Wolfram Physics Project: A Discussion with Fay Dowker

just say hello to everybody okay go ahead go live okay for people who are joining this livestream today's topic is causal sets and we're pleased to have an expert in these fade Alka with us and let me introduce so hi Faye hi and let me just introduce some other people from from our team as Jonathan gorod who is a in the same country as you are he's in Cambridge well in principle he's in Cambridge actually and we have Taliesin been on who can say hello and is in South Africa um and I think we also have max just enough max hi max has by far the most stylish backdrop mm-hmm um and and on a different multi way evolution branch max is also a causal set person uh-huh yes right right so so yeah so we I mean we think we're using a bunch of sort of causal set ideas we think there might be some sort of additional things that can be done with causal sets not only in our physics project but also in its applications and distributed computing and I don't know whether you've I don't know whether you've been exposed to this should be computing at all no I think it's a killer app for the kinds of things we're doing so hopefully we can explore that I'm glad to hear it yeah um but actually I'm curious in in the kind of the history of you know those of us involved with the speech space time how did you get into the discrete space-time business so to speak it was through my friendship and collaboration with Rafael Sorkin okay who was then I met him when he was a professor in Syracuse University in New York State and we had known each other for many years I was working on quantum gravity but from a continuum as active so from the tradition of of Stephen Hawking and the Euclidean pass integral that I suppose I I was never really sure though we could actually calculate this pass integral and that worried me troubled me for a number of years and then yeah I became aware of Rafael's work on causal sets so the idea is that instead of trying to define a path integral in the continuum you don't even you don't bother I mean you say that's that's an intractable problem and what's actually going on is that the histories in the sum over histories for space-time are discrete and you replace your intractable path integral with a finite sum over finitely so if your entity if your space time is truly discrete and there's only finitely many degrees of freedom in a finite volume and for if you want to do the physics of a finite amount of space-time then there'll only be finitely many things finitely many contributions to the to the sum over here that you're still usually thinking of it as a as a manifold underneath but you're looking at sort of the stuff I mean you can because one of the things that that I've been interested in doing and that we've done in our model is there is no underlying manifold there's no there's no coordinates of any kind it's purely everything is defined in terms of well in our case so a spatial hyper graph that gets rewritten and that the updating events in that spatial hyper graph define a causal graph and that causal graph is where I think we have contact with the world of causal sets at our um and that's common to to our programmers also so there's no underlying manifold there's no continuum there at all the causal set is the thing in itself so the the continuum exists only as an approximation to the fundamental stuff of the universe which is discrete so so indeed there's no underlying continuum the elements of our causal set are not embedded in anything they are they are reality itself and the causal relations are what what binds the atoms of space-time together and the continuum is an approximation what when it's appropriate when there's a continuum regime then the continuum is just an approximation to the right what you said is exactly the way we think no so so I'm actually curious and I don't I should know but how did Raphael get involved in discrete space-time ah so he was always discreet okay from his early days as a PhD student in Caltech he he he believed in fundamental discreteness but he started off doing reg a calculus okay yes working with at Caltech I didn't know there was anybody who did Reggie calculus laughs no no he he pursued these interests by himself so yeah well actually he he hooked up with Jim Hartle okay and they were it's a you know an important famous paper in reg a calculus together but yeah but he he was yeah he was doing it by himself at Caltech as far as going there so one of the things actually we've become interested in maybe you've looked at as well is by the time you think about everything in terms of causal graphs there's a question of you know when people do numerical relativity which is kind of what the objective of regi calculus also was you know the question is come on use the causal graph directly as a discretization method for doing numerical relativity have you guys looked at that I'm not at all but that's a very interesting question it's it it would be very challenging in the sense that when you work with the causal order I mean when you work with the causal set it you're dealing with the degrees of you know the physical degrees of freedom directly so and we know in physics modern physics that's a really difficult thing to do I mean the you know the lessons of relativity and gauge theories is that it's very difficult to deal with the true physical degrees of freedom because they're non-local so you know the gauge you know that for some reason physics doesn't like to to to reveal itself to us in it you know in the theory it seems to be easier to deal with a gauge yes yes right and then afterwards you sort of atone for your sins of having introduced this unphysical gauge and you have to make sure that your physical you know physical statements you know and so a breath you know you get rid of your gauge degrees so let me let me tell you my view of this I mean so it's okay you know we've got these causal graphs which come out in our world the causal graph emerges from you know from looking at updates in the spatial hypergraph so there's an underlying rule as an underlying dynamics and the causal graph is not a thing delivered you know it is a central thing in the theory but it is not the thing that comes out first the thing that you have first I mean is the spatial hypergraph and then it's the updates and that spatial hypergraph that lead to a causal graph but then the you know the whole question of sort of the different possible frames that you can use the different possible foliation of the causal graph that can correspond to you know the way you observe the universe the correspondence between those things is sort of what leads to special relativity eventually leads to general relativity in our models and so on and for us this question about so it's the one thing I really want to talk about actually is choice of frames and that is you know is there a theory for you know in in special relativity you know one's used to inertial frames in general activity one's used to a modest number of different kinds of reference frames and so one of the questions that I have is is is there a general theory of essentially making foliation z' of causal sets or however you want to describe it and that I mean one reason why I think that's interesting I mean it's interesting for physics it also is really interesting for distributed computing um just to explain the points of that you know when you have lots of computers that are sort of doing stuff and they don't necessarily know they may be doing things asynchronously they may not have a central clock and the way in which they are interacting is the same way that we're used to with light cones and things effectively computers are communicating with other computers and right now there really isn't any theory for kind of there is no analog of relativity and so on there's no analog of kind of picking frames it's done in a very primitive way it's the other way around and you want to eliminate the frame because the frame is on physics in physics the frame is on physical you'd like to be able to express all of the physics without referring to a frame at all so in a causal set its frame there's no frame the you deal directly with the cause of relations there's no spatial causal so it's still very badly with space and that's okay and in fact we think that's that's one of the good things about causal sets because space is not a physical quantity in general relativity so the causal set doesn't there's no sense in which the causal set has any it's not that there's something there's something spatial which is changing with time that there's no global time it's it you're really dealing with with with fully four-dimensional entity which is not it's not interpretable as something spatial in evolving in time and that's good when that be that's there you're you're you're taking the spirit of relativity most seriously so you're not you're not not trying to divide it up interest but you're not giving frames any role whatsoever in the theory so right I mean look maybe we should just show here let me just share my screen for a second and let's just look at a couple of things I to make sure we're all talking about similar kinds of things so this is some this is some kind of when you just step sorry I have a big screen here and this is confusing for me there we go um so this is kind of just a an attempt sort of overall summary of kind of our our sort of world so just just to review what what we're trying to do and then then where I think this connects with causal sets so by the way I think your causal sets live in this corner in my opinion okay so let me explain what the rest of this picture is um the okay so we start off with some underlying rule the underlying rule is operating on some graph or hyper graph and the points in that graph hyper graph are just can be thought of as points of space they don't have any coordinates they don't they're just points whose only information is their connectivity to other points okay so so there's some underlying rule and you apply that underlying rule you do it a bunch of times you will get some you know you can get some complicated mess like this in that space that space and the dimension of that space is you can approximate by just looking geodesic balls in this graph and the growth rate of the GDC ball that you know the leading term and the growth rate of that JT sit ball is the dimension is the effect of dimension of the space the first sub leading term gives you the Ricci scalar and gives you curvature of that space so that's that's the way that you get to that sort of where space comes from now when you look at these update events each update event has to know each update event uses certain points that existed certain elements that existed before and there's a there's a causal graph that tells you how one update event depends on other update events right that's the causal graph down here and that's the thing that you know we think is similar to your causal sets um now one of the things that happens in our so in our set up so it is a partial it's a partial order exact is it transitive well let's see Jonathan how would you best answer that question yes in the conventional way we we formulated the the causal network is treated as a transitive reduction of the of the kind of true causal network okay so what by what by what you mean by that is if we were to look at how one of these events could affect anything in the future we're only showing the transitive reduction of that yeah exactly it's the house diagram of the causal partial order relationship we should it's not it's not exactly at random production it's it's just more information than that because for each spatial age it shows you which events it connects it's a create event for the th and the story events for the sage which is not exactly a sense instant reduction because there could be the same different paths between two nodes okay let's let's let's put that aside for a second but but because we're going to talk about and we can talk about I mean we've studied a lot the different kinds of causal graphs that you get here and you know big okay first first big fact um okay so it's an important feature for us is this phenomenon called causal invariants and causal invariants is the statement well actually no we have to we have to say a little bit more before we get to course on variance um so one thing to say is these update rules there may be ambiguity and where those update rules get applied right so in other words there may be multiple places where this particular rule could apply in this graph so the question is which one do we do and the answer is we do all of them and that makes this thing we call the multi way system that basically says okay we starting off from that initial state then here there are multiple events that could be applied and they correspond the different branches here makes it okay I didn't understand that so so you your your spatial your spatial graphs are not are they labeled or unlabeled they're unlabeled they're unlabeled they all they know is their identity they can you can think of them as having that that's why you have this ambiguity is that right so you just knew how it's correct you have you're in one step you have one graph to start with and one graph that you end with but because they're unlabeled it you're not it's not certain where you where you've applied right so this rule for example this rule just says take any three elements that are related in this way and then replace them like this okay so those three elements could be anywhere in this graph and there may be multiple overlapping that elements there may be multiple overlapping cases where those elements can be a plot where those elements can exist does that make sense so maybe I should show I'm sure I've got some better pictures of this hold on them you could pull up a better picture this thing this thing in the middle is derived from a sequence a sequence in which your hypergraph is growing one element at a time I mean what just you know it's cardinality is increasing one at a time is that well in this particular case it might be okay so once you have that sequence then that is it that's isomorphic to this thing in the middle um there are many possible so let me try to exploit their many possible ways that that rewriting can happen this multi way graph so look ignore the orange bits for a second okay this main part of the multi way graph is showing all the possible rewritings that can happen so the thing on the right what so doesn't the state's graph on the right like oh yeah there we go yes that's the one right so so this is showing all the possible rewritings that can happen based on this rule you're at this state you've got this um and probably zoom in here oops otherwise didn't work um the No okay there we go so if we zoom in we can see that some graph and there are multiple ways that that rule can apply to that graph and so we're showing each of those possible outcomes countenance um and then so that's what this thing in the middle is this is showing all the possible outcomes now what this is also showing these orange bits are the causal relationships between those updating events so in other words in order to be able to do this update down here you may have needed this update to have happened first okay right so this thing we call this the you know so we have the ordinary multi the ordinary causal graph which is for a particular sequence of updates what are the causal relationships between the updating events this is what we call the multi way causal graph this orange thing here because it's showing for all possible sequences of updating x' it's showing the causal relations between what updating x' what updating x' are happening so from a physics point of view okay so what is this from a physics point of view that this basically is the story of quantum mechanics and this is a really amazing thing which I you know really was not expected that it would work out as nicely as this basically what happens is this multi way graph is essentially representing the sort of many possible histories of quantum mechanics and then what happens is the okay so to cut to a sort of a big big conclusion is okay for this spatial for the space-time causal graph we can show that in a bunch of limits and so on we get the Einstein equations and I can explain how that works um in this thing we basically get the path integral and the amazing thing is that the path integral and the Einstein equations are actually the same same kind of thing they they're derived in the same way but the Einstein equations come out in the special space-time causal graph and the path integral comes out in this multi way graph and it's really quite remarkable and I must say I did not see this coming at all but I think it's really cool but the what's happening here is I think some kind of generalization are probably what your causal sets have looked at I mean this is instead of looking again maybe this is something maybe somebody maybe Jonathan somebody understands this better than I do but but um I think this multi way causal graph is a I mean it it's not it's okay these causal edges can be both in a sense space-time causality and what we call branch time causality so they can relate they can be represent causal relationships on different branches of this kind of sequence of histories in quantum mechanics I know that that that may be incomprehensible and I apologize that Jonathan can you help bridge this yeah I suspect the strongest analogy is with the with with Rafael and write out sort of formulation of the you know the classical sequential growth dynamics right so so that's okay put the Stevens benefit or not for anyone elses the the CSG model is a way I think of growing a causal set basically using a classical stochastic process and my suspicion is that essentially a multi-way causal graph or something very much like it is what you get if you do if you apply the classical sequential growth model sort of a you know a statistically significant number of times so you effectively get all possible classical sequential growths in a weighted by business so I mean the classical I mean this is something like these aggregation models that we studied at one point where you where you can add a you know where you can add something to a glow at growing cluster at one of many points but you're saying that you get a particular causal graph when you do a particular probabilistically chosen sequence of additions is that right well I mean if a is probably the best person to know it does a it's a random process but you only get that running the process gives you one cause exactly yes right yes so my point is that if you wouldn't pick a branch of the multi-way evolution graph non-deterministically what you would would be something akin to a classical sequential growth dynamics model so instead it to take the converse of that if you apply a few sort of if you just apply either the CSG model multiple times so you get many many possible evolution histories for your causal set and then you and then you look at the sort of the classical probabilities that emerge from that what you would end up with I think is something not dissimilar to a multi-way cause I think that's a confusing way to say it if I might say so I mean it seems to me from what I understand I mean I don't know exactly what this let me see if I can pull up something analogous here um here let me you do that does one thing and that what I would worry about here is that you're serious about space and you as you said the causal order of your update event is derived from your theory which is really about space how space changes uh-huh and that's it's just not the spirit of relativity so you know in general tivity there's no such thing as space that concept has just gone it's not it doesn't know it doesn't it isn't there in the theory at war okay so they wanna court what the classical sequential growth model does is it grows space-time the four-dimensional space-time and it never refers to space there's nothing special about it is growing with full it's going the history of the universe element by Elman and that process of the elements so atoms coming into being we conceived of as the passage of time when that that's it that's a nice thing about all these models where there's a process so your update process each event happens and you can think of that process of the of the update event happening as being I mean for us that that that is of course that's a spacetime atom coming into being it it it's Bohr and if you like it didn't exist and then it does exist and that process is the passage of time so that's something which is natural to these sorts of both your model and and their and the classical sequential growth model right the two are not as dissimilar as I think you're making them out to be I mean so each each updating each application of these of these replacement rules is adding a new element to the space-time causal graph in exactly the same way as CSG would and the point is you could view it as the hyper graph being fundamental but actually to in order to derive something like this spatial hyper graph you you need to pick a particular evaluation order and in particular you have to pick a specific foliation of the space-time causal graph in order to say which collections of space-time updating events are treated as being simultaneous and so it's perfectly possible to formulate this model as saying that the spatial hydrograph is also a fiction and that what's really fundamental is the causal network here's another worry because of the there's this sort of beautiful work in classical from the classical era of global causal analysis by Penrose Hawking and it sort of culminated in a theorem which tells us that in a in the content so it's a continuum theorem which is that the causal order of the the space-time let's let's just say that it's a it's a globally hyperbolic space time so the causal order determines the full geometry up to a conformal factor so just local scale the local scale is missing from the conformal from the right from the order relation but the order relation contains a huge amount of information so that the conformal factors just one function so in four dimensions you have ten space-time functions in the metric so the causal order is nine-tenths of the metric and the tennis the tenth function is just a conformal factor so but once you go discrete which is what you have then you don't need your your given local scale for free because you can count so the idea of it so you have the counting measure which you don't have in the continuum so you can you can just count the number of update events and that is the volume of space-time so so that in fundamentally in a Planckian units we suppose I don't think it's totaling units but but yes I understand anyway so so you well you have so you have scale because you're discreet and you have the causal causal order which is this graph look at the bottom right hand one and that will be that should be enough so we have very strong reason to believe that that's enough to give you the full geometry but you have but you have you have the the graph the spatial ground okay well question is what what guarantees you that those two you've got an over determined system here I understand if these theorems are right then you've got enough information in this in the causal set in this look in your bottom right so the full geometry of space-time you're absolutely right yes we don't need that yeah you can throw away the spatial hypergraph if you want to okay the spatial hydrograph is merely a crutch for understanding you know for applying rules which will ultimately lead you the ultimate thing that you care about is the multi way causal graph the multi way causal graph contains both general relativity and quantum mechanics but for purposes you know for what we're doing so one one important so yes you're absolutely right we could throw away the spatial hypergraph it is useful to think about the spatial hypergraph as a way to understand to have some something we can get our hands on that looks like space now how do we get that from the causal graph we get it by making a foliation of the causal graph we can make many different variations of the causal graph those will lead to different spatial hypergraphs different sequences of spatial our eyebrows but the special hypergraph absolutely it's a fiction just in the same way that you know that space you know we can view space as we perceive it as being a fiction but yes it's a mathematically the the causal graph gives you all the information you need and so yeah sorry can I make one point about why it why I think the special hypergraph representation is kind of useful this purpose so um faith in AI and this comes into a more general question I I have but effectively defining non time like geodesics in causal sets so time like gd6 off is it very easy to define because you have a notion of a chain but if you have an on time like geodesic in particular if you wanted if you want to talk about space like gd6 it seems that you can't just do the obvious thing right you have to set space like separated elements in your causal set the obvious thing to do would be to just find like the minimum of the that the minimum distance to their common intersection either in the past or in the future right but that quantity is not Lorentz covariant so it seems that you have to do some kind of averaging process is that is that a fair assessment well has been solved although people have tried but the even in even in flat space but the the real is why do you need them you don't need them I mean there's no such thing they have they play no role in physics in relativistic physics they play no wrong so why would you want them that's well I think they look ok there's one thing we haven't mentioned yet which is this whole idea of causal invariance which is pretty important to the way that our stuff works but I think in terms of you know should we throw away space and consider only causality um the [Laughter] rule of today some criteria graphs so they wouldn't be able to generate scherzo said without yes what we can think of it as a crutch for getting to the causal graph if we want it to now by the way when it comes to the distributed computing in a first approximation all you have is the causal graph so you don't have to worry about the spatial I mean you know computers are distributed in space but in a good approximation all you have is the causal graph so think properties of the graph or what we care about there but but but 10 you know let me give you some examples of I mean think about the spatial hyper graph as Max says it's how we get to our causal graph because we haven't you know we don't have a sort of arbitrary rule for making up the causal graph we have definite dynamics for making the causal graph which happens to go through this intermediary of the spatial hyper graph um but you know the other thing is that you know in terms of just understanding the world it's useful I think to have a notion of space because a lot of our you know perception of the world involves space so for example thinking about what is a particle you know what is I don't know what what um you know well fruit for example our a as we get towards our our derivation of general relativity it's useful to think about a sort of toy version purely operating in the spatial hypergraph before we get to the full space-time version in the causal graph but I don't think this is where okay for your purposes you can ignore the special hyper graph okay will let's go causal sets all the way so to speak I mean it's it's some you know it's it's structurally useful in our model to have this and I think it's also necessary um - yeah another another point to make by the way is is that um you know one of the features of our model that is perhaps different from at least the tradition of physics in the last hundred years or so is that for us time is a different kind of thing than space so time is this sort of inexorable computational process of applying rules as opposed to space which we view as just being this thing that is you know knitted together from the events that happen you know in the system and by the way one thing that's really important is that the time that shows up in quantum mechanics the fact that that's the same as the time that shows up in general lots of it is really important and that's what in the end the derivation of so for example the relationship between yeah I mean that that that relationship is is the fact that those are the two of things at the same time is is really critical but I mean I think maybe we should say maybe we should say something about causal on variance because I'm not sure that you guys have an analog of that so cool things they do second I think the discrete general covariance of causal set theory is the analog of causal invariants in our models but I might grow okay so I don't know what the so this this requires translation because I don't know what the I could say what it is in in classical sequential growth and then you can tell me okay fine yeah so so in let's see so so the classical sequential gross a classical sequential gross model is I think Jonathan's already said it so it's a Stoke it Stoke a stick so they're probabilities in it so it grows a single causal set but a trap but it's random so you know which one you're going to get and it does it element by app so it grows the causal set element by element so it spacetime is it is growing and it's the specific so a new element is a stage which is what you would call an update event so a stage is where an n a causal set with cardinality n becomes a causal set of cardinality n plus 1 via the birth of a new space-time atom a new causal set element and that that causal set element it chooses a set of the elements that already have been born to be in its past so it's it's a it's a growth which happens it's only a future growth so the universe accretes more and more space-time atoms but only to the future that's part of the rule when there's certain probabilities as we do we have a piece of code that does this okay so I'm looking at a the first diagram in the Salkin paper from 2008 classical sequential growth dynamics and it's basically saying are you looking at POSCO that's what sort of causal set the post actors of causal sets why don't you send me send me Kelly can you send me in the zoom chat or something the the thing I'll bring it up on the screen here so POSCO is a picture although it's upside down so for us the time or you know the process goes up the page your processes go down the page okay so if you look up what I can see on the screen then top no well now I can't oh it's all right hand corner that blue thing yes if you replace those spatial hyper graphs with causal sets yep of cardinalities such-and-such right yep so the first one is just a single element the second right I get it and then you will randomly pick a path down and I randomly pick a pass through Cosco and that path corresponds to one element being to space-time now it's not you're not adding an element to space or updating space you're updating space-time and you're updating space-time only to the to its future so you add an element to the future add another element somewhere to the future so and that's choosing a round and pass through post cow this picture in in the paper that yep that is the that's the growth process and their specific okay good yeah excellent so specific probabilities so the model a particular model is an assignment of probabilities to all these different arrows there right so you can this is what we call a multi-way system sorry to interject ok great so you can see so if you see a level so the first level is just a single element causal set the second level you can either have two unconnected elements or the two I get it completely and then at levels at level three you've got five possibilities yep so you can see that there are two ways to get to the causal set which is just the to chain and a single disconnected element you know one way and you can go another way and this what Jonathan called discrete general covariance is the is the condition that the probability of going one way equals the probability of going the other way so to arrive at that to chain with single element partial causal set of cardinality three to arrive at that partial space-time it doesn't matter which way you got there it will 'ti will be this will be equal so that's like saying that the action doesn't depend on particular order in which the elements were born so long as you end up with the same thing then the probability is the same is equal and that that's true everywhere on this on this arm that is the nature of the the particular way in which you are constructing this multi-way system is such that your probabilities that the the different paths to get to this thing you're saying that each different path has the same probability that's right so it's a and it's a product of these transition probabilities so there's a transition probability on each arrow so you multiply together those transition probability to get the probability that you arrive at that particular node in Moscow okay so let me explain what what we have so in our multi way system so so I can show you endless multi way systems like this either constructed with with hypergraphs or with strings or with a bunch of other things there is an important property which is that anytime there's a branch there's also a merge the merge may take a while to happen but anytime it branches it will eventually merge and that property is a property that's been studied in a mathematical logic a lot it originally got called the church-rosser property confluence it has a variety of names and in the reason it comes up in mathematical logic is it's it comes up in ternary writing so for example simple simple example when you're doing algebra you know you like polynomial algebra you might say oh I can you know and you're eventually trying to get an expanded polynomial out there are different orders in which you can do those operations on the polynomial you can say I'll expand this piece out I'll expand that part out etc the different orderings that you use doesn't matter what what ordering you use you'll always eventually get the same answer and that's the statement that eventually the multi way graph will that whenever whenever something branches it will eventually merge again did that make sense so is that in comprehensible let let's try yeah yeah okay let's actually make um let's see um let's see we can I can either try and find this or we can just run something let's see multi light system so let's pick so we'll do it with strings instead of with them so let's say we have something like this and then we say um can we do BBB AAA something like that five steps and then we say States graph okay so this so let's see whether we see what okay you see this okay by the way mmhmm yeah um so so what this is doing this is a multi way system so it's rule is just it's just operating on strings it's not operating on graphs here just for simplicity so says whenever you see a be a replace it with a B okay makes sense so far so we start off with BBB AAA and there's only one operation we can do that the one operation is take the B a in the middle and replace it with a B and so we get this at this stage there are two operations we can do we can because there are we can either go to this state or in go to this state this seem familiar and sensible um okay so now the big property here is at the end so what is B a goes to a B going to do ultimate it's going to sort the the the um it's going to sort the string right it's just gonna take every B a and put the put the A's in front of the B's but let's say we go let's go eight steps for example here if we scroll down eventually OOP we didn't even get there I should have I should have picked a shorter string hold on let's let's take 12 steps or something um okay so eventually eventually we'll get to the string AAA BBB make sense but in the process of getting there we will have there will have been many branches we could have followed but all those branches anytime there's a branch there will eventually be a merge and well in this particular case we'll just come to an answer we'll just get the sorted string so I think this so this thing here is our multi way graph which I think is similar to your puss cow diagram um and there we have this property of course on variance which is the property that says anytime it branches it will it will later merge so now to show you um let's see just just to show you how this goes to an let see evolution events graph is that right oh for goodness sake okay so this is this is showing us let me do a simpler case um I'm just going to do a shorter initial condition okay so what this is doing is it's showing us not only okay so it's showing us here not only the states but also the events that updated from that state to this one so in other words this is your arrow this is one of your arrows but here this is an update event happening on the string right we don't have any probabilities here this is this is just saying follow all paths here now for us this causal and variance of somebody's asking us to zoom here this is some um but for us the sorry I lost my mouse there we go um let me maybe try to try a slightly longer example here um does this does this graph make sense yeah yeah I know what you're doing it seems very different though when that one thing is that when our state space is growing because at any stage the the state's state space I mean it's not it's not spatial State of course it's it's the whole universe you know it's the whole space-time up to that stage right obviously that's getting bigger because the it's the set of of causal sets of cardinality n at stage n not just getting bigger and bigger and bigger so so something like this I think just you know this postcard thing is it it's just not going to satisfy your this okay so it's ranching merging thing I mean it's just you know it's exploding so the big conceptual difference I think which I actually think is not that relevant okay but but nevertheless the conceptual differences you're growing space-time you know one step at a time you're growing the whole space-time whereas what we're doing is we're saying let's take the state of the universe and you know go and treat time as special and progressively grow States of the universe now you might mean spatial states that are right well I mean special states but it doesn't but the whole point is because that's this whole issue about foliation z-- we you can think of it as growing spatial states but those spatial states will define causal relationships and then the there are many possible choices of what those spatial States might have been depending on how we define simultaneity in this causal graph so in other words we're growing a causal graph not by okay so yeah the difference is what in your in the nodes of your graph are complete space-time histories whereas in here these these things can be thought of as what you can think of us put spatial states of the universe okay yeah maybe it's useful just so I think here's something that may be helpful so the events your events the which are the elements or the nodes of your causal causal graph here those are things that happen to space that an event is something that that that happens to space where it's a space is changing in some way yes that's what you get so the events are events are we don't need that concept at all the event is just and it's it's just the birth of an atom of space-time it just comes into being that's that's it's it's much simpler so it doesn't require any notion that something is happening to space it really is at so the hypothesis is that sufficient it's rich enough to keep right so the thing that you see one difference is so this idea of time as inexorable computation yes I like now there's really a yes right it's an unceasing process right but the fact is that what we're imagining is that there is a state and that inexorable computation is happening to that state now what you may say oh my gosh how can you get rod sophistic invariants all these kinds of things the reason those things come out is because the causal relationships between the things that happen to our state are the only thing that an observer is sensitive to and those satisfy all of the standard relativistic features so we're not I mean okay I would say historically I thought about this idea of kind of growing the whole of space-time and I kind of attempt for me the I mean there's a whole vast theory that sort of builds from this idea that time is something that is sort of progressive computation applied to the state of something as opposed to that at every moment you have the whole space-time history of the universe right I think there's a slightly different ideas now I would claim that at a sort of mathematical level most of what you're doing will probably apply to what we're doing independent of that change of kind of foundational basis for what's going on um because I think that your notion of I mean the fact that you have so what you would be doing here is you'd be taking essentially the transitive closure of for you this each one of these there's a spacetime history here right the space-time history is the sequence as a particular sequence here if I did the transitive closure of this of this graph I would get something where I basically have all of that space time history rolled up into one one node if I'm not mistaken so in other words if I take this graph I can probably just do it I just say transitive it's probably going to be a huge mess but if I say transitive closure graph of this okay there it is big yes but that I think is something closer to what you're talking about so what's been done here and maybe I maybe I got messed up but what what's been done is instead of taking instead of saying this event happens then this event in this event and so on I'm saying that event you know the whole future of space-time is kind of rolled up into that one event um and I I think I mean am I am I getting this wrong can somebody can somebody so can I ask a question the question is that the causal graphs that these models produced well for models produce seem like they're similar sorts of objects even if there's a different philosophy this so generates them to the the pro sets that occur and calls for set theory the kinds of computations that you would do fail on those on those processes similar to the ones that we doing to get sort of regeec curvature correctly Stephen that's the question I mean just just to explain in our models right we can derive the Einstein equations from are caused from features of our causal graph I don't know whether you guys do that no I think that's a non-trivial result I mean I figured that out like 20 years ago and we figured out a new version of that that includes Timmy new that includes the energy momentum tensor we think it's pretty cool it might have been something but and I think it depends on this well so they you you and Raphael and Ben Casa and people have a notion of the discrete del invasion right which you can use to define a Ricci curvature presumably yes yeah yeah if you apply it amazing it if you apply it to the constant scalar field on a causal zone yep right it actually gives you it gives you the Ricci scalar yeah we can't get oh we can't get all components of the Riemann tensor bit but they're out that's only for special causal sets so one thing I should say is that most causal sets have nothing to do with continuum space-time they don't have a proximation indeed so most of the so in the in fact the vast majority you know the overwhelming majority of causal sets of cardinality n are horrible awful claps things that last for to Planck times and are nothing like continuous basement so so we need dynamic so that we call that the in that the entropic problem the entropic problem whenever you go discrete is that you you know most of your discrete things are garbage and nothing like space-time at all so you can't you know they don't have a nice continued approximation indeed dynamics is the thing that that will have to in the end pick out the nice manifold like causal sex right there but so with we if we have a call so if we just make a nice manifold like causal set then we can we have dimension dimension estimators we have estimators for the Ricci scalar the certain components of the of the Ricci tensor so yeah we can and we can read off geometric information but only and we can't get those from a dynamical law okay if I can get I mean so for us you know we've just were very simple-minded at some level we just enumerate for all possible rules and some of those rules give us nice manifolds in the limit some give us garbage okay but we can just enumerate them so I could show you you know out of a million such rules some set of rules will give you manifold so I Croatia I was going to try and find for you a nice picture of that you just go there and there's a good example here we go um so actually here looked for this case okay so for instance this is okay so there's a trivial rule which does a trivial thing okay you can see that um so there's a that's the underlying rule I've written it in terms of symbols here you could write it as a piece of graph as well it does it just makes a polygon right this is another rule here makes it makes a tree make sense there's another rule which makes a more complicated mess that more complicated mess has a limiting dimension about two point seven okay but here for example here's another rule this one look at what that one makes that one makes a nice manifold like creature in other words that rule essentially is knitting this structure this is a spatial hyper graph okay so I'm not showing you the causal graph here I'm just showing you the spatial hyper graph um I could show you the causal graph I can look at those later and this one has a very nice causal graph as well um but so that's that's a case where that weird little rule knits itself a nice easy-to-understand manifold that makes sense um and you know so here's there's another example that makes a cone-like manifold you know here's one I mean they do weird and exotic things like here's one who'd know that this thing would make this weird three low surface but it does okay so I mean I'm I mean you've embedded these graphs they are graphs right that these are spatial graphs ooh these crap you've embedded these graphs in and 3d yeah in it just to show it just a display yeah very very suggestive that you've got a nice door treat and the continuum approximation here which is a curved geometry but but you have to show you have to prove that you've got you know you really do have something which oh yeah absolutely approximate right so let me explain I mean so we can measure the dimension of this thing right so this thing we can measure probably in the same way that you measure dimensions we're just looking at kinda factor good a flew to the GDC ball so we're not doing how stuff dimension we're doing some other kind of simpler version of dimension where you just say look at the GDC all in this hyper graph and look at the growth rate of the GDC ball right and look at the leading term and the growth rate of the G at all and if that's our to the D then D is the dimension right so I would do that for a causal step because it because of the Lorentz in nature exactly well the the the midpoint scaling dimension that's used in causal set theory is the direct analog of our G of our GD set cone construction in the causal network and so actually one of the questions I wanted to ask Faye at some point was about under what under what situations does this notion of like meh I'm Mary dimension as I think it's called count as a Maya there's this notion of counting the number of sort of chains of a particular length that exists in a region of the causal set under what situations does the dimension estimate you get from that dimension estimator differ from the one you would get by the analytic by this midpoint scaling of approximation before before maybe you answer that question could I just ask a very dumb question which is how does it how does it work at all like I'm not familiar with the way that that fair that you measure dimension yeah is there easy to explain so they the ones that we have we know how they work in in causal sets which have flat space flattening Koski spaces as approximations to them so we don't know no one has I mean the curved space is a you know as a whole terrain that we haven't yeah we haven't explored I think you could very nice methods we've used would lecture you do the same kind of thing but keep going yes the Lorentzian nature makes it tricky but anyway so no but we have we know they work so they behind maya dimension estimator has the great property that that is very efficient so it's easy to do so you need to estimate so in d dimensions that estimator needs fewer than 2 to the D elements you know to do a good job so it you take causal interval so you take a minimal element in a maximal element and you take all the elements that are in the in-between and the order and then you just from two numbers which is just n the number of elements in that causal interval which is the book which is the volume up to in fundamental union and the number of relations so you just count the number of relations pairs of related elements in your in that interval and in the continuum there's a there's just a functional relationship between those two things between the volume and the number of number of relations so you just invert that you just invert that so I have a good urge to actually run something to do this yes somebody okay and somebody get me a pro set how do I get this that's a good way of getting a place set and then if you want a really it if you want it so there's two ways to get interesting process there's a there's okay there are three ways to get posit right one is take all close sets of cardinality n in a bag and pull one out at around them let's let's just do that that's really uninteresting okay the second way is to run a classical sequential gross model and the simplest one of all is called transitive percolation that's where each each newborn element just chooses with probability P to be above in other words to the future of all the existing ones independently so it just says I'll be a built view and you just so that's that's that's transitive calculation it's super interesting is that all right so let's just read okay classical translation can we please write the piece of code I need to understand what's going on okay so so what do you thought um help me we've got many many good experts here right so we've got some let me make this a little bigger hopefully that will help I'm okay so what you're saying this is the analog this is the Lorentzian a log so at the you know minkovski and analog of the ER Disher any random graph okay you know the random graph is just you you know you put an edge between any two you know every pair with probability P okay so with n elements of you so here you just you you put an element above another seat yeah choosing a number of elements say 10 right so we're trying to make an analogue of this but we're trying to do it in a little rensi encase yes this is the Lorentzian case is it so if your elements or do you want to I have talent yes okay tell them that's right so in the Lorentzian case the order matters so right I cannot be cannot proceed J if I is bigger than J right so so node 7 cannot be cannot proceed in the order node 1 that's not that's not possible yeah right so every every other so every possible pair you put a you put a relation between them with probability P MIT I might have not been might have not been up to take any take the transitive closure so I think we could do the same for hours by just by just finding a topological sort of ordinary undirected random graph that's a let's let procedure that phase describing let's first of all do that because we can do once so we're saying for every we're going to add an element cc one thing we can do and I know I have nice there are two ways to describe it one is that you you grow the you grow your ten ten element causal set the second way is you just take the you take the ten elements already there and you just put relations between them with at random so we okay by the way it's good to be sure the one way you grow it is more it's more physical okay but so what we want is given to we're going to describe the elements the elements we want to describe each element by a unique identifier right every element has a unique identifier yeah just a number just its name is the number yeah right but but that number and and it better be the case that the sequence of numbers here that we have is a sequence which is non-decreasing it's that correct yeah what is it yeah so that element one is born at stage 1 element 2 is born at stage 2 element 3 is born at stage 3 but but what okay you know what I'm trying to consume buddy find for me maybe Matthew knows where this is the there's a lovely picture in the that I made and you know this is my problem for not being able to I made this nice picture which i think is highly relevant to what we're talking about um you see where it is drat um come on that it see this is this is some hold on oh there we go this is the one okay so this is an attempt okay so here I've got a trivial you know this is like a causal graph of someone a trivial version of a causal graph and this is a total order you know a possible total order on that cause of God yeah I have and so I have and by the way and I mean you probably know this but but in in in another world this is a depth-first traversal through this graph there's a breadth-first traversal i know whether you guys think about those kinds of things but that that's I mean in the world yeah what's that Russell has this whole library of Lisp yes yes right so in the in the universe so you know you take all the the people who know about causal sets that's one sunset and take all the people who know about Liske and you intersect them and then you get Rafael okay fair enough I I think I mean this has some um let's see what was i okay so I this was I'm just trying to find a way to generate PO sets come on we can I'm writing some code it'll be done in like two minutes okay here's a lot of peers a lovely way to generate suppose that they're two dimensional so you take take and they are flat right so take null coordinates U and V mm-hmm like this and let's take the unit interval so u goes from 0 to 1 and V goes from 0 to 1 so it's a Dimond causal diamund yep choose and choose a number random by the you know the Poisson distribution so just you know so with mean n right through okay so random variate a Poisson distribution with mean what would you like I mean 10 or something yeah right so that would give us a round out okay so then we're going to put we're going to put twelve points in this in this main cost ski diamond buy you choose the you do coordinates at random just a random number between zero and one and the V Co ordinate around a number between zero and one and you put that put that point down in that in the in the unit causal interval okay so we got that point yeah just you do that for twelve points okay and then you order them yep you order them so element P is below element Q if the you coordinate of P is less than the you coordinate of Q and the V coordinates of P is less than the V chord okay so you coordinate so what was the condition again the the new coordinate of the so yeah element P is below element Q in the order if the you coordinate of P is less than the you coordinate of Q and the V coordinate of P is less than the the coordinate of Q that's just the causal order in if you draw if you drew them out in the picture you would see this just when they're in the when when P is in the past light cone of Q I'll get sold on so this is let me just make sure that I've got the right criterion so this this should be the criterion so that's a sort of those of those things so if we now draw that if we say list line plot of that that will show us those points in they're sorted order do you think that's right so we can let's do this again so let's take I think you need to you need to rotate it by 45 to used to get it really to be all relativistic okay fine well let's do that okay so let's let's get alright so let's get these are points let's get 50 points or something here and let's say and then for each of these let's do a rotation matrix um 45 degree rotation I'm not mistaken dot that map it on to that and let's do a list let's do a graphics line I think just plotting the points is the best thing to do to start with okay well I was going to try just join them of them whoops just join them with a line because that way we can see their order but those are that we can get the points yeah I mean the points that I can I can show you the points there are the points yeah so if you yeah okay so would you like more points would it be helpful to see more points no good the the order is that is the causal order you know it's just the min-cost so you just consider the light cones mm-hmm and if a point is in the future light cone of another point then it's it's above it in the order so it's a partial order and it's just ordered by by the by the by the causal structure right to oppose a thing like I was doing when when I when I joined all these points I was picking a particular total order from this partial order right okay so okay so so now what do I do with this so I've got so I can represent the partial order how what's the best way to represent what what should I do with this partial will rather than doing a sort why not do an outer of a directed edge but in order to make the Hasse diagram yeah tell me how to do that Jonathan why do I go here so instead of doing this sort yeah so that soapy soapy and if that function P and with P and Q if you do an outer product if you make that into an outer product over the over the set of points okay so an outer of that function over the set of points but I you do it twice over the set of points right yeah they want me to I need to do hash hash here and I need to say that's at level one I need to apply that to that collection of points and let me not do quite as many as that let me do this and then I forget we want to say that there's a directed edge between P and Q if that condition holds so maybe what I want to say there is okay hold on let me get rid of all of that and then I want to say if well so for each of these actually what I really want is two poles I want pairs of points right and we could imagine how we generalize that triples of points but that's a different this year um okay so we say if this is true so you're saying if max is already written the code well okay fine no that's a cord for the different process oh okay it depends to the previous okay well let's just look at Max's code but then we're gonna write this this version because I want to understand how this version works okay max has a version that is what is this what is this do max yeah so it takes a graph and then it generates a new york tex and it attaches it to everybody exist in your exhibition probability okay so if i take the the a self loop like this and i say probability 0.3 or something here you're saying what's gonna happen here so if i do good stuff if i do multiple you should have a self loop because then it will be correct i mean it will be call Celgene okay fine all right so what I do is I do a nest list here of this growth step with a certain probability here applied to this thing let's do ten of those or something okay so this is growing so this is a way of growing this this should be growing post sets for you that right should we um now unfortunately these post sets get drawn let me let me just draw these in a better way because what I want to do is I want to let me put a frame around them all um okay so this should be hey does this seem sensible I mean this looks good all right so this should be so your errors yeah okay so the first one is just a to chain the second one of the two is what we call the L it's a t chain with a single disconnected thing and then yeah so you've drawn all the edges so if you do the transitive reduction there and you can get you'll get the hassle diagrams so in the third one you don't need the you know the extra edge there between because it's it's um right so let's do the transitive reduction okay so let's do the same thing let's take these results okay so let me just get the results here you get a set of results that we got a set of results now let's do the transitive reduction for each one of these that's compute the transitive reduction graph yep okay okay cool so what you yeah that's that's transitive percolation I think so you bad you that's right so you add a new element and it just decides to be above each already existing one with probability P what you chose 0.3 I think so yeah so the fascinating thing is that if you do this if you do the air dish Remy random graph mm-hmm for any P that's not 0 or 1 you get the same infinite graph it's good so it's deterministic in the end even though you know you you've got different probabilities and you're growing different graph table then it's it's you just get it's cool it's a Rado graph I think all the you know is the random guy but trial interpolation is super different it does what P is you get different outcomes for different Peas it's not deterministic so there's a whole range of different things you could fix P there's many different things you can get and the most interesting thing for us for cosmology is that they all the P not equal P not equal to 0 have the following property that they all have infinitely many posts and a post is one of your points Steven where everything comes together so it's an it's a causal set element for which everything is either to the past or to the future of that element so it's like a Big Bang Big Crunch Big Bang event it's where you just have one point of space if you like so everything collapses down to a single point and then and then the causal step will then will then grow again from that point so so their solution has this property that for any P that is not it no matter how small it so long as it's not zero the causal set will have this the cyclic sort of bouncing character at all but it but yeah if P is you I mean if P is very small then you won't you will have to wait a long time before you see it what value of P would you like to pick so I pick P equals point two but which I could pick P equals point seven or something here and then I did 20 steps I'll do 50 steps instead and let's make this and there'll be lots of posts I think you know what suppose oh look at that yeah see it's very chain is very very chain so it's very cheney yeah what happened so if I change the public sit down they'll be okay yes yep nice chaining yeah right I see - okay okay so let's let's take a look how this looks as a function of okay so if we'd say point five for example so you're saying it's it's a theorem of these things that every so often it will go down to a single a single for a filament so it's big it's a graph for the single okay a single edge but a single element single node yeah so like this for example is that right yes yeah there's two one after the other there right yeah so you're saying it's a good the point is that there will be a level in this graph that's it so for these I didn't make them a layers as layered graphs if I if I do this if I do the following thing if I say layered graph plot I'm then I will get something which actually shows the hazard diagram you know and you're more familiar fashion yeah yeah yeah so what you're saying is there will always be a level here whether it's just a single node at that level and then there are infinitely many of them so you can yeah you can do a simple back of the envelope calculation about what the probability is that the next element is a is a post and you find that it's it's nonzero and you just wait long enough you'll get a post so given given this um okay so so here's here's a random post that basically we got a post set here so now the question would be in you know so we have a way of computing dimensions from these things and there's a question whether it's the same as the way you have of computing dimensions from these things um and then so let's see for us I mean so we can make let's let's have a post set that isn't quite as stringy thank you get one pretty much for this this yeah well this one this one is kind of you're gonna get one very long chain II like things I think right foot for this right so we can make a light cone in here let's let's take that graph let's take one of these graphs and make a light cone out of it actually let me let me do this in a better way let me say here graph I've got to tell it um [Music] egg graph layout what is it a layered digraph is that what I wants here nobody no oops yeah there we go okay so these are now graphs and so I can pick up one of these graphs and I could put a light cone into this graph so I could say let me let me do this if I say highlight graph I'm just trying to understand what's going on here I'm highlight graph with them and so I want the vertex out component of a particular vertex oh no it's pick vertex v oh whoops I need in that graph right I think I need to say um should make you should make a function that does this yes okay well let's just do one and then we'll make a function okay um all right so that should be making the future light cone that's how do we make this sum or we might a subgraph that is that vertex out component you're right telling we should make a function let's say future light cone in that graph of this thing for this number of steps right of that that node for that number of steps make sense so what I claim that would be as highlight graph of the graph G and then what I want is the vertex as the sub graph of G that is the vertex out component of um the graph G and then the integer T if I'm not mistaken that right let's try it so this should be the future light cone of this element five ten steps okay so what does this let me just do it for two steps okay I don't understand this so this is the future light cone of that okay so that I picked that node somehow that was node five for some reason but let's pick node one maybe it'll be at the top yes it is okay so this is very confused okay this is the two-step future light cone of that node everything you can reach into two hops right okay so now the way that we would imagine computing so we can ask how big is that future light cone if we go X number of hops here how big is that future like con and I'm assuming that that is you know if we want to compute the dimension of this thing that's it's going to be the growth rate of that I've essentially that cone that gives us the if that cone grows like T to the D then that sort so for example here let's find the growth rate of that cone um the volume of it yes the number of nodes in that cone mm-hmm right so that should give you how what would you go the midpoint scaling dimensions the symmetric case of that is the midpoint scaling dimension okay should we do it just just for the sake of having something definite like my critic is a midpoint scaling you pick a time like interval so you had effectively pick a start vertex and an end vertex and the causal network and then your defining dimension in terms of the volume of the space-time that's induced between them is that a fair characterization I think that's right that's right so you yes yes so it's the symmetric case of Ockham so I think you need to take a longest you need to take a geodesic longest chain that's that that's the analog of a straight line right right and you take a point in the middle then you take the big interval from the full for the full geodesic and then you take the two smaller ones is that right okay so that's interesting because what we tend to do is we tend to start from a point and then look at the gd6 from that point so what you're saying is you take a point in the middle of the GED sick and you are because and you're going backwards up the up the post set so to speak and Falls aren't you intersecting two light curtains fair so like you're taking point in the past point in the future the past kernel the one the future kind of the other intersect them and then got their volume with respect to their distance yes the symmetric case so I suppose I mean just a ball is a problematic so we can't do anything like that so the causal interval is the closest you can get to a ball what why is the okay what why is the notion of having a GV ciccone a problem so we've already you already pointed out that we can't do space like geodesics so if you have a point right but in it but I'm like geodesics in space like do anything but the problem with time like geodesics is that you know that all the points which are proper time you know one away from a point in Mankowski space lie on an infinite hyperboloid so it's not a compact region and it has infinite volume so it's not a compact region at all so for us the way that we're computing these volumes in the space time case in the Lorentzian case is we're looking at a time like vector which is essentially a generator of this cone and we're going um I mean in the graph it's very easy to define we're just going a certain number of steps in the graph Jonathan I you maybe you understand that what's happening mathematically here better but I mean maybe we should just look at what just to understand something concrete let's just look at what the dimension estimate is for this right so we've got a thing here it's a graph so I want to know what the the fae dimension estimate of this thing is and then we look at what our dimension estimate should be for this right so but the point I was making was that okay if if I understand the midpoint scaling dimension construction at all it's that you pick an initial event a final event and then you're counting it to estimate the volume of space time induced within that interval you are counting the number of events that are a number of causal elements that exist in but it looks strictly in between those two in the partial order yes so yeah okay so if you do an asymmetric version of that what you end up with is the analog of a spacetime code right yes and that's exactly what we're those those are the constructions that we're dealing with in our formulation and so the so the the point that's worth making is and you know does it work in the continuum and is there a continuum analog of your idea yes yes just describe it for me in two dimensional Minkowski space then in the continuum in the continuum it's a light gun that how do you so what what do you do to calculate the dimension so so what so what you do is you would get you would compute the volume so in the continuum limit if if I construct a projection of a space-time cone in a particular time like direction then projection oh sorry if I if I started if I start a space-time cone sort of initiated in a particular time like direction in in the manifold right I can look at the volume of that code and how it grows with width or what do you mean space-time direction I'm sorry timeline but you've got an event that is initiating the the light cone okay what are you saying you're saying we look at a time like vector that is inside that light cone and we use that the way that we decide what the bottom of the cone is is by looking at what's transverse to that time light vector is that correct right okay so the time like that it's the cut off it's selling you yes okay actually so it's giving you the base of the cone yeah right so for instance okay in the in the context of a DM or something you would you're picking a normal direction and then you're just constructing a space-time cone in that normal direction okay so the spectacle okay so so you have some you have a space like heiko surface mm-hmm is that right and you because in a curved it does this work in curves because in a curved space of course just having a time like just having a time like vector from at a point doesn't determine the hyper surface there's lots of classroom there's lots of hyper surfaces with that normal right right but so so locally you can still locally define a notion of geodesic normal coordinates even though it doesn't even though it doesn't generalize to the global hyper surface okay right the geometry of the space in in the continuum case the geometry of a space like hyper surface is determined by your choice of gauge variables locally around each point and that's what we need to do a dimension estimate right we only need the local information right right so even in an arbitrarily curved space like hypersurface in say vADM formalism I can still specify a well defined alpha and VTI for each point okay let's let me just understand the flat one where there's no issue so they're in flat space than the normal the time like vector is normal to a flat hyper surface at that point okay and you go up you go up the time like geodesic to some future point down yes yeah right yeah sorry I can't write that so okay and then and then you define the cone okay all right so you have your cone yeah okay right and so then in in ordinary you know continuum GIS as I'm sure you know it's like the volume of that cone is going to be given by t to the D where D is the dimension of the other unseen manifold with a with a second order correction term that's proportional to the time like projection of the space-time ridgey curvature tensor in you know in that in the direction of the code yeah there's also and there'll be there'll be a term that reached there'll be a Ricci scalar term as well as a correction that goes like the Ricci scalar term and the term which goes like R 0 0 yes but but they can but if I freeze them together right right the Ricci scalar is what you get when you average out over all possible time like projections okay if you just pick a particular time like projection you just get the projection of the curvature tensor in that time like direction okay all right Piper formulas the one in either my paper or yours Jonathan there's definitely one in mine I'm not sure it's in yours you can't you know what's in mine I don't know what yeah um shall I pull up yours I you can do um let's see it's just I'm thinking of theirs yeah the doubled example of what you're doing of the formula you're talking about is in four dimensions was first calculated by young near home but then Gary Gibbons and so brings on solid you can calculate it for the full yeah any dimension yeah we for us we don't care about what us better be general for all dimensions okay Jonathan well yeah equation 83 83 yep yep that's that's the continuum equation for normal Romanian geometry right so the the the correction factor for the the volume of the of the conical region with respect to the associated volume of the conical region you would define in Euclidean space is given by one-sixth of the projection that's that's that's the discrete equation you're highlighting there the continuum one is the one above so it's given by one sixth times that the time like projection of the Ricci curvature plus in terms of order of keep a quarter hey does that make sense I'm just surprised that there's no term which is also which goes like the read the Ricci scalar times T squared there would there be if you considered all Gd sit cones right so if you if you if you integrate I don't know whether you've got that I think I have at some place yeah integrate over all directions here if you integrate over all timelike vectors that lie within that light cone then you would absolutely get a Ricci scalar there that make sense so this is this is just thinking about the result that I do know about which is the double cone so so you've got a single cone with a flat yeah just put another coat on top mm-hmm then you'll have then you'll have the double you know the causal interval do we Yuri yes we're here and I don't know whether the formula the cause line I would assume this formula has a term which goes so it has a term the volume of a causal interval is the flat piece which is your 80 to the N but then there are two correction terms one that goes like the Ricci one is like the term you have and the other is like and the coefficient of that is depends on the dimension as well right that that I think what that is I think what do you want to look at that format so there's a formula for the for the volume of a causal interval which I know should we do it cuz it would be related it's there's a paper by Gibbons and Saleh Dukan I'm sorry I can't I've lost control of my I don't I don't know how to spell that second name the Gibbons are no yes so L Oh solo solo Dukan I just said link so that there's some formula there for the cause of volume yeah there we go for the volume of the actually before I forget those you know there's this technology for determining dimension which people call spectral dimension indeed which is you know it's related to what you were saying about geode is it balls but this time you have a random Walker walking indeed and you see how far it's gotten after a certain number of steps oh but that that approach was tried with causal sets you might be interested by stomata surya and astrid Eichhorn and it it's hard to do that the lorentz your nature of the causes that makes it makes it I mean it it's ideally suited to two spaced or Euclidian Romanian geometries that yeah right anyway so no no I understand I mean those remind brain you know what we're looking at is just you know the actual judice it balls are the actual judice it cones so to speak and you know sort of fuzzing them out with random walks is okay but doesn't seem that exciting I mean we didn't we didn't find anything that useful from looking at spectral dimensions um okay let's just find the relevant formula here set of space no three volume area energy conditions looks like an interesting paper but measuring Richie and Riemann tensors okay so this is okay so the in four dimensions that's just the formula for the volume of a four sphere and that's the correction from the Ricci scalar that looks like it's in Euclidean space it is that's exactly that's Euclidean okay fine so this is yeah I mean we have we have I think I have a bunch of formulas just like these in they look just like that area right um okay so let's see they have nice looks like a nice paper um but it's something that just sort of ended here hold on a second higher order maybe it's maybe they announced this wonderfully it's equation 5 is the formula that I think they was referring to okay thank you okay yep so that a and B as these particular numbers and that's a four dimensional formula but there are similar ones in all dimensions but okay so T is the time like vector yes and what is tau that's your tea yeah see your little T okay that's the proper time so that's the proper time from tip to tip um I can't explain why the why the formula for the double cone and the formula for the single cone differs by that I mean clearly if you take essentially the if you do the integral over all of these T mu T new within the light cone you're going to get a term that's going to give you something which is proportional to the Ricci scalar and I don't know I which is what you have to do in order to reverse the current construction because in order to get in order no so my student Ian's job did he did look up cones so yeah so that's it sorry I can't search for anything because you should be able to there should be somebody can you double double click the screen that minimized oh yes okay yeah that is nothing but but you know because of this you know that this curse of lorenzi earnest that Fay keeps eluding to you right in order to construct the you want to get the blessing of Morenci in this if you want to reverse the cone construction you know effectively you you want to you want to have the the future light cone of your first event and past light cone of your second event line up right so if you want to do that by just growing the thing forward what you have to do is from the rim of the first future light cone you have to then construct the whole bunch of future light cones with respect to different time like directions and then find their intersection point I see right and so then when you pick so that then the range of different time like directions gives you up gives you a trace and that trace is what is the result of that is the is what leads to that to that Ricci scalar time to the extent that I understand this construction okay so there isn't any big conflict between these okay so volume no small causal cone okay great okay yeah I sort of recognized that that term comes from the integral term comes from the spatial integral of I thought there was a D that more or less that I didn't think it was quite that it's some cold on okay fly term plus this okay where is the relevant formula here I like you I like your students little little um subscript cone things currency yeah that's cool it's some the okay where is the relevant formula this looks like what are those what are those sectional cultures or something those K's ah so the case those are the extrinsic curvature of the surface which is the bottom of the cone okay right okay hold on look maybe with maybe it's thick yeah so it depends on the circus that you're using to cut off the bottom of the cone okay I mean these formulas so this this is presumably I mean I know I know why there isn't a leading term proportional to T to the D here that could be flat that's just in detail yet yeah okay fine right okay and so it's worth mentioning that of course the extrinsic curvature terms are the only things that aren't constrained by the gauge fair enough but look okay so there's a slight conflict that we should try to understand but Jonathan thought he had an understanding of why I mean why in our formulas we just get the Ricci tensor and here we also have a Ricci scalar term right right I mean there it's been a little while since I told about the double cone construction but at the time that was how I understood it yeah by the way this this c4 coefficient is exactly that's I'm very I mean I've seen that before so to speak that that form and that comes right okay so but okay our formulae a fairly similar let's put it that way we don't completely understand where the trace part where this trace you know the trace part is is how that trace part is working but by the way I mean just to understand how we can derive Einstein's equations I mean what we are basically having to show then is that in the limit of large T effectively that okay that the simplest case where we ignore T minou right I mean by the energy-momentum team you knew in the simplest case what we basically have to show is that in order to preserve a space with finite dimensions this essentially our arm you knew has to vanish because that's the only way to preserve as you limit to large T large you know that this that the only way to get that limit to not blow up and give you a space of different dimension is to have the army new term vanish and so that's I mean it's at least a sketch of of how we can derive Einstein's equations that's vacuum Einstein's equations and the the more non-trivial case is deriving so for us and this is something you might find interesting read for us in the causal graph we actually have an interpretation of the density of causal edges which is the the space the flux of course lodges through space like hyper surfaces is equal to the energy may seem completely bizarre and seemed bizarre to us at the beginning - but but we actually you can actually see that the only the way it works says the you know there's an interpretation of these causal edges that is this flux through space like hyper surfaces is the energy the flux through time like hyper surfaces is momentum the Lagrangian is essentially the divergence of those causal edges and the the reason that you know I made alluded to the fact that you know the final path integral is the same as the Einstein equations what happens is that the the I'm jumping many things but I mean they're these jd6 in space-time that whose whose some that are being kind of turned by the presence of T minou in the space-time and there are also essentially paths in this quantum space that are being turned by the presence of of Lagrangian in in this multi way graph there's many pieces to this sorryi that was that it's all seems rather obvious to me at this as many things I was thinking about what you said about flux and energy so yes is that and then our relations don't they're not they are not so it's a big question how to relate what's the correspondence between the discrete entity and the continuum approximation well how do you we've talked already about dimension and a little bit about geodesics or you know and you can kind of define those sort of intrinsically to be to the discrete entity itself but the you know the what we really want to know is what how the howl events you in space-time you know how do we understand it being a good approximation to the so that of all for us to the causal set the you the possibly the causal graph possibly with the with the with the spatial hypergraph as well so for us we do it by our embedding so the the idea is that continuum space-time is a good approach the renting space-time is a good approximation to our causal set if you can embed the causal set into the space-time in a way that's faithful meaning that the the density of the of the embedded elements is uniform roughly and the order relation between the elements of the causal set respects the space-time causal order the light cones it respects the light construction so that that notion is absolutely crucial to us in talking about what we mean by a continuum regime continuum approximation how we recover hope to recover general relativity so when you said when you said flux of flux of edges clocks of edges let's say across a hypersurface yep it for me you know we could talk about energy I mean you know you could talk about t00 say you'd need to define T zero zero you need a normal vector but it's a local thing right it's local so where where are these edges so for us you see the edges are not in space-time at all they're they're not sings in you know they're not like lines and space-time that they are not embedded there's no embedding of the of the of the edges the edges are just relations they're not you know they're not geometrical there's no geometry there's no line there's nothing one-dimensional about them they're just a binary relation so you know it's a bit one bit of information the way you talk it sounds like you are thinking of these edges as being actual lines embedded in space-time but then you'd need to know where to put them know know that they are like for you they're abstract things but let me take apart a few things you've said so first thing is what does it mean to have this represent a space-time you know the a picture like this suppose set like this you know what would make that be so valid space-time so what you've said is you basically wants the graph distance here to somehow agree with a minkovski type metric distance that correct not no no cuz it's more it's more roundabout than that so so the the number of elements must correspond to the to the space-time volume you know integral of root minus GD for X if it's in four dimensions so then you know that the number of elements will will give you the measure the volume measure for dimensional volume the order relation is the is the causal order relation yep and then the idea is that from that it there will be a unique if it's a good approximation if it has a good manifold approximation they'll basically be a unique way to essentially unique way to embed this thing in the in Manorville but how to actually read off the metric that's that's you know that's that's tricky so we can do we can do some time like geodesic distance in we know we can do it in flat space but yeah apart from that words you know and we have atom it we have a Ricci scalar I think you would have a much easier time if you if you broke space and time apart because I think what you're what's happening to you is that you're you're in the space and time you understood yeah that's that for you is absolutely horrifying yes it's horrifying yes what's interesting to me so the thing you know one of the meta discoveries of this project is my conclusion is there have been two major mistakes made in the history of physics which one is us you need a sometime what's that only two completing only two key ones one of them is that space and time is same kind of thing and half could bundle bundled together the other one that realized oh yeah much more recently is that in quantum amplitudes that the magnitude and the phase should be bundled together in a complex number I think that's a mistake I think the magnitude in the phase of quantum amplitudes come from completely different places that is the magnitude is comes from a thing about path counting and the phase comes with it from a thing about direction of gd6 and there really they could be they can be bundled together as a mathematical convenience but it's super confusing to do that anyway that you know so much I mean you know for us the let's see maybe we should describe how before we get on to that construction can I just quickly ask two questions about this this faithful embedding business and and also about the space-time thing so one concern that when I was first reading about this this faithful embedding idea in in causal sets was how you get around this fact that which i think is just it's now a theorem of geometric topology that you can have there exist lorenzi and manifolds where you can have two complexes that are homeomorphic but which don't have the same combinatorial structure and so the so you add but it seems to me that your statement that if you can faithfully embed the causal set then the Lorentzian manifold you get out is unique seems to depend on that not being the case I met have I misunderstood you'll have to run that result by me again because I didn't so this is about I'm sorry the the main conjecture of differential topology I think I think forgot what it's called like the help the somehow or another yes that's the one okay so that turned out to be false right right right so yeah but you can if we have our own helped for muting but it it's just we just use that term because it means central conjecture yes but but they can they're still exist you can still simplicity decompose Lorentz in space-time into two you can construct two different simplicial decompositions that are still homeomorphic but then they don't have the same combinatorial structure so that sort of mapping sorry con but that those aren't causal sets right I mean they're they're just some you have a simple ischial complex well that's what I'm not saying I mean it's some discretization of the manifold is that right yeah the manifold without the metric is that way yes sure short but but then but crucially you want your your decomposition to preserve the conformally invariant structure which is exactly what a causal set does right look at the causal set is is a discrete representation of the conformally invariant structure of some manifold unless I've completely misunderstood over Lorentz Y and then yeah that so what so what's your point Jonathan so my point is I'm struggling to see how you can make the claim that if it can be faithfully embedded than the Lorentz Ian manifold that gets implied is unique what does faithful embedding mean does it mean that you can assign coordinates Lorentz to in coordinates to every point in this post set and then have it and then what property is true well so as face edit it's then given that we're given that assignment of coordinates it's the property that the number of vertices present within a spacetime region is proportional to the volume of that region okay so it's not a cure it's not it's not the pure metric your metric and that the causal partial order is preserved which is that which is right okay so it's a pure it's a volumetric thing rather than a thing where you're actually asking for for precise metric correspondence you're just asking for correspondence at the level of of little balls around each of these points right right but I I so the thing I'm struggling to get my head around is how what is the justification in terms of geometrics apology for claiming that this faithful embedding transformation is somehow by ejected it's not of course because I mean there's if you imagine a Lorentzian geometry which is Minkowski space with tiny little wormholes you know smaller than of client scale right then the causal step can never see that so the you know the you can embed the same causal set in Minkowski space as you can in in Koski space with with structure on scales much smaller than the Planck scale so so it's not a it's not a the helpful mutant isn't precisely defined in fact part of the struggle of proving it is to actually make a precise statement so what we want so it really something like if you can embed a causal set the same cause of that into two Lorentz Ian space times those two Lorentz Ian space times are approximately isometric right it you know what that and that then you have to design what does that mean how did you know you need a notion of when to Lorentzian space times are close that surprisingly is a big big deal I'm going to define what that means is it's a big deal and in fact people have used causal they've used the idea of Lorentz T and random samplings of manifolds to make metrics on the space of the rents young manifold which sounds a bit circular but in fact it what you're saying is the causal set is a skeleton of the Sorensen manifold you're creating some kind of skeleton from this Lorenza yes at a particular particular scale so it will pick up the geometry at scale you know the not scales bigger than the you know the discreteness go but not smaller than that right so the reason I think that's right discretizing like a Reggie calculus discretization but this is another kind of discretization of the Lorentzian manifold yes you could think of it that way yeah the reason I asked of course we don't think of it that way because like you we think that the causal step is the fundamental entity and the continuum is an approximation to it rather than the other way around right but in order to make this correspondence between the approximating continuum and the real physical discrete thing we use this concept of the embedding it's just um which is just a map from the causal set to the to the manifold right faithful means that it respects the volume measure so the number of elements in a particular region is approximately given by the volume in fundamental units and the order relation of the causal set respects the other like construction but so for us we've got understand has it thinks that there's some problem I mean no I'm not infinite is is you know it's rooted in these continuum theorems that say that the the geometry is fixed by the by the by the volume measure and the and the causal order yes and here we say we have a random sampling of the causal order and we know the volume measure and therefore we should have enough information to give us back the geometry yes but the the reason I brought it up is because in the process of trying to figure out continuum limits of our structures I ran into a very similar issue one of the things that was suggested by this formalism was that in order to guarantee that this faithful embedding transformation is by ejected you basically have to make some additional assumption about that mean by ejected sorry say that again so as in you want it to be the case that your causal set that the the Lorentzian manifold implied by a causal set is unique and well not unique man that if you can embed the course on we'll set well approximate ISIL entry right right and and and somehow vice versa that if you have to you know that if you have to lorenzi manifolds that are approximately isometric they should yield you know approximately isomorphic causal sets right and the one of the things that was on one of the conjectures that kind of arises naturally out of this is that there's some kind of strength and version of a causality condition that guarantees at least approximate by Jack tivity of that of that transformation and so I was wondering you're not getting what the transformation that there's there's an embedding and then there's a skeletonization' right and they are inverse procedures what would you okay so let me tell you about the skeletonization procedure so there are lots and lots of different lots and lots of causal sets which will not be I so you know there won't be ice isomorphic order isomorphic which have them in cost these days for example as a good approximation to it and the way that you get them is is the way that we the first way we generated a causal set in from the two dimensional order interval the Minkowski order interval the one you know the the unit UV square that was by random sampling we just wrapped we just randomly sampled the the you coordinate and the B coordinate and produced ten points or however many points it was so that that's that we call a sprinkling it's a process on process so in two dimensions it's particularly simple because if you can just do it you can't do it in in any other dimension apart from two in that way but the idea is you just you you know you select points at random in your in your region you endow them with the order relation that's that's given by that by the causal order by the light cones and then that is your causal set but obviously a Poisson process is a random process you could get all sorts of different causal sets from that process so so it's definitely not by directive nuts in that way so you can yeah you can randomly sample a Lorentzian space-time to get a causal set that faithfully embeds by definition in and that you'll get many causal sets that do the job so yeah I'm just running this piece of code is that is this piece of code sorry I I is this piece of code Tali is that equivalent to the other code we were running or is it something different is it doing different cuz it could have been I because I was writing the code instead of paying attention I'm not sure this is this is a different construct this doesn't have a probability in it right this is a pure well it's half what's not so because it's into the digits of random integer it's half Oh probably is this is a rewrite that we HAP's more elegant version of this code okay so it's just growing it's growing causal sets and open in the multi way the totally multi way equivalent of this would be that you add every possible consistent X at X but yes I wonder what that looks like I mean it's it's so symmetrical it's probably not interesting but this favorite this families around that point I mean so you're picking a particular random sequence here but the talaq point tally is making you can make a multi-way version of this construction where instead of picking this at random you just pick all possible you you look at all possible causal sets that you can grow and what you're doing then like in our multi way graph you've got you've got an edge for every seat see here you're just having an edge joining these two points in your post set but you can also join this to sort of an alternative post set that you're making if that made any sense because you're for us I don't know whether I can explain this very well I mean these are like our space like space like edges and the other thing will be like our branch like edges so in other words we can just like you can how am i explaining this I mean this is saying this is this transitive percolation that you yes I think I believe it is okay you've got the two to start with and then the to chain again and then the three - yeah that's because mm-hmm so the reason that the two chain failed to change is that when selecting which whether to add in the edges so there should be an extra vertex there I'm so sorry because I did edge add and they didn't happen to be any new edges it didn't realize that should add the corresponding vertex as a singleton so in a sense there's a slight bug that showed up there in special case that you're adding a completely disconnected vertex but as long as there's at least one edge that was picked and this graph will evolve to a new graph I mean it's looking more complicated and is because you're not drawing the house of diagrams but you you have oh right right so I've got I've got say transitive reduction I mean I think that just goes to the to chain the three chain that's the four chain the next one is just the for chain I'm sorry yeah okay okay these are so those are those are similar to what we had before okay yeah yeah okay but I still want to come back to this whole question about you know sort of the correspondence between okay a couple of things that I would love to achieve because because I'd really like to talk about this choice of gauge frames whatever else you call it maybe you don't care about those things and I'd like we have that we have we have one thing which is it's not quite frame but it's gauge is gauge information so so we do you know we commit the sin of introducing gauge a gauge gauge toys which we don't have to you know get rid of it which is that in in classical sequential growth models including trunk the transitive percolation that we're running here there's a total order on the elements it's just the order in which the elements are born in in the simulation and in the rules for that total order is gauge that all some of the information of the total order is gauge the you know if you let's have a look at if you can you show one of those under the percolation things yeah you mean how I think it's the same so I think that's the same if I'm right it's the same ways as from our point of view to just be to say that the fact that you sequentially growing things is a natural fur d--ation but it's just a foliation that's equivalent to any other foliation so we call it a natural labeling so that the total order is what we call a natural labeling okay so this you know if you've got two causal set elements or two space-time atoms which are not ordered in the graph that you know they don't have a they don't have a relation and there's no fact of the matter about which one was born first physics lumber they you know they they're spaced like to each other they you know there is no odds they have no physical order even though when you ran the simulation you did it one at a time so one did actually appear first in your simulation that wasn't physical that's that's just a gauge thing completely understood yeah I think that's that's not that's our analog of your of your foliation right so the one feature of our foliation is you're defining a particular total order here which is zipping around all over the place in this picture that that total order so for example I think I had some pictures I pull them up someplace here oh come on mmm I thought I had the picture I had a picture here which I'm see the same I I usually use these giant screens but if I use a giant screen it's very confusing for um okay what am I looking for I am looking for remember that total order picture okay here a hold on a second um so relatedly this this classical sequential growth model can also be represented in terms of sort of accumulation of Ising spins is that correct that live on the causal relations yes that's right that yeah David and rock pile yeah tried to interpret their the the probabilities as as yeah something to do with yeah with with icing spins on on the on the links only perhaps rather that you know these are the irreducible relations the ones that appear in the has a diagram you know yes right right I'm remembering correct me witches but that's that seems no different to our spatial hydrograph case right it's at some level they're both just kind of fictions that are used in representing the you know the sets of relations that appear in the causal network it's just that in one it's an icing accumulation in the other it's a hyper graph transformation but I think it's the same basic idea right I think all you know it's it's whatever you feel is the right I mean the question is could you have gotten to your causal to your rules for growing the causal graph without that intermediate thing yeah that's a good question you don't really know but you know does it have its own intrinsic dynamics yeah you know or motivation let's say you know without having to introduce this right so look this question about total order so this is this is an example of a total order right so that that one is a fairly moderate you know a fairly orderly total order this is a total mess of a total order I I mean that's a possible total order on this on you know on these things labeled with these numbers right so the point that I'm what one point that I think is important and and I you know kind of hoping that there's sort of a calculus of these things is you know what are some reasonable that's a reasonable foliation that is that's something which is like an inertial frame or some such other thing it's a it's a reasonable way of foliate in this partial order hmm whereas this is a totally unreasonable way of foliate in that partial order do you need it to be an actual foliation so that you're hyper surfaces don't intersect with each other because one way to do it would be to have what we call so it's a total order on the on the vertices on the space-time atoms on the elements of the causal set but it's it's a natural labeling meaning that the total order has to be a it's a you know it's a linear extension of the partial order so I don't I don't think these are either of them that you've drawn is that right oh no no maybe the second one is yeah this was a bunch of equivalence classes right these are equivalence classes of every slice here is an equivalence class of essentially space like separated events yeah that right so you can do that you can do you know you can you you can make these anti chains you can divide up the causal set into anti chains using the concept of level which I think you mentioned already so you might you know that concept right so so it's you know that so the top one is level one yup throw that away take the next elements you know delete that and then you have a new set of minimal for me there minimal for you the maximum a new set of minimal elements that's the next level throw those away take and then you'll be left with a new set of of maximal elements right so you can you know those that I think even in you know I think they are called levels in in the language of partial orders so so that's one that's one so this type of thing would be for us like an inertial frame and we can you know we can pick for example a tip tuner shil frame and we can you know we can do all the fine relativistic things um the question that I have if it's super regular though your and then your oh yeah this is very easy this case I mean the the more non-trivial cases I wonder if I have an example here the I mean the more real cases it doesn't look as nearly as simple as that um let's see what I can have an example here there's a problem of course which is when once you once your boost gets too big then you're going to have you're going to have large voids in your you know distribution of elements indeed I mean this is this problem with the lattice not breaking Lorentz invariant right so this is let's see if I have a nice picture of this um maybe I have a picture I was looking for a picture yeah I mean that's a more realistic causal graph okay the causal graph that I showed before was just a and I wonder whether I have a picture here of what the foliation looks like I have someplace else these but that this is some I mean the thing I showed before is just a toy causal graph this is a more real causal graph where the the possible foliation czar you know it's not obvious let me see if I can find an example here's here's an example for quantum mechanics I think um hold on I need to have a better way to pull up usually much better at time when one thing I should say you obviously think about quantum mechanics and a very different way to me although I think the pass integral a concept of the path integral is floating around yes so that that we have in common that the for us the classical sequential growth models are classical they're classically stochastic there's no sense at all in which more than one causal set is real it's just that one causal set grows there's a probability distribution and it's a you know it's a random process stochastic process like a random Walker just takes one trajectory or a Brownian particles carries out one Brownian one Brownian paths and that's that's you know the Wiener measure it gives the measure on that probability measure on the path space so it's in that sense that we think of the classical sequential growth model and what we conceived of and we don't think that there's anything quantum about it or we can't get quantum a quantum dynamics from it that we think that get the quantum dynamics will have to replace the transition probabilities you know the probabilities that were attached to those arrows in Moscow with transition amplitudes so that roughly speaking I mean I you know that's how to do that exactly we don't know and we yeah we're we have yeah that's more or less what I thought about four or five months ago is that that would be the kind of way that quantum mechanics would come out hmm but what we discovered which is really kind of cool is that that this multi way graph that represents so you're thinking about it probabilistically just imagine you do you you make the graph of all possibilities okay now the what you're doing just like you have observers in space-time where the observer is sort of coordinate izing spacetime according to some foliation so similarly you can have observers in this multi way graph that are coordinate izing the multi way graph according to some foliation and what ends up happening is that so this is an example of a foliation this is actually an example of a measurement being made here these are these are foliation x' that corresponds to super positions of states these these different things here so so think about your your Pro ballistic situation and think about just following all the possibilities and each one of those possibilities each one of those paths represents a different sequence of states and then what you're saying is that if you look at this kind of slice of this collection of states this is like a superposition of those states but what you're doing here is you're making essentially a we call them quantum observation frames they're the quantum analog reference frames this frame is you're making this choice of frames that's essentially freezing this particular state and this is this is like a quantum measurement I mean there's unfortunately I realized as I'm explaining this stuff to you that there's a certain depth of assumptions about what's going on that our there's there's quite a bit to unravel here but but you know one thing that I I'm not sure that I agree with this concept of an observer associated to a foliation I mean if you really mean observer as in you know you and me or an experiment or something then that you know we're we don't have access to a space like hypersurface we don't do measurements in that way you know we it's so it not nice that this is a different situation so well even in the you know that when you were talking about relativity observers don't you know are not associated to to space like hyper surfaces well coordinate ization of the world is you know is something defined by a sequence of space like IP services I mean the observer just goes I know if I have some picture here but you know the observer is just following some timelike curve here hmm but the observers you know way of putting coordinates into space-time is defined by one of these foliation z-- and so what we're looking at is what okay so the thing that we we never said is the equivalence of different foliation x' is a consequence of this causal and variance property which i think it's similar to one of these properties that you have the fact that so I mean you know so I mean that you know relativistic invariance for example for us is a consequence of the fact that the that you can pick any one of these foliation x' and that's what's the right way to say it i mean i'm well let me try to understand your current you go ahead sorry contentment so so one way of stating that the concept of cause limb variance which is a a sufficient condition for the for the property of sort of in any bifurcation in the multi way system of eventually merges that steven already mentioned but one way you can formulate that is by saying that every multi way evolution branch corresponding to every possible updating order they eventually they all yield causal graphs that are isomorphic as directed acyclic graphs so it's some level what what that's telling you is that there's only one causal network up to isomorphism and so given that different different updating orders can be parametrized by different foliation x' of the causal network because you're saying different collections about those space like separated updating events are treated as simultaneous in this particular generation the statement of causal invariance is exactly the statements that you have invariants under you know that even though the ordering of space like separated events can vary the ordering of time like separated ones doesn't and so cause and variance naturally implies Lorentz symmetry just kind of more or less by definition Jonathan that was way too fast if they understood that I'd be super impressed it's I suppose it's correct though right I mean Jonathan's like I agree with exactly you put it at 1.5 times the speed I went to meet a lot of this hinges I mean proving deriving relativity or showing that deriving the Lorentz invariants on all those things day and then let me say what it's well it is like for us in the causes that program and then you can see how you solve this so for us it when we think about classical sequential gross models for example as a dynamics for the growing of causal set we can't even address the question of whether we get Lorentz invariance until because that's a continuum constant until we get until that we find a dynamics which generates causal sets which have nice continuum approximation okay so that's that's where we stand we don't I mean as far as we know although some of them some of the models in the classical sequential gross family are very nice and they have those transitive percolation has this balancing property it produces sort of cosmology no causal sex with called nice cosmological properties we don't know that any of them produce manifold like causal sets so it's premature to talk about relativity or general that to say we've got general tivity back or anything like that okay so that so how have you solved that problem are you claiming that one of your models produces a causal graph proximation which is of the rent seems right I understand the issue so I mean the let me let me give an analogy which i think is useful which is fluid dynamics okay so given molecular dynamics the goal is so molecular dynamics is like your causal sets or our you know hyper graphs and so on then from the from the molecular dynamics can you establish the navier-stokes equations it's a reasonable question right and the answer there I mean I happen to do this derivation back in the 1980s for some simple models based on cellular automata the the you know the fact is that the the emergence of the navier-stokes equations does not depend on the microscopic details of the molecular dynamics you can make conclusions about the navier-stokes equations by knowing general things about the molecular dynamics without knowing the precise details of which particular molecular collision rule you're using and essentially the you know the logical structure what we're doing is to say that there's sort of two branches one is to explore kind of the zoology of different actual rules and we can see that there are actual rules that give manifolds and things like this and the limits right that's branch number one so we can it's not completely crazy because we've got actual rules you can enumerate a million rules and you can find out that a thousand of them will give you nice smooth manifolds in the limit okay then branch number two is to say assume that you get a manifold in the limit now what generic things can you prove in the same way that you can generically prove things about fluid mechanics without knowing the details of about full mechanics from molecular dynamics without knowing the microscopic details of you know whether the molecule is this shape of that shape so that's that's kind of the logical structure of what we're doing but in the in the fluid dynamics case I mean that you have a fluid meaning that there are enough molecules so that you can make an a continuum approximation which we you can describe you know using density and you know so that's that that's the thing that that we don't know in the causal set case you know it we produce a causal set by running the running a CSG model and we don't have it we don't think but we can't prove but we don't think that any of them are manifold like okay so you know it's like it's like you had a molecular dynamics but it wasn't a fluid at all I mean the you know the molecules are too far apart or they're you know they're behaving in right you know so it's a different phase it's clearly a different phase of the of the you know the right okay so that's what else it's not yeah it's not liquid at all there are two branches here one is we absolutely know that we can produce manifolds from these underlying dynamics can show you all kinds I mean I showed you that example of that nice smooth thingy but there many so I would question whether they're really Lorentz young man you were not lawrencium out okay this grid here this there's two dimensional grid which is sort of disappearing off the bottom of the screen yeah that one so that you might be very tempted to think so say that was I know that's not your oh it is a little bit okay so that's gokula graph you'd be very tempted to say that's two-dimensional in costly space but it's not anything like two dimensional Minkowski space because it's the lattice if you would if you were to try to make a correspondence between that and two-dimensional in Causton space you would find that the alert that the embedding of those nodes those vertices into Minkowski space is very far from being uniform meaning if you look at it in a boosted frame you'll find that the elements you know they all bunch up in one null direction and stretched out yes like that rise right so you boost it far enough and then you'll have you know you'll have space-time regions you know with duration five hours you know and and and wit's 5 yeah why do you think my towers where there are no element no and elements embedded at all right where does that bother you actually because I mean you know for us because then that's not it's not other ensign manifold yeah but but but let me point out in the end you're trying to model physics and in the end you know for us for example we have some estimates of the size of these graphs okay so the graph for the universe the causal graph for the universe in some estimate which I don't particularly trust would have 10 to the 400 elements okay so you say if you boost it to you know velocity point nine nine nine nine nine nine nine see or something then by golly you would have you know elements that were some distance apart how do you know that doesn't what happens that is the you know at some point I mean I'm not saying that this particular this particular graph is not a serious candidate for space-time right but but what I'm saying is I don't think you should be upset by the fact that if you have a sufficiently you know you've got two different branches here one is the mathematics proving a limit of you know proving that something in a limit is precisely a manifold that's branch number one that's perfectly interesting thing to do branch number two is how does the physical universe work and the fact that in some weird limiting case the physical universe doesn't behave like continuum space it's just fine because we have no idea how the how the physical universe behaves in that weird limit okay so are you saying for example so physically there's some maximum boost with which let's say your structure is a plank plank exercise little gazelle you know across the Hubble yup you know across the Hubble scale yeah that's the maximum boost that physically we care about any other boosts we don't care about okay yes I accept that and then you can say that the density of this grid is so high that even if so if we do that boost is big but it's not infinite so we do that boost there's still enough points on your in your grid that we cover the wick that we covered is that there yes okay so it's not okay so your your scale is not clunky and then because I'm in 10 to the 400 you're right it's greater than 10 to the 2 I mean so our volume for our volume scaled discreteness you know the volume of the observable universe and Planck units is 10 to the 240 only all right Amir Amir 10 to the 240 so right so the difference for us is so our elementary length is about 10 to the in one estimate which I don't necessarily trust okay there's just one way of estimating the elementary length comes out as ten to the minus ninety one meters okay and the reason it comes out smaller is because our so the Planck energy for example is actually quite big you know it's quite macroscopic it's you know 10 to the 19th gee someone once described it as it's the rest mass of a flea okay that's good it's their energy released in a lightning bolt things like this right so for us that quantity is the thing that you get by summing over all the paths in the multi way graph in other words the Planck energy is the the energy from all the branches of the multi way graph but in our multi rate graph in this very rough estimate we've got ten to one hundred and twenty one branches in the universe today and so our effective elementary energy is ten to the hundred and twenty one times smaller than the Planck energy and that's kind of how how the thing unfolds as being you know as things being much smaller than the Planck length I mean I you know as I say I don't really trust these estimates if I mean I've got some big table of all those things but I don't really trust that this way of estimating absolute sizes but that yeah so it just so happens it is you know with these estimates it's it's significantly smaller I miss from 10 to the minus 93 meters not not a huge difference but but um you know this side quantity is this number of parallel branches in the multi way graph so I mean that that's it makes it you know it's it's less some if you're worried that you're going to run out of boost so to speak in our model you've got a lot further to go to run out of boost hey can I can I ask a related question about sort of densities and discreteness and things so it's a very very naive question just about causal sets but one which we also sort of encounter in our formalism which is inventions right at the beginning this this local finiteness assumption right so the belief that within any region of within a finite region of space-time there is there's a finite number of of causal set elements right um and you mentioned that that was you know that somehow that was integral to to this notion that space-time is ultimately discrete one question that I had was why do you go with a local finiteness assumption as opposed to a local countability assumption or to put it another way what changes in the in the formalism of causal set theory if we replace the local finiteness assumption with the local accountability assumption and I think you need you need to you need to if we're going to hang our hopes on the continuum theorems then we need to be able to recover volumes based on volume sure and unless you want to add extra degrees of freedom if you want the causal step itself to be the most austere simple thing and sufficient in itself to give you the geometry then you need to the number volume relation has to hold because number is the only thing that's going to give you back the volume measure but if your settings you can still have a number of volume relation well because you can still define you can still define a non-continuous measure over a countable set that doesn't require you're suggesting that the number is not finite it's just countable well I'm asking what changes if that if you're if that's the case right if what if the volume in a particular region of space-time if the enough elements in the post set is the causal set behaves like rational numbers rather than like integers so I don't see how the mesh you could recover the measure and under if they're infinitely many was even camp I mean say countably many elements between P and Q then yeah then volume I mean that and you want to associate that region the interval between the P and Q to a finite continuum volume how will you get that continuum bully you can define just using any any discrete measure so the monofin let me see if I understand what you're talking about you say in this construction instead of just going from integer to integer here imagine that in this construction I also go down in the denominators of fractions is that right yeah yeah so it's some something like that so in other words that in addition to I mean this is very different from both our model and phase model I'm not sure but it's I just I find it interesting that that discrete space-time people always seem to consider local finiteness rather than local accountability and I never fully understood why well then you would be attributed to each element different measures right and that yes it would it wouldn't be that each state so it you wouldn't have the you know the the equip the there's I mean our our which you could do it's like a tributing you give it a scale of it you know you put a scalar field on nth or something but you'd have to make it work for every space-time region I'm not sure I mean you right you'd have to make sure that you didn't that your measure was compatible with that every space-time it would work in every space-time region but that but you're adding this scalar field right so it's not it's not vanilla causal sets if you like right right so they I'd you know the hypothesis is that causes such as sufficient so if it's sufficient for it to be sufficient with no additional with no decorations no then you need it to be you need to have finitely many elements in a finite space-time range but what so one of the interesting features of our derivation of gr that Steven already kind of alluded to is that it gives one a way of defining a natural measure that isn't just the counting measure but it isn't a continuous measure either it's sort of it's it's this it's this measure that's given by so you know the if you have a discrete metric measure space you can define you know the natural analog of a volume measure is some probability measure that's produced by sort of rap by growing random walks from a point so if you're in a romanian manifold like you know like a hypergraph model then that's a that's a geodesic ball in the Lorentzian case it becomes a geodesic cone but produced by some by some random walk and then if you take two of those things sort of reasonably nearby you can define a transportation distance between them the Vashta stein distance that then the cut that then allows you to define a measure that sort of that's a naturally induced measure on the space that holds both in the finite case and actually in the discrete countable case and so one of the one of the reasons why we're able to make reasonably confidently these statements we were making about limits to continuum manifolds and things is because we know that the derivations are things like the Einstein equations we've come up with both in the finite case and an accountable case and so I'm curious to know in light because I Jonathan that that's okay so you claim you've proved that all right hey good well III John Olivier proved it and I used it but does okay yes but so given that the causal set program has been around much longer I I'm I'm curious to know so apart from this problem of defining a compatible measure you know scalar field thing which I think we can conceivably find ways around whether there's anything else that breaks in the formalism when you go from finiteness to accountability maybe this is a bad question I wish I I think they haven't figured it out Jonathan I don't think they've thought about that because they've had reasons to think that that isn't the direction to go I mean I think look just just to go back to the sort of logical structure question of what what are we proving and how so to speak so as I said I think there are two branches one is the mathematical limits story and the other is you know what do the actual dynamics do and you know one thing we've done which I don't know whether you guys have done you know we've done lots of essentially zoology of what actually happens with these rules right so I've looked at you know billions of these things now and so we know you know how many of them give manifolds how many of them give infinite dimensional space how many of them terminate like I was just looking actually my activity for the last night I'll show you this is my um my homework from yesterday was trying to understand space-time singularities so this that that's a that's a cosmological event horizon in our causal graph so this is time going down and this is basically two regions of the universe separating never to communicate again and so one thing I sort of understanding so there's a there's a light cones right and they don't communicate again and then so I was trying to get a oh that's at that a weird thing that happens in our situation that doesn't happen in general relativity which is that the universe actually disconnects the spacial hypergraph disconnects so that's not something you can get in in ordinary manifold you know in ordinary general activity with manifolds and things but but you know in in where I have this ok these are black hole like event horizons where basically you can have and so this is a causal graph right so this causal graph has the feature that the light cones which are rather trivial in the case of this particular causal graph can there are some light cones that spread everywhere and there are other light cones that get trapped inside a particular part of this causal graph and so you can get to some place here well that this is kind of the causal can maybe this has a name actually I'd like to know from my homework here whether this has a name this is a summary of the causal and causing calling it the causal connection graph so each one of these blobs is an equivalence class of nodes in the causal graph that have the same future that share their future light cone at infinity does that make sense so in other words if there was just one if the universe was all causally connected then every point in the universe that you pick if you go sufficiently far in the future its future light cone what will be the same as the future light cone of of every other of every other point in the universe that did that make sense have a black hole then there will be the outside of the universe where all of those where the weather light cones eventually correspond sufficiently far in the future and there will be but but those that they'll be sort of those the those points can affect the points inside the black hole but not the other way around you basically you've collapsed all the points that you've cut you've collapsed the full kind of causal network into clusters every cluster that is strongly connected you've collapsed one I see yes except that they're strongly connected with respect to this future light cone um I don't know whether this has a name I mean I've not really seen these things in in general activity they're a limited number of these pictures that would ever arise so maybe nobody gave them a name I mean I think you know in out so this is an example so that's what happens right so this is an actual rule this is its causal graph and this causal graph has the feature that it eventually develops essentially a black hole because there's a if you start from here you can the the future light cone populates the whole universe if you start from here you can't escape from this region of the universe and I mean in in I mean these are more realistic causal graphs then we were showing before that there's a rather exotic causal graph I mean again not that's a pretty simple rule as the rule as the causal graph um whether I have C what was my points here well it's kind of fun because like a space like singularity for us is just a termination in the in the in the in the course of graph that is all causal events eventually terminate and there is no future to this event so time the gd6 are in the gd6 always stopped um i know why i was bringing this up i mean that this know your models know that this is a special feature of models in which the the causal graph terminates it's a special feature that's that's not a combat ok someone's I'm doing something right yeah classical sequential growth models they go forever and they they also they have a nice feature which is that the the causal the causal set that you grow has no maximal elements when the maximals the meanings the future right so so time can't end on any on any time like pass it all there's always an element to your future so right weak or is it causal immortality it means that you'll cook your your causal influence will last forever even though i mean we may die but our future light cone will last forever yes that's exactly right yeah right now and and right but our own foliation of space-time will not last forever because you know because we die so to speak in other words we are we are defining a certain you know sequence of time time values in in space-time but yeah I mean this is an example here's an example of one which which doesn't die right I mean this is that this causal graph just goes um and about I can show you actually the statistics here this is okay these that these are more exotic causal find any ones that have that bouncing character that they they grow and then they collapse to a point but but don't end there but carrot continue on because that's a particularly interesting case for us so ok well that's that's so for example and then what we found what we found for those cases so these are this is all within the family of classical sequential growth models so there are classical sequential growth models which have these posts so they almost surely so you know that means that we probability 1 any calls you'll get dis hits random so you get different cuts any particular model is round them to get different calls or sets if you run it you know different write different runs but every time you run it you're guaranteed that there will be infinitely many of these posts that's that's there's a class of them that do that it sounds like a typical percolation Theory results yes exact so so that we can think of these you can think of classical sequential responses as generalized percolation but what happens when you have one of these posts is that after the post the dynamics of your of your growth has a different set of couplings a different set of parameters than it had before so effectively there's an effective post post dynamics so after the post the universe behaves as if it's and it's being governed by a new rule and the new rule is very simply related to the old rule by a renormalization so so it's like the universe when when the universe collapses to a single point to a post the subsequent evolution is an effective dynamics which is a realization of the of the of the dynamics it was following up to that point and it can tune itself so you get I don't people call you know self-organized criticality these sorts of ideas on now coming into play where you know the there'll be it seems as if transitive percolation is an attractor in this in this class in this class of dynamics that if you go in in the cases when you have infinitely many posts because after each post you get a renormalization of the clock things and it looks like for those those couplings are tending to transit a percolation with very small P and it and that has the consequence that each epoch between the two posts gets bigger each time and laugh for longer so you get a very suggestive kind of cosmology which may may Raphael has suggested be an start of a of a solution to cosmological fine-tuning problems like the flatness problem so I'd be interested in your models if you get do you get these sorts of collapses okay so a couple a couple of points to make so first of all the fact that your behavior after one of these posts and I'm just looking at share my other screen I have a giant collection of these things which I might be able to show you here except that I need to be able to get that screen hold on I have a solution potentially here to how to do this um I've got I've got a giant array here of 4,000 of these things showing all kinds of different behaviors but but it's on my other big screen and it's it's some no you can't see it it's on a different speed that I'm not what's that I can't read the writing on my screen it's too tight oh no it'sit's the this some sorry it's taking it's it's some it's a giant probably half gigabyte notebook of stuff so but um oh wait a minute just finished rendering let's see if I can oh come on alright here let me at the medium it's really really slow here yeah but but but the basic um so one thing about your your comment about your see for us every event here going down is the same in some sense whereas it seems like with your way of doing constraint with your construction as you go down in your construction of the post set things are changing you know in other words it's not like there's a tiny invariant dynamics it's like as you do this this some sequential growth model your your your post set is progressively getting refined as you go down and every time you hit one of these posts there's some sort of new level of refinement that's happening is that true for us I mean the the okay let me see if I could work I know that it depends what you mean by event I'm an event of the birth of the atom that's the same it's just you know an atom comes into being that's but but then you know it's the bigger the causal state at the later the stage the bigger the causal set that you have the met the more possibilities you have for the for what right because newly-born element can be joined to so in that sense yes it's you know it's it's getting increasingly more right that's your universe your whole universe in every in every element see for us these these events are just things happening somewhere in the universe whereas for you every box in your picture of your of your causal set is a whole universe is the whole history of the universe so as you go down in your sequence okay well it depends what yeah which which could there's so many calls all set so it's paska which is itself a causal set the causal set of causal sets all right you know if you have to be clear are we talking about casca where you're you know you're moving through POSCO or are we talking about the causal set which is growing no it's a different picture right so the colossal set which is growing that's the one that has the posts I get it I understand yeah so it's a the causal set that's growing is our space-time causal gruff and the post cow is our multi-way calls autograph okay there is our energy corresponds between those things why is this I'm sorry this is not going to work I'm I'm trying to why is the soul but I think it I think it is true that our mod are that CSG models are more complicated than your update rules because your update rules you know that each one is simple and you just keep doing it right yes but but the classic in classical sequential growth models there are a set of parameters and those but there's one for each stage and it's that it set our dynamical space it's the space of dynamics is the space of these parameters so it's a you know yeah right it's infinite dimensional for example it's a projective space but that's that's not relevant bit right no chemical but it's a it's an infinite dimensional space or dynamic our space of different dynamics but this this this renormalization acts on that space of dynamics you start with a particular dynamics and then it moves you - it moves you to different dynamics right but the reason that's happening to you in my view is as you go down in this in this you know as you go down you know as you go forwards in time so to speak of causal set as you go further along in your causal set you mean you will grow it as we're growing the causal so as you're growing the causal set yeah you have already accumulated a big causal set so when you get to one of these posts what's happening is surprisingly you get to the point where your causal graph has come down to this neck but that neck is what what lives in that neck the the in the in the sort of the beads that live in that neck those beads are now much bigger than the beads in the previous neck because you've accumulated more space-time right by that point and so to me it's not super surprising that by the time you neck so to speak that the this realization statement of yours is not very surprising to me and let me it's but why because my claim is it has to be very special to get to the point where you have next off the causal graph there has to be some very special property of your increasingly large sort of representation you know you're increasingly large causal set and I'm I'm what am i I'm not sure it sounds like your intuition is right so how a post is a very special thing so that in that in that imposes conditions on the future growth exactly yeah and those conditions are exactly this dis realization right so I mean for us I mean these are I'm just showing you because I didn't manage to get my whole giant collection of I could probably bring it up actually my giant collection of causal graphs but I mean these are all kinds of weird things I mean these are different this particular document is about causal graphs that show singularities in other weird behavior um so those were the ones that are shown before okay so this one for example it gets a little bigger then it gets a little smaller these ends here are like space like singularities they once you get that once the light cone if you pick that as you start for your light cone time stops immediately um and I think I might have some pictures let's see what I have anything like what you were talking about here for us a singularity would just be a part of the causal set that has no manifold like approximation so that that's all it means right so you might have a causal set that's manifold like and then stops being manifold like and that's a singularity that it might continue on so in fact we we you know a post is this is a singularity a Big Crunch Big Bang singularity it's there's no manifold like description of the causal set around around a post but it could be manifold like later a manifold like for the actors so that presumably that means you have you make no distinction between space like and time like singularities no I think so so what it would be a difference in a causal so I think you go that's right because they've got because every every point there is a whole universe yeah well let's see so timelike singularities yeah I suppose we hope that there are none but so but the way in these systems I think that I mean as of two days ago I've decided that time like singularities are what happens in a that that a space like singularity is time ending and a time like singularity is space ending basically so in other words this would be for us this would be an example of a time like singularity where check which i think is this type of singularity that they was talking about in the causal set that's what I think - yeah so which is which is why I was asking I mean - right the reason we consider this normal form case is precisely to distinguish the time like and space like cases and I'm wondering how that happens in a causal set see i think you what you've got for me the time like singularity is like what you get and you know the kerr black hole or something where you are where on the other side you've got a whole other universe coming out yeah that as far as we know that all pathological right and there's negative mass Swart shelters timer singularity but yeah very much that that doesn't happen right exactly but I mean then the ones that you know the ones in in in the time like singularities inside current and horizon Nordstrom black holes they're hidden by and when we think that they're they're not physical because they're in the region that's inside the Koshi horizon which you know most people think is unstable so that's anyway it's yeah there's no strong I mean with no kind of really definitive confirmation that that time like singularities you know can be physical also right right right but I think as Jonathan saying and I think that's right what you're seeing in a Big Crunch is essentially the emergence of a universe after a Big Crunch it's kind of it's you know it's the other side of a time like singularity wouldn't that be space like when the Big Bang is us wouldn't you call that a space like thinking about the Big Bang is it it depends what you mean like means it's sort of space like means time ends or begins yeah the Big Bang is both the space like and the time like singularity uh okay because right so be my understanding of the distinction was that if you have a space like singularity it's saying you know what once once you're at a certain point you know space like distances don't really matter that there's that in fact there's only one point in space-time and it's in your future and this is the singularity and the so it's it's all the really matters is your time like motion and then the time like singularity cases that is the sort of 45-degree inversion of that right you're saying that that you reach a certain point and suddenly in all possible spatial directions there is the time like singularity and so suddenly that the space like motion dominates completely over the time like motion and so in the Big Bang case both both things happen at the same time it's obviously hard because they are singular so it's hard to describe in but wouldn't you know wouldn't if you just delete a time like line through Minkowski space that would be a time like singularity or the you know R equals 0 in negative math short so I would call that time a time accident cigarette because it it's kind of as if it's on the time like line R equals zero you know and then obviously it's singular so you know yes that Riesling brain but if you do that if you do a timelike line is called the Big Bang singularity space like yes you're deleting both time like ants before the Big Bang you are deleting both time like and space like lines you're losing all lines I think you guys are convincing me which I had already come to the conclusion this distinction of time like versus space like singularities is a bit of a fraught distinction that that you know it's some I mean oh by the way this this might just a music this is this is looking at 4702 possible rules of a particular simple kind this is essentially the the aggregate causal connection graphs so these are universes which have no black holes these universes that break into a cosmological that have a cosmological event horizon and break into two pieces this is universities that have a one black hole in them those universes which have two black holes in them you really break into two pieces it's more like the trousers than the cosmological event horizon it doesn't imply that space is disconnected this isn't space being disconnected this is just causal gulf being disconnected we can also get trouser universes and that they're a special case of this thing okay what actually I don't even know in general relativity is there a notion of disconnection of the you know of two separated pieces to the manifold yes but you have to you have to violate a lot of global structure considerations about hypervelocity of your manifold but if you do that you can say mentioned those this class of solutions which in these trouser universes well I mean if you're just messing around with the metric how do you get the thing to separate into two pieces I don't met that you'd have to change the mouthful that the thing is sitting on right but it's just it's not all dislike a pair of trousers you have to oh I see I see a Lorentzian metric becomes degenerate at least one point oh I see I see so the the underlying manifold the topology of the underlying manifold you've determined to be the trouser yeah yeah as long as the connected components have uniform dimension you're yet like that's that's perfectly consistent with your money in geometry right so one thing by the way that can happen for us that a bet can happen for you as well is that space may not be fixed dimensional so for example for us one of the things we've been spending a bunch of time thinking about is you know normally you formulate the Einstein equations as being something in you know three plus one dimensional space or something but what happens if it's you know deep if there's a if there's a perturbation in the dimension of space-time how does that what does that look like and you know for example how do you formulate Ricci curvature and things in fractional dimensional space and then more exotically what happens if in the universe there's a mixture of dimension if most of the universe is three plus one dimensional but there's a lump of 3.01 dimensional space somewhere in the universe um and it's a nice thing about fundamental discrete structures like these you when you don't have you can hope to show to explain what by our universe is 3 plus 1 dimensional rather than having to put it in as an input absolutely no I mean it you know I'll set up right it had better be the case that we find ok so the situation right now is we have all these properties that we can sort of prove in the same kind of way that you might prove flow dynamics from molecular dynamics so they're in a sense generic proofs then we've also looked at lots of specific rules and we have a bunch of properties that we would like those specific rules to exhibit like that it limits to a three plus one dimensional manifold things like that um and for different rules we've managed to find different rules that have desirable properties but we've not found one rule that has all the desirable properties so to speak so we can find limiting two manifolds we can find causal and variance we can find various other properties but we don't have one rule that has all the necessary properties but you know that the structure of you know what we're trying to figure out is is a remarkable number of things and I suspect this is true for your causal sets as well okay so you I think are concerned that your causal sets might just completely fall apart they might not limit two manifolds at all if you know for us we can definitely find things that do limit two manifolds so that gives us the confidence to say assume that we limit to a manifold what can we then generically prove and you know I suspect I'm I'm much less pessimistic actually given the constructions you were showing about some about causal sets that you can't find reasonable smooth limits but but maybe you can't I don't know I would have you have you done experiments I mean have you actually just done the you know yeah people have done experience but we need more experiments so the sort of the sort of zoology you're doing we haven't done but people have done so David right out and I think Maqbool Achmed they studied transitive percolation and they they showed that it had some characteristics of dissatisfy in when they were cleaning it was dissatisfied but they looked at something like the the some some statistical right no you know the number of some abundance of links compared to something or other I can't remember exactly what it was but they did some they looked at statistics of something and they found that it it looked like what you would expect from if these had been random samplings of de sitter space in I mean so all of that kind of stuff needs to be done more that was just rattlin it was just transitive calculation and there's this whole family of CSG models that need to be looked at and this renormalization need to be studied so this is a I admit that why don't we see the computational zoology seems to be less commonly done than you might think I mean I've been doing it for years on different kinds of models these models are really quite difficult to do it on because you know I've done it a lot on models where the visualization is kind of self-evident and these models the visualization is non-trivial because you know you're doing graph layouts and things and you might you know this graph layout is somewhat arbitrary um but but um you know I just want to bring up one more thing which is this this whole question about the sort of calculus of reference frames which which maybe you guys don't care about but I'm I'm kind of assuming that you must be interested in making foliation z-- of these causal sets is that a true statement or do you not care about that for some reason it would be nice to make contact with what other people do in quantum gravity I mean so much of there are many approaches to quantum gravity which really very much focused on space so loop quantum gravity is you know it's it's so come on it you know it's based on canonical quantization you know you take the hamiltonian yep you you know you forced General tivity into a funnel tone Ian's system you right so you quantize it allowed Dirac you know you find that the dynamics is all pure constraint you have the problem of time anyway so but their dynamics is all about the degrees of freedom are all spacial degrees of freedom so in order to make contact with what people are colleagues do and other approaches it would be nice to you know to say well if you want to think about space you could do such and such and such and such but causal sets really do very very badly anything that's to do with space it you know that so that I'm into the simplest possible thing you could think of in doing what would be a space like hide the surface in a causal circle it would be an antichain it'll be a set of elements that have no causal relations between them the trouble with an antichain is that has no structure so all it has is cardinality any anti chain of cardinality n is the same as anything other NZ chain of cardinality n so there's no geometry in in an anti chain so people have done work so summer T Surya a David ride outs s des major have studied taking an T train and sickening it slightly so going up and down a few steps in the in the order to make a thick and 9g chain and from that you can glean information not we don't know whether we can guilt glean the full geometry the full intrinsic three-dimensional geometry from it or spatial geometry n minus 1 dimensional tree but they showed you can glean spatial topology so they have a way of reading out homology groups for this for these thick and 90 change which is very clever very sweet that's interesting yeah but yes it but basically yeah because an anti chain has no structure but it is the sort of obvious thing that should correspond to a space like hyper surface and of course also you see just doesn't do space well I know but so they might my point is that's why you know that's why our model makes sense because our model has something that that's why just producing the question believe it you don't believe in space yeah well I I mean I understand that but but my point is you have a construction method for your causal graph right which is your you know your progressive construction method so think about everything we're doing with the special hyper graph is just another construction method yeah I can see that give you an interesting class of yeah ways to then you can filter it's deterministic as I understand it gives you the same yes right it just but eh it'll give you each model gives you a causal set is that nice about that each model and each evaluation order gives you one causal set remember there's this multi way graph that represents all possible orderings of the underlying updates so on a particular branch of the multi way graph or with a particular strategy for which updates you choose to do you will get a deterministic causal set but the set of all causal sets which is and actually they all get knitted together in this multi way causal graph is something more complicated but the space-time causal graph is what you get on a single branch of the multi way system that means a single sequence of updates and and you know often like like in the in the code we have for doing this like max wrote you know version of this which amusingly has that uses this method that we call Max scan which could have stand either for max or for maximum scan which is saying at every step try to do as many updates as you can try to do as many non-overlapping updates as you can that's a particular that's a particular sequence of updates that you're doing and that will generate a particular causal graph so given a particular strategy for updating for the updating order you will get a particular causal gulf and get the same on every time because just a deterministic rule but if you choose a different updating strategy which corresponds to a different branch in the multi way system you'll get a different causal graph but Crush critical fact if the system is causal invariant every one of those updating orders will give you an isomorphic causal graph so in other words you you can say well it's terribly arbitrary which order you do the updates in but it turns out it doesn't matter if the system is causal invariant then it's deterministic it gives you the second yes absolutely yes yes so I should have said that I'm sorry I should have said the correct answer to your question is if it's causal invariant if it's causal variant then there is a single causal graph that comes out in general there could be this whole Multi multi way collection but but yes and so you can think of what we're doing as you know for your purposes okay since you're a course of golf person causal set person for your purposes everything we're doing with special hypergraphs you can just ignore it's just a construction method um then look at the causal graph that comes out but one thing that is convenient about the causal graphs that come out is they do have a natural foliation to them because because of our for example that that max scan updating order we have something where where we can essentially have where there's a natural layering of our events so to speak so our you know there might be many possible anti chains but we have a natural collection of anti chains that are defined by by this particular evaluation strategy order and so and then so the thing that I was interested in is if you look at the space of all possible foliation z-- right there are some foliation z-- that are more reasonable than others like you're saying you can't tell which are the reasonable affiliations we have some ways to tell what are the reasonable foliation x' but a question of great interest actually for the distributed computing case is to be able to characterize what are the reasonable foliation x' i mean in other words you know you you could make an anti chain here that's incredibly weird complicated anti chain right and you know that doesn't seem to be have any regularity or any sense to it um and that's and you're saying it's very floppy because you can pick any of those anti chains um what we're saying is there is a natural because of the way we constructed our partially ordered set there is a natural foliation of that partially ordered set one thing I should mention is so I meant so I said that my colleagues had done this work on sick and auntie chains so that's you know similar tea cess and David and they what they found was that if the causal set is manifold like so if it's not manifold like none of this is relevant anyway because you won't construct any any nice geometry or topology or anything so but in the case that the causes that is manifold like if you choose a wild auntie chain to start with and then what they did simulations on a 2d cylinder for example so that has a natural foliation which is just the one you know the horizontal ones the circles they're you know the homogeneous services so if you choose a wild auntie chain meaning I sort of really zig zaggy one to start with they're sickening procedure smooths out it so u s-- let's say you sicken to the future you sicken to the feet which for me is upwards you sicken to the future and then that the maximal elements of your sickens auntie chain tend to the natural foliation so it it so they have one they have a way via their thickening procedure of making you know a more natural foliation one that's you know smooth and in this case it picks out the one that's really that really is physically preferred so physically you know special I would think of thinking they cost in Minkowski space it would it would pick out the planes as well but so the thickening procedure is essentially making JD sick balls or geodesic cones is that right is doing a little bit of that yes it worked it so thickening works because you have a whole anti chain so sickening around a single point doesn't work because of the Lorentz youngness it basically because it's this whole thing about the number the points which are one you know one second of proper time away from a single point or lie along a whole infinite volume hyperboloid made so that that's the which which asymptotes to the like currents so that's so that yeah Judy's it time like ball just doesn't you know it's not a nice thing at all but when you have a whole a whole anti chain then you can thicken to the future so you just take you take all the points such that there's a fixed number let's say five elements in between it and to the future of this anti chain because you've got because it has to be a maximal anti chain one that you can't you know you can't add any more elements to and it remain an antichain but that that that procedure is possible because you have this whole anti chain now so you just take all the points which are you know far such that the volume of the causal interval it's fast causal interval to the future of this anti chain is five or ten or fifteen or something and then that sickening the the maximal elements of that sickening tend to a nice smooth type of anti chain hypersurface yeah I think I mean so that for us our max scan procedure is similar to the maximal anti chain business I think okay I think that's that's what it's the same same idea and so what what we're doing then is when we're making our what we would call our cosmological rest frame you know foliation I think that's the same type of thing as you're talking about that by getting a sequence of maximal anti chains um but I mean the the thing for what it's worth I mean the the you know so you know this picture okay you could make if you were doing distributed computing I hate to come back to this but I'm just trying to understand it you know it might be something interesting because there really are people who really care about this and they might really care about the things you guys have figured out about causal set um you know each one of these nodes could be some operation happening in a computer and what's happening is your causal connections are you know this particular operation here needs input from this operation here I mean that's kind of the way of but but you have these different operations which can happen in many different orders the only thing that we know about them is that this operation has to wait for this operation to have happened because it needs to take its its input is the output from this operation and so people have a very hard time understanding how to do programming in a situation where there's sort of asynchronous things happening so what tends to happen is people try to resolve things to the point where there is where you can say Oh everybody already finished you you've got a foliation basically we're like what did one of your max Monte chains where every everybody already finished their work and now we'll go on to the next step and so on but it's of great interest to try to figure out is there a different way of essentially foliate in or organizing this partially ordered set so that you can think about what's happening and the partially ordered set without without having to for example say everybody already finished their work now go on to the next step and so one of the things I've been hoping is that we'll be able to formulate essentially a set of primitives that describe possible ways to do this foliation that will be useful for thinking about distributed computing and I guess I had wondered whether there was sort of technology for that that had been developed by the causal set folk and I'm kind of getting the can getting the impression that there really isn't at this point that you might find that useful if it existed but I also would like to point out you know if you if you find your average distributed computing person and you show them your causal sets they will care about them that's great and uh it's some Jonathan Oh max do you want to add anything about this some distributed computing correspondents and things I I don't have anything to add and what Roadmasters I'm pushing to tell go telly telly really is very much involved in this stuff I mean this has been this problem of distributed computing and how to think about it is really an you know it's a it's a sort of major unsolved problem and I don't know I've I've worked on it off and on since the 1980s and I think finally with this sort of you know the understanding of these ideas about reference frames I finally feel like there's a chance that one can actually make progress on it um but and I think for example your notion of posts I bet that that has an analogue in distributed computing I bet that that is what the heck would that be in distributed computing um if you're both linic what's a low essentially a bottleneck it's a bottleneck oh I see it's a bottleneck we're all we're that's right that's right it's a case where most of the processors can't be doing anything right right you're all depending on one one person right so that's it so in in in distributed computing your your posts will be considered a bad thing because because what's happening is instead of all these computers being able to operate in parallel you're down to just everybody you know that one computer has to do its thing nothing else can happen right yeah and so but you know so in a sense one of the things you might ask is what's the density you know as you go through one of these one of these partially ordered sets you know you're yeah you're parallelism you're you're parallel efficiency would be the the width of your partially ordered set in some sense and where you get to the post you've got them you know the maximal inefficiency in your parallel computation um and things are known for transitive percolation so the width the typical width is one over P if P is the probability then the typical width one over P and the the time between post is some-something exponent you know e to the 1 over peel e to the PI of appeal something I mean you know if P is small then it's a very long time so you have wait a long time for the you know the post so you can be going along quite happily for a long time without the post coming and ruining everything but the post essentially what you're saying is so you're saying the average width is 1 over P then is that D so if I look up at one of these pictures that we made you know this average width if we change that point five the average widths will will scale roughly like 1 over P so that that would tell us there's probably obvious that that term as that's the parallelism that's the amount of parallelism that we can get so though P is telling us as we as we decrease P that that's probably obvious that you can get that the okay so that's a that's a measure of the amount of parallelism and the probability so so is it is it the case that these posts occur basically if we just look at the number of possible post sets that the posts occur well let's see you're saying they're exponentially rare so it's something where there's some ordering that you know we're only one out of that possible ordering we'll give you this post somehow yeah anyway and it's me that you know it's it's it's it's jogging along the number of maximal elements is roughly one over P which is you know so the new element is born and with probability P it attaches itself to you know to two to one of them and it's it's likely to attach it so it's more than likely to attach itself just to one of them if there are one over P of them and it's attaching itself with probability P to you know to each one in turn then you know it's so that you know it's just replacing one of the maximal elements with itself so right it's you know it's roughly so it just it and then just at random you know there's some small probability but finite that the next element will attach it will will be above all of the maximal elements that are already there and then that's that's opposed but and then subsequently the you know the other of the elements above muscles must must choose the that post to be in its future as well so you can show that there's a finite probability of that happening you know we are probably about to run into a space like singularity asset it's it's some this was very interesting and you know I mean I I think there's probably more that we can learn from causal sets I suspect that causal sets could learn something from the fact that we actually have concrete models that generate these things I mean it's sort of an old you know think of it as just an alternative generate ignore everything about our model except the fact that it can generate causal sets um and I think I mean we didn't really come to this whole question about how energy works and path integrals and things that's a whole nother level of discussion but i i'm and and by the way that i think there's another interesting we also didn't touch on which is about locality so as I understand faith at the the phase transitions that you were talking about with respect to these cosmological models they depend on sort of relationships between the temperature of the effect of temperature and the degree of nonlocality is that right Jonathan let's not start another is that I think it's very process and I think when I mentioned very briefly in an email to Steven that it that the I think this is something which physics lacks it lacks a sort of this active dynamic aspects of our experience so we you know we don't experience the world it all laid out once and for all the experience time passing things happening and the fact that there are these processes these event that happen for you these these space-time atoms that are born for us I think that that's that gives physics something which has the potential to go physics something that it lacks which is something to coordinate with this experience of time like with time as the passage of time which I think that's yeah that's something to write my own highlight I think for all of us right my only comment is don't insist on making the whole universe at every instant I think that's a mistake I think that they you know in other words the the your construction way you make a whole universe at every at every moment I I think that unruly whole universe at every moment because we just make the new single new element no I understand that but but but you're imagining that you're imagining that every at every step you've kind of got the whole space-time universe you're not you're not imagine nothing has ever forgotten in your models right the past is real yeah right whereas for us in a sense this spatial hyper graph is getting rewritten and the fact that there used to be some spatial hydrograph that had some different form well that's really in the past for us and it's forgotten in some sense it's not forget it we don't forget the past it has it influences are still so surely it's real exactly but but that's exactly what the causal graph is showing you is those influences of the past but you don't have to have preserved every aspect of the structure of the past all you have to have all you all you keep is the causal graph is my claim so in other words your memory well that's all we have the causal graph is all we have so no I understand that's real right so that's the past is real the well in any case it's some yeah no I mean in terms of the the dynamics of you know I have to say I mean I worked on this back twenty years ago or something more than that and somehow you know describing all the stuff to people in physics community for example I my statement I wrote some historical thing in my statement was you know I've known lots of physicists right and and so the my statement was in the period of eighteen years I do not believe I managed to keep anybody's attention and describing these models for more than fifteen minutes and so what's that what time did we start no I'm saying I'm saying that was the past of this story right so I'm saying the the good news is that I think now you know we have enough stuff that sort of assembled that all fits together that term you know and I'm I'm I'm really quite comp there's there's a huge amount of actual physics that's possible to do from our models now now some of those results may be also completely valid for causal sets and it may be that you know that once you know we have a as I say a construction method for these things it may be that you can get many of those results just by looking at causal sets of a particular type let's say but Tim I mean right now we have a really wonderfully fertile sort of opportunity and you know it's so by the way if you have any students who should come to our summer school please send them because we're going to I I'm pretty sure we're going to be able to know there's a lot of wonderfully low-hanging fruit in our models so to speak to work out and and meanwhile by the way I'll send you this notebook and I do encourage I mean you know in terms of this some if you guys maybe will see if some students at our summer school would like to look at sort of the zoology of causal sets because that seems to be a I think there's there's clearly there's there's low-hanging fruit in the zoology I think if people haven't done that many simulations and so on but Tim anyway we should it's very late for you and I'm sorry we went on much longer it's great it's it's fun to just been a pleasure yeah really fun yep there right well you should we should wrap it up here and thank you and as I say well actually this number will get posted because we do all the stuff in this very open way you know I was not paying attention to him oh my gosh it's some there were all kinds of questions on our live stream which we're not going to get to I'm sorry about about thermodynamics of black holes and how that relates to all the stuff sorry we didn't get to that we were paying too much attention to each other we didn't we didn't we ignored the the people on the livestream apologies for that anyway well thanks a lot I'm nice - nice to meet you yeah nice to talk really fun and to be continued another time yeah take care everyone good night

Channel: Wolfram

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Keywords: Wolfram, Physics, Wolfram Physics, Wolfram Physics Project, Stephen Wolfram, Science, Technology, Wolfram Language, Mathematica, Programming, Engineering, Math, Mathematics, Nature, A New Kind of Science, NKS, Computer Science, Philosophy

Id: -3YLS9Y8eZE

Length: 211min 32sec (12692 seconds)

Published: Tue May 19 2020