Harvard Black Hole Initiative: A Surprisingly Promising Approach to a Fundamental Theory of Physics

say a few words of introduction it is a great pleasure and privilege for us to host this stephen wolfram for a colloquium at Harvard University's black hole initiative an interdisciplinary center focused on the study of black holes that brings together astronomers physicists mathematicians and most importantly philosophers and we're the first image of a black hole was announced a year ago a few words about Steven Steven was born in London 1959 to German Jewish refugees as we just discussed with with him but but his heritage is close to my home where my grandfather left Germany around the same time he is known for his brilliant work in computer science mathematics and theoretical physics early on Steven was educated at Eton College where he was taught mathematics by Norman Rutledge a friend of Alan Turing he entered the st. John College in Oxford at age 17 but according to records he found the lectures awful and so he left in 1978 without graduating to attend the Cal Tech the following year where he received the PhD in particle physics at age 20 Stephens thesis committee included the Richard Feynman and Peter Gore drive whom we all know Steven published a widely cited paper and heavy quark production at age 18 and 9 other papers shortly afterwards and continued research and publications on particle physics into his early 20s following his ph.d he joined the faculty at Caltech and became the youngest recipient of the MacArthur Fellowship 1981 at age 21 1983 he left for the School of Sciences at the Institute for Advanced Study in Princeton where he conducted research into cellular automata mainly with computer simulations he produced a series of papers systematically investigating the class of elementary cellular automata in 1986 Steven left the hood for Advanced Study for the University of Illinois at urbana-champaign where he founded their Center for complex systems research and they started to work on the computer algebra system called Mathematica which was first released in 1988 when he left academia I 1987 he founded the world from research which continues to develop and market the program Mathematica in March 2009 he announced Wolfram Alpha an answer engine I first heard about the Steven when he went when I arrived this longtime postdoc member adeana stood for Advanced Study at Princeton in 1988 people refer to him as a wunderkind a young genius who decided later on to leave academia nevertheless Steven had a huge impact on academic life and I witnessed that on a daily basis when all of my students and postdocs use Mathematica without an exception but recently Steven came up with new intellectual insights that addressed not only the way we do science but also the content of what we do and he will tell us today about his exciting theory of everything Steven okay good well so I was realizing that I do not think I have given a something which can be described as a physics seminar for basically 40 years so this is I have many invitations actually right now to give these you guys happen to be the first one I'm giving so there'll be terrible things that happen here that that's that's how it goes so I want to tell you about something that as far as I'm concerned is is very surprising and it has to do with a very promising path to a fundamental theory of physics you know when I was doing physics back 40 years ago and more I just used existing theories and I did things in QCD and cosmology and all these kinds of things and I kind of took quantum field theory and general relativity just that's the way things are that's the lowest level infrastructure now I'm going to compute things based on those the the huge surprise now is I think we actually have some way of understanding why those things are the way they are and sort of a a complete theory of how physics might work and I have to say I'd been super surprised at how how well things are going and how easily things seem to be going so I want to I want to kind of tell you what we figured out so far it's certainly not finished but I think we're on a really interesting path so you know what I'm talking about builds on a whole bunch of ideas that are partly from physics partly from mathematics partly from computer science partly from things that I've discovered from kind of exploring the computational universe of possible programs and a bunch of kind of new things but it's a it's a fairly big stack many of these ideas are probably quite unfamiliar to most physics type people and it's actually one of the biggest surprises recently has been how good the contact appears to be given that the the stack of ideas is very unfamiliar how good the contact ends up being with a lot of modern familiar mathematical physics kinds of ideas and I'll try and talk about those things as I get on with this needless to say in an effort to get to sort of a fundamental theory physics things are pretty abstract but with it turns out one can get some level of visual and other representation of what's going on so let me let me give you the the kind of the high level sketch of the high level sketch and then we'll start going into more detail so the basic question an idea is can there be sort of a simple underlying rule that can generate all the physics that we know so let me let me just give you a little tour of this so this is kind of roughly this sort of represent one of the kinds of rules that we're interested in it's a rule that basically rewrites pieces of a hyper graph and the rule gets applied many times when you apply the rule many times you generate this large hyper graph that sort of in its continuum limit can behave like space like something that's a manifold has a manifold like structure so this is kind of the this is sort of where where space comes from then what one's interested in doing is looking at the sort of the relationships between the different updating events that are happening here and that produces this kind of causal graph over here and that sort of a representation of space-time and as I'll try and describe one of the really cool things that happens is that with certain assumptions and things I'll describe special relativity turns out to be something that you can then derive as a property of these systems and more than that you can derive first the vacuum Einstein equations and then the full Einstein equations with energy momentum tensor and so on from this which i think is is pretty amazing and cool so it turns out that the this this kind of picture of there's this big hyper graph that's getting rewritten in all these ways that's one of the things that happens is there's ambiguity in those possible rewritings and so while here I just showed a particular hyper graph coming out the truer picture is that there's a whole kind of graph whole first tree but eventually graph of of different forms of this hyper graph that can come out with different kinds of rewritings that can occur and it turns out that that those kinds of different rewritings that happen make quantum mechanics not just something that you sort of put in but something that's kind of inevitable in this set up and what turns out to happen so this picture in the middle is showing kind of the sort of a combination it's we call this multi way system that shows these kind of different quantum branches and shows these causal edges that connect different term different quantum states it then turns out and this is again really cool and not what I expected at all that the you can understand quantum mechanics as kind of you can understand space-time as coming from this causal graph and you can understand space as being these kind of slices through this causal graph you can understand basically quite the space of quantum states as being slices through this multi way causal goth and so there's this thing we call branch real space which is essentially a space of quantum states that represents kind of the and the the connections in here essentially represent entanglements between quantum states and it turns out and this is again one of the super cool things that I did not see coming at all it turns out that the path integral is essentially the path integral is the analog of the Einstein equations but not in ordinary space-time instead in this kind of branch time multi-way causal graph world so kind of the analog of what what we'll see is the analog of of generality in in this multi way causal graph is is the path integral so lots of lots of cool things that happen there so the I mean this is sort of the very high level ok that's good um the that's what I get for just stick you lating wildly um the UM let's see what the water um destroys my computer that will be the next the next next thing to happen here all right so anyway that's that's the that's the story here the so let's see let me maybe tell the backstory of how I got to this whole whole adventure so I mean back 40 years ago as I said I used to do particle physics and cosmology and so on having done that I got involved in using computers to do that better and that got me involved in the whole issue of kind of building computer languages too and sort of building artificial worlds and computers and then I got interested in the early night it is in the following question so I'd been been interested in various problems in science where typically systems behave in complex ways and I had kind of first thought was oh let's use standard physics and differential equations and all these kinds of things to analyze these these complicated systems that didn't work very well so I got to thinking you know what's a way that I can generalize this notion of modeling that's been so successful in physics this sort of have an abstract model of a system and see what's going on what's a way that I can generalize that notion of modeling to something beyond sort of the confines of the particular constructs of mathematics that we happen to have used so far and so that got me involved in the idea of kind of using the computational universe of possible programs as raw material for making models of things and so that led to these things called cellular automata which are we mentioned and see whether I am in a position to share more windows here and whether I am trapped I may be trapped inside them let's see what happens if I go if I create a new tab here let's see what I can successfully improvise here all right so the thing I got interested in is when you have a a simple program here's an example of a cellular automaton this is this is the program for the cellular automaton and then we're running the cellular automaton by just applying this rule over and over again in this page the surprising thing is even when the rule is very simple the behavior that you get can be complicated so for example this is probably that that's that same cellular automaton rule just run four more steps so the big surprise is even though the rule is simple the behavior can be complicated and that kind of gives one a slightly different intuition about how one might make models for things in science and that led me to a big adventure that eventually ended up with this book of new kind of science which is about this idea of using computation as the sort of raw material for understanding the world and for making models of things and I would say that so that book came out in 2002 and I would say that the the overall picture of what has happened there has been spectacularly successful I mean after 300 years of basically the dominance of mathematical equations as the way to do exact science in the last you know decade or two we basically seen this transition to programs being the raw material that gets used for lots of kinds of modeling of things and it's kind of remarkable how quickly and completely that transition has happened it's happened in a great many areas but one area it has not happened in is fundamental physics and indeed when my book came out there were lots of people who are excited about in lots of different fields I would say the response in fundamental physics was more a pitchfork oriented response well then in any other field so so the question then is can one use this idea that simple rules can produce complex behavior to do sort of the ultimate version of that and find out whether there's a simple rule for a whole universe and so I got kind of interested in that I got interested in that when I was working on my book and so here's kind of how I would think about that so the question is in in creating kind of in finding a simple rule that could produce our universe we kind of have to go to a very low level we can't expect that the things were familiar with in space and time and so on play directly in that rule we have to have a rule from which all of those things are going to emerge because there's no way we'll have a small simple rule if we're going to pack in you know the mass of the muon and things right into that rule so so this is some so kind of the thing you realize is you have to kind of go underneath things like space and time you have to have some more fundamental structure from which those things can emerge and so the the thing that I had I had kind of come up with a particular way of doing that using networks and so on thinking about some I'll describe in more detail in a moment I'm been anyway I had sort of created that structure and I wrote about 100 pages about it in in this new kind of science book and I even described things like the derivation of at least the vacuum Einstein equations and so on there um but then after that book came out I thought gosh shall I go on working on this kind of thing and my my pauling of physicists said nobody wants me to work on this it's like we're happening we're doing our string theory we're doing whatever it's like we don't really need a new new approach to physics that was in the early 2000s so I decided well gosh there are lots of interesting things to work on and they're things that I was interested in about sort of trying to build full-scale computational language which is what our or from language system has turned into building things like Wolfram Alpha so I went off and did those things and I had always been intending to say let me come back and look at physics again and try and see whether it you know I don't know how hard it is to figure out fundamental theory of physics let me that if it's easy let me try and do it but I hadn't gotten around to it and then about two years ago now and the other thing that bugged me was that the particular model that I had set up for these networks and so on there were things about it that were not to me aesthetically that satisfying there were things about it sort of a model where to even get started you have to kind of get a model with certain characteristics and I would have wanted something where I could have this sort of space of possible the sort of computational universe of possible universes that was a bit more free and open so it kind of bugged me and so a couple of years ago I was thinking about this and I realized you know I've been working on computational language design for a long time which is about kind of how you make an abstract representation of things like computation and I realized actually I was kind of embarrassed because I realized that a fundamental idea that has to do with the ways Wolfram language Mathematica my predecessor of those things called SMP work they all work the same way and they all work using essentially transformational rules for symbolic expressions and I realized gosh the core idea that's underneath that I can use for this model of for thinking about physics and so I came up with this kind of very minimal very structure less way of thinking about the creation of structure and you can talk about it very abstractly I'll show how it works it's just a thing that involves elements and relations and then you build everything from that and it starts very abstractly and sort of the goal is to connect that to what we know in physics and other places so just in terms of the the history of this of this whole project this was some every year for the last 17 18 years now we've been having the summer school about science and technology and so on and every year a few people would say oh we want to work on the stuff that's in chapter 9 a new kind of science which is about physics and I would sort of you know do some projects and things but but a few years ago two young physicists max Baskin off and Jonathan Gerard at the summer school and they said you know this visit stuff you're doing is really interesting you really should actually do something with it and we'll help you do it so so last fall we got started trying to actually see whether whether we could make progress with with the basic ideas that I had had a long time ago and it turned out to be a lot easier than we expected and we figured out a lot lot more than I expected I expected that we would be kind of dealing with um sort of maybe working out what might happen in the first 10 to the minus 100 seconds of the universe and not really knowing whether we got the right answer and so on let me explain another point the one of the phenomena that one learns by studying the computational universe is something that I called computational irreducibility and it's it's sort of it I think it's a pretty fundamental science idea that that creeps into lots of different places and people are slowly starting to understand it but it's basically this you know once used to in science the idea that you know you have a model for something that allows one to make predictions it's all good but when you have a system like like the one I'm showing here there's a question of even though you know the underlying rule it may be it may be difficult to predict what it's going to do so for the system itself it just follows the rule applying it you know many many times and getting to the results here but the question is can we jump ahead can we reduce the computational work that's needed to figure out the answer and the answer is that there are many of these systems where that is not possible where you are stuck with sort of computational irreducibility it's kind of a wall it's closely related undecidability and girdles theorem and things like that but it's this kind of wall of we're sort of jump ahead predictive science doesn't work and I thought that we would be completely mired in this from the very beginning in our efforts to study fundamental physics the thing that I didn't see coming is that it turns out there's a kind of layer of reducibility on top of a sort of underlying computational irreducibility in these models and that layer of reducibility turns out to map almost perfectly to what we already know in physics and that's so let me let me start talking about that collections of relations between elements example here these are the elements just integers this is this is just in this particular case you can represent that collection of relations between elements as a graph so now the you know what happens to this graph well what we say is we're going to have a rule that says for any elements x y and z any time we see a pair of relations XY and relations XZ we transform them in this way so we can represent that as a kind of a rewrite rule for that graph so this is saying we're just putting on labels here for the labels don't really mean anything it just like dummy variables we're just saying whenever we see a graph that looks like this transform it like this okay very simple very trivial okay so what does this actually do well let's say we start off with with this graph here and we just apply that rule wherever we can a bunch of times this is what we'll end up getting so even though the rule was very simple we started off from just a very simple graph we quickly get this not so simple graph um so what will be saying is that you know in our actual universe we might have from that a rule like that applied oh I don't know let's say 10 to the 400 times and the result is a very big graph now I want to say that in fact it is convenient to think not just about graphs but about hyper graphs so up here I just had binary relations but I can have any relations here the NRE relations are ordered NRI relations and we'll just have a rewrite rule that operates on on ordered enery relations um and we get then hypergraph rather than graph as the results here okay so first question is all right so I should say this is just talking about the fact that there are many possible rules so if you if you try different rules you'll get different behaviors so you can get all kinds of funky things happening by just changing though that very simple underlying rule ok so the first kind of first big question then is okay we're just we're generating these graphs why is that anything like space because we're thinking of these these graphs as being hyper graphs as just having points in them we're not saying where those points are in any kind of coordinate system or anything like that we're just saying there are these points and all we know is the connectivity of the how those points are related to other points essentially the connectivity in the hyper graph between those points okay so here's an example of a particular rule it's this rule here and this is what it nets it knits this thing which you can kind of see here is is kind of something which which looks like a simple two-dimensional mesh other cases bit more complicated they knit more complicated more elaborate things um oh here's an example of a rule where the thing that it creates we can kind of recognize as being something like a you know we might say that looks like a three-dimensional object um okay and there's that same thing rendered as a 3d object but remember the only thing that's really defined in these rules is the relations between these underlying elements there's no the fact that we can draw this in three dimensions and it's a nice geometrical shape it's just a matter of our rendering all that's really defined is these underlying connections okay so the next question is well how can we recognize when we have some some weird thing like this how can we recognize is that like n dimensional space okay well there's actually a very simple criterion for understanding that so imagine we're in this graph or a hyper graph we start at a point in the hyper graph and we just build this expanding ball around that point so we're just going to two nodes that are distance one graph distance one away graph distance two and so on we're building a geodesic ball around that point in this in this hyper graph and so what is the volume of that GDC ball in other words how many points does that Judy's hit ball have after you've gone our steps in the hyper graph the answer is the leading term if if the thing corresponding to D dimensional space would be R to the D so so there's kind of a test for how what what dimensional is this space so okay has an example so this is showing the the number of the the growth rate of the number of nodes that you reach after going a certain distance and these different curves corresponds the different steps in the evolution of the system and we see them in this particular case this thing is essentially converging this particular rule is converging to something whose effective dimension of its hyper graph is about two point seven so so this is a matter of fairly typical things so for example I think I have an example here if in the case just to check that we know what we're talking about if we do the same test on this some on this shape here we find out that this will show up as being dimension two in this test okay so oh and there you know we do the same thing for some kind of recognizable fractal thing it'll have its standard house stuff dimension and so on okay so the so that's kind of our test for you know we grow this very big hyper graph and we ask what is its effective dimension by looking at the growth of these TV supports okay so we can go a little bit further than that we can look at the growth of geodesic balls and we can say can we say more about this effective space that we've got now I should explain that if you run a million different rules many of them will produce things that do not limit to manifold like d dimensional space they produce much more exotic kinds of things there might be exponential dimensional they might be zero dimensional they might have a lot of other exotic kinds of things I might show a few things actually I've just figured out in the last few days about black holes and singularities in space-time from this I might have a chance to show that later and you can see some of the really exotic things that can happen but in you know in in in certain cases at least you get a limit which is kind of a something which when you take the limit and you say let me run this thing for a very very long time and then let me look it's sort of look at a medium scale between the scale of the of the individual edges in the hyper graph and the scale of the whole system and we'll have something which looks like a looks locally like you could in space looks like a manifold okay so so next question would be well what more can we say about this manifold well we can we can say things about its curvature so imagine we're looking at the the growth rate of the volume of this GDC all the full formula which I might or may not given here let me pull it up from somewhere I'm the full formula for that involves R to the D and then there's a correction term I think I have it here um there's a correction term which is proportional to the Ricci scalar curvature okay so there's an us a little R squared correction term that's proportional to the scale of curvature and you can kind of see that here so this is this is my same sort of plot of the volume of the GD growth rate of the volume of the GDC all on this curved surface here and you see that instead of it being flat as a function of R it changes as a function of R and that's the sign of the curvature of the surface so that's how you kind of get to measure Ricci scalar curvature in this case so just to give some indication of how things go a little bit further than that we can also look at the Ricci tensor we can just we we find a JD sic in this in this hyper graph which is just a graph JD SiC and we look at sort of a tube around that graph JD SiC and we ask these same questions about growth rates for that and that gives us essentially the the Ricci tensor the projection of the Ricci tensor along the direction of that JDC um so okay so now let me explain another point before we get get further here so in these models the only thing that exists in these models is space so you say that's no good you know we've got matter of all kinds in space how does that work well the answer is that it works by having by the all the matter that in the universe has to be essentially just features of space and so what might those features be so I'll give you sort of a a toy example of that imagine that the background space was a planar graph and imagine that all the rewrite rules preserved finality but now put down that little lump of non-linearity that protoss Keys theorem tells you has a very definite form put down one of those lumps that lump you can never get rid of it it's but it's this sort of pseudo localized lump that can move around in this Network preserving its identity and so that's kind of I think how one should think about particles we'll talk a little bit more about that later we don't know yet the full story about particles I'll talk about what we do know later but that's kind of the way to think about matter you know you have this sort of background of space that is created by the surviving hypergraph and matter is certain features of that space so particles might be quotes topologically stable localized structures in that space what we mean by topologically stable is not ordinary topology again we can talk about what we know about that and you know to just give a sense of the mathematics of this what we would really like to be able to do is to understand things like curvature and fractional dimensional space we'd like to understand what's a good model space to use to understand the limits of these hyper graphs because you know you put in space in D dimensions where D is an integer isn't going to cut it and so there are they're sort of a bunch of things about that and we figured out a certain amount about that but I would say that we've in a rather typical physicists way we've kind of hacked through the mathematics of this there's beautiful mathematics to be done but we haven't done it okay so that's a little bit about about space let's talk now about about time and one feature of these models that is kind of a departure from what has been the tradition in physics for the last probably a hundred years or so is space and time aren't fundamentally the same kind of thing in these models you might say oh my gosh you're never going to get relativistic invariants don't worry one will get relativistic invariants um but the the conceptual feature here is time in these models is kind of the inexorable progression of computational processes and space it's just a feature of this hyper graph and as we'll see there are other kinds space I mentioned this bronchial space that represents a space of quantum states there are other kinds of space that show up here but physical space is just essentially the extension of this hypergraph okay so let's talk about what time is and how time works so as I say we're just dealing with back to that same rewrite rule we're just dealing with applying this rewrite rule and sort of every time we apply this rewrite rule that's kind of like one moment of time has gone by so so here we're we're applying that rewrite rule and so on but here's a key point there the where we apply this rewrite rule is ambiguous we don't know there are maybe many possible places where the rewrite rule could apply so you say well is this a deterministic model well yes it is but it's deterministic in the sense that we imagine applying the rewrite rule in all possible places where it might apply we can get a particular history by choosing a particular strategy for where to apply this rewrite rule by the way for people interesting computational kinds of things this is basically an evaluation strategy this is so you can think about you know depth-first breadth-first this is a possible strategy for essentially evaluating expressions corresponds to the the the strategy that you use for applying these rewrite rules so and I also might mention by the way that these models that we've constructed these sort of elements and relations underlying models one thing that's interesting about them because they're so minimal and structuralist it's kind of an inevitable feature that they don't just apply to physics and one of the really interesting applications is to distributed computing and maybe I'll have a chance to talk about very deep connections that seem to exist between sort of core problems of distributed computing and core problems of physics and there's there's actually more to say about where these models can be applied but for these purposes were just noticing that the model is ambiguous in its application and there are many possible places where this rewrite could be applied so let's make a graph we call it a multi-way graph that shows all the possible things that can happen so this is this is that multi weight loss so you start from this initial condition and then there are two possible rewrites that could be applied then then there more and more and more and more okay so what one might say is oh this is like kind of a many-worlds quantum mechanics type thing but there's a key observation here which is not only do these things branch they also merge that is it could be the case that there's a sequence of rewrites that two completely different sequences of rewrites that end you up at the exact same state and that phenomena is really important it's okay so that's just another representation of the multi way graph showing the sort of the Treeing out in some sense of possible states okay now we have to talk about that phenomenon that i talked about about the merging of states and this is going to connect with sort of an old idea of mathematical logic sometimes called the treach'rous or property or confluence that we have a slight generalization that we call causal invariants but let me explain what it is so so here I'm going to use not a not a hyper graph as the raw material that just strings and these are just strings of characters so we're just saying there's a rewrite rule that says a gets turned into bbbbb gets turned into a so now we start off with just an a we apply the rule we get BBB now with BBB there are two possible outcomes we can get either a B or B a okay so we're building up one of these multi-layer graphs so I have to tell you one thing that I think is kind of amusing is you might ask the question you might wonder what is the continuum limit of these string rewriting systems they're just a sort of toy model for what we're doing but what is the continuum limit of these I don't know for sure but I have a suspicion that the continuum limit of string rewriting systems is string theory and I leave that to somebody else to to figure out for sure but that that's a kind of a pun turns into physics type phenomenon but let's come back to these multi-way graphs so okay so we have this multi-way graph it branches it merges and so on okay there is a let's let's look at let's look at what's going on a little bit more detail so the blue picture the blue things here are states they're strings the yellow things here are events they're updating events that this is every yellow event says this is an update that we're going to do on the string so we're here we're we're illustrating both the the states and the updates that that transition from one state to another okay okay so next critical question is what is the causal relationship between those updating events in other words a particular updating event will have certain output another updating event might need to use that output as its input or it might not it might be operating on different parts of the string and might not care whether whether that previous update has happened or it might care and so what we can do is represent we can make a graph a causal graph that shows the causal relationships between updating events okay so it gets a little bit more complicated here so the orange lines here show the causal relationships they show which updating event is causally connected to which other updating event okay so that that's this this thing so the underlying thing down that the blue thing makes a multi-way what we call multi-way graph this this orange thing is making what we're calling the multi way causal graph okay so um let's see the okay we're just showing what what what that multi way causal graph looks like okay key fact the there exist certain rewrite rules which have the property that the network of causal relationships between events do not depend on the particular branch of the multi way system that one is choosing in other words so let me give you an example I think I have an example here yeah here we go okay so here's an example where of a rewrite rule that's just sorting a string so what it's doing is it's saying whenever it sees be a turn it into a B so we all kind of know that in the end after we've turned ba into a B enough times we're going to get a sorted string at the act in other words D but but there are many possible ways that we could do that sorting there are many possible places where we could apply those rewrite rules and this multi way graph represents all the possible choices for how we can make apply those things um but the it's some um so gosh we really only have ten minutes I'm not even close to well whatever okay so so in any case the the so these some so what we're seeing here is that even though there are many possible paths we could follow we're always going to get the same answer and what kind of familiar with this from things like I don't know elementary algebra with polynomials it doesn't matter which thing you expand or whatever first you'll always get the same answer so this property of Koslow invariance is an important property it's not true of all underlying rewrite systems but it's true some of them and it's true of some hypergraph systems as well okay what does cause one Barents me cause alone variance means that in this big mess of the multi-way causal graph every possible branch of this multi a causal graph basically decomposes into a collection of identity identical individual causal graphs that correspond to essentially different branches of history that we're essentially the causal relationship between events is independent of which which particular sequence of updates you did in which order so we have sort of an invariant causal graph that comes out independent of the updating order ok so what well so let me show you an example of how this works so this is kind of at the underlying level this is showing us strings of A's and B's and these yellow pieces are a particular choice for where we do those updates this is the causal graph that represents the causal relationships between those updates but what we see is there are these are different possible choices of which which updates you do when but the causal graph is always isomorphic ok so what does this have to do with physics well imagine that we have our hyper graph and imagine then that we construct the causal graph that represents the relationship between updating events in our hyper graph now imagine that we are an observer in so another thing I should say and I'm mum is that sort of an important feature of trying to model the whole universe is that the observer has to be themselves part of this universe and has to be governed by the same rules that the rest of the universe is governed by but let me I'm going to try and speed up a little bit here but but let me um so this is a very idealized version of a causal graph the real causal graphs I might have an example of one mmm one yeah that's what a a more real causal graph the very beginning of a more real causal graph for a hyper graph system would look like but just as a toy example let's um let's go up here and we say this is the cause of graph so every event here is affecting two other events and we can conveniently draw our causal graph like this okay well now let's say we're an observer trying to understand what's going on in the universe well we might want to coordinate eyes this this causal graph and to do that we would put down some kind of we would to try and foliate this causal graph we put down some kind of reference frame so let's say we put down a reference frame like this this is a reference frame we're allowed to put down what does it mean we're allowed to put it down it means this reference frame is compatible with the partial order that's defined by this causal graph the causal graph defines a partial ordering of events and what we're saying here is the equivalence class of events that live in this slice of the foliation are all ones that can be in the same slice of affiliation there are no timelike sequential events in there they can all be viewed a space like separated events okay so that's one possible foliation we can imagine looking at other foliation x' in particular let's imagine that we just sort of we're an observer and instead of kind of going straight down here we we go at some angle we then say well what how would we draw the foliation if we were to be at if we were to sort of assume that that we have these lines of simultaneity and needless to say you get the exact thing that you would get in the standard derivation of special relativity um but what's happening so why can you do this why can you make this change to the to the reference frames and expect that you're going to get the same answer the reason is because of causal invariants so let me show you an example of that so here's an example of our same string rewrite system with a particular choice a particular choice of frame we could think of this as a kind of cosmological rest frame in which as many events as possible are happening at the same time but now let's boost the frame by essentially being the we're going to tip the frame in the same way that we took the frame here we can tip the frame in the string rewrite system and now the events won't be happening on these kind of cosmological rest frame lines they'll establish a penang on these on these skew lines and we can ask you know do we still get the same causal graph and the answer is yes because of causal invariants so this is we're just making a different choice of which updating events happen but we still get the same causal graph essentially the same physics now you might notice that we've essentially got time dilation in this picture because what's happening is because we're looking at this in this different reference frame the sort of the number of steps it takes to get the answer in the end is increased by exactly the standard time dilation factor okay now things in reality aren't quite as simple as that because the actual causal graphs that we're dealing with are considerably more complicated and there are lots of issues about showing that that these limits really work and so on but let me completely gloss over those let me move to another question so so we've got some idea now that special relativity is going to work now another thing that's an important piece of the sort of picture of relativity is energy and momentum um and by the way notice that at the beginning space and time are completely different but yet they ended up through this causal graph being the same kind of thing and having the usual kinds of relations okay so let's talk about energy momentum so here's a big surprise to me I thought that to get an idea of energy we'd have to start talking about particles and counting particles and things like that but actually there's a bulk view of energy in this in these models and the bulk view of energy is simply the flux of causal edges through space like hyper surfaces so remember in this in this picture what we're thinking of these slices are basically spaced like hyper surfaces and we can ask the question if we look at these causal edges and we look at them slicing through the space like hyper surfaces what is the density with which they slice through the space like hyper surfaces now it's a slightly tricky thing to talk about because our space is made up from precisely the same kind of thing so the notion of density is slightly tricky to understand but basically we can say how many how many causal edges slice through these space like hyper surfaces okay it turns out we can i den by that as the bulk energy of the system if we look at the time like hyper surfaces we can identify the momentum as being the the flux of causal edges through time like hyper surfaces then slightly surprisingly we get exactly the usual relativistic transformation of you know relativistic momentum that comes out it is the same transformation as for space and time but it's not trivially obvious that it would be because it is something that has to do with transforming this that this this flux in the causal graph as opposed to just looking at essentially coordinate izing this causal graph so that's how energy momentum comes out crowd-pleasing Li you can indeed derive e equals mc-squared there's a notion of rest mass that has to do with essentially the when you're looking well it's it's not too hard to see but but when you're looking at essentially what okay what what is the meaning of energy in this system the meaning is the flux of causal edges through space like hyper surfaces is kind of each each causal edge each each node in the causal graph represents an updating event and the causal edge is going through space like hyper surfaces that's kind of communicating activity in the network sort of through time so in other words in a sense energy is the there's the communication of activity in the network through time momentum is the communication of activity in the network through space in there in the end through a time like hyper surface and so mass ends up being kind of the propensity of activity to not be transmitted that rest mass to not be sort of spread to other parts of the network to be something that is that is localized in a particular region that ends up being the thing that corresponds to rest mass okay so that's a little bit of okay so now we can start constructing given that we now have an idea of what energy momentum are we can construct an energy momentum tensor and we can ask how does that relate to Einstein's equations so where I talked about these geodesic balls in the spatial hypergraph in the if we look at the the causal graph we can also create not GDC balls but now they're essentially GDC cones we start from a particular event we look at essentially the future light cone of that event it's a little tricky to measure volumes of these cones we're basically putting down a timelike vector and we're asking kind of you go down that time like vector a certain distance we cut off the bottom of the cone what is that volume so the answer is that volume is T the length of that time like vector to the D plus 1 if D is the number of space dimensions times something which is proportional to the space time Ricci curvature multiplied by those timelike vectors okay so so the actual so that's the that's the that's essentially the volume of these cones okay so now we ask the question so now we're interested in in what can we say about about from knowing but that's the growth rate of the volumes of these cones okay so the next statement is if we're going to have a space that is finite dimensional then it follows that essentially this R IJ this space time retreat okay first approximation let's let's ignore that the matter for a second assume we just have uniform sort of energy in space and then the first statement would be if we want this thing to limit to finite dimensional space then our minou has to be equal to zero then space time army nu has to be equal to zero and so what assumptions went into that well one important assumption is essentially an agar DISA T assumption so the derivation I do is being very much analogous to the direction of fluid mechanics from discrete molecular dynamics you're assuming in order to get continuum results you're assuming that you can sort of average over all of these statistical microscopic updates and so on in our world computational irreducibility pretty much guarantees that you will have that kind of microscopic randomness and that you'll be able to take these averages but the the full proof that you can do that well nobody's ever been able to do that for molecular dynamics going to fluid dynamics the full pure mathematical proof is certainly not something that we have there there are interesting pieces of it we certainly don't have that that pure mathematical proof but we can at a physicists level of mathematics we can show that if you're going to have finite dimensional space and you have causal invariants and there is this microscopic computational irreducibility microscopic randomness then it follows that the aggregate effect on this large scale limit must correspond to satisfying the Einstein equations so we can look at all kinds of things in this in the space we can look at GD 6 for example GD 6 are very easy to define and in these in these causal graphs and we can look at um you know all the standard equations of generality in terms of Gd 6 and so on and they all work very nicely um and you can also look at okay so how does energy momentum come in what you have to think about in terms of energy momentum is kind of the difference between the energy momentum associated with what we consider to be matter and the sort of background energy momentum that represents space so so one weird feature of these models is that because they are making their own space so to speak the whole idea so one of the things that will come out is we talk about quantum mechanics is that so the the the the super virtual particles that we are used to seeing in quantum field theory in the vacuum those are those exist in these models but their role is to make space so that their role is to kind of knit together the structure of space and so when we're looking at in fact you know in a rather rickety estimate that I have about 10 to the minus 120 one of the activity of the universe is something other than creating space so one part in 10 to 100 and twenty one is everything that we know about matter the rest of it is just to do with the maintenance of space basically so um the let's see um so that's so that's a little bit on the Einstein equations I can talk a bit I have just done some more recent work on on black holes which I'd like to show you but but this is some the way that um basically in these causal graphs there are ways that you get horizons and they're rather easy to see horizons in these causal graphs this is kind of a cosmological event horizon as the universe kind of splits into two pieces that aren't going to communicate at least to the classical level by the way when we talk about the multi way cause we'll see how there are essentially quantum connections that developed between things which might be classically correspond to event horizons okay and really funky things can happen in our universe is like that the bits of the spatial hypergraph can actually break off so it's more than just that there's a causal horizon there's actually space basically a piece of space can break off doesn't always happen but can happen oh and we can okay all right we could talk about cosmology let me let me skip that if people are interested I'll talk about that about the possibility that in the early universe one can have because one of the features of these models is they don't have fixed dimensional space so there's nothing that says that the the dimension cannot be a dynamical variable in in the evolution of the universe and so for example it could start very high-dimensional gradually cool down to be lower dimensional all sorts of funky things can happen it's not even obviously can't reformulate general relativity as a change of dimension instead of a curvature because in these models you can see the kind of trade-offs between dimension and curvature and what happens okay oh I mentioned already a little bit about particles and I'm sorry it would have been better if I'd been able to use actual slides but oh yeah let me mention one thing about particles so so one of the features of thinking this is just a cellular automaton showing kind of a a sort of how you get particle like things in a system where every every piece of the system is being updated in the same way I mean this is very familiar from from from sort of topological excitations in condensed matter physics and everywhere else and so on this is this is the same kind of thing you're getting particles that are localized structures emerging in this in this otherwise uniform system but um so one of the features of this of these models is that um you can you can ask questions like okay well again again sort of rickety estimates of parameters which I'll talk about a little bit more later but in those estimates you can do things like ask you know how big is an electron answer 10 to the minus 81 meters so very small the actual limits that are known right now are like 10 to the minus 22 meters electrons are otherwise thought to be point particles but in this model they would have a size might be - 81 meters one of the features of that is it allows a lot of room for particles that are much lighter than the electron so one of the kind of pseudo predictions of this kind of model is that there should be I calling them all Egan's things with with a small number of edges in the hypergraph involved in in these particles but they can have masses of like 10 to the minus 20 electron volts and things and that might be sort of interesting for astrophysics okay let's talk about quantum mechanics um the so the question i mentioned before that there's this ambiguity that can happen in that does happen that generically happens in in these rewrite rules so instead of just getting a single path of you get this then this and this you get this whole multi way system of all possible paths okay how can we think about that quantum mechanically well the thing we realized is that we can actually think about quantum mechanics is very much the same way that we think about relativity we can think about instead of reference frames we can think about what we call quantum observation frames and essentially what we're imagining is each one of these each one of these nodes here is a quantum eigenstate and what we're seeing here is we're choosing what we're making a superposition of these two quantum eigenstates by choosing this quantum observation frame we're saying we're looking at a superposition of those two eigenstates and so what we're then asking is when we when we start thinking about quantum mechanics and about how things work and we say let's imagine we're going to make a measurement so making a measurement in quantum mechanics is we can think of as picking a quantum observation frame that has certain properties so for example you might say I want to decide that the outcome of my experiment is I want to measure this particular thing I want to decide the outcome is this I'm going to pick a quantum observation frame in which essentially time freezes after that state has been produced it's kind of like a black hole being produced but it's a coordinate singularity that one is generating in this sort of quantum space that is what corresponds to the sort of the act of measurement here and what happens is once you've tried to force that coordinate singularity it doesn't quite work it um in order to keep consistent with a partial order that's been defined here one ends up having to sort of spread the region that um that one is freezing so to speak and that's essentially quantum decoherence happening um so in other words that's a so this is a view of quantum measurement where you're essentially thinking of things I'm I'm kind of a lighting a bunch of stuff here so so one important point is that in this in this model there's this whole multi way graph for possible things that are happening and so the question is why does anything definite happen in the world well ultimately there's causal invariants and that causes there to be in a sense of definite outcome except that these things run to infinity so the notion of a definite outcome is to be a little more nuanced but the other thing is that the observer themselves is one of these multi way systems so you're essentially saying how does an observer who is themselves a multi way system observe what's happening in this multi way system it's similar to the kind of logic that you would use in in relativity to understand how an observer with some limitations about speed of light and so on can understand understand the world there okay so in more realistic cases things get much wilder than that and these quantum observation frames the possible foliation x' that correspond to these quantum observation frames get really complicated I would show here this is an example of a qubit basically you're trying to make essentially a black hole like thing you're trying to freeze time for this particular part of your quantum system so that it cannot evolve so it stays in a quantum pure state um and the way in this particular case there's a very simple rule easily was able to freeze this in more realistic cases it's considerably more complicated okay so so can we understand um let's say there's there's much to say about quantum mechanics what what what we can think about is the and see where to go next um try and bring up something else I'm sorry I don't have quite the right material there because I had beautiful slides all right let's see if I can bring this up um II that's the right thing that might be the right thing okay so this is this is one of these multi based systems and it's representing essentially each one of these nodes is a quantum state each here we're looking at if we foliate this thing by just slices across here we're looking at a series of super positions of quantum states so one thing that's interesting to look at is what do these what do these quantum states how are these quantum states in a particular slice related to each other will be like saying if this was a causal graph and we took slices will be like saying that's a particular view of an instantaneous instantaneous configuration of space and then we can ask how the different points in space are related but here were asking how are the quantum states related and so what we can do here is we can draw what we call a branch eelgrass branch he'll grow is basically just showing two nodes are connected here if they have a common ancestor in this multi-way graph and this is essentially a map of the entanglements of quantum states so two quantum states that are that have sort of recent common ancestors are close in in this branch field graph are close in branch field space um and - and so what we're seeing here okay so now now you can ask the question okay look I mentioned the path integral let's talk about the path integral so essentially what you're interested in is you can think about jd6 through this multi way graph and you can think about how the Howard GD sik can go from state to state to state in this multi way graph now imagine a bundle of jd6 so we're trying to compute let's say an S matrix and we say we've got an initial state that's defined by the starting points with a bunch of jd6 we have a final state defined by the end point of a bunch of jt6 and now there may be many jd6 that wind their way through this multi way graph so a question is what can we say how can we compute essentially the transition amplitude the S matrix from that initial state to the final state okay so here's the thing that we only realized quite recently and I'm sort of embarrassed I didn't realize a long time ago so in quantum mechanics we talked about quantum amplitudes and you're used to the idea that a quantum amplitude is a complex number so how the complex numbers sharp I think that the packaging of quantum amplitudes into a complex number is kind of misleading I think they're actually two pieces to a quantum amplitude they're important its magnitude and its phase you say that's trivial those can be just put together in a complex number but I think that the magnitude and the phase come from different places and so here's roughly what happens in these models the magnitude is essentially a path counting as you go down here so what's happening here in this branch eelgrass the measure that the question of how much measure there is associated with a particular end point in the branch field graph is essentially just path counting in the multi-way graph but there's another thing which is where the particular state that you were trying to measure or the particular you know superposition that might be a combination of points in this in this branch of graph where are they that's essentially corresponds to the phase of the quantum amplitude and so what's happening is as you go down this multi-way graph these gd6 are turning and where they end up in branch field space is telling you essentially the phase associated with the quantum amplitude and so here's that has the really cool part as far as I'm concerned so the turning of jd6 in the multi-way graph is a consequence of the presence of updating events and that is what causes the jd6 to turn every time there's an updating event there's a branch player-created that's essentially a instantaneous that's a that's a an elementary turning unit turning of the GDC of the of the JDC and so so every so what's happening is updating events essentially caused the turning of jd6 but the updating events as I mentioned are connected to the density of updating events I mentioned is the density of updating events is connected to the density of causal edges the density of causal edges is connected to in the simple case I was describing before to energy the flux of those causal edges through space like hyper surfaces we identified as energy so now we're in this sort of quantum situation and we again can talk about the density of essentially updating events or essentially the the flux of causal edges which is the equivalence of the density of updating events now what turns out to be important here is the divergence of the of these the the number of these updating events that divergence plays the role of Lagrangian density it's kind of the relativistically invariant analog of energy and momentum that were between space like and time like hyper surfaces this is sort of the relativistic our log of that and so what's happening so okay end end picture sort of energy roughly energy actually Lagrangian density is represented by the density of causal edges in this in in fact the multi way causal graph that's the causal graph that connects not just space like separated things but also what we can call branch like separated things things that went on a different branch in this quantum space so the the sort of the turning of gd6 is related to the that the rate of turning of gd6 is connected to the Lagrangian density as measured by this divergence of causal edges in the multi way causal graph and so that's pretty neat because it means that basically the phase e to the I something is basically e to the I times that an action and that that is giving you a path integral and what's cool is that the it's just a geodesic but that it's just this bundle of jd6 but instead of thinking about it in space-time this is a bundle of jd6 in branch field space in the space of quantum states now branch field spaces are wild a much wilder space it's some kind of projective hilbert space thing it's it's probably an exponential dimensional space it's probably not a finite dimensional space but it's you can still think of it as space you can think of it about it as branch field distances which are some kind of entanglement distances and so on and a lot of the apparatus comes over but what to me is really cool is that you're ending up with a situation where the the sort of the conceptual foundation of where the paths undergo comes from is exactly the same as the conceptual foundation of where we're in Stein's equations are coming from so okay so that's that's a little bit on on quantum mechanics says there's more to say about quantum mechanics we can talk about for example the uncertainty principle okay so the uncertainty principle is we can look at the curvature of Branchville space so in physical space we are you know we're measuring curvature by it by looking at parallel transport or something we make this little rectangle and we ask you know how does the rectangle match up and that gives us a Riemann tensor okay what is the analogous thing in in branch field space well the answer is that Riemann tensor that little rectangle is like a commutator you're doing one operation or another operation in two different orders you can do exactly the same thing in branch field space except there that commutator is giving you essentially the commutator of things like position momentum commutator and so on and you can you can see exactly how position momentum commutator z' end up when they are as a result of curvature and branch hill spaced those commentators end up being non-zero and that's sort of another another piece of the quantum mechanics story okay so let me see when I have this um so that's that's okay that's one of these another one whose branch field graphs a few other things - there are lots of other things I could talk about but but let's talk about motion and branch field space so in physical space we have this idea of the speed of light as a limit to the rate of motion the speed of light is kind of the constant of proportionality between an elementary sort of evolution in one one step in the ghazal graph and how far that goes in physical space the identification of that in physical space well in branch real space we have the same kind of thing so we have actually there a single branch field a single branch pair has a certain extent in branch field space and that extent in branch channel space is just like we have light cones in physical space were from an event we affect a certain increasing size of region and physical space so in branch real space we have the same kind of thing we have we can call entanglement cones and so entitlement cones tell you there's a maximum rate at which you can entangle new quantum states in this branch heel space of quantum states so actually you can get all kinds of all kinds of cool things because what happens is you you end up with this multi way causal graph which has some causal edges that represent spatial connections and some colleges that represent branch field connections in other words some colleges that represent two things the two events that are separated in physical space are the ones separated in in branch real space in the space of quantum states and so there's a there's a whole bunch of things one can do in looking at kind of the interplay between those things and I think those are relevant in in black hole horizons and so on I mean let me see if I can pull up something there you see um um well see if I can pull something up here um yeah let me roughly say so you can have a physical event horizon um that is in physical space but and that corresponds to a okay what is an event horizon I just literally just posted I wonder if it's actually been posted yet I just posted something I just wrote in honor of you guys because I I'm writing this whole series of kind of um see of great of um kind of bulletins about things happening in our project I would say that one feature of this project is we're sort of running it as a very open science project so what kind of live-streaming even our internal discussions about it okay so you can see they're related live streams and related notebooks so we're but posting all of the kind of all of the kind of notebooks about everything okay pathologies in space-time let's see what I can find some okay so that's kind of a cosmological event horizon um which has two light cones that are disjoint um we can kind of map out this is kind of the we can kind of map these kind of causal connections between different regions of space-time this is an extremely toy black hole so extremely toy because there's a piece of space here on the side where you can go backwards and forwards but there's also a place in this causal graph where once you go there you're stuck it's a one-way trip into that part of the causal graph so we can represent by this kind of causal connection graph we say this is the rest of the universe this is what's inside the black hole you're you're on a one-way trip there and you can't you can go and you can look at different kinds of things that can happen and for example it's very easy to understand something like a space like circularity is is basically just time terminates there's a place where all the light cones just don't splat and everything stops um that's a corresponds to a space like singularity in our models so you could ask the question if we find a black hole like singularity these are more exotic kinds of things that can happen if you find a black hole like singularity these are ordinary well these things here are sort of ordinary causal graphs and they're showing what happens instead of in space-time oh this is this if you look at different possible rules you can say what happens well there a lot of universities that don't have any horizons at all there are some universities that split into two sort of cosmologically separate pieces there are universities that have one black hole two black holes weird kinds of nested black holes and so on um but so the the thing that I didn't manage to finish as I was thinking about this but that you can ask the question oh actually this is an interesting phenomenon this is a kind of classical version of Hawking radiation which I can talk about if people are interested which I I think may happen um but okay so this is um okay the quantum case so this is an ordinary course of graph which shows in this case two very trivial well these are basically two timelike singularities two separated time like the ER is very trivial case okay but now let's ask if we instead of just taking a single path in the multi way graph let's look at all paths in the multi way graph in other words let's look at the malt and that's from that generate a multi way causal graph and so what this is showing is that instead of just having two separated pieces two causally separated pieces there and this is not a good picture yet but there are two roughly causally separated pieces except that there are in branch real space there are connections between what's inside the here and what's not inside here so I think the way to think about this is that because we have this maximum entanglement speed there is not only a causal event horizon but there is also an entanglement horizon and that represents the which is not spatially localized in quite the same way but you can have a situation where essentially the entanglement horizon is outside of the causal event horizon and so you can essentially preserve for example quantum information in that in the zone between the entanglement horizon and the causal horizon um but what what's what's kind of interesting about this is I haven't made them yet but we should get to be able to actually make pictures of how that works which is really pretty pretty interesting and we can see kind of how in these branch pairs we can see how one member of the branch pair ends up going to the other side of the causal horizon and other ones don't and all those kinds of things um so so one of the things so one thing you might ask is ok what is this maximum entanglement speed what's it equal to we don't really know because it's hard to set a scale in these systems but I think my my very rough estimate is maybe it's 10 to the 5 solar masses per second ok might that might be completely wrong might be wrong by many orders of magnitude but that's a least a rough estimate of the of the maximum entanglement speed which is measured in energy because it's sort of related h-bar but it's it's some it's it's the so that's the maximum speed of entangling quantum states and that gives you so it tells you that if you have you know the merger of of two black holes with with 10 to the 5 solar masses that happens in a second or something you might start to see effects that are associated with the fact that there's a maximum entanglement speed a maximum speed at which these things can sort of combine in quantum space as well as the speed of light so it's an example of those kinds of things all right well so let's see um so question might be so where have we got to so far in this whole you know what what's what's really surprising to me is what we could look at all these different kinds of things it's kind of a laundry list of different things in physics like cpt invariants ok we realized recently CPT if you look at the multi way causal graph parity is inversion of the space directions here time is inversion of the time direction and see charge conjugation is effectively inversion of the branch real edges so essentially cpt invariance is essentially a pure inversion of the multi way causal graph well one thing you might also ask is what is the what is the continuum limit of the multi way causal graph we know something about the continuum limit of this spatial hyper graph we hope it's something like a manifold there is some vague idea that maybe the continuum limit of the multi rate causal graph is like a twister space not sure that's a that's a thing for mathematicians we have not figured that out for sure that's an example so you can you can go on and look at different kinds of things you can ask questions about what is on your momentum how the spin work how is the quantization spin work we don't know for sure we have some ideas about that but the thing that to me is really really significant about this this model is it's often hard to figure out how things work but so far nothing we've run into we've had to say oh my gosh to fit how physics works we have to add some weird Cluj quite to the contrary I mean it's like sometimes it's hard to figure out how it works but when we see it it's like an obvious natural thing couldn't be any other way we couldn't avoid having quantum mechanics for example we can't just take quantum mechanics out of the model um so that to me is really encouraging now there's a question of okay so let's go actually searching for the ultimate rule you know can we just enumerate possible rules do we have any guarantee that the rule for our universe is a simple rule okay we don't have any guarantee we we know um and in fact as we start looking at these individual rules we are throwing right into the middle of computational irreducibility because we run one of these rules and we can run it for a billion steps or something and it will be complicated to see what it actually does what what I thought was going to happen is that we were going to be mired in computational irreducibility all the way down the surprising thing that's happened is there's this kind of layer of reducibility that's all associated with all of these very floppy things that have to do with things that have to be represented by reference frames and so on all these very floppy things that have a lot of essentially gauge degrees of freedom associated with them by the way I could talk about how local gauge invariance happens to but separate thing um the you know all these various sort of very floppy features which seem to float on top of computational disability and those floppy features are exactly what we know in physics basically and in a sense I should have known this is how it was gonna work out because if our whole world was sort of full of computational irreducibility we would have a hard time figuring out what was ever going to happen in the world because if everything we want to know involves running this rule ten to the four hundred times or something we'd never know anything we know there has to be some layer of computational reducibility that allows us to sort of be able to predict anything about how things work in the world now it turns out that layer of computational reducibility seems to map really nicely i can talk about how it maps into different kinds of approaches to mathematical physics and so on but but that's some okay so one last thing I'll talk about and then wrap up here um okay this is this is now going into it yet another level of abstraction so we went from the the hyper graph that represents space to the multi-way graph that represents the space of quantum states and this Branchville space there is another level of generalization okay and that other level of generalization sort of is the following thing so you say gosh you know let's find the rule for the universe okay let's say we have the rule for the universe we might be it'll be a very confusing thing the day when we might have this rule and we say here's the rule for our universe why is it this one why did we get this particular rule why do we not get another rule and it's kind of a you know if that rule is simple it's kind of a very anti Copernican moment because you know post Copernicus we've kind of had the point of view that there's nothing special about us yet if we end up with this particular very simple rule it's like oh my gosh we got the universe that has this simple rule that's very special for us okay how does one understand that okay so here is my current understanding of this so here's the idea so I've been talking about how at every at every step there's a particular rule applied but there are many possible places where that rule could be applied to a given hypergraph and so on but imagine you took something more general than that imagine that at every step every possible rule can be applied so there's a there's a countable number of rules in fact for a universe of finite size there's a finite number of rules that can be applied but at every at every step or possible rules can be applied okay so now we have a sort of an ultra multi way graph where every every node in the multi way graph every every event not only represents this particular place in this particular hyper graph in this particular place in the multi way graph but also which rule got applied so we end up with this with this kind of ultra multi way graph okay so an interesting fact about the ultra multi way graph is that it is inevitably causal invariant so that means that it doesn't matter in some sense it doesn't matter what path we followed through this and that allows us to basically do play the same the same that play the relativity game essentially again in this space of rules and so we have this thing we're calling rule II old space which is the space of all possible rules and you can essentially imagine making reference frames kind of observation frames in that space of all possible rules and essentially the atypical choice of reference frames might be reference frames where you describe the world as working according to a particular rule that's a sort of a possible choice of reference frames and so what what ends up happening I think is that you you end up being able to say well and this is abstract and not completely worked out yet um the essentially the instead of having to say we've got this rule and no other what what you're saying is we've got all possible rules operating but the reference frames that we choose to use based on our way of interpreting the universe correspond to essentially this particular rule so it's not like the universe fundamentally has that particular rule the universe fundamentally has all possible rules but our particular choice of reference frame so you know I used to say when people talk about extraterrestrials and you know communicating with the aliens and things I used to say well at least the aliens will have the same physics because they live in the same universe but I now realize that just isn't true because there can be completely incoherent views of that same kind of ultra multi-way graph that corresponds to different sort of foliation zuv that graph that correspond to completely different views of of what what we consider important in physics so okay so it turns out that you can look at for example the same thing I mentioned light cones and entanglement cones there's a similar kind of cone in rural space and there's a maximum speed in rural space and these have to do with essentially the maximum speed of translation between different rule systems in this space and I think that there's a I mean I think independent of anything to do with physics this study of kind of the space of all possible rules and so on gives you a very geometrical view of things like computational complexity theory but in its own right is pretty interesting and when you have things like for example computationally computational reducibility is the analog of a black hole in rural space and that term that's so you can kind of apply these same ideas and okay the the the challenge question is what is the analog of Einstein's equations in rural space and I have some ideas about that and the sort of continuum limit of rural space actually plays somewhat with ideas about so the mathematical versions of machine learning different topic and I I did mention distributed computing I might say that this whole question about about how you how you organize distributed computing distributed computing is about a bunch of computers doing things potentially asynchronously that with certain partial order relations about who has to do what first and one can start thinking about sort of programming in different reference frames corresponding to different ways to view distributed computing but anyway so so on lots of lots lots of interesting stuff there and in the end one of the things that comes out from this whole coeruleus space story and so on is that in the end the one definitive fact about the universe ends up being that it is in a sense computational that it can emulate that it is equivalent to a Turing machine it satisfies my principle of computational equivalence and so on and in a sense one way to think about these different things in rural spaces to say well you know once we can simulate the the universe for the Turing machine we can emulate any other Turing machine with that Turing machine that's kind of what's happening in these different term in these different kind of reference frames in rural space and there's things to say about so how we think about the project of I knew fundamental theory of physics given that there's this of all of this freedom and I sort of view it as being kind of like a language design problem so you know I've spent a lot of my life kind of designing computational languages which try and connect kind of human thinking with what computers can do now we've got kind of a third piece which is physics and the question is can we have something which which when we have human thinking we can sort of understand what's going on implement it through a computer and have it kind of play to what happens in physics so I sort of view the problem of finding the right primitives so to speak to describe these models are something like a language design problem so in terms of our project as I say we've been doing this is a very kind of open project and sort of interesting to see how that works like for example you know we have sort of archives of every Wolfram language notebook that I've done in this project since 1994 it's kind of cool that the ones from 1994 still run today but you can find all of these including probably the ones I did last night about black holes will be somewhere here yeah probably black holes 12 or something that will be the thing I I just did last night and you can you can pick this up and just run it um all of the programs that do all of these simulations and so on are we they're all available and they all just run and if you take that that term that document that I was just looking at you can just click one of the one of the pictures in there and just put it into a notebook and in the cloud on the desktop it'll all just run there's a whole bunch of functions that we've created specifically for this project um and we've also been doing kind of the unusual thing of live-streaming our kind of internal discussions about this project and discussions with outside experts we just did one yesterday with a causal sets person and it's been an interesting process because it's allowed us to kind of engage lots of people in the general public so to speak and when I say general public we find a lot of very technically sophisticated people some of them physics sophisticated some of the math sophisticated some of the computer sophisticated who are joining our live streams and so on and making all kinds of interesting and good suggestions and I think it's sort of a for me this is a you know I view this project as being kind of an attempt to climb a certain kind of sort of Mount Everest of science physics whatever and you know we may or may not get to the summit anytime soon but at least the the the trip is is kind of interesting to see and you know I don't know how difficult it will be I mean what's happening right now is we're figuring things out I would say very rapidly and lots of different as I said this is kind of laundry list of can you find this this this in physics and maybe some people will ask about different things here you could talk about what we know about them but it's it feels like we've we've got a you know we've got a good paradigm here that is allowing us to make a lot of progress and as I say one of the things that's really nice about it is all of these ideas whether it's you know you know ER equals EPR whether it's you know 80s safety and things I think those play directly into a bunch of things that we're doing they don't turn you know that making that bridge is non-trivial and will be significant work for people to do but I think in the end you can think of what we're doing is providing sort of a new machine code for thinking about physics it's not it doesn't you know it doesn't remove the the high level languages that have already been built but it's a different machine code and there are things that you get to see and you get to ask about based on that sort of different machine code that I think are pretty interesting and hopefully people will will find it fun to be involved with and I I mentioned we're doing a summer school that starts in what end of June I think that we'll have a bunch of physics people there and people are encouraged to to consider coming if you're interested all right I should wrap up there thanks I went on way too long sorry about that but I'm happy to try and answer any questions people have um thank you very much Steven for a wonderful talk out-of-the-box thinking that I very much salute I mean the mainstream in physics has not been very innovative in this way and it's very interesting to hear a completely different perspective to immediate questions that come to mind if there are young people in the audience that would like to help the project what would be your advice for how what would be easiest way for them to engage and help come to the summer school we have a bunch of physics graduate students physics undergraduates and so on coming to the summer school that's some it's so that that's that's kind of the you know because what you know the things I've told you here I've given you sort of a high-level view there's considerably more detail there's a thousand pages of stuff already to read about this we're sort of trying to figure out what's the best way to to present it now I might say that on our website and we have a just a very however how you can help a section so I think some of these some of these things like working out the connections between our models and existing theoretical frameworks that's a really valuable thing to do and people have started on that understanding what is the relationship so for example category theory ok categoric asian so for example we already know our main evolution multi-way graphs are like one level of category are causal graphs are like higher-order categories sort of two two-dimensional categories then the the structure of frames on top of those causal graphs is like a next level higher-order category but understanding that and understanding how one can use ideas from category theory to help is highly interesting basically any sort of standard theoretical framework whether it is conformal field theory whether it's some something about singularity theorems and general relativity the question is how do they play in our model now you know we might find one day gosh it doesn't play at all we know it's true in physics but it isn't true our model that will be a problem but so far it's going really well and the main thing that's interesting like like looking at the singularity theorems for example I was just looking at these and it's it's pretty interesting how there are things which way you can directly see how they correspond and there are places where our models suggest a different thing to look at right I mean you're thinking mostly about physics but in principle that there could be applications to biology for example where the rules that create systems that you are describing and and that could be quite right okay so the other there's another branch to that okay so this is a highly speculative thing but I have a suspicion that this class of models and that's way of thinking in terms of reference frames and things might be applicable to make a general mathematical theory of biological evolution right essentially this multi way graph is some kind of rep know how this works yet okay so I'm this is very vague but um you know one of the problems in biological evolution is that it's been hard to make a general theory you know you say well I can do a simulation of natural selection it's easy enough but like how many how many fittest organisms do you keep do you keep the top 5 where you keep whatever how do you think about that I have a vague suspicion that these that this kind of approach to modeling might give you something interesting there I mean Plus also the detailed underlying model is has you know you can you can make very biological looking things with it I don't know whether that's significant or not I'm just mention two other areas that might be of interest one is mathematics the other is computer science um you know I think there are you know these models as I say we've hacked through the mathematics we have these things that for example we really need to know what the rotation group outside of integer number of dimensions is like but we have an idea what that's like these models give us a suggestion of how you can take a limit and get something like a li group that is just like we can limit to space we can also limit to an internal space that's like a li group but we get to look at sort of different kinds of things like fractional tensors with fractional numbers of indices and things like this we don't know how that works there's a lot of interesting mathematics I think to be done around that there is of course I mean the criticism that someone may have a raise is that at least in the context of physics you draw analogies with known results and known derivations and non conclusions but the test of a physical theory is by making new predictions that you know can be experimentally that are new that were not derived before and that experiments can write it verify so do you are you where's that how is that going to happen right so I think that the couple of things to say first of all given a theory figuring out the precise experimental predictions isn't the world's easiest thing I mean look at quantum electrodynamics for jerrod 70 you know these are there it's it's non-trivial work to derive these things um however the thing that I'm most encouraged by is what I might call theoretical predictions so in other words we've got this whole giant list of things that should be true about physics you know existence of fermions they you know all sorts of things can we reproduce those things naturally but do we have to sort of bolt on some new features of the model to reproduce those so far the theoretical predictions are going just great now when it comes to sort of practical predictions let me give you some indications of how those might work so one of the things that's that's difficult for us is we don't know the overall scale of things we don't know the elementary lengths we suspect you know as I said I have a very lousy estimate that says the elementary length is about 10 to the minus 93 meters um but that might be right might be wrong about hundred orders of magnitude for all right now it's pretty clear it's much smaller than the Planck length but beyond that it's not clear um and so not knowing that scale kind of you know when I say there might be something happens in a merger of two ten to five solar mass black holes I don't know if it's ten to the five solar mass or ten to the three or ten to the 200 solar mass black holes so that's a problem that we don't yet know that scale we will start to have a pretty good idea of that scale once we know more about how particles work and that's a you know clearly if we can find a rule where we reproduce the mass ratios of the other particles I think we are we're in you know I don't know whether you think that's significant I think that'll be super significant you know that we generate you know we have an indication that there might be particles with ten to the minus twenty electron volt masses and things like that most other theories haven't really had a natural reason to think there might be such like particles that might be relevant to dark matter and so on over the masses of particles are given in the standard model they're not derived and so if you were it's sort of like a fear of epicycles you know that used to explain the planet motions but with finely tuned parameters right that's the situation we have right now with masses so well then came newton's theory that from a fundamental point of view was able you know just with a small number I mean there wasn't any free parameter just but you could explain all these orbit so if you can explain the masses of particles in a simple way that that would be an amazing so you know we can have various check boxes it's like get the path integral get general relativity get you know get you know gauge invariance things like this and one of the ones which we really cool is okay so the one that I think is going to be easier is getting the local gauge groups okay I'm thinking I think so one question is what's generic and what's specific right so what we've discovered to my surprise is that general Atilla T in the path integral are generic they happen in a wide class of models it doesn't matter what the underlying rule is it's similar to what happens in molecular dynamics going to fluid dynamics it doesn't matter that you're dealing with water versus air they both satisfy the navier-stokes equations but now if we want to ask what's this what's the viscosity then we need to know is it water or is it air and that's the situation so the question is what's like the navier-stokes equations and what's like the actual value of viscosity and I don't know yet for the you know local gauge groups I have this slight suspicion that we will be able to say something generic there and that that won't be specific I'm guessing that the particle masses will be specific but I don't know for sure and and so that's but then there other predictions so here's another one that has to do with quantum computing so here's the thing that some so our model has a very definite view of how quantum mechanics works and actually one of the ways that we sort of prove that we reproduce in quantum mechanics is an amusing thing it's what I might call proof by compilation and so what we're doing actually might even have this done by tomorrow you can watch the live stream tomorrow and see whether we managed to make it work is a compiler that goes from a quantum computing framework that we've been building standard quantum computing framework with standard quantum gates and and so on goes from that framework compiles it into our multi way systems okay so in other words it's taking the exact formalism of standard quantum information and compiling it to a multi-way system except there's a big difference which is that we get to actually describe the measurement process normally in quantum you know quantum mechanics quantum information it's just like you do all these quantum things and then there's a measurement and that sort of magic um but for we get to have a micro description of measurement and so it's quite interesting from a quantum computing point of view because as you think you're doing you know some Shor's algorithm factoring thing and you're Treeing out all these possible sort of quantum paths in in our multi way system you can actually see them being tree doubt then when you do the measurement you have to corral the whole thing back in again to make a definite measurement but we actually get to see that process happen and so I think there going to be some very interesting theoretical results again how that theoretical result translates into decoherence times for practical quantum computers I don't know but being able to make theoretical statements about what can happen I think will be possible well why don't we open up the floor for additional questions from other people in the audience so you can raise your hand on the participant list or you can just speak if you see that you're not getting the attention you need Andy you want go ahead go ahead Andy you need to unmute recede yes yeah okay got so two related questions one are these hydrograph updates are they are they reversible can you go in another direction and nope so so how do you see unitarity so unitarity must emerge it's some kind of approximate so let me explain reversibility and unitarity of two completely different things so unitarity in these models is actually trivial because as I mentioned the magnitude of these quantum amplitudes is determined by path counting and unitarity is essentially the statement that again some more details but basically unitarity the quantum unitarity is trivial reversibility of microscopic laws you know any kind of microscopic reversibility in the classical sense is non-trivial and that has to emerge from essentially an equilibrium state of these microscopic updates and so on that's not something that's built in which is probably a good thing because we have time reversal violation and reality and things like that but unitarity the pure quantum unitarity um is is essentially a necessary feature of the fact that the magnitude of quantum amplitudes is given by path counting if you go backwards in time eventually get to the first point right the first class yep so how do you how do you have unitary transformation to something because because because you've got because what you're looking at unitarity applies in the multi way graph right so you're saying there's a sort of a single path at the top and that that you know that magnitude of Parthi Ness is spread among all the other paths as you go forward so that's always that always gets that counting always always works out it's you know you're concentrating all that all that you know probability or something back into that single initial graph but that's how it works okay other questions don't be shy there are lots of people on the call but apparently I did mention a physical phenomenon we can talk about how it might work in these models or not come on there's lots of us one comment while people are thinking that I wanted to make is that if you take a solar mass collapsing on on its Schwartzel crossing time that's roughly 10 you know 10 to the minus 5 seconds so you can get your 10 to the 5 the solar masses per second even for a stellar-mass black hole that's interesting see see one feature of all of this is there are all kinds of cosmological things that might come out of this but actually doing the astrophysics and so on to work it all out that's non-trivial so for example I wouldn't be surprised we were talking I was talking to Andy a few weeks ago about this in the case of photons essentially in orbit around a black hole I suspect that our entanglements arising business has different implications for for example photon photon correlations and if you're looking at you know the photon photon correlations and for photons orbiting a black hole but to actually work that out there's a whole bunch of physics that we haven't done but speaking about that you are probably familiar with the shapiro delay do you know that it's in here it's a phenomena in general relativity were you know if you move through I mean near an object that time is ticking differently and so there is a time delay associated with it and the person that I mean already served only means something different is actually Erin Shapiro who raised his hand right now so we should ask him to speak I mean he had some delay before he raised his hand but he raised a sensor okay go ahead oh we should unmute you I was curious about something like the maximum tangling speed nope you make a definite prediction of what its magnitude is yes we don't know it's like can you tell me what the speed of light or Planck's constant is you know it doesnõt them right you don't you your procedure your structure would enable you to make a prediction okay if we can get if we can find a specific underlying rule we will absolutely have a prediction of that it's it's on givin a particular underlying rule you absolutely nail it every everything has to come out of that but if we just say looking generically at all possible rules we can't tell you what the actual magnitude that I look I do have a derivation okay so it's not not completely um let me let me show you that so um let's see this is this is an attempt to understand that the scales that occur but I would say that this is a um what you find I said okay so I mean there are there are definite things here there's there's a quantity that I cook sigh that so why do we not end up with Planck units the key reason is because a multi-way system what the Planck energy as people know is actually quite big the Planck energy is you know the the energy of a lightning strike or something about 10 to the 19th electron volts or something um that's a surprisingly big number but the in our models the the relevant elementary energy is the Planck energy divided by a divided by essentially the the number of parallel threads that exist in this multi-way graph which is very big number and so you end up being able to give it but we don't know what that number is so call that number sigh then we can work out um you know a bunch of relations between elementary lengths Elementary times and so on um there is a possible estimate for psy that turns out to involve a transcendental equation which has the nice feature that it is very insensitive to specific parameter values it's not completely insensitive but it's as you as you vary parameters this quantity varies from 10 to 100 and 14 200 117 so it's not a big variation whether those that range a parameter is correct I don't know for sure but if you if you you know we're we kind of lucked out that this transcendental equation doesn't depend much on the value of those parameters with those assumptions we can then compute what the elementary length the elementary time whoops all those kinds of things are but you know I don't know if those assumptions are right and you know we will start to know if those assumptions are right if we can connect anything to you know to some actual physical observation then we'll know if those parameters are right and even if you put these numbers on the table I mean immediately one can think of constraints from astrophysics that one can absolutely yeah right so that's and so for example the you know the existence of particles with masses so this says there's an elementary mass unit this says that mass is quantized in in units of 10 to the minus 30 electron volts so if you find a particle that has a mass of 10 to the minus 40 electron volts these numbers are wrong right so I mean it this says and the question is are there particles which are a small number of units of the elementary mass maybe there are you have to recognize that the debroglie wavelength of such an or the Compton wavelength corresponds bigger than the size of the universe so indeed indeed yeah right so I mean as I understand it I'd like to know I mean as I understand it the constraints on dark matter particles by the tiny are 10 to the minus 20 electron volts you start to not have as many constraints I mean the main question is what's the decoupling of these particles in the very early universe that depends on things like interaction cross sections that in turn depends on whether you think that they decouple when the universe is three dimensional or do you think they decouple when the universe is exponential dimensional if they decouple when the universe is exponential dimensional they get very very very cold by now and that's but I mean you know there's a lot of detailed stuff to work out and I haven't done that that's because it gives opportunity for people to contribute to your project yeah no I mean this is this is what I'm I'm I'm excited by that because you know there's a huge amount to do and you know as I said I feel very good about the framework but you know there's lots of actual work to be done yes so let's say let's make a deal that we will hear from you again I mean we can we can set a time for that but it may not be prudent because you know it all depends on how quickly you converge to something much more significant so I suggest it's undecidable this is one of the problems it's not a let's say the following that once you you know double the amount of knowledge that you have we will invite you again to learn more about what you've learned and it's really fascinating and I encourage the young people to learn more about it at the summer school that you mentioned and that we thank you so much the fact that the over 100 people stayed for two hours I mean that's said twice the typical colloquium duration that's coming through the more often than once every 40 years it's a great compliment to the intellectual excitement that you bring so thank you very much right okay nice to see you all and by the way just if you want to get in touch with us please just go find the website and go find the the contact us thing on the website

Channel: Wolfram

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Id: Ps5wovm3OQ4

Length: 106min 45sec (6405 seconds)

Published: Thu May 21 2020