you see this very complicated behavior. How do you characterize it? You could just say, oh, it's very complicated. And in fact, a lot of the people had seen these phenomena for centuries, actually, of, you know, fairly simple setups like, you know, the digits of pie.

Specifying them is quite easy, yet once you've generated the digits, they seem completely random. And people would typically say, oh, it's very complicated. There's not much we can say about that. And they would try to find, you know, like the distribution of prime numbers or something. They worked for a long time to try and figure out what regularity can we tease out of this thing that seems quite random. And so what I was interested in was trying to understand how do we characterize this

complexity that gets generated. And so the thing that I realized is the way you can think about what's happening is any process of generating some pattern, for example, you can think of as a computation. And then the question is, how sophisticated is that computation? So, for example, it might be that computation is just like adding numbers together. It's something very simple. Or that computation is something more sophisticated. And the question is, can you rate the sort of sophistication of computations. And so the big discovery of the 1930s and so on was that you might have thought every different kind of computation you want to do, you'd need a different machine to do it. But then with Turing machines and combinators and a bunch of other ideas, this notion of universal

computation arose that says you can just have one piece of hardware and just feed it different programs and it can do different things. And, you know, that's been a pretty centrally important idea in the technology of the last 15 years, that's the origin of software, the sort of developments of the computer revolution. But for science, the implication there is, well, perhaps these simple programs might actually be, for example, computation universal. They might be able to do essentially any computation you want. So this principle of computational equivalence of mine is basically the idea that when

you pick a program, if the program doesn't do something obviously simple, it will be doing a computation that is as sophisticated as the computations that anything can do. So in other words, what that's saying is, you know, you might think you start up with an incredibly trivial program, it only has trivial computations. As you make the program a little bit more complicated, it would gradually do more and more sophisticated computations. And as you make a really, really, really, really complicated program, it would do really, really complicated computations.

But the somewhat surprising claim of the principle of computational equivalence is that's not true. Once you get above some very low threshold, you're immediately at the max. You're immediately doing computations that are as sophisticated as anything. And that principle, I mean, that principle has many implications. And I mean, probably the most important for immediately for physics is this phenomenon I call computational irreduced ability. That was the next one. So go ahead and say that, yeah. Right.

So, I mean, the question there is, if you are running a program, can you tell what it will do? One way you can tell what it will do is you just run every step just like the program would run itself. But another thing you can do is say, I'm much smarter than that program. I can jump ahead

and it's going to run for a million steps but I can just jump to the end and say the answer is going to be 42 or something. And so, you know, one of the ideas of sort of the exact sciences, the mathematical scientists for a long time has been sort of a sign of doing wonderful things is that you can jump ahead.

like that. You can readily predict, you know, where will the planets be at some time in the future and so on. But so that's what I call computational reducibility, the ability to reduce the computational effort necessary to find the answer to jump ahead. So the claim is actually there are lots of systems that are computationally irreducible in the sense the only way to find out what they'll do is just to run every step or in effect just to observe what the system does. And the reason that happens is because the principle of computational equivalence. Because if you think you're the observer, you're the predictor, you are a computational system as well. And the question is, how do you as a computational system compete with the system you're trying to predict? So if it was the case that you could really be smarter than the system you're trying to predict, then yes, you could potentially jump ahead. But what the principle of computational equivalent says is, no, actually that won't be that way.

you will be exactly computationally equivalent to the system you're trying to predict. And so the system you're trying to predict its behavior will seem to use sort of irreducibly complicated. So that's one of its implications. So in some sense, this is saying, both of these principles,

the equivalence and the irreducibility, there are computers, as we've known since touring, that are universal computers. They can calculate any function you want to calculate. And what you're saying is, in the space of all computers, computations, Computations with that capacity are generic in some sense. They're all over the place.

They're much easier to find than you might have thought, and there's no way to simplify the typical one into something that you can just leap ahead. Right. I mean, principle of computational equivalence, one of its predictions in effect is that universal computation will be ubiquitous. It actually goes considerably further than that

because at a technical level, one of the things that one talks about in universal computation is, It is possible to program this system to do whatever computation you want. It's a little bit of a different statement to say, if you are presented with this system and it is fed, even this very simple program, then it will already do something complicated. So it's a couple of additional steps. But one of the key predictions is computation universality is ubiquitous.

And that has many implications. I mean, it has implications in terms of the practicalities of building computers. It has implications in terms of what aspects of physics we can expect to understand. It has implications at a more philosophical level about things like free will and so on, which maybe we'll talk about. Well, and one of the implications is that if we buy into the philosophy of it, it gives us some optimism for this program of starting program in the human sense, not the computer sense, for this project of starting with some very, very simple rules and getting the actual universe out, right?

if all sufficiently complicated computations are at the same level of ability in some sense. Yes, that's true. I mean, one of the things that is sort of a key, I would, you could even say prediction, statement of our physics project is what can be done in our universe is just computation and not hypercomputation.

That is, it's the kind of computation that can be done by a Turing machine. It's not the kind of computation that can be done by a thing that is infinitely more powerful than a Turing machine. that you can conceptually imagine, but you can't, you know, the claim is no such thing exists in our universe. And in fact, you can get a bit more technical about it.

And the claim of these models is even if it exists, it will be hidden behind a cosmological event horizon from us. We will never be able to observe it. So it's kind of like, well, you can say it exists, so you can say it doesn't exist, it doesn't really matter because we'll never be able to interact with it. But so, you know, that's one of the key statements, yes. And the idea that it's sufficient to have something that is a computational system that we humans can kind of understand how it's constructed, like a Turing machine or like one of these hypergraphy writing systems, that is a, you know, that's a supposition of this project. And it's a supposition that, in a sense, is sort of encouraging for us humans that it might be the case.

there's something we can kind of hold in our hands that represents everything the universe does, even if the implications of that thing we can hold in our hand are irreducibly complex. And even if we couldn't tell, you know, given that you give us the rules for the universe, if you say, can I make a warp drive, the answer might be, it's really hard to tell. It's an irreducibly complicated thing. It's like, you know, recapitulating the whole history of mathematics and so on to get to the point, you know, can we prove this theorem that there is or isn't a war? warp drive. So it's, but to me that's, yes, that's, that's, that's the concept.

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I'm all on board with that. So good. So with those in mind, I mean, let's go back to the physics that you're actually constructing here. We have a hypergraph. We have some rules for updating it. Now, should I read the specific, is the idea that there is a specific correct rule

for our universe, and that rule basically is the fundamental laws of physics? Okay. So this is where things get a little bit more complicated. So I think that it is probably the case that there will be a rule that will be able to hold up and say, with this rule, we can reproduce what we observe in physics. Now, footnote, which is really interesting footnote, I think, you might say, why did we get this rule and not another rule? It seems very, particularly if the rule is simple, that's like Copernicus was wrong, so to speak. You know, there isn't nothing special about us.

We got the simple universe, so to speak, or we got the universe with a simple rule, not the universe with the, to us, incredibly complicated, seeming rule. So the thing that is then very surprising is that, and this is, you know, we probably have to go a few more steps before we can really, really dig into this properly, is the idea that actually you can think of the universe as running all possible rules.

And we, as observers of the universe, are essentially exist in a particular, in a sense reference frame, in a particular position in rule-ial space, as we call it, that essentially gives us a particular sampling of this sort of all possible universes. And that particular sampling has, is given our way of parsing what happens in the universe, that is something that we could attribute to a particular underlying rule, but actually we can also think of it as just a slice of this kind of universe of all possible universes. So it's a little bit more, but I think, you know, operationally, I would hope that we will

be able to produce a rule that will have the property that if we work out all its consequences, and we cannot, because remember, one of the very tricky things is we as observers of the universe are embedded in this universe. And so as you change that underlying rule, you know, you know, it's a very tricky. You not only change the rule for the universe, you also change the rule for us. And so those two effects, in some very, very vague sense, cancel each other out. So that there is a large class of rules for the universe and rules for us, which all seem the same, and which all have the same laws of physics, effectively.

So to give an analogy, to make this a little bit less bizarre, so let's go back to fluid dynamics, for example. And, you know, we have the laws of fluid dynamics, the Navi Stokes equations, all these kinds of things, and you might say, how do we derive these? Underneath there are a bunch of molecules bouncing around. Well, there can be water molecules, which have their particular, you know, H-2O form. There can be, you know, air molecules, there, you know, N2 or whatever it is. These different kinds of molecules bouncing around, all of their physics are quite different. All the details of what happens when those molecules collide is quite different.

Yet on a large scale, the laws of fluid dynamics are the same for air and for water. They have some different parameters, but they're basically the same laws. And it's the same kind of thing that I think happens in the derivation of laws of physics. That is, certain aspects of the laws of physics are quite generic. I think general relativity and quantum mechanics are quite generic. And they are things which, if you are an observer, even vaguely like us, they are inevitable that you will observe. And if you want to home in on, well, do we have the fine structure constant with a particular value it has, you probably have to home in on more features of us as observers.

Right. I mean, do you think that, do you anticipate that at some step in understanding the relationship between these rules in our universe, there will be an anthropic move of some sort, where you say there's a lot of things going on, but intelligent observers will only observe this subset of things? Intelligent observers, no. observers with... So the question is,

what is the right idealization of the human observer? So I'll give you a couple that seem to be enough. Okay? So one important one is we're computationally bounded. We don't get to observe.

So let's take the gas molecule example again. If you are, you know, you have this gas, it's got a bunch of molecules bouncing around. If you're a sophisticated enough observer, you can see every single molecule, you can work out all the collisions

and in particular, that will allow you. So, you know, a big principle of statistical mechanics is second law of thermodynamics, which says, you know, typically the sort of typically things get more random as the molecules bounce around in a gas. But if we are not computationally bounded observers, and we can figure out what all these trajectories of all these molecules are, we don't get the second row of thermodynamics.

As non-computationally bounded observers, the second law of thermodynamics simply isn't true. And so the first, and that, And that same idea of a computationally bounded observer is necessary, I think, for us to believe that space has a continuous structure and various other things about the universe. So that's kind of step one. So we're not Laplace's demon. What's that, sorry? We're not Laplace's demon.

That is correct. We're not Laplace. We're not Maxwell's demon. We are thoroughly non-demonic in that sense. And finite and bounding. We only have this sort of bounded ability to. to look at the details of the universe. That's step one.

Step two I've understood more recently, actually. And step two is something which I think is sort of the physics essence of consciousness, which is that we have this idea that there is a single thread of experience that we have. It might not be that way. That is, in the universe,

and we didn't talk about this in detail yet, but in the universe, all these little updatings are happening all over the universe. But we think that, in a sense, there's sort of microscopic pieces of time happening all over the universe. But we believe we have a single identity through time and that there is a single thread of experience that we have through time. Now, it's by no means self-evident that that would be the way it is, because in our model of physics, everything about the universe is being rewritten, you know, 10 to the 100 times a second or something like this. And so the fact that we have the idea that we are still the same us a second later is non-travial. And so I think that an observer who has that idea that there's a single thread of time has that is one of the things that you need in order to get the laws of physics that we humans observe.

And I think that's a, so there's this idea of we maintain our identity through time, which we might not. And the thing I realized actually even more recently, although I should have realized this earlier, is there's a similar idea for space. It is not self-evident that there is a notion of pure motion. So in other words, you take an object, you move it somewhere else. In our models, you know, I take an object, I move it somewhere else. It's made of different atoms of space by the time it gets to its destination, probably. And so then the question is, how come there's an idea of pure motion? How come a thing can just be moved without changing?

Now, obviously, we know in general uncertainty, you know, you're close to a black hole, funky thing can happen to you if you move because the structure of space is changing. But this is a very basic fact that most of the time, you know, something moves and it is still the same thing. And so this idea of this sort of this single thread of experience in time, the idea that it is possible to have pure motion and it's possible to maintain your identity as you move around in space, those two attributes seem to be what you need to derive generic logophysics. That's what you seem to need to derive general relativity and quantum mechanics.

Now, if you start asking, why does the electron mass come out to be the actual value it has, for us humans, we probably need a little bit more information about us humans to be able to figure that out. I think. I'm not sure. I mean, for example, one of the things that's a current challenge is the local symmetry groups, local gauge groups, it is looking very likely that it is an inevitable feature of our models

that there will be local gauge invariants. And the thing that looks conceivable, I don't know yet, is that the actual gauge group will be determined. So in other words, even though we don't yet know why the universe has three dimensions, we might actually know why the internal symmetry group is maybe a subgroup of E8 or whatever it is

before we even know this thing about space. And so that would be an example. But we don't know, if you ask, what's the electron neuron muon mass ratio? You know, we're still a fair distance away from figuring that out. The classic question. My expectation would be that there'll be a particular rule that will find, which will have where a whole bunch of those things will come out. And one of the features, if you believe that the rule is simple, is if you move from one

simple rule to another nearby simple rule, everything's going to change because there just aren't very many rules that are that simple. And so instead of having, you know, electron muon mass ratio, it'll be, well, there actually are an infinite number of generations of particles and there are, you know, a bizarre number of dimensions of space and all this kind of thing. It'll just be completely different. But that's the current expectation. I think you raised the anthropic principle. And to me, the anthropic principle is sort of a story of lack of imagination, so to speak, because it's saying, you know, the only way that we can have life, intelligence, consciousness, whatever, is the particular way we've seen it. And, you know, one of the consequences of this principle of computational equivalence is that actually something like intelligence is ubiquitous.

That is, you might have thought to get something as computationally sophisticated as us humans with our brains and all this kind of thing, you need the whole process that's led to us humans. But what the principle of computational equivalence says is that's not true. Even these very simple, the systems were very simple can do it. And that has lots of consequences. You know, if you're, if you're worrying about

extraterrestrial intelligence, for example, that tells you it's everywhere. It's a question of whether we are sufficiently aligned with that intelligence to be able to recognize it as something that, for example, has purposes that we can understand as a sort of human-like purposes. So, you know, it's a, and I think this idea, intelligence requires liquid water is almost I'm on your side when it comes to that. But intelligence might require space time in some sense. So let's at least try to get that. I mean, do I, is this naive picture that I have in mind where you have the hypergraph, you update, it's a discrete updating. Can I think of the graph at any one update as space and the update itself as time, or is that too simple-minded? Okay, so it gets a little

complicated. And in fact, the complexity of the arises is quantum mechanics, I think. And so it's, in a sense, you try and make it that simple. And you, okay, so the basic point is the sort of, the rule says, if you have a lump of atoms of space that are connected in this way, transform it into a lump that's connected in this other way. And it basically, the rule just says, that's what you do. It doesn't say where you do it. It doesn't say when you do it. It's just, any, Anytime there's a lump that looks like this, you can transform it into a lump that looks like that. And so those transformations can be happening all over this hypergraph.

And so it is not at all obvious. That is the only thing that's defined is these can happen. The question of when they happen, what counts as the sort of simultaneity surface? What counts as that moment in time is something that's really in the eye of the observer? Okay. But the updated graph is supposed to represent space-time and the things within it, or is it a more subtle map there? No, no.

So at any given, what's happening is this graph is getting updated, and there are lots of little places where it can get updated. And you can say, okay, I'm going to consider the graph with this collection of updates having been done, I am going to consider that as time T equals zero, let's say. And then another situation, you're saying, I'm going to say, now I'm going to say this collection of updates is time T equals one, for example. And each one of those time slices, at each one of those sort of, well, in the language of physics, space-like hyperservices, that represents an instantaneous structure of space.

But it is somewhat arbitrary what you consider to be this instantaneous sort of structure of space just as it is in general relativity. Well, sure, right. That's very familiar from general relativity. But I'm just saying, is the collection of the whole shebang space-time and the things No, it's just space. A single hypergraph. The collection of all the updated hypergraphs, that's what I'm asking. Oh, yeah, yeah, right. The sequence of updates, the hypergraph, together with all its updates,

is supposed to be space time. Okay. And one of the things that is interesting and non-trivial here is, you know, a lot of most traditional views of physics have thought of space and time as being the same kind of thing. In this model, they're really not. Sure. Spaced is the extent of the spatial hypergraph Time is the computational process of updating this hypergraph So time is a time is the is the progression of a computation Space is just oh you follow these these connections in the hypergraph

Okay, and so that makes it not at all obvious that you're going to get things like relativity out of the model because it's some you know because one is breaking apart the sort of traditional connection between space and time But you say that the whole shabang is spacetime, one of the reasons I hesitate a little bit about that is the shabang is made by a computationally irreducible process. So we don't get to, it is a, you know, to say let's just unroll the whole structure of the universe forever. Well, we don't get to do that in the universe. I mean, the universe will do that as the universe does it, but we can't just say this is what's going to happen because there is this phenomenon of computational reducibility. But I do want to just distinguish between this idea and something that someone like myself would be very fond of,

which is some combination of quantum gravity and ever ready in many worlds, where there are different branches of the wave function where the geometry of space time is just different, perhaps radically different. It sounds like that is not something that is coming out of your hypergraphs. Oh, no, no, that will come out. I was simplifying a bit. Okay. So what's happening is at every, you know, these updates, you're just saying, whenever you see a piece of hypergraph that looks like you can do an update.

There are many possible sequences of updates, there are many possible sequencing of these updates that could occur. Each one of those sequencing defines a different path of history, effectively. Now, the non-trivial fact, another additional piece is this thing we call causal invariance. And so what's happening is if you look at this, this, okay, so if you look at this hypergraph, and you imagine you roll up the whole universe into a single lump that you can kind of hold in your hand, and then you say that single lump is going to evolve to another whole universe. that process of evolution is not a single thread. That process can branch because the whole universe,

if you did two updatings in different orders, for example, you would get different outcomes, you would get different whole universes. And that's very similar to the many worlds story. And one difference is that in our models, we're really taking seriously this idea that there's branching as a result of these different possible orders of updating and so on, and that there's this thing we called it a multi-way graph, which is just this description of all these different possible paths of history.

But for us, it's very important that the parts of history can branch, and they can also merge. And they merge when two outcomes for the universe end up, you apply two different rules, and it so happens that they end up with an identical universe. They end up with a hypergraph that is isomorphic that you can rearrange to be exactly the same hypergraph. And so that branching and merging process is really important

because one feature of many of our models, and in fact, in the end, it's a ubiquitous feature of all of them, but that's not how it initially looks. Initially, it looks like you have to pick it specially, but in the end you don't, is this thing we call causal invariance. And so essentially what causal invariance says is every time there's a branch, there will be a subsequent merge. So in other words, even though you thought you were going to go in two different branches of history,

even though for a while you've got two completely incompatible structures of space time, in the end they will merge. In the end, they'll be, and that's some, and so that's why, in our models, that's in the end why you get objectivity in quantum mechanics. Organic Valley Protein Plus ultra-filtered milk is pasture raised from cows that might more steps than most people. That's a plus. And it has 50% more protein and 50% less sugar than regular milk.

That's a big plus. And the fact that it didn't make a pun about moving, also a plus. Organic Valley Protein Plus ultra-filtered milk. Protein plus pasture raised. Learn more at Organicvalley.com. Well, let me see. So if I'm just a conventional many worlds person, there's perfectly deterministic evolution

of the entire wave function it will branch and part of the reason why it branches and operationally they never merge is because there's some environment which is huge numbers of degrees of freedom and the different states

in the environment will never exactly line up so eventually you might imagine them merging back together 10 to the 100 years from now is that the kind of thing you have going on or is you're remerging much more rapid? No, the remerging is faster than that. But again, it's a little complicated, but the fact that there is divergence and merging is basically, so what in the end happens is that

the uncertainty principle is a story of essentially curvature in this basically multi-way graph space. And it's just like in space time, one of the sort of, you know, the classic feature is you go around a square, you know, you go around a square with all the right angles and distances, does it close up? If space is curved, it doesn't close up. And what's happening in our models is it's the exact same thing, although the space is more complicated and more like a Hilbert space, that what's happening is the same exact kind of thing is what leads to the uncertainty principle in quantum mechanics. It's that same failure to close up. And now, what ends up being the case, and we don't understand this as completely as I would like yet.

So just to give some sense of the situation with respect to what we know and what we don't know, so these multi-way graphs, okay, so first things to say about multi-way graphs is they represent the sort of branching and merging of parts of history for the universe. So one thing you can do is you can say, at any given time, what is the structure

of all of those possible branches of history? So just like in space time, we're going to say, let's pick a space like hyperservice. Let's pick a simultaneous moment for all these different parts of space. What does space look like? Or we can do the same kind of thing for this multi-way graph. If we do that, we get this thing which has all these different, all these different sort of possible states of the universe. And how are they connected?

Well, they're connected in what we call a branchial graph, a graph that represents these different branches. And so one convenient way to construct a branch of your graph is to look at the common ancestry of those states. So if two of those states just branched one step before, they're close together in the branchial graph. If they branched many, many, many steps before, they're further apart.

And so you can construct, just like from this sort of structure of space time, you can construct a sort of instantaneous structure of space. So from this multi-way graph, you can construct what we call branchial space. This is the instantaneous representation of the possible branches of quantum history. And in a sense, it's a map of entanglements because every one of these branchings, the fact that they had a common ancestor and that they will, is a story of entanglement. That's saying that these two branches of history are not independent.

They are connected by the fact that they are entangled. So in a sense, this branchial space is a map of quantum entanglements. And so it's really, I think, incredibly cool because Bransheel space has many of the same attributes the physical space has. It's more like a Hilbert space. It's a little bit complicated and we don't fully understand it yet. It's something like a what is the analog in space time. We have this idea of Minkowski space, which has both sort of the usual Euclidean type space for space and then this notion of time being a separate thing. and exactly how that works for a Hilbert space,

the analog of Mikovsky space for the Hilbert space of states, we don't understand. It may be something related to Twister space. That's a special case of it, but it's something we don't yet understand. But one thing we do know is, you know, in terms of what we understand, what we don't understand, we do know that our multi-way graphs reproduce an approach

to quantum mechanics called categorical quantum mechanics, and we can just show that. that they are equivalent, basically. And it's known that categorical quantum mechanics is equivalent to standard quantum mechanics that people studied from the 1920s and so on. And not only can we show that they're equivalent, one of the recent things in the last month or two

has been we can actually use our models to model quantum circuits, and we can essentially compile a standard quantum circuit into a multi-way graph, and the big excitement of the last few weeks has been, and we can do better circuit optimization in multi-way graphs than you've been able to do in standard quantum circuit formalism. So in other words, we can, if you want to simulate a quantum circuit, it's better

to do it through our model than it is to do it through the standard mechanisms of quantum mechanics, but we know the answers have to be the same. So that's what we know for quantum mechanics. We don't yet know fully how that works for quantum field theory. Quantum mechanics, just for other people's benefit, I know you know this very well, but quantum mechanics is a story of what to individual particles do. Quantum field theory is a story of even if you can make particles and the particles are sort of spread out through space, how does that work quantum mechanically?

And that's a more difficult thing, which we're just starting to understand. I do want to dig in more specifically to mapping your approach onto the different interpretations of quantum mechanics. But let's just back up a little bit and keep it simple for the folks who are trying to stay with us here.

if I look at the pictures of the graphs of the hypergraphs that are following the rules, very naively, it looks super duper classical. There's a graph. Something is happening at every point. So could you just explain why you even get something like the double-slit experiment where you can get interference, but then if you look at it, you don't get interference? Is that something that pops out of your model? Yes.

Yes. I mean, by pop out, there's a little bit of effort. to make that pop happen. Sure, of course. But the, okay, so when you see pictures of these hypergraphs, they are pictures drawn for a particular branch of history, particular branch of quantum mechanical history. That's what we're easily able to visualize. Now it's been a big project to make what we call local multi-way systems which mix spatial

extent with branch-hill extent, that is mix things which are moving around in space with things which are going to different branches of history. And this has been a big challenge because it's pretty hard for us humans to visualize that stuff. And we've been working on it for the last year, actually. We're getting fairly close to having pretty good visualizations of that. But those are significantly more difficult to understand. But the typical pictures, you'll see, our project is full of graphs.

So there are many different kinds of graphs that show up. And they're kind of in all the, one of the things that's been a big kind of practical win is that we, at some moment we kind of did the colors for our model and picked this graph is going to be this color. Oh, good. This graph is going to be this other color. And so as soon as you see one of these things, you can tell that's a causal graph,

that's a branch of graph or whatever. That's just a, and, you know, because all the code for our models is all openly available, people have been just using that code. And so people have been following kind of the color standards, which really help. But so there are probably six different kinds

of graphs that show up. The ones that are the most common and colored bullish are the ones that are the hypergraphs that represent the instantaneous structure of space. So there are then these multi-way graphs that represent the kind of the branching and merging of quantum histories. Then there are branchial graphs which are traditionally pink in our world that represent these slices through the multi-way system. They represent, so in space time, we're taking a slice through space time and finding this is what space

instantaneously looks like. In the multi-way graph, you take a slice and you're saying this is what the instantaneous map of quantum states looks like. And in fact, what happens is in relativity, we're used to having this idea of reference frames of how do we label what space and time are. How does an observer decide what counts of space, what counts as time, and so on, if the observer's going at different speeds, you know, you're tipping the reference frame, all those kinds of things. We have exactly the same kind of thing in quantum mechanics. We have what we call quantum observation frames, which are basically the thing that represents, this corresponds to a particular state of the quantum world.

And you can think of these in branchial space, you can think of these points in branchial space, which are histories of the universe, as being different quantum eigenstates. So you can think of them as being different basis elements, so different things from which you would make up a quantum state. And so then what happens is, when you have one of these quantum observation frames, you've sliced this multi-way graph, you've got this whole collection of quantum states,

and now you ask, for example, so now, okay, this is one other thing I have to explain. In quantum mechanics, the big idea is this notion of quantum amplitudes. That is, you just get to know there's this object, which has complex numbers associated with it and all this kind of thing, that is the quantum amplitude. So the question is that quantum amplitude, the complex numbers that are associated with it, you know, one of the things that we've really really, is the idea that you can think of a quantum amplitude in some of the complex numbers is confusing. In fact, the right way to think about it is

it has a magnitude and it has a phase. And in our models, the magnitude of the quantum amplitude is the result of path counting in the multi-way graph. So in other words, if there are many ways to get to that state in the multi-way graph, the magnitude of the quantum amplitude is determined by the number of ways to get to that point in the multi-way graph.

the phase of the quantum amplitude is determined by the position in branchial space. So in other words, the idea is as you move around in branchial space, you are changing the phase, the coordinates of branch of space are essentially phase coordinates in quantum mechanics. Sorry, you might need to tell me what branchial space is again. Okay, so, okay, so then we've got this all possible histories of the universe, which correspond to this multi-way graph, they branch, they merge, it's a complicated thing. Now we're saying, let's, let's, and that graph is being formed through time. So every, every, and so now we say let's take an instantaneous time slice across that graph.

It is the quantum analog of a reference frame in relativity, okay? And that, and by the way, those reference frames can get quite wild in quantum mechanics. They're somewhat wilder than the ones you would typically use in general relativity. At least that's our view of how this works. So those reference frames are sort of the choice of the observer. They're not something that is intrinsic to the physics. They're just, oh, we've got this multi-way graph. Now we're going to say, for our purposes,

we're going to say this is what the time slices of that will be like. So this branchial graph is the structure you get by slicing through this multi-way graph. So you're asking the various points in this multibreact graph, how are they arranged? Well, we can say they're arranged nearby if they correspond to something which just had a common ancestor.

Just one step back, they had a common ancestor. And other ones, you know, you have to... So basically what happens is the simplest way to make these branch of graphs is you simply join by an edge every pair of nodes that have a common ancestor. Okay?

And then... So what that means is if there are... If there are... things are far apart in the branchial graph, it means their common ancestor... ancestor was probably many steps from the past. And the idea then is that that, okay, so the claim, which is, again, mathematically quite non-trivial, is that the coordination of branchial space, that is if you try and assign

coordinates, just like in physical space, you'd say there are all these atoms of space and they're just connected in certain ways, but now we say, based on those connections, it is, we can say this one has coordinate position, blah, blah, blah, and it's next to this other one which has a very similar coordinate position. You can do the same kind of thing in branch of shell space. You can try and coordinate eyes branchial space. And the claim is the coordinates of branchial space are the phase of the quantum amplitude. So in other words, what that means, so just to sort of perhaps for people's benefit, maybe

we can explain something about, so, you know, I don't know how to, we can try doing, let's see, you probably can do this better than I can, but explain quantum phase. But, you know, the idea. Sorry, the way that I just say it is that, you know, the wave function for a single particle is like a wave. Sometimes it's positive, sometimes it's negative or plus or minus I. So different contributions can destructively interfere as well as constructively interfere. Right. But the critical idea in a sense is as you have that wave that's going up and down, the phase of the wave is at a particular moment in time, are you up or are you down?

Right. And the phase is kind of a shifting. of that wave relative to a particular moment in time. So what we're saying is this branchial space represents kind of where that thing is shifted. A position in branchial space corresponds to a certain shift of that thing. So here's the place where it's actually my favorite result so far in our physics project, which is worth explaining because then we can talk about double stuff experiment. a little bit more easily.

Okay, so let's talk about physical space. One of the questions is, if you have an object, a particle, for example, in physical space, how will it move in physical space? And the standard idea of general relativity is, and of actually Newtonian physics, is it moves according to a shortest path from one point to another. So it's like whatever the shortest path is in space, it'll go along that shortest path. If space is flat, the shortest path will be a straight line. If space is curved, the shortest path might be deformed.

And the big idea of, you know, in Einstein's equations and so on, is the defamation of space, the curvature of space is produced by the presence of energy momentum, by the presence of mass in space. So essentially what we're saying is GD6 in space are deflected by the presence of energy. GD6 shortest paths in space are deflected by the presence of energy. That's kind of a version of what's happening in general relativity. Okay, so now let's look at these multi-way graphs. So the derivation of the Einstein equations and the deflection of GD6 by the presence of energy

is something we can do in terms of our rewriting of hypergraphs and a bunch of mathematics associated with that. So it's sort of a mathematical result from that. Now, something very analogous to that mathematical result is also true for the multi-way graph. And what then happens is the GAD6 essentially in branchial space, the shortest paths in this branchial space, are also deflected by the presence of energy momentum. Okay? So this is where it gets, that's really fun because the sort of, you know, the sort of a good core formulation of quantum mechanics is the Feynman path integral.

And the Feynman Path integral says, you look at all these possible paths of history and you say the phase, the phase, the quantum phase associated with a path of history is, you know, E to the I.S over H bar, where S is this quantity of the so-called the action, which is an integral of Lagrangian density, which is essentially the a version of the amount of energy momentum in a particular region of something. Now, in our world, that's a particular region of branchial space. effectively the density of energy momentum in branchial space leads to the deflection of JD6s in branchial space and so what does that mean?

If the position in branchial space is the phase of the quantum amplitude, that means the deflection of a JDIC is a change of phase. And the Feynman-Path integral is precisely telling you that the presence of energy momentum leads to a change of phase in a quantum amplitude. So what it's saying is the bigger picture, here is that the same mathematics that leads to the Einstein equations in space-time, leads to the Feynman-Path integral in multi-way graphs and branchial space and so on.

So in a sense, these two sort of fundamental principles of 20th century physics, general relativity, and quantum mechanics, and our models are basically the same thing, except that the general relativity is a story of what happens in space time, and quantum mechanics is the story of what happens in what we sometimes call branch time, the thing you get, which is the sort of cross of branchial space together with time. So this might be a very unfair question, but I think there's something super simple that I am missing here, which is I just don't see where the negative numbers come in. I hear you saying that there's this position in branch of shield space is a phase, and I get

that if that is true, you get the result you want to have. But why are there both... It's a little bit more complicated. Just as simple as possible explanation for the negative numbers would make me happy. In other words, the reason why you can interfere as well as add your probabilities. Let me explain.

So let's say you're doing a double-slut experiment. The photon can go through one slit, it can go through the other slit. Those two possibilities, the question is, where do those different possible parts for the photon, where are they in branchial space? In physical space, they're going through two slits. The question is, where are they in branch of shield space?

If those two possibilities wind up at essentially the same point in branchial space, that they end up with the same phase. If those two things wind up at opposite ends of branchial space, those two possible paths of the photon end up in two radically different pieces of branchial space. Okay, so what? So the issue is, how does an observer observe what's going on? So this is again, I'm afraid there's another level of concept we have to have here, which is how an observer makes sense of what's going on in the quantum universe.

Sure. So one of the issues is when you are an observer embedded within the universe, you, well, we can talk about space-time and causal, let's forget space time for a second. In this world of multi-way graphs and branching and merging and so on, one of the things that's a little bit brain-twisting is you as an observer are also branching. You're part of this multi-way graph. So the question of quantum mechanics is, how does a branching brain perceive a branching universe? So in other words, your brain is branching just like the universe is branching.

And so now what you have to do, and this is where, for example, the idea of a single thread of experience, a single thread of time is important because the whole notion of us as observers is we have at least the fiction that our brain doesn't branch. We have the fiction that definite things happen. And so the question is, how do we implement that fiction? And then is our implementation of the fiction consistent with what we actually observe in the world? So the way that we, a convenient way, and this is an idea of Jonathan Gorod, who's one of the people who's been working on this project, is what he calls

the completion interpretation of quantum mechanics. And what it has to do with is imagine we've got a branch, but then say, but the observer has to corral those branches together in order to believe

that something definite happens. So how is the observer going to do that? What the observer does is to just say, I'm going to imagine that these two branches are I'm just going to make a mathematical equivalence between them. And then I'm going to see what consequence does that have for my view of the universe?

Okay? And so that's the thing. So it might be the case that the view of the universe knitting together these branches would rapidly become incredibly inconsistent. But this causal invariance property implies that eventually these branches will merge. So eventually it's going to be okay. But the question of whether the observer can kind of speed that up, that up and say, no, I'm going to insist it's okay right now, even though it might take a century for

these branches from different parts of the universe to merge, it's okay to think of them as merging right now. So that's a consequence mathematically of this causal and variance idea that it's possible to do that. But so in that interpretation, what you're doing is you're essentially making completions that try and corral back together these different branches of history. Okay, so what happens if the two photons wind up at opposite ends of branchial space. Basically, what happens is you as an observer have to construct completions that basically complete the whole universe because you're trying to get from one end of branchal space to another.

You're trying to knit everything together. And that means basically that photon, as far as your concern, isn't there. So essentially the reason that you're getting destructive interference is that you as an observer simply can't knit it together. to say there really is a photon here. It's like the photon has been just smashed into pieces that are scattered all over Branch Hill space. You know what? I actually did understand that,

even though I wasn't necessarily expecting to, but it seems, and maybe this is, again, an unfair comment, but I'll let you riff on it. It seems like just like we thought we needed interpretations of quantum mechanics in the 1930s, that somehow we need interpretations of hypergraphs or growing hypergraphs. I mean, the story you just told,

which I'm perfectly willing to grant gets you the double slidics, et cetera, does seem to involve some metaphysical interpretational steps along the way. Fair point. I mean, this is why, this is a fundamental point. In our models,

if we were not roughly the way we are, we would not conclude that the laws of physics are the way they are. So, consider the case of the gas molecules again. If we were one of these little demons that's able to compute everything about the gas molecules and do all those things, we wouldn't conclude that thermodynamics and hydrodynamics work the way they do. That's right.

We would say, oh, actually, we noticed that there are these little correlations of molecules here and there and more and the other, and we'd get a completely different conclusion. So the fact that we derive quantum mechanics is a consequence of the fact that we're that we are roughly the way we are, not in detail with two eyes and all this kind of thing. But in, you know, so in particular, the fact that the branching brain perceiving the branching universe has the idea that definite things happen, that's critical. If we didn't have that notion, and you can think about it in terms of how our brains work, you know, we got all these neurons firing all over our brains.

And in a sense, there wouldn't be any real reason to think that. that there will be a definite threat of experience that we have. It could just be, you know, there's all these different things happening all throughout our brain, and it's, you know, this is a fundamental feature

of our experience of the world is that we believe there's a definite threat of time. And so, yes, that is necessary. And, you know, I think without that, and it's interesting to think about it, you know, what do the aliens, so to speak, think physics is like? Because, you know, for example,

it matters probably, that we are roughly the size we are, it matters that we have this notion of single thread of time. You know, I don't think we would recognize the aliens, so to speak, if they didn't have something like a single thread of time. It just doesn't, you know, I'm fond of making the comment, you know, the weather has a mind of its own. You know, it's got all this complicated hydrodynamic processes

happening in the atmosphere and so on. But yet, we don't have the idea that it has some sort of, you know, human-like consciousness, usually at least, depending on our animism, our level of animism. Pan-psychism, yeah. But would I be then correct to conclude from that that you think that from this kind of fundamental setup with the hypergraphs and the updates and so forth,

we could not get a universe that was fundamentally classical and had observers like ourselves in it? I think our models are very fundamentally quantum mechanical. I think that you can't really get, you know, what you can, okay, so what you would have, have to do is say there is only one, okay, here's a way you can get something that's like a classical universe. And at first it sounds completely crazy, but let's say the universe was built using a touring machine. So a touring machine is this model of computation that has, you know,

it's tape that has all this data on it and has this head that moves up and down along the tape. And at any moment, it's only doing one thing. It's only at the position of the head that anything changes. And so you could say, imagine a universe where the touring machine does a definite thing. Let me think about this for a second. Yeah, let me think about this. Is that right? Well, okay, so you can certainly imagine setting it up so that there is only one possible thread of time because the turning machine is an ordinary turning machine. And for every state it gets into, it has only one possible successful state. And so you might say, and so that would be a kind of degenerate form of model like ours, that

is classical. Yeah, okay, that's fair. But usually what's happening is there are many different places where things can happen. So there are many choices of what can happen, what you do now, what you do next, et cetera, et cetera, et cetera. And I keep on, I'm sort of dancing around one issue which we haven't talked about, which is the notion of causal relationships between updating events. So we've got two different updating events,

and we could say which one happens before which other one. Well, if one of them, specifically, if the output from one of them is used as the input to another one, then it better be the case that the one whose output is being used occurs before the one that needs that output as input. So that defines a certain ordering on certain events have a definite ordering. And in physics terms, those are the time-like separated events. And those are events that are necessarily time-like separated. So then there are other events that could happen, you could decide that you're going to say these two events happen simultaneously.

The spacecraft landed on Mars right as I was eating lunch, so to speak. And those were considered simultaneous events. So one of the issues is, given this set of events that are happening, there are certain causal relationships. That is, one event must occur before another event. And you can make another kind of graph, a so-called causal graph, that represents that net of relationships. And just to make things more complicated, in the multi-way graph, there is a multi-way causal graph which says there are these events happening, and events can be separated.

They can either be time-like. That is one event must occur before another event. They can be space-like in the sense that they can be at different positions in physical space, or they can also be branch-like in the sense that they occur on different branches of quantum history. And so all three of those things are possible, time-like, space-like, branch-like separation. And in fact, the interplay between those things gets pretty interesting when you start talking about, you know, things in black holes and ADF, CFT, and all this kind of stuff. That's how those kinds of things seem to come out, is the relationship between time-like, space-like, branch-like, separated events.

I mean, you hinted at this a little bit, but maybe to concisely say how you would characterize the solution to the measurement problem of quantum mechanics in your view. I mean, there's a standard list of options with many worlds and hidden variables and dynamical collapses. Are you different than all of those? Probably. I mean, you know, I'm not sure because I think that, let's see, we have a much more microscopic view of what happens in measurement. Because we're trying to describe the whole universe. We don't get to say, here's my quantum system, and then outside of my quantum system is my measurement system. Everything has to be part of the description that we're giving of the universe.

And so that's the first big difference. Now, I think it depends what you mean by what the quantum measurement problem is. So one thing that I think is a significant piece of the quantum measurement story is why do people think that definite things happen? And so what essentially is going on is different quantum frames cause people to believe at some moment that different things have happened. but there's some sense in which there is a global consistency to what must happen. So, again, in other words, so the fact that there are two different outcomes from this measurement is a consequence of the fact that the observer can have defined two different quantum observation frames, and that is what causes those different possible outcomes from the experiment.

It sounds like your closest in spirit to many worlds in the sense that someone who believed in hidden variables or dynamical collapses would say that there just is one outcome for every measurement. You don't need to think very hard about it. Right. No, that's correct. I mean, this does not have that idea. This has the idea that the outcome is in a sense based on the frame with which the observer observes it. So in a sense, it's kind of like a many worlds thing where, you know, which branch did you go on? But the difference is in a sense that in our situation, those branches always eventually merge. So, in other words, even though you've picked a particular,

this observer at this moment has got this outcome, that there will be eventual consistency in how the universe behaves. Now, have we figured all of this out? Not completely. Let me give you an example of where, sort of the frontier of what we understand and don't understand. So an obvious question is, what does it mean for quantum computing? So in quantum computing, there's kind of this notion

that has been around since I first studied this stuff around 1981 or so, that, gosh, in quantum mechanics, there's this, you know, all these different possible paths of history. And I happened to study this with Dick Feynman. So he was very into the multiple parts of history idea because that had been his idea for a way of setting up quantum mechanics. It does make sense. It's like, just think, you know, if you could get this computation to not just go in one, not just do one computation at a time, but follow all these different possible.

paths of computation, you could in parallel work out all these different results and you'd be able to factor in just fast. We didn't know about that at that time. But those kinds of things. So the idea of quantum computing, you know, the big hope is just use all those paths of history to do a different computation on each path. But the next question is, in the end, we have to observe what the answer was. And so in a sense, we've got all these threads of history that have all split apart, they're all doing their different things, we've got to corral all those threads of history back together again to make a definite measurement and say what happened. And so in our models, we can actually ask what is microscopically involved in corraling those threads of history

back together. And so in Jonathan's setup of this completion interpretation, we have at least a model of that, which is it involves these completions, these places where you assume that two branches are connected. And by the way, the mathematical underpinning of that, somewhat interesting. The mathematical underpinning comes from the theory of automated theorem proving, which might seem irrelevant, but it isn't. And so the point is when you have in mathematics, you say there's some axiom system. You say, you know, X plus Y is or equal to Y plus X.

You know, X plus zero is equal to X and so on. And then you say, given those rules, is it, you know, can you prove that X, I don't know, some result in algebra? Can you prove that results? By just, and you say, can I, one way to think about that is can I get from one algebraic expression to another algebraic expression just by making this series of moves that correspond to applying particular axioms? So then if you say, well, I'm starting off from one expression, now let me apply those axioms in all possible ways and build up this whole graph of possible outcomes. Well, what you build is precisely one of these multi-way graphs

and the space of possible results that you can get, starting from one expression, what can you get is a multi-way graph. And so automated theorem proving is the question of the proof of a theorem is a path that goes from one expression to another expression. So automated theorem is about finding that path. And so completions

become the formation of lemurs. So in mathematics you would say, let's make a lemma that where we prove that this particular thing is equal to this thing and that's a piece that we'll use in our big proof.

And so these completions are essentially lemurs. They are assumptions about how the universe works that we as observers are making. And so one thing we can then ask for the quantum computing case

is how many such lemurs do we have to produce? How much effort of knitting together the structure of branchial space do we observers have to do in order to conclude the result of the quantum computation? And it seems we're still nailing this down, it's still a little bit messy, but it seems that you can never win with quantum computing. That is that when you branch out in all these different ways,

the effort to corral things back together is at least as great. as the gain that you get from things branching apart. Now, to really nail that down, we have to get ever better models of the observer, and that's just not something that traditional models of quantum mechanics, traditional interpretations of quantum mechanics, they are out of that business,

because they just say, and the observer works like this, external to the quantum system, but we have to be more microscopic and say, no, we're actually going to describe how the observer is working, And so our best model so far is Jonathan's completion interpretation that involves these potentially lemurs being produced. I'm sorry, there's a traditional understanding in, well, there's an understanding in traditional quantum computing that there are some problems for which quantum mechanics certainly gives you a speed up. Are you saying that's not true in your model? Yeah, I think it's not going to be true. That's my guess. I think what's going to happen is, you know, if you take, like, Shaw's algorithm for factoring, right, which is primarily, a quantum Fourier transform.

That Fourier transform is done beautifully quickly because there are all these threads that are running in parallel. The problem is every thread is somewhere in a different place in Brantial space. That thread, but us observers, we have to corral all those things back together again

in order to tell what actually happened. And that's, you know, so there's a, you know, usually in quantum computing one just says, and then there's a measurement. Now, in actuality, when you have an actuality, device, you have all kinds of issues in making that measurement, all kinds of how quickly does it decoher, all these kinds of things. There are all these kinds of very practical experimental features.

And I think people have generally said, given the formalism of quantum mechanics, it's like, well, all this quantum stuff happens, and then boom, we do a measurement. And the boom, we do a measurement is actually pretty difficult in practice with actual experiments. But people have said, but, you know, if we do these experiments well enough, it will become the mathematical idealization that, you know, Voen-Nyman and others made about how measurement works. And I don't think that's going to be true. I mean, we're not sure yet, but it seems likely that there will be no way to do to sort

of, if you're honest about how the measurement works, the measurement takes effort. But are you saying that that's going to be a generic result about even conventional quantum mechanics, or is this specific to your approach? If you could study how measurement works well enough in quantum mechanics, yes. The problem is in traditional quantum mechanics, it's like this is all the quantum, these are all the quantum amplitudes, and then the hammer comes down and it's the born rule or whatever, and that's the end of it. And so you don't get to look inside that measurement process.

And when people, you know, I'm probably not, you know, one of the strange things that's happened to me in this physics project is that, you know, I knew about every, you know, I knew about every everything that was going on in physics, not everything perhaps, but a lot of things 40 years ago. And then I kind of went to sleep for 40 years and woke up again. And it's remarkable, it's really cool actually to see, you know, a lot of stuff really didn't change. And some stuff advanced. And it's like, you know, I used to know the masses of all these elementary particles, you know, the mass of the lambda is 1-115 MEV. And now I look it up again. It's, you know, 1114.962 or something. It's like that's, so, you know, there are things like that, and there are also mathematical ideas and so on, but I'm not sure that I would know everything that people have done in kind of sort of models of quantum measurement, but certainly back when I last knew about it in great detail, people were sort of starting to say, and then we take the small number of quantum degrees of freedom that we're looking at as the quantum system, and we let that sort of spread out in a larger number of quantum degrees of freedom, where eventually it was,

become unambiguous what happened for essentially thermodynamic reasons. Because it's more or less like what happens in the gas getting randomized. And in fact, this was a thing Dick Feynman and I worked on back when we were both consultants for a computer company in Boston and we were, I was insisting that we tried to do something useful and we tried to invent a quantum randomness chip. And so we tried to think about how would you make, you know, in those days one gigahertz seemed like an impossibly fast clock rate for this chip. How would you make quantum randomness at 1 gigahertz?

And what we realized is, you know, you have this little quantum effect, but then when you measure that effect, you have to amplify it into a sort of classical number of degrees of freedom. And that process, in order for you to have an unambiguous result, in order for successive bits that you measure to be uncorrelated, you have to essentially have essentially sort of a thermodynamic phenomenon going on. You have to be, you know, it has to randomize, then that randomization has to not be correlated with the next thing that happens. And that was, to me, the first sort of sign that there was going to be a cost to measurement, so to speak.

Well, if you start a podcast of your own, you can have me on and we can talk about interpretations of quantum mechanics, and that would be fun. Okay, you can teach me all about it. But, I mean, I think that that does help me understand a little bit because what you're saying is that it's possible that your approach has a different, attitude toward, let's just say, the measurement problem of quantum mechanics. So the next obvious question is, do you have a different attitude toward the Schrodinger equation, towards the unitary evolution? I mean, the conventional Schrodinger equation is continuous in time, and you're not continuous in time. Are you getting the Schrodinger equation as an approximation, or is there some sense of which it's exact?

Yeah, right. So it's very similar, and we don't know all the details of this yet. But let's talk about the space time case. In the space-time case, we've got these atoms of space and with connections and so on, and we're looking at things on a large scale. And what should happen is that just like in a fluid where the molecules are bouncing around, but on a large scale we get the fluid equations, on a large scale, we should get the Einstein equations. And that's what theoretically we should get.

Now, when I say theoretically, this is, you know, when physicists do mathematics, they're really sloppy. And, you know, there are many limits. have to be taken. You know, there's a certain amount of computational reducibility, which leads to a certain amount of randomness, which leads to certain statistical averages, working out in a certain way. When you look at the, you know, the distances have to be large compared to the elementary distance between atoms of space and small compared to the size of the universe, and the curvature has to be small, blah, blah, blah, blah, blah, a whole bunch of conditions.

But with all of those conditions, in physicist-level mathematics, we can derive Einstein's equations. Now, it's worth pointing out that the effort to derive the fluid equations from molecular dynamics has been unsuccessful for 100 years. That is, with a level of, you know, we can derive Einstein's equations

with the same level of kind of mathematical rigor that you can derive the fluid equations, which is to say it isn't really mathematically rigorous, but if you take all kinds of limits, that one advantage that we have, actually, one big advantage that I always forget is that we've done endless similar, So we actually know what limits work and what don't.

Right. So it's not like, it's just let's hope this works based on some mathematical sort of hope. It's like we ran an experiment with a billion cases and this is what happened. So we can kind of check those assumptions. But okay, so what then happens is what we are saying is that the, from this kind of microscopic atoms of space thing, when we do that, when we look at that on a large scale, we reproduce produced Einstein's equations. So one of the things that we've been doing recently, Jonathan has been much involved in this, has been saying, well, what about practical numerical relativity?

So people study black hole mergers, and they do that by taking the structure of the black holes, and they say we have Einstein's equations, which are partial differential equations, and then we will. But in order to make those work on a computer, we have to discreetize them because a computer deals digitally with sort of digital elements. And so then you make numerical relativity where you've made some kind of mesh that represents some approximation to space time as approximating the space time of Einstein's equations.

So that's kind of the traditional approach. Now, what we're doing instead is to work upwards from the atoms, so to speak. So we're saying, instead of going down from the Einstein equations, just come up from this kind of structure of this hypergraph and look at

what is the aggregate behavior of all these atoms of space. The neat thing is that it actually works and it's a practical method for general numerical relativity. And in fact, it's looking really promising to and what's interesting, so

this is now a general relativity, just a side comment for general relativity. So one thing you can ask is so obviously in our simulations, we have these atoms of space and we simulate on a computer. I think the extreme kind of amusing version of Jonathan has been doing all these simulations on his laptop

just to prove that that's possible, which I think is totally crazy. So he has these simulations of black hole mergers and things done on his laptop using our models. You could do a lot better if you weren't just using a laptop. But anyway, the thing that is a... So it's not obvious that we would be able to get enough approximation to continuum space

just kind of on a laptop from our atoms of space rather than having to go many, many, many orders of magnitude. But it seems that that works. So the next question is, in physical space, where might we see evidence of discreteness in spacetime? So in other words, is there a way of making essentially a gravitational microscope that will see into the structure of space time?

And so one of the things we've looked at is rapidly rotating black holes, right, close to the critical, you know, close to J equals M. What happens? And it's pretty neat because in these simulations that Jonathan has, what you see is the thing, eventually as you go above the critical angular momentum, critical spin rate for the black hole, a piece of space simply separates. So, okay, another thing to explain is in our models of physics, This network doesn't have a built-in notion of dimensions. So, you know, our experience is space is three-dimensional, but this network doesn't have any particular number of dimensions that are built into it. It can be the case that on a large scale, it approximates three-dimensional space. But because it doesn't have a fixed number of dimensions,

because it isn't microscopically a manifold, so to speak, it can do things like it can change its topology. Things can, there can be tears that form in it, there can be holes that form in it and so on. or more extremely, there can be pieces of the universe that just break off. And so what we see happening for critical, supercritical black holes is a piece of the universe just breaks off. And as the critical and momentum, what you see is that a piece of the universe is hanging

by a thread. And so there are these small number of cordial edges that essentially connect one part of the universe to others. And so it's conceivable that we may get some kind of signature, maybe. being gravitational radiation of that discreetness, you know, in physics terms, probably shot noise associated with the fact that there is a small number of causal edges that are associated with that situation. So anyway, that's, but that's just the general relativity case.

In the quantum mechanics case, you're asking about the Schroenger equation. So the Schroenger equation has to emerge as a pile of limits from the things that we're doing. Okay. because the Schrodinger equation is a non-relativistic limit of, et cetera, et cetera, et cetera. So the answer is, I think, I mean... Well, I'm sorry, just to be super clear. I'm using the generalized version of the Schrodinger equation that would be just as true for quantum field theory and relativity and everything.

Just H-SI equals I, D-by-T, psi, unitary evolution of the way function. Right. So we are, from the setup of these multi-way graphs from the interpretation of quantum amplitudes, we are guaranteed unitary evolution. 100% not as an approximation. That, you think about that for a second. It's a little tricky because you have to define what the initial and final states are.

And if you say I have the whole multivory graph of everything, I think you're guaranteed it. But if you say, I'm going to look at a particular region of space time, I think it's not so obvious. I think there it's an approximation. That's just true. and even in the regular Schrodinger equation, so that I'm perfectly happy.

I think that, let me think about that for a second. I think that, let me see. I mean, quantum measurement is non-unitary. That's a thing that we have to deal with. But a many-world person says that's just because you're only seeing part of the weight function in exactly the same way that you are. Right, right.

But I'm a little concerned about the statement that the, see, the statement about the way the Hamiltonian appears there in the non-relativistic showendure equation is certainly not what we're seeing. So, because we have a, you know, our thing is sort of relativistic from the get-go. And so what I think, let me think about this for a second. Well, you could define it in terms of the path integral. I mean, is any unitary evolution from one, one wave function to another, one quantum

state to another? Right. I think between an initial hypersurface and a final hypersurface, I think you inevitably get unitary evolution. But you have to define what those hyperservices are, as you do in, you know, as you do when you merge relative to quantum, general relativity in quantum mechanics. So I think it's, I'm, I think that's right. I would have to, I mean, the part of this, yeah, I think that that has to be that way. Okay. I mean, the part that we've been interested in recently is, can we make a practical scheme for doing quantum field theory?

in the same way that we can make a practical scheme for dealing with Einstein's equations. And so in a sense, one of the things that encourages me in this project is what I might call proof by compilation. That is the fact, you say, do you really get Einstein's equations? Well, we can do numerical relativity

and it actually works, and we can do black hole mergers and things. So if Einstein's equations, I'll write them, what we're doing is the same. And in quantum mechanics, we've been able to do this compilation of quantum circuits. The next big challenge is to do quantum field theory and to do and to be able to do things like few particle quantum mechanics and those kinds of things. You know, for example, let me give you an example of how things get a little bit complicated.

So in quantum mechanics, you know, a very traditional thing is the harmonic oscillator, just kind of the analog of, you know, things bouncing on a spring back and forth. In order to get that in our models, you have to have closed time-like curves. So, in other words, to get a perfect harmonic oscillator, it's very easy to have a perfect harmonic oscillator if you have closed time like curves. In other words, if the history of the universe recapitulate, if the history of the universe repeats itself, that's kind of how you get something like a harmonic oscillator. That's how you get the idealization of a harmonic oscillator.

Now, in the real situation in physics, we probably don't believe that there are a perfect harmonic oscillators. We can decompose quantum fields into, you know, as, let's think of it as a sum of harmonic oscillators, but we're not imagining there's a perfect harmonic oscillator here. But if you say, you might say, and this relates to the kind of reverse engineering versus building from the ground up, so to speak, you might say, gosh, how can you think you have a model of quantum mechanics if you can't even have the harmonic oscillator? But, well, we can have a harmonic oscillator, but it's a weird idealization, which

in our models shows up with closed time like curves. And so that's an example of how things get a little, you know, if you insist on these idealizations, you end up sort of stressing, as I think you should, you kind of stress the model. So it's similar to, you know, if you said, you say, I'm studying molecular dynamics, okay, so then somebody says, well, why can't you get rigid body mechanics? Why can't you get, you know, this thing with a solid object that just moves around and doesn't distort. Well, that's a complicated limit of molecular dynamics that involves, you know, inter-atomic forces are large compared to the, you know, forces on the whole object,

blah, blah, blah. You know, it's, it's some, that's a complicated story. And I think we're seeing some, one of the things that's difficult about our model is that it attempts to be a model for everything. So you don't get to say, we're going to have a, this piece we're going to make a model for, that piece we're going to idealize away. You have to kind of, everything has to come together. And that means that seeing things like, how does the harmonic oscillator work, it's a weird idealization. Yeah. That does make sense. If you get exactly unitary evolution, that is to say you get something like the Schrodinger equation.

I mean, a many worlds person will say, all there is is the Schrodinger equation. You just need to pick the Hamiltonian. And someone like me, if I can indulge myself, you know, our project is to start from a nothing but a Hamiltonian expressed as a list of energy eigenvalues and show how there's an emergent geometric structure that looks like a graph, you get Einstein's equation, and the whole thing. It's similar in spirit in that sense to what you're doing. But so then the question is,

if you exactly get unitary evolution, what is the benefit to not just starting with quantum mechanics in the Schrodinger equation versus starting with hypergraphs? Well, because you have a whole structure that you're building. So I mean, for example, when you say, what's the Hamiltonian, you know, that better come out of our model.

So, for example, you could ask, which we haven't discussed, what is energy in our model? This is one of the things that surprised me. I thought energy was going to be a very difficult to derive idea that was going to require notions of what particles are and so on and so on and so on. It seems that energy is very simple. It's basically the amount of activity in the hypergraph.

And a little bit more formally, it's the flux of causal edges through space-like hyperservices. And it's concerned? And it needs to be said, so a causal edge is a causal connection between two updating events. And so the question is, you have to say it a little bit more carefully because if you say it's the density of activity,

it's what the heck does density mean when we don't have a notion of space yet. So the more formal thing is energy is the flux of causal edges through space-like hyperservices, momentum is the flux of causal edges through time-like hyperservices. And the... Actually, you now make me

wonder, and I don't think I know the answer to this, what is the flux of edges through branch-like hyperservices? I should know the answer to that, and I don't immediately know. And that's related to, okay, so the point here is there's a, I think, see, one of the things that's interesting in our models is the relationship between quantum energy and classical energy. Because in our models, it's this notion of in classical energy, it's the, you know, you've got this single thread of history, you've got this causal graph associated with that thread of history, and you're asking this flux of causal edges. But now you can also have this multi-way causal

graph, which not only has space-like separation of events, but also branch-like separation of events. So events can be separated in branch of space and in physical space. And then the idea is that quantum energy is that activity in this whole multi-way graph. So it's the amount of activity in the multi-way causal graph. And so then, like the Lagrangian is this relativistically invariant, this is the, you know, the Lagrangian density is that kind of, you know, relativistically invariant notion of the amount of activity in this multi-way causal graph. And so, for example, one little mystery that this seems to clear up is the thing, if you look at the plank length, you know, the characteristic length people often talk about for where sort of quantum mechanics and gravity have to kind of come together. In our models, the elementary length is much smaller than the plank length.

And in the, a typical thing is the plank energy, whereas the plank length is, you know, 10 to minus 34 meters or whatever it is, the plank, the plank energy. is 10 to the 19 electron volts, if I remember correctly. GEV. What's that? 10 to the 19, GEV. GEV, okay, thank you. But it's about the energy in a thunderbolt or something, a lightning strike. It's huge, not a very small quantum thing.

In our models, the elementary energy is also small, because the elementary energy is an individual causal edge in the multi-way causal graph. And the fact that what you observe is this thing, the analog of the plank energy is something that has been averaged, that has been sort of summed over all possible branches in this multi-way causal graph. This is maybe this is a footnote. Forget this footnote.

This is just a part of the story. But so you're saying, why do we care about this whole structure that we're building? Why not just put a Hamiltonian in there? And what I'm saying is, for listeners who don't know this, Hamiltonians and Lagrangians are closely related. Sorry, I think for you and me, that's pretty obvious. Right. That may not be obvious.

But in some sense, like we said, there's a simple set of rules that you start with. And I mean, maybe I can just jump. I'm going to try to guess what you're thinking here. And then you can tell me whether I'm right or wrong. there's an infinite number of Hamiltonians I can write down, but at least for simple ones, there's a finite set of little rules that we can write down for our hypergraphs.

So maybe there's a map from the set of rules to Hamiltonians, but it's a very, very, you know, one, you know, few to few rather than covering the whole space of possible Hamiltonians. Right. Well, so this is a good question. As you look at simple, you know, rewrite rules for hypergraphs, you get some set of Hamiltonians. If you say, oh, I'm going to have,

a million line program to represent my rules for the hypergraph, then you're going to get some other, you know, you can kind of custom engineer whatever Hamiltonian you want probably. Got it. So it is not self-evident. What kind of Hamiltonians you get from simple update rules, we don't entirely know. So for example, it looks like we get ones that show local gauge invariance. And then we have to ask, you know, and that's a, you know, if you just pick your average Hamiltonian There's no particular reason for it to have local gauge invariance, but in our models it probably does and so that's a good sign now

You know, you know, what other attributes will we like our Hamiltonians for the universe to have? Well, I mean, you know my approach here is to just say let's see what we get and for example. Okay, so here's one of the complicated things So the Hamiltonians that you will typically write down and in quantum field theory they are the things that are in those Hamiltonians, Lagrangians, whatever, are quantum fields. They're quantum fields that represent, you know, field operators that correspond to particular particles, right? So there'll be an electron quantum field, there'll be a photon quantum field, and so on. Now, in our models, we're not yet talking about particles. So those Hamiltonians have to be, what we have is a description.

of things at a much lower level than a place where you've got actual field operators for particles. So for us, what are the particles? The particles are locally stable kind of deformations in this hypergraph. So for example, an analogy in fluid mechanics, if you say what's like a particle in fluid mechanics, well a vortex, a kind of whirlpool-like thing in a fluid, that is locally stable and it can move through the fluid and it can track with other vortices. It's something vaguely similar to that in our hypergraphs, more a kind of topological structure

that is some kind of not yet very well understood feature that is locally stable under the rules. And so when you talk about field operators that would occur in your average communal garden particle for its Lagrangian, those field operators are effective descriptions of these defamations, these topologically stable objects in this hypergraph. The hypergraph is a lower-level description. And so it takes, you know, it's not obvious, and it's a very good question, and we don't know, you know, how does the thing that looks like it operates in terms of particles,

how does that emerge, you know, what kind of things that look like there in terms of particles emerge from this lower-level structure? Because in our models, you know, all we have at the beginning is the structure of space. Particles are just features of the structure of space. And they are, you know, so there's an underlying dynamic. I mean, one of the things that sort of a little shocking in our models

is that most of the activity of the universe is associated with the maintenance of the structure of space and has nothing to do with all the particles that we observe in space. So that's a, and so what, I mean, so it's a very reasonable. question how the Hamiltonians that we're used to in particle physics emerge as effective Hamiltonians from this underlying dynamics, and we don't yet know the answer. Well, you know, you mentioned the particles as something that does come out as these persistent structures. So, but as you certainly know, quantum field theory, the usual modern physics way of saying

things is that fields are somehow more fundamental. When you perturb them and excite them, you get particle-like things. there are circumstances, you know, like the condensation of the Higgs field where the particle language is not appropriate. Is there a view toward a more field-like ontology coming out of this? Well, I think that, okay, so again, this is, if we say, when we talk about particles in particle physics, they are, as you say, sort of, you know, what we imagine is, oh, there's a particle, it's not interacting with anything. It's separated from everything else. It has a definite momentum.

It is a definite mode of the quantum field. And when it comes close to other particles, then it gets more complicated, and there are other modes of the quantum field that get excited. And eventually some other particle goes out and you observe it in a particle detector. So for us, we have to think about what do those scattering experiments look like? What does an experiment where you have that separated particle? And this is one of the cases where, you know, we are building everything from the ground up. So we don't get to say, oh, it's just a separated particle in the vacuum, so to speak. It's an excitation of some kind embedded in the space that has a lot of activity in it.

So, for example, when we talk about, you know, so one of the things that I'm wondering is, will there be bizarrely different kinds of stable structures in our hypergraph that, aren't like the particles we're used to. So in other words, you know, you mentioned the Higgs field. The Higgs field is a case where we have this condensate, where we have a sort of uniform value of the Higgs field throughout space. And that's, you know, that's a simple case. There are, you know, people know about, you know, Solton-type solution to field equations. And, you know, there are other kinds of things like that. And one of the possibilities is that in our models, particularly because we have this dimension variation, that there might be

bizarrely different stable structures that could exist in the context of our models that don't seem like particles of the ordinary kind of Depplementinza and things like that might be something bizarrely different. Maybe it's even related to something, you know, maybe it's related to dark matter for all we know. I mean, it's just something that could exist as a sort of stable feature of our universe that goes beyond just the structure of space, but it has a form that we don't yet understand. And I think, I mean, in the other thing to realize, so in standard particle physics, we talk about virtual particles. You know, you can, they're always these particle antiparticle pairs that are sort of popping into existence in the vacuum for a very short time and then annihilating.

So, and one of the features of that, it's always been somewhat mysterious in quantum field theory, because quantum field theory says there's an infinite number of these virtual particles that are occurring all over the universe. and that corresponds to effectively a very high energy density of stuff in the universe. How can that be? How can it not, in Einstein's equations, say, you know, whenever you have energy, you're going to curl up the universe with gravitational attraction associated with that energy. How can the universe not curl up into a tiny ball because of these vacuum fluctuations of virtual particles? Well, in our models, the sort of limiting case of those virtual particles is, these actual rewrite rules in our hypergraph.

So in other words, the ultimate version of all of those virtual particles, it's just all these little rewrite rules in the hypergraph, which in our models are exactly what create the structure of space. So in a sense, that makes it a little less mysterious that all of this virtual particle stuff doesn't have this terrible effect,

this terrible gravitational effect on the universe. Now, you can ask things like, you know, can you imagine how the Higgs mechanism works and all these kinds of things? And yes, I've thought about that. And there's a certain amount you can say. I don't think we can't, I mean, I was very disappointed when the Higgs particle was discovered. I kind of knew it was going to happen. But it was kind of like, it's kind of a hack. And it's disappointing that, you know, it looks, so far the Higgs mechanism looks like a hack. And the question is, can we make it look less like a hack and that will be really neat.

So one little footnote is that you might be interested in work by Tom Banks and Willie Fisler who have emphasized within the context of more or less conventional quantum gravity, the point you just made about the cosmological constant, that we should not think about it as the sum of all these virtual fluctuations. We should think of it as an input, a boundary condition they call it, that just sets the size of space in some sense. So that's more in line, I think, with the philosophy that you're thinking of. I knew Tom Banks.

when we were both at the Institute for Advanced Study in probably 1980 or something. I haven't, great, I'm glad he's, I'm glad he's doing interesting things. And then the other point was, I mean, I don't want to dwell on all of these, but I presume that if your program pans out, we will want to find solutions to the classic puzzles of particle physics, so not just the cosmational constant problem, but the hierarchy problem, you know, the strong CP problem, bariogenesis, are these the kinds of things?

I'm guessing that I would have heard already if you would solve them, but maybe are they solvable, you think, within the program? You know, eventually they should be. I mean, we can start sort of teasing apart what each of them involves. I mean, you know, the, oh, there are three generations of particles

and, you know, the question is what's generic and what's not. So the surprise for me is that, General relativity and quantum mechanics seem to be generic, just as thermodynamics is generic. And the question is, how far does the generosity go? Because I wouldn't have expected, well, for example, I wouldn't have expected the detailed form of Einstein's equations to be as generic as they seem to be. So, and, you know, it could be, I have this vague suspicion that the structure of the local gauge

group might be generic. I might be completely wrong about that. Yeah. You can hope. I don't think the three dimensions of space are generic. I don't think the three for the three dimensions of space is generic. Whether there are bizarre possibilities. Like maybe the two threes are related.

The three generations and the three dimensions of space. Maybe, I don't know. I mean, these things are all certainly possible. I think that when it comes to things like, Well, one example is CPP invariance. So that's the question of why is there CPP invariance, and I can explain a little bit about that. We don't know completely. I mean, so in sort of in the stack of things you need in particle physics, a big feature of particle physics is ferviomps, you know, particles like electrons with a fermion.

Then have the exclusion principle don't want to be together versus photons, which are bosons, and are very happy to be together. and those are distinguished by the, you know, the spin statistics theorem says, if you are an, you know, if you are half-interdispin particle, then you will be a fermion, and if you're an interdispin particle, you'll be a boson. So one question is, what are fermions and bosons in our models, and can we derive the spin statistics theorem? And that's an example of a sort of a, you know,

one of the achievements of quantum field theory is the derivation of the spin statistics theorem. And so for us, fermions versus bosons, We have a guess. We're not sure. The guess has to do with, in these multi-way graphs, we suspect that bosons are associated with, in a sense, branchings that merge very quickly, and fermions are associated with branchings that do not merge quickly.

And so that's what leads you, in a sense, the branchings that don't merge quickly wind up with opposite sides of branchial space, things that can't happen effectively, or things that are not observed to happen. And in fact, at a mathematical level, it is the, you actually will then sort of see the fermions are kind of the square root of the bosons, because the fermions are it just branches, it doesn't do the, it doesn't merge as well, whereas the bosons are both the branches and it merges. And so now the question is, can we relate that to rotational symmetries that are associated with the spins and so on? And looking promising, not done yet, but looking promising. So then if we can do that, then we can derive the spin statistics theorem.

And then the next question is, well, CPT invariance, just to explain for people, this is a fundamental symmetry that's believed to exist in quantum field theory is if you turn particles to antiparticles, you reflect space and you reverse time, then you'll always get something where the laws of physics work the same way. So, for us, again, it's kind of neat because it seems like C, charge conjugation, which is particle to antiparticle, is effectively like it is an inversion in branch hill space, it seems to be. In other words, it's a reflection in branch hill space. Parity is a reflection in physical space, and time reversal is a reflection in time. And so what this is saying is, in the end, that our multi-way causal graph is invariant under a reflection in all three of those directions, so to speak. Now, so it's at least a very clean formulation of what it means to be CPT invariant, whether we have not yet shown that we necessarily get CPT invariants, but at least we know what it corresponds to. But that gives some sense. So when you ask about CP violation, for example,

that's still little ways away because we don't yet understand CPT invariance. In regular quantum field theory, CPT invariance is tightly connected to Lorentz invariance, the invariance under boosts and so forth. I'm guessing that in a discrete graph model, Lorentz invariance must be, once again, approximate, right? And there'll be at some point,

even if it's very, very experiment, environmentally inaccessible where it will break down. Absolutely. There is a maximum boost. You can't get, you know, there's a, in other words, as eventually you start seeing the granularity of space time. Yeah. So you can do, and so, yes, the relationship between, that is, that is in fact the derivation

path that we're aiming for is causal invariance in the ordinary causal graph leads to ordinary Lorenton variance. And the guess is that some kind of causal invariance in the multi-way causal graph is associated with the essentially extension of Lorentzian variance that is CPT invariance. So that's the guess about how it works. Right. That's, yeah. We don't, we don't know if that's really, and, you know, obviously CPT invariance is a complicated story in general relativity and in non, you know, non-s, you know, non-flat space time and so on. And, and no doubt, and, and, no doubt, we will run into those issues. There are very good experimental limits on Lorenz violation from different propagation speeds of photons of different wavelengths from gamma ray bursts. They can put limits up to the plank scale,

but as you said, your scale is much tinier, so it might not be anywhere close. Right. I mean, I think the thing that I'm most interested in with respect to those propagation experiments is finding kind of dimension fluctuations in the universe. Because in our models, you know, there's no reason for the dimension to always be three. In fact, one of the things that's a possibility is, so one of the more bizarre possibilities, again, not properly explored, is that the Einstein equations are usually formulated in terms of change in curvature of space. One possibility is you might be able to say the curvature is always zero, but the dimension

changes. And it might be that there is some mathematical equivalence between those for small curvatures, small dimension changes. But one of the things that we suspect in our models is that the universe started infinite dimensional and only gradually sort of cooled down to be three-dimensional. And so that then a lot of the problems about these horizon problems in the early universe and so on disappear because when it's infinite dimensional, everything is closely connected. The graph distance between things is small.

Now, the suspicion from this is that as it cools down to be three-dimensional, there may be dimension fluctuations left over. And so obviously, you know, the question is, would those be visible at the time of recombination? Can we see that in the microwave and the cosmic microwave background? Of course, we don't know. But the first question is, what would a dimension fluctuation do to the cosmic microwave background? And then if there was a dimension fluctuation that exists in the current universe, what would it do to the propagation of a photon through that region of dimension fluctuation?

And this is a piece of, in a sense, classical physics that I keep on meaning to work out, and I keep on hoping somebody else is going to work it out. Because it's kind of like I've been collecting this inventory of laws of physics and D dimensions. Well, I think, I mean, I'm just guessing here, because I haven't thought of it myself. But if the effective dimensionality space is not exactly three, Kulam's law would not be exactly true. And wouldn't that have dramatic effects on QED scattering rates? Well, the question is something like the Thompson cross section, for example. How does that depend on? So I am, well, you're correct that in D-dimensional space.

So the way to think about, I think, photon propagation is using Hewigen's principle, where you're saying that every, on this wavefront of a propagating, a propagating, you know, light wave or whatever, every point on that wavefront is a source of a spherical wave that is going to build up the next piece of the wavefront. And if you are in D-dimensional space where D isn't three, those little spherical waves have a different, you know, there's a different density on those spherical waves, so to speak, when you project them into three dimensions. That's at least my thinking about how to do this. Now, the question is, so the question is, so in particle physics, back when I used to do particle physics, it's a long time. ago, I was a big enthusiast of this method called dimensional regularization, which is the idea

that, you know, well, you're doing an integral, and if the dimensionality of space time is four, the integral diverges horribly, but if we can only say it's four minus epsilon, we can do the integral, and then we take the limit epsilon goes to zero at the end. So I'm sure I've worked out back in the day what the low energy electron photon scattering cross-section is in D-Dimensions, and that would be a good, that would be a good, that That would be a good case to look at. I mean, I think that I will not be surprised if there is some thing that one can, some fairly sensitive test of the dimensionality of space.

For example, I don't know, and I've been meaning to look it up, what the anomalous magnetic moment of the electron or the muon, you know, it's one minus alpha over two pi, but that certainly depends on the dimension of space. I just don't know how. Yeah, there's probably some sensitive particle physics experiment that, that is sensitive to that. So, okay, I think I only have two questions, but they're both huge questions,

so I suspect they're gonna take more than two minutes each. But one of them is you referred to this idea that in the early universe, maybe we have some bundle of dimensionality or dimensionality is not well defined, and that sort of comes out in the wash as the evolution happens. So I've thought very much about the arrow of time problem,

but I've thought about it in the context of fundamentally reversible, reversible directionless laws of physics, which is a little bit different from where you're coming from. So for me, the huge question is, you know, we're evolving toward an equilibrium, empty space situation where all the galaxies are all dissolved around and we have nothing around us but the cosmological horizon, and that's the highest entropy state. But we came from 14 billion years ago, a very, very, very low entropy state. And in conventional cosmality, there's just no good explanation for that except the papers I've written, and even those are not universally accepted yet.

But it sounds like you have a possible completely different point of view where it's just natural that you start with what we would expose facto label low entropy. So I think the discussion about entropy is kind of confusing. Okay, that's my first statement. I remember, actually, I remember having this vigorous arguments with Dick Hyman about this very topic of whether the universe was, you know, he made this claim, the universe is a fluctuation. You know, the fact that we are in the state where we're not kind of, you know, where we aren't at sort of the maximally disordered state. And I kept on saying, that's just a completely crazy claim because it's like everything we know is a fluctuation.

So, you know, I think I would be more eloquent these days than I was back. I'm on your side. Yeah. And explaining why that was such a, such a wrong idea. But anyway, no, the first thing to understand is that, People say, oh, eventually the universe will kind of unwind and there'll be nothing interesting in it, there'll be no structure in it, it will all just be in a high entropy state. Okay, my claim is, the reason you say that is because you're not an adequate demon. That is, in fact, in the future of the universe, if it turns out that it's all just seems like a gas in equilibrium and so on, that you might say that's completely boring.

That's all just random. It's the high entropy state. but to another kind of observer, they might say, but look at all those little details about what's happening. This is the most amazing complex structure that you can imagine. I mean, I happen to think if we ask, what will we look back on this time in scientific history and say, how could those guys have missed this point?

My guess is one of the bigger things will be when people describe something like a gas in thermal equilibrium, they just say it has a certain pressure and temperature. They don't consider any other features of the gas. They don't say, oh, it's got these little microscopic, you know, things that we can measure in this way and that way. But my guess is that there are a lot of other features. Now, this question of, are they computationally boundedly observable? Are they measurable by a computationally bounded observer? I don't know the answer to that, really. But the point,

the claim I'm making is that when you say it's going to a high entropy state, you're saying, oh, all you're saying is, so I as an observer of the kind of observer I am, can't make anything out of it other than to say it's uniform and random. I think I'm actually saying more than that. I think in this case I'm going to disagree with you because I think that it actually goes to the vacuum quantum state,

the bunch Davis vacuum of decider space. It's a dissipative system because there are, there's a horizon around us. And within our horizon, all the excitations above the vacuum, vacuum go away. And there's no structure there to be seen by any observer computationally bounded or other ways. What do you think is in the vacuum? Well, it's a single quantum state, thinking in quantum mechanical terms.

It's a unique quantum state, the lowest energy state. So I would claim that's the mistake. And it may be interesting to try and take that apart. So in our models, the very fact that there's space there, space is a thing in our models. You have to make space. You don't just get to say, oh, there's this big area of space. Space is made of something. And in our model, space is made of this sort of teeming collection of atoms of space doing all these complicated things.

So our vacuum cannot be just the boring, there's nothing there vacuum. Our vacuum has to have a lot of structure in it. Now, if you ask the question, if you say every particle excitation has gone over the horizon, that's what you're basically saying. So let's think about that for a second. What does that mean? In our models, so horizons in our models are visible in essentially breaks in the causal graph. So the causal graph says this is the, you know, causal effects can propagate between these,

between this event can affect this event, can affect this event, and you make these light cones where one event is having an effect on a whole sort of increasing size cone of other events. So what is it, what is an event horizon? Event horizon is, and you literally see this in our models when you run them, and like these black holes, you can literally see the event horizon. And it's a place where the causal edges, for example, in a black hole, the causal edges go one way. The causal edges go in and they can't come out. And that's, it's very concrete. And so a cosmological event horizon is something where you're separating in the causal graph.

You have two different regions of the causal graph, and they are, you simply have no causal edges that go from one side to the other. So what you would be saying is, what you're claiming is every particle excitation has somehow is in a region of space that is causally disconnected from the region of space that we care about, so to speak. So I think my statement would be, I mean, your claim is an interesting one because basically what you're saying is that all the big particle excitations, all these sort of identifiable topological excitations, will. be gone and outside the, you know, outside of this event horizon.

Now, what I would be saying is that in order for there to be space inside there, there still has to be stuff going on. Now, what, so what this turns into is an argument about is there stuff, is there useful stuff in that region of the universe? And by useful stuff, you might say it's this complicated topological structure that represents a photon. But that depends on, you know, and so you say the observer, an observer that you know about right now is observing things like photons, is observing these big, complicated topological structures. But, you know, in terms of the pure underlying structure of the universe, you could still have all these little atoms of space doing their thing.

It's just maybe, maybe it's smoothed out enough conceivably that there aren't any of these sort of identifiable topologically stable structures that still exist there. That's a possible kind of, and then the observer would have to be at the demon level, so to speak, in order to observe that there's anything interesting happening there. And not at a level. If the observer is made of particles, for example, is made of photons, that might be the case that you couldn't get, with an observer made of photons, you simply couldn't get information about that more microscopic structure. So, yeah, I think we should postpone this to either an in-person conversation or me appearing on your podcast because it does get kind of technical at some point.

But I don't want to not give you an opportunity to talk about the beginning of the universe. You talked about the end there. But I'm going to guess that you're going to say that there is a principled reason within this approach to start in what looks something like very roughly the Big Bang. Well, okay. So the question is what, you know, in these models, okay, this gets us into another. This is a deep rabbit hole that we're about to enter, but let me say a few things about it, because it's something I've recently been thinking a lot about. So, okay, so in these models, you might imagine that the universe starts as a fairly small hypergraph

and gradually expands. And so one of the mistakes I've been trying to not make is kind of a repeat of the Einstein mistake, so to speak. Because in our models, the most natural thing is that this hypergraph just expands dramatically with time. And, you know, I'm not, I have a great tendency, and this is kind of a reverse engineering type tendency to say, no, no, no, there's got to be these other rules that reduce the size of the hypergraph so that it doesn't expand insanely. You know, just like Einstein was saying, well, you know, even though the Einstein equations predict the universe should expand, of course we know it doesn't expand, except it turns out actually we discovered it does expand.

So I'm trying to not repeat that mistake. But, you know, I think that the current expectation would be that just as there's a simple rule, so also there's a simple initial condition, and that is a small hypergraph, which is something that doesn't really even define a dimension, because it's just a tiny little hypergraph and it starts expanding and at the beginning that expansion is like, you know, is something which affects it has high dimension than it goes down. So one of the things we've been trying to do, which is probably more in your day job wheelhouse, so to speak, is we'd really like to know what the analog of the Friedman-Robertson-Whorpe

model of the universe is for dimension change. So we were pretty sure that there is an analog for a homogeneous isotropic universe that there's something just like the usual model, just as it's homogeneous and the universe is homogeneous and isotropic, so what does the scale factor of the universe? How does that work with time? We're pretty sure there's a directly analogous thing that describes dimension change in the universe.

And that will be the effective equations that describe in our models the very early universe. Yeah, I think there's actually, I'll just sort of mention a couple of things for your benefit as well as for the audience. audiences, if you want to imagine you have a bunch of dimensions that start small and some of them start expanding, there's a very set of cute models from Brandenberger and Vafa, based on string theory, but they have a way why only three dimensions of space would start expanding. And it's

basically because you can tie your shoes in three dimensions, but not in four dimensions, right? Strings will generically intersect in three dimensions, but not higher numbers of dimensions. So that was there. I wrote that down. I'm going to look that up. Yeah, that's a fun one. The dual of that, the converse, was actually by me and Lisa Randall and Matt Johnson, where we showed how if you start with higher dimensional decider space, you could actually create a sub-universe with fewer macroscopic dimensions by spontaneously compactifying some dimensions.

So I think dynamically, given curved space time, you're allowed to go either way. You can have them all small and let some grow, or you can have them all big and let some shrink. Oh, that's interesting. I mean, you know, the question is if it's homogeneous and isotropic, whatever the generalization of that for dimension changes, what I expect is that there's going to be some equation that eventually will have the dimension and the scale factor in it. And it will be only those things. So you don't, in the things you were just describing, there's presumably non-homogeneity

because you're saying that there are some, you know, pieces, some dimensions that grow, some that shrink and so on. This would be kind of the homogeneous isotropic analog of that. I mean, I think this is a, in a sense, this is a quintessential, not particularly easy mathematical physics problem, but it is in a sense it's the reason, one reason it's not easy, and perhaps interesting to say, is that calculus, which is what all of this stuff is based on, is a story of changes with respect to a variable like time. But in our models, and when people do multivariate calculus, they're doing calculus with

with two variables or three variables, but they never get to do calculus with two and a half variables. That's not a thing. And nobody knows how that works. And so in our models, what we effectively have to do is go below the level of where we talk about variables. And we're actually in our hypergraphs, you can think about a direction in the hypergraph as being like a variable.

So as you move through the hypergraph in a certain direction, that's like a variable. But if the hypergraph is a big mess, it's not obvious what you mean by a direction. If the hypergraph was a nice orderly, like, crystal-like thing, then you'd have a well-defined notion of direction. But in the actual hypergraph, you don't. And that's kind of where you begin. I mean, there needs to be a generalization of calculus made that works in fractional dimensional space on these hypergraphs and so on.

And that's kind of the technical difficulty of this. But that's interesting. I'm going to look up that reference. That's very useful. Well, good. And I think this leads actually very naturally into the final big issue I wanted to address, which is locality. And, you know, you've talked a lot, I think, even though I never got to sort of specifically

ask, I think you've done a good job of explaining how space time arises and Einstein's equation and general relativity. And as you said, three dimensions doesn't seem to magically come out, but maybe we can figure that we get it with the right rules. So that's future work. Good. But locality is an interesting thing. By locality, I just mean the fact that in space,

the laws of physics, for the most part, have the property that if you poke the system in one location of space, the effects ripple out locally, right? Only to nearest neighbors as you go forward in time. So that's one thing you might want to get out of an approach like yours, is that you have this notion of locality. Quantum field theory is based on locality. But in the real world,

there's at least two interesting places. where we think locality breaks down. One is in quantum measurement with Bell's inequality, right? And the other is potentially in quantum gravity, with the holographic principle, complementarity, ADS, CFT. So what is your feeling about how you can thread that needle of getting enough locality out of your rules

but also slight deviations from locality when you need them? Okay, my current guess is the following. In space and physical, in these hypergraphs, These hypergraphs, we are saying there's a certain clump of nodes in the hypergraph that have this structure, they get transformed into one with that structure. That's an essential local in space, in the sense that it involves this local region of nodes in the hypergraph. Now, there's two kinds of locality.

One is spatial locality, and the other is branchial locality. In other words, is it, and so my guess is, and the way that things like Bell's inequality, the violation of Bell's inequality come out, is through non-that, that Brantial space is differently laid out than physical space. Physical space, we know in this hypergraph that it's, you know,

we are choosing to a rule that is local with respect to the hypergraph. But the question of what counts as local in this, in Bronshiel space, locality is a different story. Locality is a story there of, well, it's just a different kind of thing because it involves these ancestry and the multi-way graph and so on. So I think that the story there is locality in Bronsheel space is a different issue.

And for example, you can ask what's the analog of a light cone in Branchield space? So in physical space, you know, you make a perturbation somewhere. It expands at the speed of light. What happens in Brantial space? Well, it also expands at finite speed in our mind. models, and that means there is a maximum entanglement speed in quantum mechanics. And the value of that, we only have one parameter we really need. So if we knew the elementary length, or if we knew the elementary time or the elementary

energy, or if we knew the maximum entanglement speed, any one of those parameters is enough to determine the length scale for our models. But this maximum entanglement speed is saying, in branchial sense, you know, in branchingle space, you have a, one vague estimate that we have is the maximum entanglement speed is 10 to the five solar masses per second. I don't know if that's correct. That's based on a very hokey calculation that I did, which makes a bunch of assumptions. But it gives you some sense of scale.

If that was the right value, we would observe, we would observe limitations on black hole mergers of big black holes or very rapid mergers. that are associated with maximum entanglement speed rather than speed of light. So in a sense, it's a limit of the rate of quantum measurement or the rate of quantum effects. And so one question is, which we were just thinking about recently, is can we actually bound, in other words,

in these violation of Bells and Equality experiments where you're looking at propagation in physical space and you're saying it seems to happen instantaneously in physical space. Well, the reason that's happening is because that propagation, so to speak, is actually happening in our models in Bronshield space, which is a different story than physical space. But if we knew the maximum entanglement speed, we might be able to make bounds on that speed in physical space, which would

mean that even though, you know, these various Bells inequality, you know, entanglement experiments and so on, are much, you know, they seem instantaneous, so to speak, that actually there might be a finite bound. We're not sure about that yet. I think it's probably not exactly the same, but this is certainly spiritually a similar answer to what an Everettian would give to the whole Bell's inequality question. There's not a unique measurement outcome in space. So the rules that Bell was assuming do not necessarily apply. Right. I mean, I think that so in so by the way, this maximum entanglement speed has all kinds of consequences. So for example, you can have entanglement horizons, so you can have a essentially a horizon in branchial space.

And so one of the things we suspect, but we don't yet know, is that there's an entanglement horizon outside of the physical event horizon in black holes. And so what that means is so far we can get black holes in our single thread of history, you know, not dealing with multi-way causal graphs. It's just a technically difficult thing to do the analogous, calculation for a multi-way graph, but we think we'll be able to do that. And at that point, we will see black holes that have an event horizon in physical space and an event horizon in branchial space. And so the question is, and that is an interesting thing, because, you know,

if you think about the propagation of information in the history of a black hole, there's a question of can information be trapped between the entanglement horizon and the physical event horizon? And so one of the questions is, if you're an observer falling into a black hole and you're, you know, when you're an observer falling into a black hole to the outside world, you know, time has stopped for, you know, your, and so the question is what's the analogous thing that happens in the entanglement horizon? And I think what happens is kind of cute. I think what happens is that the observer at the entanglement horizon cannot form a classical thought. So in other words, it happens. From the, from the observers. point of view, they just fall right through, though, in classical GR. Yes, yes, right. But to a person, correct. So that is, to a person far away from the entanglement horizon, they would conclude that the observer at the entanglement horizon could not form a classical thought.

So that means they could never, in the sort of traditional quantum mechanics world, they could never collapse the wave function, they could never. And so what would happen is for them, you know, they're observing, you know, some virtual particle pair, you know, did that electron go into the black hole or not? Well, they don't know because they can't form the classical thought to determine whether it went into the black hole or not. And so that's how kind of at least qualitative thinking about the sort of black hole information paradox kinds of things in our models. I would suspect that that, I know, I haven't worked on this, but Jonathan Gorod has worked a bunch on the ER equals EPR story in our models

and claims that he is about to produce an actual sort of explicit simulation of an ER equals EPR thing. So that term, which has been, I mean, it's an imminent claim. So it will have a ride. That's fine. Maybe by the time the episode comes out. But it does remind me of a potential, and I promise is the last thing, A potential quantitative question. The entropy of a black hole is its horizon area divided by four measured in plank units, according to Hawking and Beckenstein.

Someone like me would interpret that as saying that the black hole is described by a factor of Hilbert space with a dimensionality approximately E to that number. But the plank length is not fundamental for you. Is there any chance that you would be able to calculate the Beckenstein-Hawking entropy and get the right answer? Yeah, that's an interesting question. So, I mean, in, you know, the notion of what black hole entropy is, what is entropy? Entropy is basically a measure of the number of states, the number of microscopic states of a system consistent with the macroscopic thing that you identify as being the system, so to speak. So this is basically saying, given this thing that we say as a black hole, how many possible microscopic states are consistent with this macroscopic observation that it is a black hole? So I would think that we should be able to compute that.

And let me think. How would we? Yeah, I mean, I can imagine how we would compute that. And I think it will, you know, what we have to do, okay, so in space time, we can see these event horizons in the causal graph. You can explicitly, I've even studied them a bunch and you can see all kinds of exotic black holes inside black holes and all kinds of interesting and wild things that can happen. But I started looking at the multi-wave version of that. It's just computationally hard because what's happening is you've got these, you know, it's for the exact reason you've got this big exponential in there. You've got this, you know, all these different threads of history that you have to account for. And as I say, there is an optimization of this that another person working on this project, Max Piskinoff, has been working on for most of the last year, actually, which is the theory of local multi-way systems, which are systems.

So the idea is, okay, so roughly the idea is the following. So you have in when you're looking at this causal graph that represents all these different events and they're happening through time, and you say, I'm going to take a slice of this causal graph at a particular, what I consider to be a particular moment in time, then I've got a structure of space associated with those causal connections and so on. that is getting us a version of space. When we also have this multi-way graph, there are many different configurations for space. So there's this notion that I was calling multi-space, which is this thing where at every place in space there's also essentially this stack of multi-way possibilities.

And what's difficult is to understand we can say, okay, there's this multi-ray graph and the whole universe is one node in this multi-way graph. All we can say, we've got one path in the multi-way graph, and now we're going to look at what happens in space. But the thing we're trying to do is to see how do we get a representation that nits those two things together, where we are looking at locality in space and essentially locality in branchial space

and being able to sort of have those two things combined. If we can successfully do this, I'm pretty sure we can do essentially numerical quantum field theory, and we can do it in the traditional, way this is done, it's either Feynman diagrams or its lattice gauge theory. Latus gauge theory is an imaginary time, which is kind of a, you know, not great. And in this setup, we will be able to do it in actual time. And we'll be able to do it. And then, you know, the question of whether we see sort of perturbation theory in Feynman diagram is a story of what happens with our, you know,

with our topological excitations and so on. That might be a more complicated story. But anyway, That's a, yeah, I mean, the thing that's really, to me, you know, just tremendously exciting about this whole thing is, you know, there's this underlying structure and we keep on being able to figure out more and more stuff. It's not easy to figure out these things, but it's like nothing is, we're not ending up running into something where we say, oh my gosh, that's, you know, we just can't explain that. That's just a thing which just is, you know, totally orthogonal to what our models is. saying. And the other thing that's happening, which I think is that we keep on running into things which say, oh, that's just like causal set theory. Oh, that's just like, you know, constructor theory or whatever. Oh, that's just like one of these other theories that's been developed. And what's, you know, to me, both intellectually and sociologically, a terrific thing,

is that, you know, the results that people have developed in these theories, like take causal set theory, for example. There's been kind of a mystery. You're just throwing down events in space time, and you're saying, just look at the relationship between those events in a causal graph. But there's been kind of a mystery. Why are the event thrown down in just such a way that you get Lorentz invariance? Well, in our models, we have an algorithmic procedure for generating those events, which immediately gives you Lorentz invariance. But yet, all the mathematics that people have done in causal set theory, which works on arbitrary collections of events. events in space time can be applied to our models.

And so that's a really neat thing. And I'm, you know, it's looking and the same thing, I mean, we've been in higher category theory is another, you know, very elaborate area of mathematics, which is, I mean, our stuff is a concretization, is a concrete instance of higher category theory and all kinds of rewards that exist in higher category theory that have been extremely abstract. Basically, my test is I can now understand a bunch of these results, which I've never been able to do before. Because they have a concrete instantiation, and one of the things which we are not going to have a chance to talk about here, but the question of which we sort of started to talk about is why this universe and not another one,

you know, why does this universe exist at all? That we seem to be able to have some things to stay about. And it turns out that it's all to do with this limit of higher category. theory, which turns out to also be this limit of running all possible rules in a multi-way graph. And these are things related to this thing called the infinity groupoid, which was introduced by this chap Grothendique, who was a very abstract, abstruse mathematician. And I never thought I would have any reason to care about the infinity groupoid, except that

it seems that it is the limit of a bunch of our models, and it seems that it's deeply related to a lot of things about sort of why this universe and why does the universe exist anyway. And just as a teaser of something we're not going to have a chance to talk about. But one of the things that I've been excited about recently is just as we are making this sort of foundational model for physics, I would like to make a foundational model of mathematics. And we're talking about the set of all possible rules for the physical universe. You can also talk about the set of all possible rules in mathematics. And one of my recent conclusions is, if we believe that the universe exists, which presumably

we do, because that's a notion of existence that we are pretty familiar with, then it necessarily follows that mathematics exists in the same way, and that our doing of mathematics is kind of carving out pieces of this kind of structure that is all mathematics in much the same way that we're kind of carving out pieces of the universe to observe. But anyway, that's a coming attraction. as yet, I mean, what I want to do is to find a bulk theory of metamathematics in the same way that we have bulk theories like relativity and quantum mechanics and physics. Well, I can tell you that it was just a couple months ago

that we had a podcast episode with Emily Real, a mathematician from Johns Hopkins, a category theorist, and we talked about the infinity groupoid. Oh, good, very good. We talked about the infinity groupoid, which tells me both that there's a unity to all of knowledge, but also there's plenty of room to go, which I think that everyone listening to this episode will definitely get that impression. So Stephen Wolfen, thanks very much for being on the Mindscape podcast. That was fun. Good stuff. What if you could have even more and more and more help to pursue your goals?

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