The Wolfram model - Jonathan Gorard

okay so it's a pleasure to introduce our today's speaker Jonathan Gordon from the University of Cambridge and the Wolfram research company and today we stage is yours thank you so much so your first of all thank you so much all of you guys for coming thank you to the infn for actually organizing these or for hosting these wonderful at Newton 1665 seminars and thanks so much to Alfredo and for all of the organizers for actually inviting me to come and speak I feel very much out of place being here so not least because I suppose I should apologize at the beginning I'm very much not a high energy physicist I am a mathematician but but nevertheless I hope I can I can say something that might be of interest to you guys and my apologies if not so this is intended to be a very short necessarily rather terse but hopefully it's self-contained introduction to this thing called the Wolfram model which is a new discrete space-time formalism that I guess you can think of as being kind of a sibling of things like the causal set theory and causal dynamical triangulation programs and arguably a more distant cousin of the whole kind of loop quantum gravity spin Network spin foam story so it's the discrete space-time formalism based on what are called hypergraph transformation rules or set substitution systems and I'm going to try it first of all to it to explain exactly what that means so but first things first like I should state right off the bat that all of the original research that I'm going to be presenting here was was done in collaboration with Stephen Wolfram and max Fisk and of this particular talk will be based on primarily on two of the preprints by myself and Stephen covering some elementary properties of the model and some of its relativistic and gravitational implications but there's you know there's much more to say than I can reasonably fit into a 50 minute talk and so for those of you who are interested in finding out more you know that there's you can download all of our research materials our papers our preprints everything from from the Wolfram physics research website ok anyway let's start with let's start with the the basics so what is a hyper graph that part's easy so a hyper graph is a generalization of an ordinary graph in which one has hyper edges the rather than connecting exactly two vertices can connect any arbitrary non-empty subset of vertices so in the in the case of a directed hyper graph or an ordered hyper graph you can think of this as being just a collection of ordered relations between elements so on the left here we have a collection of ordered arity 2 relations binary relations that therefore corresponds to a directed graph and on the right here we have a collection of arity 3 relations which therefore corresponds to a - graph in which each hyper edge connects exactly 3 vertices now to be to be a bit more precise for the purposes of this talk I'm actually going to be discussing the case in which E is allowed to be a multi set and in which therefore we are actually considering multi hyper graphs in which hyper edges are allowed to have arbitrary multiplicity but that's that's a somewhat irrelevant technical point so I should say sort of at the beginning that you know the fund one of the fundamental intuitions behind us model is that space far from being a continuum is just a very large collection of discrete points with a with an adjacency structure defined on them by by the the relations of ordered hyper edges and so that gives us a kind of an underlying statics and then the dynamics of the model is defined in terms of hyper graph transformation rules so we have rules like this that say basically you if you have a hyper graph if you have a piece of hyper graph that looks like this then you can replace it with a piece of hyper graph that looks like that so you're you're matching sort of the left-hand side of some some rule pattern to to a hyper graph that's isomorphic so that pattern and replacing it with a sub half some hyper growth that's isomorphic to this pattern so set theoretically you can formalize that as a set substitution rule so you're basically saying in your collection of ordered relations between elements if you have a sub set that matches this pattern then replace it with a sub set that matches that pattern so then we can start from a very very simple initial hyper graph something like this that consists of three nodes and two hyper edges and then just by applying this rule wherever it can be applied but you know by attempting to update every possible vertex by applying this rule in all possible places then we can obtain a an evolution of this hyper graph through time so if we apply it for five evolution steps you can trace the evolution we get this thing over here if we apply the same rule for ten evolution steps we get we get this more complicated evolution over here if in fact if we apply the if we applied this rule 5415 evolution steps we would eventually get this rather magnificent looking creature so this is a very beautiful object with a bunch of really interesting sort of mathematical and combinatorial properties and I could give I could give a whole series of lectures on the other sort of mathematical properties of these hypergraph systems just in their own right but you know the question you might be asking at this point is what does any of this have to do with physics so um the the bit you know what the question that we've been interested in is could it be the case that hypopnea formalisms based on hypergraph transformational dynamics of this kind could they be a reasonable kind of minimal underlying candidate for the actual dynamics of our physical universe seems like a slightly you know bizarre sort of idea but nevertheless it's actually it's a hypothesis that we can test empirically because um you know in effective if we think of the if we think of this as being like the evolution of some you know computational universe then each such hypergraph transformation rule effectively corresponds to a different candidate universe and then we can just enumerate the space of all hydrograph transformation rules of a particular size rules of a particular given signature and effectively therefore it enumerates the space of all such candidate universes and we see things like this so that this is a small sample of the space of all such hypergraph transformation systems so you see the hyper graphs we obtain at the top here and the hyper graph transformation rules that produce them at the bottom and so effectively the question we wanted to ask was could it be the case that you know somewhere in this space of all possible such universes in all possible candidate universes that there exists one but you know how's the same features as our physical universe like I say it seems like a slightly crazy idea but the remarkable thing that we've discovered is that there actually exists large classes of hypergraph transformation rules which we can prove are imply in the large scale limit known features of sort of our physical universe including but not limited to special relativity with both Lorenz and Planck area invariants following from this general property of these systems that we call causal invariants they also turn out to imply general relativity and aspects of gravitation as a consequence of both causal invariants a weak Oh Gerda City hypothesis that we make about the microscopic dynamics of the hydrograph transformation rules and this property that lead this thing that we call the causal network which I'll introduce in a moment limits to something that has fixed mentioned and is therefore kind of manifold like that it turns out that the most general set of constraints that give you those three axes that are consistent with those three axioms are exactly the Einstein field equations in the continuum limit and I'd like to kind of walk you through that derivation today there is a lot there are lots of other things that I probably won't have time to get to in this particular talk but have relevance for things like cosmology so in particular it turns out that a fairly generic feature of these models is that they start out being the hypergraph start out being very densely connected initially and then gradually converge down to something that's that more sparsely connected you can think of that as the the the spatial hydrograph corresponding starting off in effectively infinite dimensions and converging down to something finite dimensional and that turns out to be consistent with our formulation of GR and it also implies that effectively there's a kind of phase transition in the effective speed of light during the early universe which we can show that it's not too difficult to prove that that implies a conformal structure for space-time that is actually consistent with this and predictions made by inflationary cosmology and in particular you know constitutes a valid solution to both the horizon and flatness problems of lambda-cdm cosmology it's also possible to derive many features of the standard mathematical formalism of quantum mechanics and quantum field theory including but not limited to the path integral the derivation of the particle by the way integral is it turns out to be very sort of mathematically beautiful it's kind of the it's the direct and so if you consider I mentioned the Einstein field equations kind of occur in the continuum limits of these causal networks the causal partial order for one of these for one of these hypergraph systems there in a way that I'll explain momentarily but there's also this thing called the multi way causal network that contains causal relationships between events not just on a single branch of evolution history but actually between branches of evolution histories and it turns out applying exactly the same argument that gives you the Einstein equations so the multi way causal graph gives you the path integral with the standard path integral action taking the place of the Einstein Hilbert actions in the continuum limit and there's a bunch of really cool geometry and mathematics that kind of underlies that with the Frobenius 2d metric tensor on projective Hilbert spaces taking place in the standard space-time metric tensor and we have a nice interpretation of things like the canonical commutation relations as effectively the analog of of Riemann curvature as the analog of the failure of commutativity of the covariant derivative operator in the context of the multi way causal network so I if I have time I'll try to give some hints about the quantum mechanical side of things towards the end of this talk but in the interest of time constraints I'm gonna try to be disciplined and restrict myself just to talking about the relativistic and gravitational properties of the model if you want to find out more about the other sides of the formalism please feel free to read our papers or I'm also happy to give a follow-up talk a part two to this at some point in the future if you guys are up for it ok anyway let's get to some actual content so um let's start with sr because that's actually you know that's pretty easy so um one concern that you might very reasonably have if you're first confronted with this formalism is about the uniqueness of the evolution history the uniqueness of the updating order because if you're given a spatial hydrograph like this and a rule like that there will generally be many possible places in which you could apply that rule there's no canonical updating order there's no canonical first place where that rule should get applied by the way I'm skipping ahead a bit but that that statement that there's no canonical updating order for hypergraphs turns out in the continuum limit to correspond to the statement that there's no preferred reference frame in the universe so and so effectively if you if you consider that the the sort of sets of space like separated updates or updating events that you could apply in other words the sets of updating events where the inputs don't overlap they don't you know make use of the same collections of nodes there will in general be many different possible sets of space like separated updating events that could be applied in parallel without conflict and each one of these corresponds to a different possible updating order for the hyper graph and in general these different updating orders will yield different evolutions they will yield non isomorphic spatial hyper graphs so you might very reasonably be concerned that the evolution of one of these wolf remodel systems is non-deterministic and that's a very very good concern and so and in fact it's true so in general um rather than having a single evolution path rather than having a single thread of time the evolution of an arbitrary wolf remodel system actually corresponds to a directed acyclic graph that we call the multi way system or the multi way evolution graph so here's a simple example for this very simple hydrograph transformation rule here so the multi-way evolution graph is as Theodore ected a cyclic graph in which every node is as a particular hyper graph corresponds to a particular state of the universe and every gray edge every directed edge here corresponds to a rewrite operation so it corresponds to an application of this rule and so the multi way system a multi way evolution graph effectively parameter eise's all possible evolution histories all possible updating orders for this Wolfram model system and so again we come back to this question of how can you get a deterministic evolution well it turns out there exists a large class of these hyper graph transformation rules that satisfy a property we call causal invariants which I'd like to define for you in just a moment and causal invariants effectively says in a very precise sense that the order of evolution doesn't the the the order of application of these updating events doesn't matter and and so therefore it it's causal invariants that allows you to obtain effectively deterministic evolution of the Wolfram model system rather excitingly it also turns out to be a sufficient condition for proving for proving both Lorentz and Planck are a symmetry which is the basis of our derivation of SR and I'd like to kind of walk you through how that works now so before I can define causal invariants I first have to introduce a deeply related concept from mathematical logic so in the theory of abstract rewriting systems in logic there's this concept we call confluence or global confluence which is in all the texts I think it's referred to as the church ross of property because of its relation to the untyped lambda calculus that was investigated by Alonzo Church and J Buckley Rosser back in the 1930s and so an abstract rewriting system or an ARS is just a set of elements which we'll call a equipped with some binary relation which we'll call arrow and sort of intuitively you know that set a is the set of all objects in your rewrite system and the binary operation is the rewrite relation so in the context of a wolfram model system if we expressible for model system as an ARS the set a is the set of all possible hyper graphs and the rewrite relation tells you that one hyper graph can be transformed into another so in in this multi-way evolution graph here each of these each of these nodes is an element of a is an element of the the rewrite system and each directed edge corresponds to an instance of that rewrite relation so that so a arrows B indicates that the the vertex and the multi-way evolution graph corresponding to hydrograph a is directly connected to the vertex corresponding to hypergraph B they are connected by a single directed edge so then once you have a notion of the rewrite relation you can define it's reflexive transitive closure which will call arrows star which is the transitive closure of the union of the rewrite relation with the identity relation so so effectively what that's so a arrow star B implies either a and B are equivalent they are they are isomorphic hypergraphs oh yeah I should have mentioned I mean the the the the merging criterion for these multi rate systems is hyper graph isomorphism as you might expect so either a and B are isomorphic or there exists a finite rewrite sequence that connects A to B so in other words if you consider the vertices corresponding to hydrographs a and B in the multi way system you're saying that they're connected by some finite path or more precisely that the vertex corresponding to hydrograph B exists in the out components of vertex of the vertex corresponding to hydrograph a in the multi revolution graph so effectively that this this reflexive transitive closure is saying a can be rewritten as B by some sequence and so then we can define this notion of global complements which is exactly the statement that for a triple of elements a B and C in your abstract rewriting system if it is the case that a can be rewritten as B and a can be rewritten as C then it is also the case that there exists some common element D such that B can be rewritten as D and C can be rewritten as D so effectively intuitively what that's saying is any time you have an ambiguity in your updating order anytime you get a bifurcation in the multi way system then it's au so that modulus has been bifurcates into two independent paths then it will always be possible for those two paths to reconverge on some common future elements that you know for any elements B and C that exist on those two paths there's a common there's an element D that exists in the kind of common future light cone so to speak of the elements B and C so global confluence is one very precise sense in which we can set and one very precise context in which we can say that the updating order doesn't matter because it's always possible to get back to it to a common shared state so global complements as I say is deeply related to in fact we can prove is a necessary condition for there's much stronger condition that we call causal invariance to explain what that means I have to introduce this notion of a causal network so I'm given a multi-way system I'll get or more precisely get in a path through the multi-way system given a particular evolution history you might ask you know so you know each of these each of these directed edges corresponds to what an application of one of these rules it corresponds to a single updating event and so you might ask what orders of updating events are and are not permitted or more to be more specific what what are the causal dependencies between updating events because it could be the case that some updating event depends on a previous updating event having already been applied and that's what the causal network is trying to demonstrate so the causal network is a directed acyclic graph in which all of the vertices are updating events and in which the directed edges are causal relationships between those updating events so here's a very simple example so each yellow box here is a representation of one of these high-profile dates updating events and each orange edge is a causal relationship between the two so the directed edge a to B exists if and only if a causes B or event B could only have been applied if event a had previously been applied and the way we can determine that mathematically is we say that the input for event B has a non-trivial overlap with the output of event a in other words if a event B makes use of nodes that were produced by the output of event a then we know that event B couldn't have been applied event a hadn't previously been applied so that's what the causal network is trying to show you so in effect we can think of the causal the updating events that are separated by directed caused legends we can consider to be time like separated and conversely any any updating events that are not connected directly connected by causal edges we can consider to be space like separated and therefore they correspond to non overlapping updating events within the spatial hydrograph which can therefore be applied simultaneously without any kind of conflict and so in effect my statement earlier that there are different updating orders corresponds to the statement that there exists different possible sets of space like separated updating events that we can consider to be simultaneous and that will be important in just a moment and by the way I mean again I'm going to say this kind of rather suggestively about hinting at something that's about to come one can think of the transitive reduction of the causal network as being the hussa diagram for the used causal partial order for these hypergraph transformation systems or to be even more suggestive you can think of it the transitive reduction as being a representation of the conformally invariant structure of some kind of skeletonized Lorenzi and manifold of some discrete approximation to a Lorentzian manifold and that's going to become and also very important in just a moment so then I finally I'm able to give a precise definition of cosmic variance so causal invariance is precisely the claim that the independent of which updating order you choose the causal network you end up with there's always the same up to isomorphism so in other words we say well for a model system is causal invariant if and only if all of the different possible evolution histories all the different possible paths through the multi-way evolution graph yield causal networks that are all ultimately isomorphic as directed acyclic graphs so down here you can see an example of a multi-way system that is explicitly causal invariance and so so in this case you can see all of the each blue box corresponds to a different high program for different states of the universe each each yellow box corresponds to an application of an updating event and each orange edge as usual corresponds to a causal relationship between those updating events and in this case here you can just see by tracing explicitly this for the four possible paths through this system that that all of the causal networks that you obtain all the relationships between the updating events that you see or all of those different paths are ultimately that the same that in fact in this case all of a call for causal networks are actually trivial they're all just there they're all just path graphs that consist of exactly for updating events connected by three causal edges so I already mentioned the global confluence is a necessary condition for causal invariants we can prove that if the system is causal invariants then it's necessarily globally complement but perhaps more excitingly we can also prove that if the system is causal invariant then it also obeys Lorentz symmetry in fact we can prove it in potato-based prime carry symmetry and in a slightly strengthened form we can even prove it obeys local Lorentz symmetry which forms part of the basis of our GG our derivation that that I want to show in a minute okay so I already mentioned this this rather suggestive comment about you know thinking of the the transitive reduction of the causal network as being the hussar diagram for some you know for the conformally invariant structure of some discrete approximation to a Lorentzian manifold and and in particular all of the standard algebraic and differential topology sort of techniques that you could apply in in the context of you know conformal structure of Lorentzian manifolds you can also apply to the combinatorial structure of the causal network so here's a is a trivial example you mean we can define you know notions of causal future causal past and in the causal network so if the causal future j+ for some updating event x just corresponds to the out components of that vertex x the causal past corresponds to the in components in that vertex x so we get an immediate kind of combinatorial interpretation of the past and future like cones of an updating event okay so as I sort of mentioned a couple of times now that you know one can think of these different choices of updating order corresponding to different choices of space like separated updating events to consider to be simultaneous so each one of those can effectively be thought of as a different choice of foliation of the causal network into simultaneity surfaces into space like hyper surfaces into sort of collections of updating events that are that are faced like separated in the sense that they do not they're not directly connected by by causal edges and so each such foliation of the causal network corresponds to a different possible updating order for the spatial hydrograph so if we construct a if we construct this if we consider this very simple hyper graph transformation rule here we can do a layered digraph embedding of the causal network like this and then we can pick the kind of default foliation choice which happens to correspond to our default choice of updating order for the hyper graph and so you can consider this as being that you know the causal Network foliation as seen by an observer in some rest frame and then the so that observer sees this particular sequence of hypergraphs in their evolution history but the point is there will generically be many differents foliation of the causal network into simultaneity surfaces of events that are nevertheless consistent with the causal partial order and so here's a different foliation we could have chosen it's slightly more complicated one and so but this foliation is still perfectly compatible with with the with the causal partial order and so an observer in this foliation would see this very different sequence of hydrograph so in particular you know for instance you can see here the third hydrograph are seen by this observer is clearly non isomorphic to the third hydrograph are seen by this observer so but this is still nevertheless a valid choice of updating order here's the key point though causal invariance guarantees that the conformal structure the destroyed that the the combinatorial structure of the causal network is unchanged between different choices of updating order and so in particular guarantees that if you make a pair if you make some parameterize change of foliation of the causal network then even though you're ordering of space like separated updating events will change the ordering of your time like separated updating events ie those not taking events that are specified by the causal partial order that are linked by directed edges we will always be preserved and so in this in the particular case where your foliation czar are spatially flat in a sense that i'll define in a minute this corresponds precisely to the state so the statement of the Lorentz symmetry and therefore is the thing that guarantees that these models are compatible with special relativity and in fact if you think about it that's actually pretty obvious because as I mentioned you know that the causal network is some discrete approximation to the conformal structure of a Lorentzian manifold and so in particular it's the causal invariants the statement that there is just a single causal network up to isomorphism that's defined uniquely corresponds to the statement that the combinatorial structure of the causal network is invariant under the action of the conformal group and so because both Lorentz imprecatory groups and subgroups is the conformal group you know the rents and the rent and Poincare symmetry follow follow pretty trivially and if you want to you can then go in to write all of the standard properties of the Lorentz transform kind of from first principles so like you know here is a one-dimensional hydrograph evolution that whose causal Network happens to be this rather simple grid like structure and so we can construct a default foliation that we can think of as corresponding to the to the updating order as seen by an observer in a rest frame but if we wanted to we could construct a boosted foliation in which we now get this different evolution order that we can think of as being the evolution order of the hydrograph as seen by an observer moving at some finite velocity with respect to the rest frame and so in particular you know but by sort of elementary geometry we can see that whereas this observer takes 14 evolution steps to move from the initial hydrograph so the final one the observer and the rest frame takes only 11 and so doesn't say with sort of basic geometry all the standard features of you know time dilation and length contraction relativistic mass transformation those can all be derived from first principles for the for the full gory details of all that you know go and see our papers okay there's one crucial sort of mathematical subtlety that I've kind of glossed over in in doing this which you may have noticed which is that I kind of introduced this concept of a causal network foliation by a sort of sleight of hand move and so those of you who have more mathematically inclined might very reasonably be asking you know what what exactly is a causal network foliation how do we think about them how do we parameterize them and so on that's a very legitimate that's a very good and legitimate question so it turns out it's a slightly mathematically non-trivial story but actually we've been able to make extensive use of the techniques of geometric quantization and canonical Hamiltonian quantum gravity to effectively derive a discrete approximation to the Adm formalism in standard general and sylheti so in all of the causal networks that i've shown so far and in fact all the ones I'm going to show later on we're assuming that they are all globally hyperbolic and global high publicity in our context corresponds to a very precise combinatorial statement it corresponds to the statement that the causal network can be that we can construct a layered digraph embedding of the causal network into the plane in such a way that all of the causal edges point monotonically downwards and if we construct Slayer digraph embedding in which a causal edge does not point monotonically downwards that corresponds to a local failure of high publicity so anyway assuming a globally hyperbolic causal network that can be embedded in this way we can define a universal time function on that causal network that map's each updating event to an integer corresponding to the index of the space like hyper surface on which that updating event lies such that delta T is non zero everywhere effectively such that between any updating event and it's corresponding future updating event on the neighboring space like hyper surface there must exist at least one causal edge there has to exist at least one elementary time interval between those between those two events so there's two space like hyper surfaces don't sort of trivially intersect in which case this definition of the universal time function immediately Fowley AIT's the causal network into this parametric family of non overlapping space like hyper surfaces that are the level sets of that universal time function and so then we can ask the question of all how do we parameterize this foliation and to do that as I kind of alluded to we can just do the discrete analogue of the ADM gauge choice so in particular start by noticing that each of these spatial hydrographs has a natural distance metric defined upon it which is the combinatorial distance so if you have a pair of vertices we can ask what is the graph what is the length of the graph geodesic between those two vertices what is the length what is the you know it for the shortest path between those two vertices what is the number of directed hyper edges that we have to cross in order to get from one to the other that induces effectively a discrete spatial metric tensor which I'll call gamma IJ and so now at each for each updating event in the causal network we can define a lapse function alpha and a shift vector beta so that so that so alpha is effectively defining the number of causal edges separating an updating amount from one hyper surface from its corresponding updating events on the next Tyco surface so it's indicating that the elementary time like separation between those those two points on the neighboring hyper surfaces and the shift vector beta is indicating the spatial hypergraph distance between the location on the hyper graph where the first updating event got applied and the location in the hyper graph where the second updating about got applied and so so by defining a lapse function and a shift vector for every updating event in the causal Network we can completely determine its foliation into space like hyper surfaces and combining all of that together it allows us to write a line element for the overall causal network that actually has effectively the same form as the standard space-time line elements in in the ADM formalism okay that's a bit about s art but we'll come on to we'll come back to some of the ideas in just a moment but I'd like now to talk about something that's a bit more complicated which is how we go about deriving gr in the context of this formalism so um because this is a slightly more involved procedure I'm going to start for the purposes of kind of pedagogy I'm gonna start from a toy case which is considering curvature just in spatial hypergraphs before moving on to the full space-time case which is considering curvature in causal networks so um now first very obvious thing to say is of course these these networks that were considered to be hyper graphs and the causal networks that we're considering are obviously not manifolds they don't have manifold structure they don't they don't satisfy the axioms of a manifold um so we can't just define something like a Ricci scalar on something like a spatial hyper graph which might seem like a like a problem but nevertheless we could we can ask the question can we do fine a quantity on the special hydrograph that preserves are the usual geometrical intuition for what the Ricci scalar is in the romanian case so just to remind you of something you all know perfectly well you know in the static for the in a standard Romani and manifold the usual geometrical intuition for what the Ricci scalar represents is if you pick a point P in that manifold and you grow out a finite ball of radius epsilon which we'll called B epsilon P in that manifold the Ricci scalar is what determines the ratio of the volume of that ball to the volume of a ball of the same radius but in ordinary flat Euclidean space or more precisely it's the Ricci scalar gives you the second order correction term in that in the discrepancy between the two volumes so a mathematically equivalent way of making the same statement that turns out to be slightly more directly Ameena ball to the hypergraph case is if you could consider a finite GDC ball B epsilon P centered at Point P and you parallel transport all the points from that ball to it to a neighboring point through it to a ball centered on a neighboring point which we'll call B epsilon Q then you can ask the question what is the average distance or what is the ratio of the average distance between a point on B epsilon P and its corresponding point Maps off to parallel transport on to the epsilon Q what is its ratio to the actual distance between the metric distance between points P and Q which we'll call Delta and so again that that ratio turns out to the up to second order determined by by the Ricci scalar at at Point P and this is in the limit as both the radius epsilon goes to zero and the separation distance between the centers Delta goes to zero so so then you might ask what can we define an analogous quantity on one of our hyper graphs and the answer turns out to yes so this is a mathematical problem that's been worked on by many people including folks like Robin Forman Juergen Yost and others but in our particular research we've made use of and extended many of the mathematical constructions that are due to yan Olivier so Olivier in particular has constructed a generalization of the Ricci scalar that applies to arbitrary metric spaces including direct including discrete metric spaces which include of course directed hydrographs so the kind of the big-picture intuition behind oliviers construction is that you generalize the notion of a volume measure given by a GD SiC ball in a manifold to a probability measure in your metric space and then this concept of the average distance between a point on the geodesic poor and its corresponding point after parallel transport then becomes the measure of the vash lying distance the transportation distance from one measure to another measure okay so just to remind you all of a bit of elementary measure theory if you have a Polish SpaceX that's equipped with a metric D and it's also equipped with a bore L Sigma algebra then we can define a set of for each point in the metric space X we can define a probability measure MX that has finite first moments and where the map going from the point X to the measure MX is itself a measurable map in which case we call the set of all such probability measures a random walk and so this random walk then allows us to define that the Vashta stein distance metric on the metric space so okay those of you who know about measure theory will know that there are actually infinitely many different from dr. stein distances all kinds of different orders but actually we're just interested in the boring case the first order case which is the one batch the stein distance so if you have two probability measures NX and NY you can ask the rancher stein transportation distance is the optimal transportation distance between those two measures so the kind of the standard intuition is it's like an earthmover distance so if you think of the measure MX as being like a sand pile you know heaps at position x then you know the faster stein distance is telling you the amount of work that you need to do to disassemble the measure at position X transported over to position Y and reassemble it at position wise and so we can express that formally as an infimum taken over the set of all probability measures defined on what Cartesian product of the metric space with itself which effectively designates the couplings between those random walks that project onto the measure NX and those random walks the project onto the measure my that gives us the the one basha stein distance and so then as I say we think of that as being like this notion of the average distance between points on two geodesic balls in which case the Olivier gives us this construction of the of the scalar curvature so if you have two points P and Q that are nearby that are separated by some distance epsilon in the metric space then the Olivier Ricci scalar at Point P can be thought of as being one - the ratio between the Vashta in transportation distance from measure MP to measure MQ to the actual metric distance between the points P and Q and so this is nice because in particular you can see in the special case where X is a manifold and where D is a you know it is an ordinary Romanian metric and M becomes the standard Romanian volume measure in terms of you know finite finite geodesic balls then this Olivier Ricci scalar is exactly you know that the Romanian Riccio at Point P is exactly the Romanian Ricci scalar at Point P up to some arbitrary multiplicative constant and so this gives us the formal justification for interpreting geometrically the is probability measures as being the natural generalization of the you know of the volumes of finite GDC calls and some generalized manifold so that's what we're going to use to define curvature in the context of a hyper graph and later on in the context of a causal network so given so we can apply this specifically to the case of a discrete metric space in which case obviously this integral reduces to a sum and we obtain what's called the multi marginal optimal transportation distance between two discrete probability measures and now that the projection condition that defines that this the set PI now becomes the the condition that these two sums are satisfied and we can specialize even further and take this to the case of a hyper graph I'm aware I don't have a huge amount of time so I'm not going to take you through the complete details of this but the basic idea is you can consider a directed hyper graph to be a collection of edges of directed hyper edges where each hyper edge maps from a tail set consisting of vertices x1 to xn to a head set consisting of vertices y1 to yn in which case the Olivier Ricci scalar on the directed hydrograph becomes 1 - the master stein transportation distance between the probability measures mu a in and mu B out where these measures are where mu a in is the sum over the probability measures for all of the incoming hyper edges to the tail set a and mu B out as some of over all of the outgoing of all the probability measures for the outgoing hyper edges from the head set B and if we want to we can write these measures in full explicit a gory detail but basically all of this is trying to formally justify this intuition you probably already have that if we want to generalize the notion of a finite ball in a hyper graph then we can just do that by if you want to consider a ball of radius epsilon and I put in a directed hydrograph then you just pick a particular node then you look at all of the nodes that are adjacent to that node then you look at all the nodes through adjacent to those nodes and so on and then you effectively grow out this ball of radius epsilon and then the your your volume measure is just a counting measure it's just a count of the number of nodes that lie within a distance epsilon of that initial node and so that and this derivation shows one that in fact it does indeed satisfy the axioms of a probability measure and therefore we're correct to interpret it as being something like a volume measure and so then and then what once we have this notion of a gd6 boiler and the hydrograph of course then this notion of the average distance between a point on the geodesic ball and its corresponding point after parallel transport simply reduces to the one basha stein distance between those two between those two probability measures nu a N and mu B out with respect to the standard induced hypergraph metric that I mentioned earlier that you know the combinatorial metric between between any pair of vertices the number of directed hyper edges you have to traverse to get from point A to point B and what one thing that's worth mentioning by the way is that actually Olivier's construction is a little bit more general than we actually need it to be because in particular it allows Y by a definition of this function Epsilon it allows one to consider hyper edges that have arbitrary weights associated with them turns out for our derivation of GR we don't need that we actually consider only the case where each hyper Edge has a unit weight and therefore corresponds to a single units of the kind of elementary spatial distance but it's worth mentioning you know if you want to derive kind of torsion metrics and things like that then you can also define hydrographs where we're different hyper edges have different weights okay so the reason for introducing all of this complicated formalism is now to be to be able to rigorously justify us doing effectively dimension and curvature calculations in the context of directed hyper graphs because now what we can do is we can say take a special hyper graph that corresponds to flat n dimensional space and let's grow a geodesic ball of radius R and ask how many nodes lie within that geodesic ball of course if it's if it corresponds to flat n dimensional space that function will grow exactly like our to the N we now know that if it corresponds to a curved n-dimensional space that there will be a second order correction factor that is now proportional to the Ricci to the olivier Ricci scalar curvature and so this information allows us to do dimension curvature estimates on arbitrary hydrographs so here for example is only obviously these hydrographs are very simple they just correspond to flat grid like two dimensional and three dimensional space this is a more complicated kind of hyper graph this is a hyper graph produced by one of our rules that we actually know limits to an asymptotically flat structure that that is therefore kind of like a two dimensional manifold in fact we can see by doing this by doing a century a logarithmic difference estimate of the dimension that it is approximately two dimensional but with certain higher order Corrections that are proportional to the curvature moreover even in cases where there isn't an obvious manifold like structure we can still define it in a meaningful sense notions of dimension and curvature so I like here we take up one of these hyper graphs did you know the geometry of this thing is much more complicated it doesn't doesn't doesn't seem to have a meaningful and emergent geometry doesn't seem to correspond to a meaningful manifold structure but nevertheless we can still define its dimension and curvature and so in this particular case it appears to correspond to something of dimension roughly 2.5 which of course indicates that because we're using an approximation to the house-door dimension as opposed to the topological dimension there's no rule that says it has to be integer it has to be the structure has to be integer dimensional and so actually I might I might talk towards the end about some of the implications that that has for a kind of generalization of GR that applies in in fractal dimensional manifolds but that's that's a topic for another talk oh yeah so these dimension estimates we just computed using very simple logarithmic just a different estimation um one really important thing that you will have noticed is that when we because unlike a manifold which has a fixed dimension or at least the connected component has a fixed dimension in for some manifold these spatial hyper graphs don't have a dimension that's defined a priori the dimension and the curvature both emerge from the combinatorial structure so in particular there exists this subtle trade-off when we compute this function n but for the volume measure there exists a trade-off between the exponential contribution from the dimension and the quadratic contribution from the curvature and one interesting feature of that is that if we want this if we want something like the special hyper graph to limit to a fixed dimensional manifold like structure it played that place as certain constraints on the growth conditions for the curvature because effectively if the curvature is allowed to grow without bound then in the continue if you try to take an infinite limit a continuum limit then there will be no way for you to distinguish between a monotonically increasing curvature and a global increase in the effective dimension and so in order for you to have a structure that limits to something that's finite dimensional that places certain conditions on the degree to which the curvature can grow and you know spoiler alert as I say in the most general case it turns out that those conditions are the Einstein field of equations but we'll get to that in just a moment so in order to derive the EF ease we have to consider curvature not just in space but in space-time as defined by causal networks so that involves going to introducing what wanting you one new piece of sort of technology so I want to ask a related question to the to the Ricci scalar question that I asked I asked earlier which is we enter in romani and geometry we have this notion of sectional curvature and you know the normal geometrical intuition for the sectional curvature just to remind you is you know you pick a point X and you pick projection directions that are just you know linearly independent tangent vectors or in the simple case they're just orthonormal vectors U and V and you ask what what is the what is the sectional curvature projected in directions U and V at point X and what the sectional curvature is measuring is if you take a nearby point Y that's within a distance epsilon and then you you go you go along these two units BG&E 6 u and V by a distance of epsilon then the sectional curvature allows you to measure the ratio or the discrepancy between the distance between the endpoints of those two unit units BG v6 and the actual distance between the points x and y and and so then I'm going to ask you know very similar questions on the one I asked before which is can we define a meaningful notion of sectional curvature in the context of hyper graphs and causal networks and again the answer turns out to be yes this again was is based on the work of yan Olivier so if we have an arbitrary metric space X then we can define an analogue of the sectional curvature that preserves this standard of geometrical intuition so now instead of having sort of orthonormal vectors we have unit speed graph gd6 or units be gd6 in the metric space and we're asking if we take the the exponential map from the tangent space onto the onto the sort of manifold so to speak that gives us the the endpoints of those those two unit speeds ud6 as a distance of epsilon away from their origin points and then we can ask what is the average distance what is the discrepancy between the distance between those endpoints D and the distance between x and y which is Delta and just as in the reach the scalar case you know that discrepancy is is determined up to up to second order by the by the sectional curvature in these two in these two directions okay and we can validate that actually this debt this definition is indeed compatible with the with our prior definition of the scalar curvature because in just the same way as if we take an index contraction of the sectional curvature we expect to obtain the Ricci scalar we can we can verify that if we take the average of this discreet potentially discreet sectional curvature over all such vectors over all such geo d6v then then effectively what we're doing is we're constructing the set of all tangent vectors of length epsilon centered at point X which is like a finite geodesic ball and similarly for the set sy send to the point why and then we're measuring the average distance between the corresponding points on it on SX and sy after parallel transports and we can indeed verify that the average distance between those points and the ratio of that average distance to there to the distance between the Centers is indeed given by the Ricci scalar up to second order as epsilon and Delta go to zero so now we have a mathematically consistent way of defining sexual curvature over discrete metric stick this is including hyper graphs and causal networks and so and that's quite exciting because of course the the sectional curvature in romanian geometry completely determines the components of the riemann of the riemann tensor and so this means we can define some generalized analogue of the Riemann curvature and in particular we can define things like the region the Ricci curvature tensor and other higher order contractions so the reason I bring that up is because if we now want to consider how do we assess the dimension and curvature in a causal network um we have this new problem which is the unlike in the hyper graph case you know the causal network because of its because of its Lorenzi Earnest it's a directed graph with with all edges pointing monotonically downwards and so we can't grow a finite geodesic ball in effect we have to grow a finite judy citko we have to pick a GED sikh direction through the causal network and then grow a finite cone centered around that geodesic of length T that's our analog of the judaistic ball construction and so then again in exactly the same way for a causal network that corresponds to flat n dimensional space-time the volume of this of this discrete space-time cone will grow like T to the N but for a cause which we can see here for this very elementary flat causal network we can see it's we can see it limits to something that's that's tuned up that's flat and two-dimensional but for formal causal networks that corresponds to curved space-time that is now going to be a correction term that we now know is proportional to the projection of the Olivier Ricci curvature tensor in the time like direction so in other words when we picked that geodesic direction through the causal network on which to Center this space-time cone we're now projecting the curvature the Olivier Ricci curvature tensor in that time like direction and that projection is giving us the second order correction to the volume of the of the space-time cone in the causal network so here's a more complicated spatial hydrograph in which induces this more complicated causal network and we can see that the hyper graph limits to something that's roughly two dimensional and the causal network limits to something that's that's roughly one dimension higher as we'd expect okay so then the this projection that we're doing I mean this there's a lot of mathematical detail that I have I'm having to skim over here but it's really it's quite nice that you know effectively this notion of a tensor index defined on a causal network or a hydrograph is given just by a gd6 direction in the hydrograph all the network and that's that that's the sense in which we can define these these projections of curvature tensors and so on the the time like direction into which we're projecting the Olivier michi curvature tensor is exactly the same as the time like vector that we defined earlier on in the special relativity case when we're talking about when we were parameterizing these foliation so in other words this this time like projec in the direction given by alpha by V by alpha times the normal vector plus beta okay so one thing that you might immediately notice is that many of the causal networks that we're considering here have a property that's quite undesirable so here's an example of that property so you can see in this causal network as we go down these these different layers the number of updating events that exists on the next layer actually increases exponentially and so in particular if we try to grow a finite GEDs at cone in this causal network as the causal network gets bigger the the exponent in our in our volume measure is going to grow and so effectively if we take a continue if we try to take a continuum limit so the causal network like this it's going to end up being infinite dimensional it's not going to converge to something like a finite dimensional space-time and so that's kind of bad if we want this to be a representation of space-time and so one question that you could ask would be what are that you know what are the constraints that we would have to enforce on the actual hypergraph dynamics to ensure that the causal network has a well-defined limit as a finite dimensional space-time like structure and okay so one thing we can do is I mentioned earlier that one condition in which that's definitely not going to happen is if the curvature is allowed to grow without bound because if the curvature grows without bound you're continually adding these higher sort of quadratic corrections to the volume and so if you're given that we don't have you know we don't have the dimension information we don't have the curvature information we have to that we have to that all has to emerge from the dynamics of the causal network if the curvature is allowed to grow without bound then then the dimension then we can't distinguish that curvature from a from a global change in dimension and so one minimal condition that we need in order to us in order to ensure that this causal network limits to something that's finite dimensional is that the second order correction which we're going to call the dimension and normally when averaged out over the over the group over the overall causal networks with a global dimensional anomaly for the whole causal network that quantity can't be allowed to grow without bound because if it can grow without bound as I say it will be impossible in the continuum limit to distinguish between its growth and an increase in curvature so how do we how do we determine this global donation anomaly well we average out over all events in the causal network we start out by averaging out over all possible time like projections for these GDC cones and by taught by averaging out of all these time like projections that's equivalent to taking a contraction of the ritchie curvature tensor between the first and third indices so that gives us the space-time Ricci scalar we're constructing a volume average over space-time over the causal network and so we we wait the space-time Ricci scalar by the elementary volume element in the causal network which is expressible in terms of the standard volume measure that we introduced earlier D mu G and so now our condition that the causal network limits to something finite dimensional becomes the statement that as the causal network grows the rate of change of this global dimension anomaly quantity the rate of change of the global second order correction to this to this volume measure should converge to zero as as the causal network becomes infinite so now if we make a weak arrghh audacity assumption in other words if we assume that the actual hypergraph transformation dynamics are sufficiently random or sufficiently a Ghatak so as to analytically justify exchanging for this sum for a for an integral as we take the continuum limit then we obtain of course the classical vacuum Einstein Hilbert action and the condition that the global second-order dimensional normally converges to 0 as the causal network becomes infinite becomes exactly the statement that the vacuum is a Hilbert action is extra mised and so of course we can do the standard thing we have a standard relativistic low energy and density we take a functional derivative with respect to the inverse metric we assume 0 surface terms we get this and so as I say that the statement that that Delta s by Delta GA be converges to 0 as the causal network becomes infinite becomes exactly the statement that the vacuum Einstein hill but the vacuum Einstein field equations hold in the continuum limit now of course this argument only gives you the vacuum Einstein equations to derive the full internal field equations you have to go a step further which I'm I'm already nearing the end of my time so I'm probably not gonna have time to to walk through that full derivation here but loosely speaking the way that that works is we give an interpretation of upper index T mu nu as the flux of course ledges through a hyper of constant ex-new in the causal network that's the direct sort of translation of the statement that upper index T mu nu is the flux of relativistic P mu through the surface of constant X nu in space-time so effectively you know energy in our model corresponds to the flux of causal edges through a space like hyper surface momentum corresponds to the flux of Couples ledges through a time like hyper surface and then we note that so some a large fraction of the energy is associated with just maintaining the background space just maintaining the background hydrograph then on top of that there can exist local persistent topological obstructions that we have a combinatorial classification for in terms of this thing called the Seymour Robertson theorem and hours of conjecture is that these localized table logical destructions in the hypergraph and in the causal causal network corresponds to baryonic matter correspond to elementary particles and so these topological obstructions correspond to locally increased connectivity in both the hyper graph and the causal network you can see the pre can see an example of one propagating through a manifold like causal Network here so this is kind of a minimal model of baryonic matter so then if we subtract off all of the causal edges associated with the maintenance of space and we consider only the causal edges can involved in the maintenance of these baryonic matter contributions these locally increased regions of connectivity in the both the causal network and the spatial hyper graph have also have to be accounted for when we can when we're doing this dimensionality calculation so in addition to the second-order contribution from the curvature there's also an additional contribution from the from the matter density from from the density of these topological obstructions and so that has when we run through the argument again that has the effect of adding in a matter field term to the relativistic Lagrangian density and so then you know doing the same extra my zation argument yields the full line interesting field equations ok so that's that's a little bit about some of the sort of relativistic and gravitational properties that we now kind of have proved to be the case for for these moral model systems there's many many more things that I like really would love to talk about but which we we probably don't have time to cover here one of the things that I've kind of mentioned briefly a couple of times is that you know because unlike in conventional differential geometry our manifolds which are these networks these hydrographs and causal networks don't have a fixed dimension dimension can actually vary both locally and globally and so this implies that actions are okay our derivation of the Einstein field equations has assumed that the causal network has a fixed dimension in the limit and that the curvature is allowed to vary locally but the point and that's of course that that's how we get correspondence with standard gr but of course we could just as equally have said that actually the curvature is fixed globally and the dimension is allowed to vary locally and we could also have we could formulate a version of the Einstein field equations in terms of local dimension change that would also be perfectly valid and we could also consider the free decoupled case where both curvature and dimension are allowed to vary and they had it turns out we've done a few initial computations about this and they turn out have a bunch of really interesting interactions with each other some so that this implies that kind of a grand generalization of general activity to the very to the condition in which dimension is treated as a dynamical variable this also as I mentioned right at the start has implications for cosmology because if if as seems to be the case in in a large class of these modules one starts from a spatial hypergraph that exists and effectively infinite dimensions and then converges down to something finite dimensional the conformal structure of that of such a universe turns out to be equivalent to a universe in which the speed of light is a dynamical scalar field variable and in which there exists a phase transition in the effective speed of lights during the early universe and we can make correspondence would be with the so-called Petitte Albrecht model of a variable speed of light cosmology which yields you know which yields a valid solution to the horizon of flatness problems and sort of correctly predicts the large-scale homogeneity and isotropy of the universe without the need to postulate an infra tom field that's another kind of runner interesting Avenue of Investigation I also mentioned right at the start that there's a generalization of this whole argument that gives one the path integral and so the way this works is you know we've so far considered only these causal networks that correspond to a particular updating order but right at the beginning I showed you an example if I can find it here of a factory a multi-way causal Network a causal network that has updating events not just for a single branch of evolution history but actually between branches that has causal relations between different branches of evolution history and so this multi way causal network is a structure that's like a an ordinary cause on that work.but has you know interactions between different branches of history different branches of the multi weight system and then it turns out if we apply the same argument that gives us the Einstein field equations in the continuum limits of the causal network in space-time if we apply the same argument to the multi way causal Network we get the path integral in fact so what happens effectively is that the wetware in this argument we get the space-time metric tensor in the context of a multi way causal graph we get the Foo be nice to D metric tensor on projective hilbert space and and then the instead and then if we try and make the statement that the the multi way causal Network limits to something that doesn't end up being infinite dimensional then instead of getting the Einstein Hilbert action in as we do in the space-time case we actually get the path integral action and this has a bunch as I kind of very briefly mentioned towards the start this has a bunch of really cool this gives you a bunch of very cool geometrical intuitions for kind of standard aspects of the quantum mechanical formalism so in particular you know it in our model the canonical commutation commutation relations and the uncertainty principle turn out to be the direct analogue of Riemann curvature in the multi way causal Network because you know in exactly the same way as the Riemann tensor tells you the failure of commutativity of the covariant derivative operator the failure of geometrical intuitive information to be preserved as you parallel transport that parallel transport some vector around around a closed curve the analogous concept in the multi way causal graph tells you sort of the again gives you it gives you a failure commutativity of some generalized covariant derivative operator which yields the the canonical commutation relations so if X ability uncertainty principle becomes the analogue of curvature in the context of the multi-layer causal Network and all of the stuff that I've mentioned here actually has a bunch of really nice neat implications for other areas like quantum information theory quantum computational complexity theory and things like black hole thermodynamics and so again I unfortunately don't have time to discuss it here but it turns out if you consider when you consider the multi way causal Network there exist things like event horizons apparent horizons trapped surfaces not just in space-time in the context of a causal network but actually in the multi way causal Network we have these these analogs of trap services that we call entanglement horizons and one of the neat features of our interpretation of con mechanics is that if you have in falling sort of information into into a black hole the that information in the multi way causal network actually never passes a true space-time causal event horizon it gets pretty it instead gets preserved at the entanglement horizon which exists just outside and so we can make us a direct correspondence with standard like holographic and ATSC ft resolutions to the black hole information paradox those kinds of things that's all still a very speculative direction but that's that's another thing that we're super excited about so I'm already over time so I'm sorry for speaking so long so thank you very much all of you for listening and I'm very happy to take any questions thanks is a very nice crash course and the gravitational properties of the world I think you won't be out of the audience so let me see whether there are questions from the audience so maybe I can ask a very curiosity so the Einstein gravity is not the the most general construction because I told from equal to zero by and basically so in your language is impossible to to describe also these kind of effects yes absolutely so okay that's a really good question so um I smuggled in the torsion freeness I smuggled in the lobby Savita connection before when I said that this we're going to assume this per at this function epsilon is unit it has a value of unity everywhere so effectively if you have a spatial if you have this if you have a hyper edge that goes in one direction there's also a hyper edge that goes in the other direction and and though those two things have the same weights if instead relax that condition and we said that this function epsilon for you know for the hydrogen acting vertices U and V that that waiting can be different to the weighting of the hyper edge connecting vertices V and u and if we do the same thing in the causal network then when we actually get either we get a torsion metric and as you say we get these these more general sets of Einstein equations that those are not strictly torture and free thanks oh by the way sorry just on the on the context of subjects of generalized satellite equations another neat thing is that one so the the obviously the derivation that I showed you implies that there are condition that there are some second-order conditions on the curvature but actually if we depending on exactly how how quickly we want the DeMint we want the causal network to converge down to something finite dimensional we can actually uncertain can under certain convergence assumptions we can also derive higher order terms of the Einstein equations effectively we can derive certain conditions on the viol curvature we can derive conditions not just on the volumes of geodesic bundles in the causal network but on certain convergence of assumptions we can derive conditions on the shapes of those geodesic bundles that give us sort of contribution to the higher order contractions of the Riemann tensor that's another kind of generalized gravity story that we're also very interested to see how you know how that works out from the audience if I may introduction of the hyperbolic nature of the network you did at the beginning yes so I want you to understand if this is necessary to do and you have can only go on if you introduce this by hand or at some point in not really emerges from something else which I've missed in your talk is it stick requirement you have to put in at the beginning of your construction or you can go and do it without that's a really great question and so I mean the honest answer is we don't yet know um it so this construction we don't know how strict the causality conditions have to be in order for you to do these kinds of derivations the derivation is the easiest if we assume the strictest possible causality condition which of course is global high publicity but you know whether we could get away with just having you know future distinguishing or past distinguishing space times or you know totally not ambitious space times or something we don't know if that's gonna work out so to answer the the sort of the second part of your question about you know how this how this condition emerges so obviously if you if you have um there are kind of there are two related instances in which you can get a failure of high publicity in one of these causal networks either you have a loop in the causal network where you know you have a you have a directed edge that goes back up again in your direct in your lair dygraf impending then of course corresponds to a CTC and we know that they trivially violate go bhai purple ersity but actually you can also have the case where you have a causal edge that connects two points on the same layer that connects two edges on so sorry two events on the same layer and that you can kind of think I was being like a degenerate case of a CTC at CTC that doesn't take you that takes you exactly zero distance back in time and that also as I mentioned corresponds to a local failure of hypervelocity so we do in fact you know this is not a natural feature of these causal networks we do have plenty of examples of causal networks that have CTCs and which violates you know high publicity and other kinds of ways like really related to these to these pairs of updating events that are connected by bike course Ledger's it just happens to be the case that you know it's difficult to define to introduce a well-defined foliation if you have a violation of high publicity and given that the arguments I constructed about you know growing finite judy SiC cones and things like that they all kind of depend on this notion you can define a well-defined phone you can introduce a well-defined foliation you know the problem arises of course if you if you have a loop in your causal network then it's kind of that there isn't a straightforward way you can introduce a foliation you can introduce a space like hyper surface that still respects the causal partial order and that's a problem we don't yet know how to solve but you know in I'm not ruling out the possibility that we may be able to do the same constructions as you say by assuming kind of less you know less less rigorous causality conditions I hope that hope that answers your question I do have a mention that a good portion basically banerjee goes into maintaining the structure of space-time itself but on the other hand we have the equivalence principle essentially tells us that all forms of energy are the same basically so how do you reconcile this with that right so that okay great question and and so it's at the level you know at the level of the actual hypergraph in the actual causal network the equivalence principle holds almost by definition because as I say you know if you interpret energy as just being the flux of cause alleges through a space-like hypersurface then you know the flux of those calls Ledger's doesn't care whether the causal edge is associated with the maintenance of the background space or if it's associated with the propagation of an elementary particle that's that's the distinction that we impose kind of after the fact and so the point is that the if you want to make connect you know in conventional gr you know you generally make a separation between the cosmological constant term and and the and the stress energy term and so we have to have a way of doing that if we want to connect it to the known formalism and so to do that what we basically do is we take the causal network and we divide it up into the background space-time and these localized topological obstructions and we consider the fluxes of course Ledger's that correspond to both things separately but of course in reality there's no such distinction they're all just causal edges they're all fluxing through the same space like hypersurface they will contribute in the same way so the equivalence principle holds it's just our interpretation of the energy momentum you know it's it involves explicitly violating the equivalence principle so I hope that how'd that make sense yeah thanks [Music] you you you just one final comment maybe there is a construction that is similar to you know so ye cannae Amanda who uses a cluster algebra simple rules algebra to construct the complex structures research so I I'm aware of the research i I don't understand it well enough to be able to speak I don't understand the formalism of cluster algebras as well you know as well as I should and so I can't really I don't want to kind of speak about that specifically I don't think I necessarily have anything sensible to say um one thing that is worth saying is you know I completely agree that there is a there is clearly a great deal of overlap between the stuff that Nima has been doing with with cluster algebras and the answers you Adrian program and all that kind of stuff and we'd be really excited in fact we're currently in the process of trying to convince Nima to come in to come in to get kind of give us give us on the team a sort of crash course in what he's been doing and how it might connect to what we're up to but so it's a good question I don't know the answer but I I agree it's almost certainly going to end up being related ok well I think you again would be very nice talking thank you for your time and see you soon oh it was my pleasure thank you so much for inviting me it was it was really great to be here

Channel: Newton 1665 physics seminars

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Keywords: physics, particle physics, high-energy phenomenology, high-energy theory, cosmology, webinar, seminar

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Length: 67min 27sec (4047 seconds)

Published: Tue May 26 2020