This discussion explores Stephen Wolfram's "computational universe" paradigm, which posits that the fundamental nature of reality is computational and that complex phenomena, including the laws of physics, emerge from simple underlying rules. It highlights how this computational perspective offers new insights beyond traditional mathematical physics.
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I was a pretty successful young particle physics operative when
I was a late teenager and so I had the feature- Wait, you're doing particle physics as a teenager?
Yeah, I wrote my first paper when I was 15.
15? Yeah.
What age were you in graduate school? I got my PhD when I was 20.
Oh, so you finished at 20? Yeah.
Wow. Hey, everyone, thanks for joining us. Today's conversation, a little bit different from some of
the things that we've been focusing upon of late. We're going to talk about computation,
the computational universe, a different way really of approaching science which may provide
insights that the traditional methods are unable to reach. For that conversation, I suspect many
of you've already guessed who our guest will be. That is Stephen Wolfram, founder and CEO of
Wolfram Research, creator of Mathematica, Wolfram Alpha, the Wolfram Language, and I should say,
from a more personal perspective, someone whose contributions have really radically changed the
way we theoretical physicists do our calculations. It really has had a profound impact. Welcome,
Stephen. It's so good to have you again part of our conversation. I just wanted to jump right in.
Your career, and correct me if I'm wrong, but your career has been a blend of fundamental science,
and then creating this wondrous, marvelous technology of the software that really has changed
the way we do our calculations and really even approach calculations. Do you see these as two
separate threads that every so often you go from one to the other or are they part of a coherent
approach to try to understand how the world works? At first, they seem to be just separate things,
but increasingly I realized how coherent they actually are. In nominal terms, I started out
when I was a kid doing particle physics and using computers, and I've done about five iterations of
going between really concentrating on doing basic science, really concentrating on doing technology
development. What's happened is the basic science has shown me a bunch of conceptual things that
have allowed me to do the technology development. The technology development has provided me with
tools that allow me to do the basic science. It's turned out that the way of thinking about
things that's relevant in doing the technology is surprisingly similar to the way of thinking
about things that's relevant for basic science. I think to me it's, in the end most of these
things are about foundational thinking about things. Whether those foundations are what's
really underneath biological evolution or machine learning or physics, or whether
that's what are really the primitives that we should be building in this technology system.
The other thing which is perhaps another perhaps surprise is that I make a living running a tech
company. That might seem like a very different kind of activity from doing basic science, but
actually it's more similar than one might think because it's really, as far as I'm concerned,
it's all about the strategy of what you should do. For me, I spend my time making lots of decisions
about technology and so on, and that's a very useful thing to know how to make strategic
decisions. Then one once doing science it's like, okay, can we push this direction? Do we
push that direction? It's surprisingly similar and to me it's very invigorating to see that
same kind of thinking play out on both basic science and in technology development. Again,
this thing that... well, for example, when it comes to fundamental physics, I had thought I was
developing a different way of thinking about how the universe is computational all the way down.
I just realized long after the fact that some ideas that I developed for technology development
having to do with transformations for symbolic expressions, which is the foundation of a lot
of our technology, that is embarrassingly similar to the foundational kinds of things that I think
about for physics. Even though I imagined that they were coming from two completely different
lines of thought. Actually, the most bizarre thing is that I, back in the early '80s, when I was
doing a bunch of technology development trying to understand things about recursive function
evaluation for technology development, I thought I was doing that. I was also working on gauge theory
and QCD and quantum field theory and so on. I thought that was a, and also in general
relativity and so on, I thought these were completely separate activities. Only to discover
recently that the questions about how you choose simultaneity surfaces and so on were actually the
same story in both cases. 40 something years ago I thought I was working on two completely unrelated
things, and now, about a year ago I realized it's actually the same foundational problem.
I'd love to get into some of the details there, but I do have a follow-up question,
and I really do mean this in the most positive light. I'm just wondering,
do you ever worry or consider the possibility that because you have been looking at the world
from a computational standpoint through all the technology that you have been developing,
that you have this powerful hammer that you've developed and in some sense you're looking for
other nails that it might be able to hammer into. Could it be that it's really your mindset that is
creating the synergies as opposed to there being some objective connection between these things?
Well, look, as a practical matter, I talked about strategy for doing things. I like to find
nails that hammers that I have can actually do something with. Yes, there is some part of that,
but another thing to say about the computational way of thinking about physics and the world is,
if we lived in a different time, I probably wouldn't use the word computation to describe
what we're doing. It's really a question of, you have definite rules, you're applying these rules,
you're seeing what their consequences are. In modern times, for most people, computers are the
quintessential example of that. In another age, it might've been, I don't know, telling army troops
what to do or something, follow these rules and make this, taught us configuration or something.
It's just computation is the metaphor that is the useful one today for thinking about
understanding systems by just seeing, what consequences do these rules have
when you apply them over and over again. Right. Now famously, you've shown over the
years how very simple rules that allow you to update a system, say a cellular automata,
can take you from a very simple initial condition and simple rules to very complex
output, complex behaviors. As a flat-footed physicist, I wonder about the following,
I can always consider the laws of physics to be a rule that takes the universe at time T and evolves
it forward to time T plus epsilon. Of course, people actually put the laws of QCD, for instance,
the laws that you mentioned. Lattice QCD is a system where we don't exactly do it, but you
can always imagine doing a version of what I just described, and people do that. In what sense would
that not be a version of what you're describing? Okay. Well, so the first point is you said T plus
epsilon. Right there you lost me because as soon as you're saying something like time is
a continuous thing, where you can pick any possible value of time, that's not something
where you're just following discrete rules. Right there you have to bring in all the apparatus of
continuous mathematics, which blows you out of a lot of the phenomena that are the most important
ones that we learn from computation. In other words, to be able to apply those methods from
continuous mathematics, you simply can't have a lot of the structure that is going on in these
computational systems. That's one thing to say. The other thing to say is, a lot of phenomena
that I've understood by doing computational experiments, thinking about their results
and so on, those phenomena absolutely existed in simulations people had done for other purposes,
but they ignored them. In other words, there's a rich literature of places where people had studied
systems similar to ones I've studied and the main thing they said was, oh, there's some noise in the
system and it's a nuisance and we don't really care about it. What they concentrated on was the
things where there was enough regularity that they could have something, a simple narrative
that they could tell about what was happening. That's been a narrative that's typically a
narrative that's based on mathematical methods and it's things that can be reached by using
mathematics, so to speak. The big surprise, which surprised me a lot, was when you just do computer
experiments, you just see how the chips fall, you see what actually happens. There's this
big surprise that even though the rules you put in may be very simple, the thing you get out is
very complicated. Is such a phenomenon going on in simulations in Lattice QCD? I'm sure it is. Has
anybody looked at that? Probably not. Even though I've been telling people to do that for decades. I
also would say that we've made a lot of progress actually recently in understanding quantum field
theory in the context of our physics project. One of the hacks of Lattice QCD is time isn't
really time there. You have to make this, time is like space, and I think we can finally fix
that and that makes some of these phenomena a bit easier to understand. But yes, I think the
question is, for example, let's take Lattice QCD as an example. The big question in QCD remains,
how come quarks seem like they're permanently confined inside things like protons? We've never
seen a free quark. How does that happen? Well, there's a certain amount that's known about that,
but what's really going on there? We really need to know things about what the configuration of the
QCD fields really is. What are they really like? Probably they're really complicated, but that's
not what people measure. What people do is to measure some particular number that
represents some correlation between this and that or something of this kind. My guess is,
if you really want to understand what's going on, you have to do things like look at what's the
actual very complicated configuration of fields. From doing that, you will get different intuition
about how things work, but that hasn't been the traditional direction that's been going in doing
mathematical science, because mathematical science is, it's fantastic when it works
because you get to have some sort of, here's a formula. We're done here because we have this
formula that says what's going to happen. But that's not, you are ignoring a lot of
the structure of what's going on. Now, when it comes, we talked about time a little bit,
I think it's a place where one really sees this difference, and so the traditional mathematical
physics approach, time is at some level just a parameter. You can say, I've solved this system,
I've got this T parameter in my formula, I can pick any value I want for that T parameter. In
the computational way of thinking about things, time is the result of the continued application of
rules, and there's no guarantee that you can just jump ahead and say, oh, I know what the answer is
going to be a million steps, a billion steps. In the traditional mathematical approach,
going a million steps and going a billion steps is just changing that T parameter. But
in the computational approach, it's really applying this rule more and more times,
and there's this phenomenon that seems to happen quite a lot, which is this idea of
computational irreducibility. You can't actually just figure out immediately how to jump ahead.
You are forced to live time as time actually progresses, go through the steps one by one.
I think that's a sort of important distinction between the computational way of thinking about
things and the mathematical one, where it's just like there's going to be a formula for the result.
That's such a vital idea, computational irreducibility. Again, just to reiterate what
you said, when we solve a physics problem we have the answer, maybe it's a function of T and we just
choose whatever T we want and we get the state of the system at that particular moment in time,
that as you're saying in many systems you can't do that. You have to actually go step by step by
step. Can you prove a system's computationally irreducible? Is that an airtight mathematical
proof? Are there conditions that need to be satisfied where you can say that is
definitely one of those problems where you can't have a T parameter and jump ahead?
This is a complicated story. It really dates back to things like Gödel's theorem from 1931, which is
a version of this kind of statement. What one is saying is, so there are versions of the statement.
For example, if you say, can I know what will happen after an infinite time in this system?
That's a sharper distinction than can I know quickly what will happen after a million steps, a
billion steps, whatever? Basically, there are ways one can poke at this question where one can say,
yes, there is undecidability. There is computation universality, there are these properties that all
point in towards computational irreducibility. Really, the bigger principle is this thing I
call the principle of computational equivalence, which is a guiding principle for a lot of what I
think about and so on. It's an interesting, well, let's say what it is first. Basically,
you have some system, it's got some definite rules. You can think about the progress of that
system as being like the doing of a computation. Then, the question is, how sophisticated is that
computation? Is that computation one that I could get some other system that's much
simpler to do it, or is that computation as sophisticated as it could be? The principle
of computational equivalence says, when you get above things whose behavior is obviously simple,
you will generically have computations that are as sophisticated as they could be.
One of the consequences of that is computational irreducibility, because computationally,
when you're trying to predict how a system is going to behave, it's like the system is
doing its thing and computing what's going to happen and you on the side are trying to say,
I'm going to outrun that system. I'm going to be smarter than that system. I'm going to do my
computation more efficiently than that system. Principle of computational equivalence says in
general that's not going to be possible. But now, what is that principle? In other words, is that
a statement? Is that something you could prove from the laws of physics? Is that a statement
about defining the notion of computation? What kind of a statement is it? It's a
complicated mixture of those things, very similar to, and actually it's closely related to things
like the second law of thermodynamics, which it's not clear what kind of a statement that is. Is it
a statement about the laws of physics as they are? Is it a statement about the ways that we
can prepare states and so on? But there's a long way of saying. What we're seeing is,
this is a guiding principle for thinking about things and we keep on seeing it. It
has predictions about what should happen. It says, in this particular class of systems,
if you just enumerate those systems as soon as you get past ones whose behavior is obviously simple,
you should show the features of being as sophisticated computational as anything can be.
Like for example, computational universality. Very nicely one can check off in a bunch of these kinds
of systems, yup, that theoretical prediction is satisfied. Took a lot of work to prove it,
but yes, it's satisfied. You can keep poking in at different features of this overall intellectual
framework and seems to keep on doing very well. But it is a conceptually complicated
thing because it is all tied up with what you even mean by computation and so on.
It's not something where you can say, we'll do an experiment and we'll just prove this principle,
because it's a principle that is also related to the definition of things. But it's very nice
that one can make theoretical predictions and they keep on checking out, and that's
very encouraging for an intellectual framework. Yeah, and so I want to turn to the physics project
that you mentioned just a little bit ago, and just by way of introduction, most of our audiences,
of course, are familiar with the fact that the big discoveries, if you will, of the last 100, 150
years, we physicists focus usually on two or three of them. We always mention quantum mechanics. We
always mention the general theory of relativity, and sometimes we throw in statistical mechanics
and so forth as another fundamental pillar of our understanding. My understanding, again, which is
rudimentary of some of the work that you've been doing recently, is that you have suggested that
your approach can yield at least some of these key ideas from this computational standpoint.
But before we get to that, I just want to ask the following question, which is related to
what I asked before. Can you reverse engineer the systems that you build in the following sense? Can
you start with the laws of the general theory of relativity, discretize them in some clever way,
map them to a cellular automaton in some clever way, have clever rules that approximate Einstein's
equations? If I challenge you not to find the general theory of relativity, but rather to
implement the general theory of relativity in your computational paradigm, is that always doable? Is
it tough to do? Is it trivial to do? Well, that will be the traditional
kind of natural science approach, is drill down from what we already see.
There I am. That was the thing,
you asked about my own personal trajectory, that was the thing that I learned to do when
I was doing things like particle physics and cosmology in the 1970s. The thing that for me
was a very useful personal experience was, in building tools, partly to do that, I tried to,
I built my first big computer language computer system and in doing that, the way I approached
doing that was a drilling down process. I was saying, I've got one to do all these different
kinds of computations, let me drill down and find the atomic primitives from which I can
build that up, traditional natural science thing. But then, I had the experience of going from these
primitives, and in that particular case I just completely made up, and seeing how one built
from that into this whole structure that did lots of interesting things. That gave me the
impetus a few years later to say, well, maybe I should try doing something like that for science,
much later for physics of just saying, let's look at all the possible rules we can imagine. Let's
drill down to the simplest possible rules and just build up from there and see what happens.
The big surprise at first was, you get all these kinds of complex phenomena that seem
relevant to lots of everyday features of physics and biology and so on. It took me another decade
to see how that might be related to what was underneath space and time and things like that.
But the reverse engineering approach is, it's much harder work than... The big surprise is that
you can do this thing where you start from the simplest foundations and end up getting things
that relate to what we already know is true. It's not at all obvious that something like that would
work. To give you an analogy, let's say that you were starting from laws of fluid mechanics,
and somebody said, I want you to figure out, how do you drill down from that? Well,
it turns out you can get the laws of fluid mechanics by thinking about lots of discrete
molecules bouncing around. But it would be pretty hard work I think to figure out from the laws
of fluid mechanics that, oh, you could actually represent that by a bunch of discrete molecules.
You'd probably end up with some very complicated setup that wasn't actually, in that particular
case, what we know is going on. The approach that I've taken is a much more extreme approach that
on its face has absolutely no reason to work, which is just, start with the simplest things,
build up from there, see what happens. The big surprise is that it does seem to work.
In other words, and for example, when it comes to, well the Einstein equations for example,
I've just been looking at this recently because I'm trying to get, I'm chasing my beliefs about
what may be happening with spacetime and dark matter and so on. To do that I actually have to
have good accurate reproductions of things that one can compute with the Einstein equations,
starting from these very simple underlying rules. I am more acutely aware than I might be about
all of the complications of both traditional things that people do in simulating the Einstein
equations, and what I have to do. But in terms of how do you get to something where you are saying,
I can simulate this thing with the Einstein equations. The big surprise is, just like
with discrete molecules and you look at lots of them, you get something that looks like fluid
mechanics. What happens in this case is you start with this discrete underlying graph structure for
spacetime. You build up from that and we can show mathematically at a physicist level of rigor that,
yes, the limit of those kinds of underlying processes is the Einstein equations.
I say at a physicist level of rigor because, for example, in the case of even molecular dynamics
giving fluid mechanics for a hundred years or so, actually people gave up, but people were trying to
rigorously mathematically prove that that limit actually works. That's never successfully been
done. I think some things I've done recently show one why that's hard and give one a pointer to how
it might be done. But at a physicist level of this is how the limits work, we can show that
underlying rewriting of these hypergraphs and so on to these Einstein equations. Then, you have to
say, okay, let's do it in practice. Let's actually simulate something that starts off as a couple of
black holes, look at their merger and so on. Then, you have an immense level of complexity
which has also afflicted people starting from the Einstein equations and trying to
discretize and get them on computers. We have many of the same issues about what's a valid
consistent initial condition, how exactly do the updates and so on. It is very encouraging that
the things we're doing can get you, it's not difficult to get to something which is close
to what one would get to in terms of the answers with a discretization of the Einstein equations.
When you say you get to the Einstein equation, just so I understand the target, because I want
to drill down a little bit. In other words, you do some work with your basic primitives
and out at the end of that is Rμν-(1/2)gμR, because (8πG/c4)Tμν? That's what pops-
Yep, that's the idea. Let me explain what's involved in that. Remember the derivation,
if you think about the derivation of fluid mechanics from molecular dynamics,
this is a very similar kind of thing and you might say, oh gosh, we got the Navier-Stokes
equations with all of their detail and so on. It turns out that detail is just the mathematical
physics representation of something that at some level is a much simpler story. I think
it's the same thing here. The fact that there's the (1/2R)gμ so to speak, that's a detail of,
it has to be that way given that we're making this thing that's traceless and-
Well, that's what I was asking. For instance, are you imposing a certain kind of local energy
conservate? What are you putting into the system? Right. It's surprisingly little. The underlying
idea is that space and everything in it is represented by this hypergraph,
just a thing where you have these atoms of space that are just discrete points.
Those are just nodes in this hypergraph? Yes, and all you're saying about them
is how they're related to other nodes. The relation is whether they're touching
each other by some link in your hypergraph? Yeah, whether they're linked, whether they're
connected, when you draw it as a graph. Then, what you're doing, you're saying that's the structure
of space. Time is something very different from space in these models. Time is the progressive
rewriting of that hypergraph by using some rule that says, whenever you have a little piece of
hypergraph that looks like this, transform it into one that looks like that. It turns out,
very similar to the case of fluid mechanics, it doesn't matter so much what that rule is like.
Just like air molecules, water molecules, they have very different kind of scattering processes,
but when you look at them in the aggregate, they have the same laws of fluid mechanics.
It's the same thing here. There are some limitations on that, but that's roughly
the idea. Then, the question of how you get from that microscopic thing, what happens when you look
at a large scale? You have to ask questions like, how do you get from this network to
anything that is even vaguely like space? The first thing you have to answer is, well, what
dimension of space? The network, you can connect things any way you want. It's a friend network.
There's nothing that says that there's anything three-dimensional about your friend network,
and nor is there here. You can define the dimension by looking at, start from one node.
Scale- You ask how many nodes you get to,
and you ask what the growth rate of that is. Now, you start to see why something like the Einstein
equations might be a relevant thing to show up, because if you say, well, look at the growth rate
of this ball of nodes as you go a certain graph distance out, you say the growth rate is like R
to the D, where R is the distance you've gone and D is the dimension. Okay, so then you say, well,
what would it look like if this thing wasn't just an approximation to flat space and D dimensions?
What if it had some curvature? Well, turns out that the correction term, if you just look at
this growth rate thing, the correction term is the Ricci scale of curvature, which interestingly is
something that shows up in the Einstein equations. That's the beginning of things starting to fall
out, so it comes from something that is very simplistic. Just asking for growth rates of
volumes in this hypergraph and then you have to ask, well, how is that volume affected by
these rewriting rules and so on. The other big thing you have to deal with is the right-hand
side of the Einstein equations, the Tμ, the energy momentum story. A big question
in these models is what is energy? I had at first assumed that to know what energy is,
I would have to know what particles are. My qualitative picture for what particles are,
is they're like knots in the ether in some sense. There are things that are topological features of
this network that persist, because the big thing about particles is particles like electrons and
things, they can move from here to there and it's still that same electron when you arrive,
so to speak. It's like, what is the thing that is a robust feature of this structure and
that's a topological object in this network? I thought one was going to have to know what
a particle's like and then one was going to have to try to figure out what's the energy
of a particle like. But actually, there's a bulk definition of energy and it's basically
just the amount of activity in the network. In other words, this network is you having
all these little events where a piece of network is being rewritten into another piece of network.
Well, to be a little bit fancier about it, you can build up a causal graph that represents the causal
interdependence of those events. If you have one event, it produces a bunch of atoms of space,
another event is going to come along and it's going to consume some atoms of space.
These are now the rules that you're putting in for the updating of the hypergraph?
Yes, that's right. Then, one has one event, it produced some atoms of space. If the next event
that's coming along needs atoms of space that came from the first event, then you
say there is a causal relationship between these events. These events have to be in physics terms
time-like separated. It has to be the case that one of them has already happened by the time the
next one happens. You build up this whole graph of causal relationships like that. Given that graph
of causal relationships, you can say, well, I can trace what corresponds to a time-like path in this
network of causal relationships. If I want to know what is space like at a certain instant of time?
Then, I'm going to make a transversal to that. I'm going to make a space-like hypersurface in
the language of relativity, that is in the more mathematical language, you can say that there's a
partially ordered set of these events and this so-called anti chain. It's a thing that goes
across, where every event that's in this kind of space-like hypersurface can be said to be
happening simultaneously. Then, there's another space-like hypersurface where those events are
all time-like separated from the ones on the previous space-like hypersurface. Okay, so it
turns out that energy in these models is the flux of causal edges through space-like hypersurfaces.
In other words, you've got the space-like that represents the instantaneous state of space.
You're asking, as we look at events that happened before that space-like hypersurface and after,
we're saying that the flux of causal edges through that space-like hypersurface, that's our way of
computing energy. Momentum, for example, ends up being the, if you have a time-like hypersurface,
it's just the flux of causal edges through time-like hypersurfaces. By the way,
one of the nice things that comes out is when you unravel all this stuff about space-like
hypersurfaces and so on, you can understand why special relativity works in these models. But one
of the things that's always been a bit mysterious, at least to me, about special relativity is that
there are these transformations between that link space and time and special relativity.
There are also transformations that link energy and momentum in special relativity. The question
is, why are they the same? That was something where they just happened to be the same.
Mathematically, it's very nice that they're the same, but why are they the same? In these models,
it is absolutely inevitable that they're the same. In other words, they have to,
because of the way that one is defining energy momentum, and what one is considering to be
energy and momentum, the transformations have to work the same way as they do for space and
time. That's just one side feature. Sure, but there are some questions in
there that perhaps you can illuminate me on, because where does the causal structure come
from? Because just naively we could live in a Newtonian universe in which there isn't a speed
of light limit on the speed with which signals can transmit from location to location. How does
your model eliminate that logical possibility? Or, is it simply that's another set of rules
that do not yield the Einstein equation? No, I don't think you could make a version
of the models that we've been dealing with that have an infinite speed of light. I just don't
think that would, because what's happening is as soon as you have the idea that space
is this discrete network of atoms of space, there is a sense in which... and then you
have the idea that you are updating that network using rules that have finite information content.
They're not rules that reach across the whole universe. They're rules that somehow involve
only a finite amount of information about, there's this network and it's a little piece
of network and whenever it looks like this, it gets transformed to one that looks like that.
As soon as you have that finite information content of the rule, and you have the idea
that things are made of this big graph of discrete structure, then you would never- Off of discrete
structure, then you inevitably have ... That implies finite rate of information transmission.
But the speed of light in our models is just a matter of units. In other words-
I understand. There are these updates and if you say,
"Well, what is the transformation between the time between these updates and the distance
across the network that it can go," that constant proportionality is the speed of light. It's just a
feature of units. So I think in a model like ours, infinite speed of light would require
infinitely complex rules. It is a fundamental fact about science that the universe doesn't seem to
have infinitely complex rules. If the universe had infinitely complex rules, every different particle
in the universe so to speak, will be doing its own thing. And we wouldn't get to talk about laws of
nature and have a way to take nature and crush it down to a narrative that we can understand. So a
fundamental fact about sciences is there is some finiteness to the rules that are being applied.
As soon as you have that idea in the context of these models, I think you inevitably have what
will work out in physics as finite speed of light. Right. And so the other ... When we teach the
general theory of relativity, some of the things that we usually focus upon are the fundamental
symmetries of spacetime, ideas of general covariance. We like to emphasize that the
get-go, that your T parameter and your spatial parameters naturally are knitted together into
a mathematical structure by which we can do the local Lorentz transformations that allow ... So
if your approach from the get-go singles out time is just the updating parameter
and space is quite different, it's built from this causal network. Is there a tension there
or does it somehow all work out in the end? I think there was a mistake made back in 1919
when Minkowski said, "Let's take space and time and isn't it cool that we can make a quadratic
form?" Minkowski had been a number theorist studying things like that before. Isn't it
cool that we can put space and time together in this mathematical structure that says
T squared minus x squared more or less? And that looks like this nice mathematical thing. I don't
think Einstein actually had that point of view. He didn't like it at first, but he did warm to it.
Yes. I think it was a mistake. I think it's mathematically elegant,
but I think it's as physically missing. You mean a mistake for science to think
that way? There was no error obviously in
that formulation. Yeah. Right. You're saying it's just the wrong direction.
It's a wrong direction because it packages together two things that are really ... Space
and time in our common experience don't seem like the same kind of thing. You can go to a
lot of effort to make them seem like the same kind of thing by telling various relativity stories and
so on. But the fact is our common experience of space and time, they're somewhat different.
And in our models, the fact that T squared minus x squared and know Lorentz transformations and all
that kind of thing show up is an emergent feature. It's something which is true only in the limits of
a large region of space and things like this. And for example, by the way, when it comes to speed
of light being maximum speed of transmission of information, even that is only an emergent
approximation. Because what happens is that in ... Okay. So basically what you realize is that if you
could do infinitely good engineering, which you never can, then you could go faster than light in
these models. So let me try to explain that. So there's an analogy which happens when you think
about gases and things like that with molecules. In the rooms that we're in, molecules are going.
A micron before they hit another molecule. It's not like a molecule that starts here immediately
zips at the speed of sound to the other side of the room. They're going in this random walk.
But if we were able to piggyback on a molecule and if at every collision of those molecules we were
able to figure out which molecule should we choose to piggyback on next, then we could go across the
room at the speed of sound. We wouldn't be just defusing across the room like this random walk.
Okay. So there's an analogous thing that happens in spacetime in our models. So
we have this structure of space that is this network and we're saying ... Sorry,
this is a little complicated so it takes me a moment to think through how this works. What
we're dealing with. We have this causal graph that represents all these different events that
we can think of as being events in spacetime. And each event has effects on subsequent events. So
there's a cone of effects. So we start off from this one event and we get a light cone's worth
of other events that depend on that first event that are causally affected by that first event.
And so now the question is, so that light cone defines this whole collection of events that can
be affected from the first event. And that in a sense now we ask for the projection of that light
cone onto a particular moment in space. A moment of time that gets on one of these space like hyper
surfaces. So now the question is if we're asking ... Our definition of space is, we can think of
it as being the result of slicing this light cone. Well, the basic point here is just like we can do
with the molecules and the gas. So if we pick the exact right causal edge to attach ourselves to,
we can end up getting into a place in space that is further away than we should be able to get to
based on this light cone of what the effects that can be defined are. It's because of the
relationship, because of the fact that space is just an emergent thing that is knitted together
by all of these causal events that we can get through space faster than we should be able to.
Now in order to do that, we would have to know exactly which update event we should select next
to attach ourselves to. So there's two big problems with that. One is the computational
irreducibility problem that it's hard to know what's going to happen without actually having
it happen. So you can't outrun the universe and see where to go next. You just have to wait and
see what the universe does. That's problem number one. Problem number two is a critter like us,
that's some observer of the universe certainly doesn't fit through one or two atoms of space.
In other words, nothing like us can be transported that way. Just like in the case of molecules of
air in a room, nothing like us is transportable on molecules of air. But it is an interesting
thing that at some level the problem of going faster than light is in some sense an engineering
problem. It's not in some sense a feature. It's not something that's burnt into the structure
of the model. It's a feature of what you can actually get done for example, as an observer
or experimentalist like us in a model like this. Yeah. It's interesting. I don't know if you've
been following much about some of the recent work emerging from holography in ideas of
quantum gravity and string theory, but there has been a suggestion by a number of physicists
that if you could undertake infinitely complex operations that you'd be able
to effectuate non-local properties, non-local interactions. So maybe there's some resonance.
I think it's very closely related. The whole holography story ends up playing out in an
interesting way in our models. The I's are not dotted and T's are not crossed in the story,
but roughly here's how it works. So what we've been talking about is we have this hypergraph
that represents the structure of space and everything in it, and we're talking about
the updating of that hypergraph and how that in the aggregate produces the laws of spacetime. But
one of the things that we're a little bit sweeping under the rug in the first description is we say,
"Well, there's this update." But there are actually many possible ways that updating
could be done. All we've defined is whenever you see a piece of hypergraph that looks like this,
rewrite it to one that looks like that. But actually there are many different places
where that rewriting could be done. Is it order dependent? Yeah. Okay.
Yes. So you get essentially many paths of history. So we get these things we call multi-way graphs,
which represent the different possible sequences of updating events. And what can
happen ... People have talked about many worlds approach to quantum mechanics where one says,
"Oh, there are many different possible histories to quantum mechanics." That's what's happening
here as well. But there's one big difference which is made obvious by the fact that these
models are discrete. Which is that not only can you have branching, you can also have merging.
In other words, it can be the case that you end up ... You start off with two identical states
of the universe. They end up in two different states. But those two different states can end
up both evolving to the same state. If this was a continuous system, you would say that will never
happen. The probability of the two things landing at exactly the same number is zero. But in these
discrete models it happens all the time. And so you get this complicated structure of branching
and merging essentially paths of history, branching and merging threads of time going on.
If you look across those different branching emerging threads of time and you say, what's
the structure? If you make a slice across this multi-way graph of threads of time,
what is the structure that you get there? Well, you can say, well, these two threads of time
are somehow nearby because they had a common ancestor, they had a recent common ancestor.
These other threads are further apart. You can make a version of space that isn't like ordinary
physical space that we're used to. It's just another kind of space that corresponds to the
map of these different threads of history. And we call that branchial space, the space of quantum
branches. And it has its own geometry and so on. And branchial space is where quantum mechanics
plays itself out. So we think more or less ... In quantum mechanics, one thinks about quantum
amplitudes that have things like complex number of phases and so on. We think that those phases are
roughly positioned in branchial space. In fact, my guess is that another mistake of packaging that
was made about a hundred years ago was to say that when we think about quantum amplitudes, that they
have a magnitude and they have a phase as complex numbers and that they're all packaged together as
a complex number. I think that's a mistake. I think the magnitude comes from a completely
different place than the phase. The phase is this position in branchial space. The magnitude has
to do with counting the number of paths that you can follow to get to a particular point.
But in any case, the basic idea is we've got this notion of branchial space,
which is a different kind of space than ordinary space. And one of the big claims is that what in
physical space is the Einstein equations is the path integral and branchial space. So in other
words, one of the ways to think about the Einstein equations is that the presence of energy momentum,
say the Einstein equations is deflecting shortest paths in spacetime. That's the-
Usual story. Yeah. That's the usual story. That's what
you've taught for years and so on. And so presence of energy, momentum deflects shortest paths. So in
branchial space, the same thing happens. So except that there the question of what it means ... Okay.
So presence of energy momentum is Deflecting paths, which is changing phases of things because
it's moving things around in branchial space. But now in the path integral formulation of quantum
mechanics, what that says is that the amplitude for some path depends on ... Ones changing the
phase of this quantum amplitude based on the presence of relativistically invariant energy,
momentum. The action. And so what this is saying is that in what is playing out in physical space
as being something explained by the Einstein equations in branchial space is what the path
integral is saying, which is already cool. It's very interesting and there are two
directions that I'm struggling with, which one I should go and I can't resist doing a little bit
of quantum, although I anticipated doing some more general activity. May double back. But if this is
an approach to generating now quantum mechanics from this underlying computational system,
and if you're following a path integral like approach, do you have to put in the weighting
factor E to the I action over H bar? Does? Where does that come from in your-
Oh no. It comes from the fact that it is part of the equation of motion in branchial space.
But it emerges from- Yes. It emerges. In the
same way that in general relativity. If you are simply computing ... So remember I was saying
that energy is basically the density of activity in the network. And so what then happens is that
if you ask what happens to a shortest path in the network, if there is a certain amount of activity
in the network? You can get some idea. There's a bunch of mathematical versions of this. But
the rough qualitative picture is if stuff is happening in the network, then the shortest
path from here to there is going to be affected by stuff happening in the network. And it's the same
story in branchial space for quantum mechanics. That stuff happening in the network ... And again,
there's many footnotes to this and it's a slightly ... But the qualitative picture is stuff happening
in the network, deflects paths in branchial space and the deflection of paths is a change
of phase. So what one's saying is the presence of energy momentum is changing the phase, which is
essentially what the path integral says too. And you're saying that that emerges. None of
this has been ... In other words, I guess another way of saying it ... And again,
I wish I understood the innards of this more and I'm appreciative of your patience for describing
this all to me, but if you had no understanding of the general theory of relativity, if you weren't a
physicist at heart and didn't have the training of a physicist and hadn't worked in physics and you
had no understanding of the Schrodinger equation or the Feynman path integral and you were just
taking this computational approach, is it the case that you would have extracted general relativity
and quantum mechanics from this approach? I think I would have to be pretty good to
be able to do that. But yes. Just as if you are ... You start from computer simulations,
you try and figure out what are the emergent laws. I've tried to do that in lots of places
where people didn't know what the emergent laws were and have had some success there. If you're
asking me would I have successfully been able to extrapolate from what I'd seen in computer
experiments to deduce those laws, I would say that will be a stretch for me. It's not conceptually
impossible, but to have thought of the right things to study would've been challenging.
Well, the example I could give you, and this is perhaps a poor example and maybe a self-serving
example. I don't know. But when you study string theory, you study the quantum mechanical equations
of motion of a string and it really is the case. And you can say this with confidence
and you're not hiding anything. If you didn't know the general theory of relativity, but you
knew quantum mechanics well enough to study the quantum mechanical motion of his string, you could
really extract out the Einstein equations from that knowledge. From that structure.
You now what? I would disagree with that. Because they come out
essentially as a consistency condition. The fact that there is such a consistency condition,
you wouldn't have even thought to look for that. You just said the strings flap around
and they have non-causal things happening and they have tachyonic modes and they have this
and that and the other. So what you might say. You have to know there are no faster than light
modes and so on in the physical universe. Yeah. I'm not saying that you can get it
without putting any physical insight into the system, but what I am saying is you're
putting in insight that does not require any knowledge of the general theory of relativity,
the Einstein equation. That's a better way of saying it. The Einstein equation.
At the same level, you would deduce the same thing here. It would be nice if you
knew some differential geometry because then you would think about studying these aggregated
quantities. For example, if you ask have graph theorists come up with these things independent,
they've gotten the kinds of quantities that you would think about looking at based on knowing
general relativity are not quite the same, but they're not super far away from the kinds
of quantities that people who just study graph theory have thought about looking at. So in that
counterfactual history, so to speak, I think there's a decent chance. I wouldn't give myself a
hundred percent chance of noticing the right connections. Again, I've had the good fortune in
a sense to work in a bunch of areas where I start from the computer experiments and one doesn't know
the emergent laws. And so I've had to try and deduce those and see what their consequences
are and so on. And I think knowing from both sides is convenient, but I don't think that one
was fudging it to be able to meet somewhere in the middle. I think that's an inevitable thing.
Now, there are obviously, as you're very familiar with puzzles in the general fear of relativity
such as the cosmological constant. So Einstein wrote down the Einstein field equations and
famously did not include the cosmological constant at first. And then when he found that he gave rise
to a dynamic universe that he didn't like, he put it in and later on he removed it when the
observations showed there was a dynamic universe. And then fast-forward 70 or 80 years later,
we put it back in to account for the accelerated expansion. So since you're
getting Einstein's equations extracting it from this computational framework, do you
have insight into the cosmological constant? Maybe. This is the thing I'm currently trying
to do of really tightening up how we deal with general relativity in these models is an attempt
to understand that better. So in traditional general relativity and quantum field theory,
there's always been this mystery about how can it be the case that there are all these vacuum
fluctuations that are implied by quantum mechanics that have all this energy associated with them,
and yet they don't seem to have a gravitational effect. They're not curling our universe up into
a little ball because of all that energy, so to speak. And that's been a strange and mysterious
thing. In our models that issue doesn't really come up because it is those things that we can
think of as quantum fluctuations. It is that microscopic structure of spacetime. It's the
activity that knits together the structure of space. So it is not the case that we have on the
one side thinking about quantum fluctuations and so on, on the other side, thinking about
the structure of spacetime. In some sense, the quantum fluctuations make the spacetime.
That means one isn't immediately thrust into the question of how come there isn't an immensely huge
cosmological constant associated with all of those vacuum fluctuations. But it also,
it puts one on the hook to have to derive, well, what is the zero so to speak? Because there's a
certain amount of activity that's necessary to knit together the structure of space. Then
the things that we are looking at with actual massive objects sitting in space and so on,
those things are above that zero. And figuring out where the zero is is something
one has a chance to actually be able to do. So my current guess which might be wrong is
dark energy, which is less politely called negative mass matter or negative mass stuff
is what is associated with the zero kind of thing. Whether dark energy is a real thing,
I'm not yet convinced experimentally, but that's a different issue. Then there's dark matter,
which has been known about. Some signs of it have been known for a hundred years. My own guess is
that again, a bit of a mistake was made in calling it dark matter because that implies that it must
be something related to matter. And my analogy for this is what happened in the 1800s with the
question of what heat is. People notice that heat could flow from one substance to another. What
flows? Well it must be a fluid Caloric.
To the invented caloric fluid. Yes. And of course, by the end of the 19th century it became clear,
well, that's not actually right. It's discrete molecules and so on. Heat is the motion of
discrete molecules. So my irony of scientific history is the thought that dark matter is
really the spacetime analog of heat. It's a macroscopic manifestation of the microscopic
structure of spacetime. Now have I proved that? Absolutely, I have not. But that's my-
I'm just wondering if that idea is correct, would the equation of state of that thing that you just
made reference to be the same as the equation of state, or at least in the appropriate limit that
we use for dark matter a pressure? A pressureless- Yeah. Right. The fact that it's pressureless means
it's just a bit like space itself, so to speak. That might even be a clue right there. But I
think that's obviously what one has to derive. The question is if you were just presented with
here's a solid and I'm telling you there are these discrete molecules that are running around inside
it, what effect does that have? It would not be easy, I think, to deduce laws about heat and
so on. This is the situation that we have here. It's even a little bit more complicated because
of the fact the idea of this network structure of space and so on is a little bit less concrete
than having this block of solid in front of you. Yes. The question is how does one get
accurate enough in one's going from this discrete underlying network to the continuum structure of
spacetime and so on, to be able to say, is there some extra little effect that isn't
quite the Einstein equations that gives you what we are thinking of as dark matter, so to speak?
Wait, so you're suggesting there's a correction term to the Einstein
equations when you go to the continuum limit. Do you have? Do you know what that looks like?
No. I don't know what that's like. But it's worth remembering. If we go a little bit technical,
general relativity, the left-hand side of the Einstein equations is all about the structure
of spacetime. The right-hand side is all as about energy momentum. When you think
about gravitational waves or something, there's a trade-off. You can say, well,
I'm going to stick that on the right-hand side as part of the energy momentum story,
or I can stick that on the spacetime side. So if you're asking ... In this case, you say, well,
what's the correction term like? I suspect it'll be the same kind of thing. You could talk about
the gravitational waves as being a correction term to the left-hand side if you wanted to, although
that's not the way one usually thinks about it. Right.
But if you're asking, how does this work, I don't know yet. This is just a guess.
This is a thing where ... This idea that it's a feature of the structure of space rather than
something associated with particles, one reason that's important or observable in a sense is if
it's a feature of the structure of space, that's something that's happening at the scale of the
elementary length, at the scale of the ... Which might be 10 to the minus a hundred meters or
something. It's really tiny. If one's talking about it being particles, one could say well,
people are doing dark matter searches and there's going to be one particle that comes through every
minute and it's a discrete thing. Whereas if it's a feature of the structure of space, it's like,
well, there's this tiny effect and there's a gazillion of them at every moment. So it's
really a different kind of thing. You would never expect the one particle event that you would think
of from dark matter in a particle interpretation. Now, another puzzle in addition to dark matter,
dark energy are singularities. Famously, the first closed form solution to the Einstein's
equation was not written down by Einstein. It was written down by Schwarzschild. And when you look
at that solution, people often describe it as it sows the seeds of the own destruction of general
relativity because at the center of this solution, which we now call The Black Hole Solution,
there are things that become infinite. Invariant things that can't be done away
with by a coordinate transformation. Curvature invariance and so forth. So you must encounter
those in some form and what do they look like in your language and what do you do about them?
It's very interesting. In a Schwarzschild black hole, for example, you can't represent everything
about a Schwarzschild black hole using a continuum view of space. You have to punch a hole out at the
center. There's this singularity where you say, "Well, the Einstein equations don't quite apply
here. We're going to assume that spacetime has a little hole in it." Well, in our models, it's
nothing like that. That seems like a hack. You don't have any kind of thing like that hack in our
models. First of all, in our models, things are discrete. So the idea that, for example, there is
a transformation between one kind of topological structure of spacetime and another that has to be
this strange discontinuous thing when you're dealing with traditional general relativity
and manifolds and continuous structures. It isn't a discontinuous thing in our models
because something that didn't have a hole, there isn't a notion of a distinct hole, so to speak.
It's just there's a structure of network and when you look at the emergent properties of that, you
can describe it as having a hole. Now, in the case of the Schwarzschild black hole, the thing that
happens is you can think about the singularity at the center of a Schwarzschild black hole as
being a place where time stops. Every geodesic has finite length. If you're falling in there,
after finite time, you're toast, so to speak, at the singularity. So in our models, what happens is
that you get something where time literally does stop because you are thinking about rewriting this
hypergraph and at some point you realize you've got a structure of hypergraph where there is no
more rewrite that applies. So time corresponds to the progressive rewriting of this hypergraph.
And how do you see that there's no updating, there's no application of your rule? What
does it mean that- Well, it has a-
I know what means ... Yeah, go ahead. Yeah. It has a kind of discrete structure
that is such that, depending on what the rule is, there isn't a configuration of atoms of
space that are related in this or that way. There just are no atoms of space related in that way in
the place that corresponds to the singularity in the black hole. Now, in the case of our models,
the event horizon of a black hole is very easy to recognize because it's a place where there
is a either ... Well, event horizon of a black hole is a one-way event horizon. Cosmological
event horizon will be two-way event horizon where you can get information in either direction. But
whatever it is, the question is are these causal edges that are associated with this causal graph,
do they both go into region and out of the region or do they only go into the region?
And so you can identify a black hole as being a place where the causal edges only go in,
information can get in, but nothing comes out. That's a fairly easy thing to identify. Now,
when you ask about things like rotating black holes where things get funkier in general
relativity, what seems to happen is that at the critical rotation rate ... Isn't really rotation,
but the critical rotation-like parameter rate. If you were to exceed that, you
would disconnect a piece of the universe. So in other words, what happens is that in our models,
you end up with fewer and fewer causal edges that connect these different parts of spacetime.
And if you were to have a black hole where the J parameter was greater than M, it would disconnect.
And just so I understand the nuts and bolts a little bit of this, so if I said to you,
I'd like to see in your description a black hole solution that has mass M, angular momentum J,
maybe electric charge Q, do you know how to set up your hypergraph and then you have your fundamental
rules at hand to see how that evolves through your updating through your temporal parameter?
I can't do the charge part. The electric charge. I don't know how that works yet. For mass and
angular momentum, yes, we can do something there. But you've got to understand we don't know how to
make a black hole from nothing. That takes a lot of computation to get a black hole from
nothing. So the best we can do, which is by the way, the same thing you would do in relativity-
Set it up. Is to say,
"Let's start from that configuration." And so we can start from this continuum metric, for example,
and we can say, "Let's say how we would make a hypergraph approximation to that metric."
So we can set that up and then we can say, "Okay. How would that evolve under our rules?"
And so that's the way in which you can say, "Well, what then happens? Do you get something which is a
kind of stable solution to general relativity that way or not?" It's a bit tricky as it is
in the case of numerical relativity. And the reason it's a bit tricky is, as I mentioned,
it's ... Well in our case, it's a little bit more physical why it's tricky. Because there
are these different possible updating orders that you can use, and depending on which updating order
you use, you may or may not end up with something where you're evolving time in a
reasonable way as opposed to an unreasonable way. In other words, if you said, "I'm going to do all
the updates here. I'm going to update many, many times in this place, and I'm not going
to update it all over here.", you'll get funky things happening. And that's the same exact same
thing happens in numerical relativity. And so we have pretty much the same set of issues as
numerical relativity. If you say, "I'm going to pick updating rules," so what does it mean to pick
these updating rules? Basically it's saying if you imagine an observer observing what's going on,
the observer believes that certain things have happened and other things have not happened.
It could be the case. So having an observer who has a reasonable way of probing what's happening,
you're going to end up with a reasonable sequence of updating events happening, and that will give
you the same kind of thing that you see in numerical relativity, but rather nicely.
Some of the issues you see in numerical relativity, like you say, well, in this particular
place there's high curvature, and so we have to put in more points in our numerical relativity,
and we do that kind of by hand. In our models, it's not by hand at all. The reason there's more
curvature is because there's more activity in the network, and that means there's more nodes in our
hypergraph. In other words, we're doing the analog of an adaptive numerical method is happening
because of the physics of what's going on, not because we chose to put it in from the outside,
so to speak. But I will say that as far as I'm concerned, this is subtle stuff. It took a long- I
will say that, as far as I'm concerned, this is subtle stuff. I mean, it took a long time.
It took until, what, 10 years ago or something before people had decent numerical relativity
for black holes? And that was a solid a hundred years after the original construction
of general relativity. We're way ahead of that schedule, but it's still subtle stuff.
I mean, can you see gravitational waves? Yes, one can. And one can see, when one computes,
you can compute the Ricci curvature, one can compute know Weyl curvature, the form of
curvature that lets you see gravitational waves and so on. That all works nicely.
And this is maybe the wrong question. I don't mean this to be provocative at all. But are there any
insights that you can get from this approach that people who are doing more conventional numerical
general relativity cannot get? And I know that the value of what you're talking about is you're
going deeper to the atoms of space and time. But in the larger scale limit, are there insights?
Yeah, I think so. I mean, the question is, what effects will you see here that you wouldn't see
in traditional general relativity? Right.
And so one of the more dramatic ones is dimension fluctuations.
Oh, I see. Nothing guarantees an integral number of dimensions, yeah.
Right. And that can fluctuate. And we don't know how much it should fluctuate. We would
really love to have an analog of the standard continuum, Friedmann-Robertson-Walker metric
for the early universe, a homogeneous universe, which in the traditional way of setting it up,
you can have sort of the one parameter universe where you just say what the overall curvature
parameter, the overall scale parameter is for the universe. We'd love to have a version of that
solution for a homogeneous universe that includes dimension as a dynamical parameter, as well.
We don't quite know how that works. We are somewhat handicapped by the fact that
the apparatus of calculus and things like differential geometry built over the last
300 years is all built based on the assumption that the dimensionality of space is an integer.
And it's all, you do univariate calculus with one variable. You do multivariate
with a few variables. You don't do 2.3 dimensional, 2.3 variable calculus.
Well, I think there's a thing called non-standard analysis.
Doesn't help you. That doesn't help you?
That's solving a different problem. Okay.
No, I think it is fair to say, and we've been poking at this for long enough, that I think
there's really nothing known about this issue. I mean, if you say, what's the Riemann tensor
in 2.1 dimensions, one just doesn't know. And so, we've been trying to build what
we call inferred geometry, which is geometry where the underlying structure of space doesn't
start from down in space like Euclid said it was like, but starts from this hypergraph.
And that's been a bunch of mathematical work, which we're getting further with. But we need
to have that in order to really tighten up these discussions of how exactly things work. You need
this kind of framework to think about that. Now, you described earlier on how you define
dimensionality here, which makes perfect sense. You look at the causal relationships. You see
how they scale with some notion of radius and if it goes like R to the D and so forth.
Right. But do you find that your updating
rules tend to preserve that dimensionality or do they preserve it within an integer? So,
if it's at three, it'll bounce around 3.2? Or does it go up to five or something like that?
Well, things come to equilibrium usually fairly quickly for the same reasons that they do in
gases. I mean, basically, what happens is, and it's a symptom of computational irreducibility,
that basically things evolve to the average random place fairly quickly. And once they're
in the average random place, things don't change much. Now, there certainly are rules which crush
down to zero dimensions and things like this and expand out to infinite dimensions and so on.
I mean, the other tricky thing is that it matters what the updating order is.
In other words, you can have a pathological updating order, which can make crazy things
happen. But that pathological updating order isn't the one that observers like us, so to speak,
would ever see happening in the universe. You mean it's something about us as
observers that restricts the kinds of systems that we could ever encounter?
Yeah, I mean, there are two key things about us as observers, which turn out to be crucial
in the models that we have. One is that we are computationally bounded observers. If we were not,
if we could follow every individual molecule, if we could follow every individual atom of space,
we wouldn't even believe in continuum fluids, continuum space because we'd just say, "Look,
I can just see where all the molecules are going. I can see where all the atoms
of space are going." For example, we wouldn't believe in the second law of thermodynamics.
Of course. If we were not computationally bounded.
What's happening in the second law is we're going from this sort of maybe simple initial condition
through this irreducible computation to something which we, as computationally bounded observers,
just sort of throw up our hands and have to say, "It looks random to us," rather than, "Oh, I can
see how everything went." That's one assumption. The other assumption turns out to be important
is that we believe we are persistent in time. In other words, even though we might
be made of different atoms of space at every successive moment in time, we feel that we have
a continuous thread of experience through time. And it's important that we have a continuous
thread of experience. It's actually important that we have a single thread of experience through
time. That's the thing in understanding quantum mechanics and how we deduce things from quantum
mechanics. It's important to understand that these different threads of history,
we have to conflate those together because we're going to say, "We think definite things happened."
Whereas in a sense, at the level of the universe underneath, many different threads of time are
taking place. For the universe underneath, many different things are happening. For us,
we're conflating those together. By the way, in space-time, the fact that we
are pretty big compared to the elementary length, the distance, the structure of this network,
and so on, we're big, so we just average a bunch of things out just like we do for fluids,
for example. We say, "We're not looking at the individual molecules because we're pretty big
compared to the molecules." In the case of quantum mechanics, it's a little bit more
subtle because we're big. We span many different paths of history. So, our minds effectively have
the same branching and merging of paths of history going on in them. But we are taking a
big sample out of Branchial space and that's why we end up believing that definite things happen.
Just as if it was the case that we were looking at a fluid at the level of molecules,
we wouldn't say, "Oh, the fluid is going in this definite direction." We'd say, "Oh, it's got all
these different directions it's going because all these different molecules are going all these
different ways." But yet when we average it on a large scale, we say, "Oh, the fluid is flowing
in this direction." And it's the same kind of thing with what's happening in quantum mechanics.
I mean, it's a tricky thing because, for example, you can ask, well,
why is there randomness in quantum mechanics? In these models, everything is kind of determined,
at least at the level of this multi-way graph of all possible threads of history. The structure of
all possible threads of history is completely determined. What is not determined is where
we are as observers across Branchial space and so on. Which branches did we actually sample?
And the analogy that I think is perhaps useful is if you say,
"What does the night sky look like?" You can say, "Well, there are these stars in the night sky."
And we all agree about what it looks like because we're all on the same planet. If you say, well,
"What does the night sky look like according to the laws of physics?" We can't say that because
it is a contingent thing that depends on us being in the particular place we are but the important
thing is we all agree about what it looks like. It's an objective reality for us because we all
happen to be nearby in physical space. And I think the same thing happens in
Branchial space, which is why, when we do some quantum experiment, people can agree,
oh yeah, that was the answer for that quantum experiment, not because that's the only possible
answer there could be but because in a sense, all the minds of all those people are close
together in Branchial space. And so they all kind of perceive the same threads of history.
Right. So, Stephen, there was a whole basket of questions that I would like
to ask at this point. Unfortunately, I know that we are running out of time.
I can go a bit longer. This is fun. We've got to go a bit longer.
Well, if you're up for it. I mean, the questions that then come to mind for me if we're going to
talk a little bit about quantum mechanics is, in quantum mechanics, perhaps the strangest
quality of the theory of all is entanglement. Schrodinger famously described it as not one,
but the quality of quantum mechanics, which distinguishes it from a classical worldview.
The tricky thing about entanglement, of course, is, if you're dealing with a computational system
that's local, your updates, your rules are all local updates as I understand it. So,
where from your perspective does this non-locality, this very modest non-locality
but still quite wondrous, where does that emerge? There are many threads of history. Different
things are happening on these different threads of history.
And by the way, the sort of notion of entanglement is bizarrely direct in these
models because literally you're dealing with these different threads of history and they
are entangled in the sense that they have common ancestry, so to speak. So, these things that are
happening on different threads of history are connected because they have the common
ancestors in this graph of possibilities. But there's nothing that says that...
The relationship between what is spatially distant and what is distant in branchial space,
those are two different things. Sure.
Now, those two things get knitted together in this thing we call a multi-way causal graph,
where you have this notion of two events, for example. You can say those events are time-like
separated. One has to follow another in time. They can be space-like separated in the sense
that they can both be at the same time but you can think of them as different places in space. They
can also be what we call branch-like separated, where they are occurring on different branches of
history. All three of those are possible. Now you can make this whole graph that
shows the causal relationships between things in space, in branchial space,
and through time. And so, then that whole object is the ultimate story of what are
the relationships between all these events that can happen in the updating of this hypergraph.
And so we were talking a little bit before about holography, which is this connection
between things that happen in general and things that happen in quantum field theory.
The way that works, I think, in our models, is that this multi-way causal graph is this
knitting together of things that are associated with space-like separation and things that are
associated with branch-like separation. And in a rough approximation,
quantum mechanics is one projection of this in essentially the branchial direction. And
general relativity is another projection of this essentially in the spatial direction. And so,
that's kind of how that relationship works out. And some people have looked at the ER equals EPR,
which is kind of a quintessential example of holography. And there are claims that you
can see that phenomenon in our models. I'm not yet totally convinced myself. But that will be
a nice thing. That's sort of this quintessential connection between the world of quantum mechanics
and the world of general relativity. I mean, do you see wormhole-like
solutions presumably, as well, if you try to create them, much like black hole solutions?
Yes. I mean, my favorite of these things I call space tunnels, which are these things that have
a different dimensional space. So, that's a thing that's different from general relativity. It's a
wilder collection of things that can happen than in general relativity, particularly
because of this dimension change thing. Now, you were asking what can we see that is
different from what you see in general relativity? I would say, well, the number one effect,
which doesn't depend on the elementary length scale, is dimension fluctuations because that is
a feature of models like this where you don't have to know how big that elementary length is. I mean,
Einstein was lucky that some of his predictions didn't depend on the one parameter in his theory,
namely the cosmological constant. I mean, things like the bending of light around the sun being a
factor of two different than it would've been the Newtonian theory, it's convenient that the
two didn't have a lambda in it, so to speak. Although for me, it's a major cautionary tale
that, when the bending of light around the sun was first computed, they didn't get the right
answer. And then the experiment was almost done at a time when they would've said, it doesn't agree.
But World War I got in the way, and that team was arrested as they crossed into Russia,
as I recall the history of it. Had they not been arrested and measured the bending of starlight,
they would've ruled out Einstein's first prediction of the bending of starlight.
Right. So, for me. This is a cautionary tale in terms of the theory is difficult to work out,
the experiments are difficult to do, it's complicated to navigate having to meet in
the middle. So, the next big class of things is the macroscopic effects of what one can call
space-time heat. I don't know what they are yet. So, are you calculating things like how a
particle's trajectory might be affected by fluctuating dimensions or scattering
experience? I mean, that's the kind of thing? Yeah, that's the hope. I mean, the thing with
dimension fluctuations is seeing really bizarre gravitational lensing where normally light is
bent by the presence of mass in gravity, but a dimension fluctuation would produce
some very bizarre shattering of wave fronts of light. But it's complicated to work that
out. And it also depends on exactly what the form of the dimension fluctuation is,
which we don't really know. We just know that's an effect that you can get in models like this.
Now, I mean, there are other pieces. There are other kinds of things. You talked about
particles. Well, one of the problems is we don't actually have a good model for a particle.
We know qualitatively what we think it'll be like. But I was feeling really bad about this until I
realized recently that quantum field theory also doesn't have a good model for what particles are.
Asymptotic states in the S-matrix, right? Indeed. I mean, but if you say what actually
is a proton? It's like nobody has any idea how to represent a whole proton in
quantum field theory. And so what one instead does, as you were alluding to, is just say...
In fact, here's a piece of history I learned recently. Back in the early part of the 20th
century, turns out most physicists believe that space was discrete. I didn't know this.
Is that true? I didn't know that either. Yeah. It's a very obscure thing because they
didn't publish anything about it. Because what happened is, 19th century, nobody knew whether
matter was discrete. Then it became clear through Brownian motion and so on that, yes, matter is
discrete. Einstein argued that light was discrete. And Einstein believed, even has a nice letter he
wrote in 1916, where he says, "In the end, it will turn out that space is discrete, but we don't have
the mathematical tools to analyze that yet." So, I can say Einstein believed this. Bohr
believed it. Heisenberg believed that space was discrete. Heisenberg tried to make a model of
discrete space. None of them could make a model that was consistent with relativity. They all
had this lattice structures and things and they couldn't make that consistent with relativity.
They published nothing about this stuff. But what I learned recently was that,
in 1930 or so, Heisenberg was trying to make this work and was like, I can't make this work.
So all I'm going to do is I'm not going to even think about what happens inside the region where
particles are interacting. I'm just going to talk about the incoming states and the outgoing states.
And I'm just going to put this S matrix in the middle of those. And then he never went back and
thought about discrete space again. He just was describing interactions of particles in
terms of what comes in and what goes out. And one thing I realized recently is,
one of the ways to understand that in quantum field theory is using Feynman diagrams and saying,
"There are just a very few interactions between these particles that are really
going on. And we're going to draw these diagrams that represent the relationships between these
different actual points of interaction." What I realized recently and should have
known long ago is that Feynman diagrams are causal graphs. They're causal diagrams.
They basically tell you the causal relationship between those actual interactions of the photon
and the electron interacted at this point. And the lines in the Feynman diagram are
telling you how that particular interaction is causally related to other interactions.
And this is again work in progress right now to try and reinterpret, to try and have something
that's sort of underneath Feynman diagrams. It's kind of a way of going directly from the causal
graphs of our models to a sub-Feynman diagram. In other words, it allows us to do something like
quantum field theory. In the same way that quantum field theory doesn't have to talk about particles,
nor do we have to talk about particles. We can do something that is analogous to quantum field
theory without having an explicit story about what the proton or the electron really is, so to speak.
And what's kind of interesting is that we can make much simpler models that
generate these Feynman diagrams-like causal graphs. I knew Dick Feynman fairly well. And
one of the things he was fond of saying, because one of the things I did in particle physics early
on was computing Feynman diagrams. And I would make all these computer programs to do it and all
this kind of thing. And Dick Feynman would say, "This Feynman diagram idea is just a stupid idea.
The fact that it takes all this effort to compute all these things, it's just the wrong way to do
it. There's got to be a better way to do it." So, I'm like, okay, well, what is it then?
He says, "Well, I don't know what it is." But I think we are slowly maybe getting to something
where at least in these toy cases, I can kind of see the thing that is underneath the Feynman
diagrams. And it's one of these sort of rewriting process things where you can describe the whole
rewriting process and you can break down the rewriting process into all these different
possible causal graphs of the way it might've gone. And those possible causal graphs are Feynman
diagrams, but the underlying rewriting process is sort of what really happened. And you can
potentially analyze that in a better way than just working out all these individual Feynman diagrams.
I wonder, when you say that it brings to mind, just for the audience, Feynman diagrams,
you're taking a calculation and you're finding a way of breaking it up into small pieces that,
when you sum all of the pieces together, are meant in some not mathematically rigorous way to add up
to the answer to the original calculation that you were doing. But as you're saying,
diagram by diagram by diagram, it is a lot of work to calculate these individual pieces. But
the work of Nima Arkani-Hamed and others on this amplitude hedron, if I'm pronouncing it correctly,
is sort of a way of trying to put these guys all back together. It's like you take a vase
and you shatter it and you have to put all the pieces back together. Why shatter it at
all? Maybe there's a way of dealing with it from the get-go. Have you thought about that?
Yeah, I'm sure that's closely related to what I'm trying to do.
Right. I mean, in other words,
I think what I have is, in a sense, a much more mathematically simplistic object that I
can look at, which potentially allows one to see the answer more easily. But for example,
in enumerating these causal graphs and things, one has a direct analog for normalization. I mean, one
of the problems of Feynman diagrams is, when you actually compute them, they're full of infinities.
And then you have to make the argument, "Well, there are infinities everywhere. And all we get to
notice in the world is the change in the infinity from one thing we do to another thing we do."
And you get exactly the same phenomenon here. And actually the same kind of graph theoretic methods
that one uses to do normalization in Feynman diagrams seem to work in this case as well,
which is kind of interesting. And just because you raise it,
when I teach renormalization to students, I tend to focus on the Wilsonian approach where you have
a cutoff from the get-go. And renormalization is just how the structures, the amplitudes, and the
parameters change as a function of a change in the scale of the cutoff. And that way you can
sort of get rid of the infinities from the get-go. And I think it's conceptually easier and probably
more relevant because many of us see quantum field theory as not relevant to arbitrarily short scale.
So, put a darn cutoff in there from the get-go, avoid the infinities, and now make sure that you
allow your parameters to vary where the cutoff in a way that your observables don't change. Is there
a version of that in the approach that you take? Well, I don't know yet, really.
One, there's some very funky things that happen. So, I'll mention a few things that I know and
this is incomplete at this point. One thing that happens, which is kind of funky, is that you get
these things. in these rewrites, you get these kind of loops that can happen. So, in general,
activity one always says closed time-like curves where you go forwards in time and then you can
kind of go backwards in time. We don't like these. In the way that these things are set up, you
effectively get lots of closed time-like curves. And there is a notion of what I've been
preliminarily calling sub-time, which is the thing that you get if you actually are
going through and enumerating. You're going through and you're effectively getting from
this to this to this. But then you can say, "Well, actually, the state that we got to is exactly the
same as the state that we started from." So, from the point of view of an observer, those
states should be. There is a sub-time that goes infinitely, but observable time is only finite.
And so, that's a slightly different form of infinity than one has seen. It's related to the
infinity of virtual loops in quantum field theory. The next thing that is kind of fun is the question
of, in Feynman diagrams, one of the big features is you've got a particle and it's just going
from here to there and nothing happened to it. There's this notion of the propagator,
which has a mathematical formula, 1/p-squared minus [inaudible 01:34:49] whatever it is.
Plus i-epsilon if you want to avoid the singularities. But anyway...
Well, yes, right. Hopefully, the approach we'll have will not even have to get into that story.
But anyway, it seems like the particle is just going from here to there. And in the Feynman
diagram it's just like nothing happened to it. It just went from here to there.
Well, actually, there's a thing that strangely enough is a little trick that Dick Feynman told
me one day. And I don't think he ever wrote it down, but it has become known as the Feynman
checkerboard. And it has a whole literature. And I didn't know until very recently that
anybody knew about it other than me, so to speak. But it has to do with, on a discrete lattice,
you can say you just have particles that either go left or right. And you look at
all possible paths and you say, every time a particle was turned from left to right,
there's a factor of -i. And turns out when you add up all the possible paths,
the thing you get is the solutions to the Dirac equation. So, it's like the quantum mechanical
solution to what the amplitude of a particle being found at a particular position relative
to its starting point is.And so, effectively it's like the beginnings of the propagator,
beginnings of the thing that say, what happens to a particle if it just goes from here to there.
Now, the thing that's kind of interesting is that the parameter that tells you the mass of
the particle is how many times it got kicked, how many times the thing changed its direction.
So, in Feynman diagrams, we think about these causal connections between this event that is the
interaction vertex, so to speak, and this other event, that's another interaction vertex. And
what seems to be happening is that as it's going on this thing, which was just the free propagator,
so to speak, that you can think about it as getting continually kicked. And the amount of
kicking is the mass of the particle. So, what seems to happen is that,
from our picture, it's just this causal edge, but the causal edge that is representing this
event happened and then this other event is going to happen. The particle, insofar as it is sort of
interacting with the background structure of space, in our models, it can't just go from
here to there. It's going through this. It has to interact with this network. And so,
in a sense, we're saying the amount of... Again, this is all kind of vague at this
point, but the amount of interaction of the network is the thing that tells you the mass.
And so, from this point of view, is it possible that you imagine mass emerging?
Yeah. I mean, if you had the right rules, the mass
of the electron might be something extractable as opposed to what we normally do? In quantum field
theories, we just put it in from the outside. Yeah, right. It is strange. In this picture
what's happening is that mass is associated with the amount of interaction with the background
structure of space, which is a little bit like the Higgs mechanism, which is the standard
model idea for where mass comes from, where you say, "Well, there's this background Higgs field
everywhere and the mass is how much the particle is interacting with that background Higgs field."
But in that case, there's a parameter, the coupling constant, which takes the place of
the dial by which we can set the mass of the particle. It interacts with the Higgs field,
but that coupling constant determines how much of that interaction translates into mass. Would you
have a similar kind of free parameter that would play the role of lambda, the coupling constant?
Well, if we knew what particles were, the answer is no, we would not have a free parameter.
Right. In other words, if a
particle was this topological structure that has this or that form, then we should be able to say,
well, what is the amount of interaction that's happening with this background structure of space?
Right. So, then that would be a way of
us being able to derive the masses of particles, which would be pretty cool, because as you say,
that's been a thing we've never been able to do. Now, it's worth remembering that in our models...
I mean, we didn't talk about this much. But there's a lot of complicated issues about
observers and what thing you extract from the structure of what's going on. And it's worth
that the same is true even for an electron. You say the mass of the electron is 0.511 MEV or
whatever it is. But that is only for observers who have certain properties at zero energy.
You mean, their motion relative? When you say observers that have certain properties,
what do you mean by that in this context? Well, I mean, in traditional particle physics,
you can think about the mass of the electron as being the amount of inertia the electron
has when you kick it. But the question is how hard did you kick it? And depending
on how hard you kick it, you will come to a different conclusion about what the mass is.
And so, if you are an observer, same thing. If you're looking at the thing on a certain
length scale, an observer looking at it on a very large length scale will conclude one mass, on a
different length scale will conclude a different mass. So, that's a way in which in traditional
particle physics, you have observer dependence to the so-called constants of nature, so to speak.
Right. But just so I fully understand, are you talking about the renormalization effects at the
scale depending on that parameter? Yeah, right.
Sure. I mean,
that's just an example. In traditional physics, we have more elaborate versions of the same thing.
Right, got it. But I think the hope would be that, again,
I'm sort of homing in on trying to understand the underpinnings of quantum field theory. And that
gives one a picture. And it's nice that one is starting to see how things like mass can show up.
Just to make a more general statement, because we didn't talk about this and maybe it's worth
mentioning, I mean, one of the questions is... We talked about how, if there's some underlying
rule for updating this hypergraph and so on, this and that happens. But then the question might be,
well, which rule do you pick? Yeah.
If you say, we've got the rule for the universe, it's like the next question is, well, why did the
universe get this particular rule and not another? And quantum mechanics is tip off about this
because in quantum mechanics we're saying there is a particular rule, but it's being applied in
all possible ways. So, the obvious limit to that is, well, why not just apply all possible rules?
And you might say, well, that's just going to make a big mess. But the thing you have to realize
is that it's the same thing as with quantum mechanics. There's branching when different
rules get applied but there can be merging when those different rules lead to the same
outcome. So, you have this very complicated structure, which I sort of think of as the
entangled limit of all possible computational processes. It's this thing I call the ruliad.
And so the thing that then one has to think about is, within the ruliad, there are observers
potentially like us who are made from the same stuff that the ruliad is made from, who are
looking out at the ruliad and saying, what's going on here? But because of their properties
as observers, they conclude certain things about what's going on. For example, if they were looking
at something that might be a bunch of molecules bouncing around because they are large-scale
observers who are computationally bounded and so on, they don't see all those molecules bouncing
around. They just say, "Looks like a fluid to me." And so the thing that seems to be the case is
that, to observers who are roughly like us in the sense that we're computationally bounded, believe
we're persistent in time, I believe that it is inevitable that those observers must conclude
that second law of thermodynamics, general relativity and quantum mechanics are the way the
universe works. If the observers were different from that and were not computationally bounded,
etc, etc, etc, they might conclude completely different things about how the universe works.
But the claim is that, to observers like us, it is inevitable that the laws of
physics have to be the way they are. So, it could be that some AI system in
the future analyzes observations of the world and comes to a radically different picture of
what the laws of physics are and what reality is? Yes, but there's a problem with that. We wouldn't
be able to communicate with that AI system. Right.
In other words, something which does brain-like things... Lots of systems in
nature are doing computationally complicated things, but not terribly brain-like things.
I mean, one feature of brains, which I've been increasingly realizing is, a lot of what we do,
we've got all this sensory apparatus. We've got millions of different bits of information that
are coming in every second. But a lot of our brain does is to try and concentrate that all down into,
well, what's the next action you should take? Maybe 10 times a second or something,
we are figuring out what muscle should we move, what next action should we take?
My very recent meta theory about this is, when animals first emerged, it became important for
the animal to know, where's it going to go next? And there's only one place for it to
go. It's not like the animal's going to split into 17 pieces. It's going to make a definite
decision. And so the idea of a brain, which would take a lot of input data and -
So the idea of a brain, which would take a lot of input data and conclude
one definite thing to do next, I claim as sort of something that emerged with the development,
with the emergence of animals and so on. And so we're stuck with this thing that we have just one
thread of experience. Within a single individual, the whole of society is a different story,
but within a single individual, we sort of believe that there's a single thread of experience,
and that seems to be a fundamental feature of brains, and that's also one of the pieces that's
necessary to see physics the way we see physics. I mean, I think it's a worthwhile sort of thought
experiment to imagine that our minds worked or brains worked a million times faster than they
do. So right now, I look around the room, the photons are coming to me in a microsecond. My
brain is taking milliseconds to process that. So as far as I'm concerned, it looks like there's a
state of space at a particular moment in time. But if my brain worked a million times faster
than it does, that would not be my impression. I would just be like, there's a photon from here,
there's a photon from there, the fact that you aggregate them all into the structure
of space is something that is particular to our scale and our kind of way of operating
Yeah, sort of a concrete version of that, which is maybe more analogical than literal,
but I have, at times, wasted time looking at a Necker cube and trying to see both orientations
of that cube simultaneously, and the brain, at least my brain, is simply unable to do that. I
don't know if anyone has had that capacity to be able to experience those two things. Again,
I think it's just some limitation of evolution by natural selection acting on this structure trying
to aid us in survival, and survival is a definite course of action with a definite perspective.
Yeah, no, that's an interesting example. But I think that that means when you say imagine an
AI system that could sort of think about the world very differently, the answer is it's
actually easy to have such a system. I mean, lots of these simple rules that I've studied,
they are doing computational things just like AI systems do computational things, but they don't
have a view of the world that is aligned with our view of the world. And so it's like, yes,
there's this thing, we can't talk to it and say, "Hey, let's compare our view and your view because
they're just really wildly different." Right. Right.
I think that the big claim is something that I absolutely did not see coming, which is the
claim that the laws of physics are the way they are because we are the way we are, and it is
inevitable that laws of physics will be that way. In other words, that when you say, "Why is general
relativity true?" It's like, I'm not sure. People might ask you, "Why is general relativity true?"
And I think when I was thinking about that stuff years ago, I was like, "I don't know why it's
true. It's that's what we've discovered is the way the universe works." And the big surprise here is
that there's now, I think, we have the beginnings of an answer to why. And it is a contingent fact
that we are the way we are. We might not be the way we are, but we know we are the way we
are. And given that fact, we can then deduce that physics must seem to us the way it seems to us.
And you're saying that when you say the way we are, it is simply bounded computationally and
that we have a continuous experience? Again, I don't grasp all the details.
Those seem to be the most important. But you're saying those two criterion
are enough, you think, in your structure, to single out general relativity and quantum
mechanics and thermodynamics of the law as well? Yes, yes. But what I'd really like to figure out
is I can't derive the three for the number of dimensions of space that we perceive.
Well, sometimes I like that number three to be 10 or 11. So if you can get either
of those numbers, I'd be perhaps even happier. Right. No, I think my guess is that that three
is a consequence of some feature of the way that we are that is at some level totally obvious,
but I just don't know what it is. So the fact that we parse the universe by saying space seems to be
three dimensional, it doesn't have to be that way. Sure.
We could be saying, "We are thinking about what's happening in space and we're just thinking about
it according to this on this line. We're going through a space filling curve. We're just
examining these different parts of space in that order." But no, we choose to gulp in space,
so to speak, as three dimensions. And there are things that I realized. One thing I realized
recently is that it's very important to our perception of physics that we are not massless.
If we were photons, time doesn't really progress. I always tell my students when they ask me that,
"Does time stop from the perspective of a photon? Is space infinitely Lorentz
compressed from the perspective of a...?" I say, "Yes poetically, but we can't travel and
no conscious being can travel at light speed, so take that with a grain of salt." For sure. Yeah.
Right, right. I mean, so it's important to us. It's another contingent fact that we are massive
objects, so to speak. Yeah.
And here's another contingent factor. Here's another thing that's interesting to perhaps...
We believe in doing science that we can choose the experiments we do. It's not self-evident
that that will be possible. In a deterministic universe, it could be that every experiment we do,
we are fated to do, so to speak. But we somehow have this belief that we can do these arbitrary
experiments. We also have beliefs like the belief that we can do a controlled experiment. We can do
an experiment where the thing that happens here isn't affected by the things that are happening
everywhere else in the universe. And I think one of the things that sometimes is confusing
to people about fields other than things like physics, physics has this feature that you can
do things like controlled experiments. It is the case that the simple laws tend to
be the right laws and so on. The simple laws being the right laws isn't true in biology,
for example. In biology, the typical experience is the most complicated possible explanation
is sort of the right explanation. I mean, there's always another effect.
I think I have some understanding, actually, of that now, and it's related to the whole
computational reducibility story. But there are other places like when you think about something
like ethics. I think that's a place where this idea of being able to separate the thing that's
happening in this particular situation from everything else that happens just doesn't work,
in other words. And so some of these intuitions that we have about the physical world and things
we believe about the physical world are, it's not obvious that they will be true,
but they are features of the way that we deal with the physical world. And I'm kind of suspecting
that somewhere in there is an assumption that we make that is totally obvious to us, but that
inevitably leads us through some potentially quite circuitous route to the fact that space has to be
three dimensional as we perceive it, for example. I mean there are other features like we believe in
discrete objects. It's not obvious that discrete objects would be a thing. If we were fluid
organisms, they wouldn't necessarily be a thing. We wouldn't even have the conception, presumably.
I mean, another thing we believe in is motion. We believe that pure motion is possible. In other
words, you can take a thing and move it somewhere else and it'll still be the same thing. That's
not obviously true. I mean, in other words, if space were dotted with spacetime singularities,
it wouldn't even be true in general relativity. But the fact that... For example, one can think
of particles as being essentially carriers of pure motion. Particles are things which have the
feature that the electron just moves and stays as an electron. Curiously, black holes are the same
kind of thing. You can take a black hole and move it around and it'll still be that same black hole.
And maybe there's some... My guess is there is a close relationship between black holes
and particles like electrons and so on because among other things in our models,
it's all just features of the structure of space. I can't help but just jumping in. I don't know if
you're familiar with this paper that I wrote with Andy Strominger and Dave Morrison some years ago,
but we showed, at least in certain string theoretic models, which, of course, you can claim
we don't know that those are the right models of the world, but we had a parameter that we could
continuously vary in which a black hole mass got smaller and smaller and then it transmuted into an
elementary particle. So a direct smooth connection between black holes and particles does exist...
In string theory. In string theory. Yeah.
That's cool. Yeah. Yeah.
I got to look at that paper. That's cool. No, it doesn't surprise me, but it's great. Anyway,
so I mean, the thing for me, this whole idea that one might be able to derive physics is just
completely... For me, I'd never imagined that something like that will be possible. I mean,
sort of a clue... The second law of thermodynamics is kind of a clue because it is something which
for 150 years people have kind of thought might be derivable, but I don't think anybody thought
that about general relativity, for example. I don't think that... But just to fill in,
and before we have to wrap up here, but just to fill in, I've been having a lot of fun recently
because a bunch of these ideas seem to be applicable to a lot of different fields, and so
it's been interesting to see their application, it's been interesting to see sort of the common
themes in these different areas. So one that I looked at a little while ago is meta-mathematics,
the space of all possible mathematical theorems. And there seems to be a very sort of close analogy
between how theorems get derived from other theorems and how time progresses through the
physical universe. And so for example, one of the meta results is the idea that higher mathematics
is possible. So for example, you might say, well, we've got these very axiomatic details
of mathematics and we can, based on our definition of real numbers and this and that and the other,
with a lot of effort, we can get to the point of defining what a triangle is,
for example. And then with even more effort, we can get the Pythagorean theorem. But the fact is,
most of the time when you think about the Pythagorean theorem, you're not having to think
down to that level of those microscopic axioms. And it's the same kind of thing, I think, as in
doing fluid mechanics. A lot of the time you can just work in terms of fluid mechanics. You don't
have to dive down to the molecular dynamics level. So I think that there's sort of a nontrivial fact
about mathematics, that higher level mathematics, not going down to the molecular dynamics level, is
possible. And I claim that that's a thing that's very similar to the fact that we believe in things
like continuum space. And I think it is another feature of being observers like us with certain
computational boundedness characteristics and so on that we can sort of think about mathematics at
a higher level. And then if you really want to go sort of funky in the mathematics direction,
my claim is that black holes in physical space are like decidable theories in meta-mathematics.
So I got to unpack that one. That one is... So one feature of black holes is,
at least in the simplest case, is that they're a place where time stops. So that means that any
path in space, you're going through this path, and it always comes to an end if you run into
a black hole. So in mathematics, those paths are things like proofs. From this theorem,
we derive another theorem, we derive another theorem, and so on and so on and so on.
Gödel's theorem showed that there are mathematical theories, and it's in fact the generic case, where
there are proofs of arbitrarily great length. There are essentially infinite paths. So when
you have an undecidable theory, you have that no upper bound on the length of proofs. They can just
wander around through proof space for arbitrarily long. But if you have a decidable theory,
a theory where every statement that you might make can be decided by a proof of bounded length,
then in effect, that isn't true anymore. So the paths always have to be finite. So it's
as if time stops. So what you end up with is what in a black hole is this physical path stopping,
in meta-mathematics is the proof stops. So QED is like a singularity.
Yes. QED in the sense of at the end of the proof.
Right. And so then it gets even funkier because in general relativity, we know
that there are singularity theorems that tell one, when there's enough energy and momentum,
it's inevitable you'll form a singularity. Okay, so the analogous thing, not completely worked out,
but the picture in meta-mathematics is when there's a high enough proof density, you
will inevitably form a decidable theory. So when there are enough things that you can prove, you'll
inevitably be able to prove everything in finite time. And so then it gets even funkier. If you
say, "What's the future of the physical universe?" Well, there are lots of possibilities, but one
thing is that you form a bunch of black holes. When you have quantum effects, you get more than
that, but roughly you form a bunch of black holes. And so then the picture is the future of physics
is like the future of mathematics because what would happen in mathematics is there
inevitably would form these singularities that are decidable theories. So just as the future
of the physical universe might be a bunch of black holes, the future of the meta-mathematical
universe might be the mathematical analog of black holes, namely decidable theories.
But I can't help but think, we also know that you wait long enough and those black holes
will Hawking evaporate and you'll just not have black holes and just particles wafting through
space. What's the analog on the proof side? Yeah, that's a good question. That's a good
question. Let me think. I think that the... I don't know. I'll give you another example
of a thing that you can see on the two sides. In physical space, one of the notable facts
is that space is somewhat homogeneous and things like motion is possible. Space is sort of the same
here and there, you can move things around, they're sort of the same. In mathematics,
in meta-mathematics, different places in meta-mathematical space are different kinds
of mathematical theories, like an algebraic way to represent things or a geometrical way
to represent things or a category theory way to represent things or something. And one of the
things that's been found in modern mathematics is these remarkable dualities between different areas
of mathematics where you can rewrite them in a different sort mathematical language.
So the claim would be that the homogeneity of physical space is analogous to a homogeneity
in meta-mathematical space, and that the possibility of motion in physical space is like
the possibility of making these dualities, these translations between different mathematical areas.
So that's an example of another correspondence. The question about quantum mechanics is an
interesting one in meta-mathematics. Let me give you a little bit of an indication of
something. I don't know really how this works. So in mathematics, you're often asking the question,
can I get a proof of this thing? So proof is a path from one set of theorems to another set of
theorems. It's a way of going from this theorem, prove the next one, prove the next one, so on,
get to the one you want. Okay, so people say in mathematics, "I want to see a proof of that
thing." So it's saying, "I want to find a path." But actually there may be many paths. There may
be many proofs. There will be, in general, many proofs of the same thing. So in a sense,
the analog of quantum mechanics is that there are many proofs of the same thing. So then the
question would be, what can you say about the space of possible proofs of the same thing?
And that's a thing which people have sort of slightly started studying in homotopy type
theory and so on, or they didn't really get very far. The topology of proof space is something
rather interesting and rather unstudied. There is an analogy to that in studying games. For example,
let's say tic-tac-toe. You've got the state of the board for tic-tac-toe, you make a move,
you turn it into another state of the board, you can make this game graph of all the possible moves
you can make in tic-tac-toe. And you can ask, what is the structure of that graph? And you can ask,
does that graph, for example, have sort of continuous deformation from one path, one
possible sequence of moves in the game to another sequence of moves? So for example, one case I know
is the Towers of Hanoi puzzle where you're putting these disks, stacking these disks up. If you make
the game graph of the Towers of Hanoi puzzle, it is a fractal Sierpinski kind of structure.
So it has a great big hole in the middle. So in other words, you make a commitment. You're
either doing it this way or you're doing it that way. And that's analogous to... You can think of
that like a proof space. And so I think my initial take would be that the story of quantum mechanics
in meta-mathematical space is related to a story of thinking about the multiple possible proofs
that you can make of a particular statement. And we don't have a good intuition about that.
I mean, I think that's just not a thing because in mathematics it's just like, "I just want a proof.
Give me any proof." So this whole thing about... And I think it's also related to our fundamental
single thread of experience. We're very bad at thinking about these kind of multiple paths of
things happening. It's a thing that's afflicted, sort of distributed computing where you have many
computers working on some problem and it's like, well, they could all be doing it, and they're all
sort doing it in parallel, and we don't know how they sort of merge with each other. And that's
a thing, we just find it hard to think about. Sure. One of the quick things, Stephen, before
we wrap up, it's so fascinating, but we spoke about black holes, the center point being like
the end of time, the Big Bang we can also perhaps think of as the beginning of time. Your approach,
does it say anything about the beginning? Does it give us a way of thinking about it?
Yeah, yeah. No, so I think one of the things that is... The general picture would be the
beginning of the universe, it's probably infinite dimensional and it probably, you have some little
tiny network and it's effectively, everything is connected to everything else. And gradually as the
thing expands, you can kind of start thinking that things are less connected, and so the thing is
sort of cooling down to being, well, we don't know why it's three, but a finite dimensional universe.
Now, what implication does that have? Well, it means you don't have to worry about inflation,
all these other kinds of things, because you know why... The fact that if the universe was infinite
dimensional initially, everything is kind of causally connected to everything else. You
don't have to explain why one part of the universe is similar to another part of the universe even
though it hadn't had a chance to see that other part of the universe during the actual evolution
of the universe. So for me, this is... I see. So the horizon problem you say
you solve from the get go because causal connections are almost part of the initial
state in that way of thinking about it. Yes. That would be the idea. I mean,
I think that it's ironic because I... You may know I wrote this paper with Rocky Cole back in,
when was it? 1978, 1979, where we were looking at the expansion rate of the early universe, and we
realized that there could be an exponential phase of the expansion of the universe, and we realized
that that would have all these consequences... Wait, so you were at inflation before inflation?
Indeed. I didn't know that.
Yes. Yeah. No, but okay, this is one of these stories about doing science that's
sort of interesting, which was, so I worked out in order to get any significant effect that one
would care about, the universe would have to super cool by 10 orders of magnitude. And I was
looking at nucleation in liquids and all this kind of thing, and I'm like, this is not going
to happen. The universe is not homogeneous enough for that to be a thing. So it's like we relegated
to a footnote this statement about what happens if you get this period of exponential expansion,
et cetera, et cetera, et cetera. And what sort of, to me, interesting about that is, it was
sort of the fact that when people really trotted out inflation, what was happening there was people
said, the result is so worthwhile that even though it requires a slightly implausible assumption,
it's worth making the implausible assumption because the payoff is so big. And I think for me,
that was an interesting realization about the meta structure of science that that's
a reasonable thing to do, to say... But anyway. There's one thing on that that's interesting to
say because Andre Linde also found inflation in the mid-seventies, but couldn't solve the
so-called graceful exit problem and didn't think that the payoff was large enough,
worthwhile enough to write a paper on it, and he basically put it in his desk draw and only
revisited after Alan Guth wrote his paper, and Linde was like, "Oh my God, it's out there in the
world and people are taking it seriously." Now he went back at it with a vengeance.
Right, right, right. No, I remember Linde's work, I think, from... I mean he worked on related
things actually around the same time. So yes, it was one of those things where the most significant
thing was deciding that it was as significant as it was and taking it that seriously, so to speak.
Exactly. Now, it may turn out that it's all nonsense
anyway. That would make me feel better about my footnotes, so to speak. But I think it's still
kind of a... But that's sort of the picture of the early universe, and I think one of the things that
I would love to be able to do is to work that out in enough detail to be able to see things like,
would I expect that dimension fluctuations would survive past the time of recombination where the
universe became transparent? And would we be able to... Could we see kind of inhomogeneities in the
cosmic microwave background or something that could be associated with dimension fluctuations?
Just to add another little curve ball to the whole story. In general relativity, we usually think
of things in terms of, well, we've got three plus one dimensional space, and then we have
curvature of three plus one dimensional space, but actually the mathematics is rather similar,
at least for small gravitational fields, to us to saying we have zero curvature, but we have
deviations in the dimensions. In the dimensions. I see.
Because if you just think about the volume of... Actually, I can picture that. Sure,
of course. That makes sense. So that throws yet another curve ball
into the whole thing of... So I don't know. For example, one of the things I looked at a little
bit was dimension waves, which are the analog gravitational waves for dimension fluctuations.
Again, I don't know how they work, and it's... I mean, I would say that one of the issues is we
have to rebuild, basically we're sort of rewinding a bunch of mathematical science all the way back
to Euclid. So it's kind of a heavy lift to get from Euclid to where we are today with a different
set of assumptions. However, we have at least one really big advantage, which is we can simulate
stuff on computers, and that means that we can get an idea, which is, for me, the kind of meta
approach that I've taken. You simulate things on a computer, it does a lot of stuff you don't expect,
then you try and figure out what the big picture is, which is almost a philosophical activity.
You've got the computer experiments, you've got the philosophy, and somehow in the middle you
hope to get some sort of results about science. And as an added bonus, every so often you can
throw in some fancy mathematical physics to help you see how those things connect.
Yeah. One final counterfactual that I can't resist asking. So you mentioned the footnote,
the paper Rocky Cole and you wrote, just wondering, if that hadn't been just a footnote,
but it had really been a description of an inflation-like theory, and it took the physics
world by storm the way Alan Guth's paper did in 1980, just a couple years later, do you imagine
your career would've gone in a radically different trajectory, or would it somehow have found its
way to pretty much what you do now? That's an interesting question. Look,
I was a pretty successful young particle physics operative. And it's kind of cool because I was
involved in a field that was sort of the hot field of its time when I was kind of a late teenager.
And so I had the... Wait,
you were doing particle physics as a teenager? Yeah, I wrote my first paper when I was 15.
15? Yeah.
So what age were you in graduate school? I got my PhD when I was 20.
So you finished at 20? Yeah.
Wow. So no, I wrote a bunch of stuff about
QCD and about cosmology when I was 17, 18 years old that have been, I'm happy to say, seem to have
been surviving papers and things, people still care about them. And that stuff with Rocky, I was
probably 18, 19 years old or something. I think I was a decently successful operative in that area.
And I think... I mean, I kind got interested in more and more foundational things. And also
particle physics, frankly, it kind of went into cruise mode, to a large extent. My last papers in
particle physics were like 1982 or something like this, which was... In another piece of the history
of this whole thing, I was a faculty person at Caltech when I was 21, 22 years old, whatever.
And the other thing that happened was I developed this big computer system, and in those days,
being a professor and doing things like starting companies, they were not compatible activities,
and so that ended up with me having this sort of big flap with Caltech, and so I quit.
I see, I see. Oh, I didn't know that. And had that not happened, I might still
be professoring, so to speak. I mean, I then went to the Institute for Advanced Study in Princeton,
which had the feature that its then director said, actually then chairman of its board said,
"We gave away the computer when Von Neumann kind of died, so all this intellectual property stuff,
we're just not worried about that." I see.
I don't know whether they've changed their point of view since.
Yeah, I'm sure they did. That was a convenient feature
at the time. But anyway. I mean, the fact is that certainly in terms of I might have stayed being
a physics professor type thing, I'm not sure that I would've been a perfect fit because I
think I'm a bit... I know that my life leading companies and things like that has the feature
that companies are a very efficient machine for turning ideas that you have into real things,
and I think academia, for me at least, was not as efficient, and I think I would've found
the path of least action or something somewhere along the way. I mean, also if you ask the things
I've done recently in fundamental physics, would I have gotten there if I'd still been doing physics,
done inflation and done more? The answer to that I think is absolutely definitively no.
Right, because you wouldn't have had the free... Yeah.
I mean, without question, because the fact is that as I think about the physics project,
which I think it seems to have gone really well, I say, what was necessary for that project to
happen? And the answer is a whole series of things like the fact that I used to do standard particle
physics and cosmology, so I know that stuff pretty well. That was a necessary feature. The fact that
I built a bunch of tools that let one explore the kinds of things that I wanted to explore,
and the fact that I got a bunch of intuition from looking at experiments that I looked at for quite
different reasons. And the whole series of these things that ended up being sort of,
you have to have that whole stack to get there. In other words, to me, I see the things we're
trying to do as being a different paradigmatic direction for things like physics, and it's
kind of like, to break out of the a hundred year tradition of the particular way of doing physics
that comes from mathematical physics and so on is, you kind of have to, A, you have to have
seen the outside, so to speak. I think another feature is you have to not care that much because
in other words, in some sense, the fact that our physics project has worked as well as it has is
really cool, I'm really excited about it, but it was not what I was expecting to be doing at this
point in my life, so to speak. It was just sort of a bonus feature that was possible because I'd
built a bunch of other things and then I kind of decided, what the heck, I might as well try doing
this because I'd been meaning to try for 30 years. A couple of young physicists sort of got me said,
"You've got to do this." So I'm like, "Okay, fine. Let's see whether this actually works." And then
it went a lot better than I expected it to go. Wow. Well, Stephen, it's always a pleasure
speaking with you, and I hope we can do it again at some point in the not too distant future. But
thanks so much for spending the time today. Yeah, happy to.
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