Eigenbros ep 117 - Jonathan Gorard (Wolfram Physics)

welcome welcome back eigen families so we're glad to see you guys once again and first off if you guys have not done so yet make sure you like share comment and subscribe um check us out of course the websites eigenbros.com uh eigenbros on twitter again roles on instagram i can bros two on tick tock and then also guys thank you once again to the patrons we really greatly appreciate it um we know you know we couldn't do without you guys and also if you want to become a patron uh make sure you check out patreon.com eigenbros so today guys you uh we have a special guest today you may even recognize him uh from the wolfram physics project um he is the research and research fellow at cambridge university um and he also contributes to world from physics.org and he is the co-founder of the wolfram physics project uh welcome everyone it is jonathan gerrard cool hi thanks so much for inviting me yeah jonathan it's great to have you on um let's uh kind of get into it here yeah we were saying uh yeah how you basically reach out to us through youtube because we did a we did a um a kind of real i'd say you know kind of loose episode on the the wolfram physics project and you know there were there were a few maybe gripes you had i guess about the uh some of the some of the things we said about it maybe some things you can clear up for us here today yeah but i think uh yeah overall you were pretty nervous about it jonathan yeah i think i i left basically quite a rude youtube comment on the video uh which i which i later somewhat regretted but yeah you guys were grateful enough to invite me along to to be interviewed so that was thank you so much yeah we greatly appreciate uh you coming here man yeah um so yes so i guess jonathan to start it off we kind of wanted to um ask you a little bit of basic questions so just like you know we're physicists um you have i do you have a math or a um computer science background jonathan uh so i i have a math background right it's applied mathematics correct uh yeah i sort of drifted in that direction by accident my i mean like my my bachelor's degree was in pure mathematics and then i did a masters in phd in sort of mathematical physics so i i drifted more applied was there a reason why you chose to go that direction rather than the pure math side yeah it's an interesting question um i guess it would okay it was somewhat accidental but it was also just the the um the kinds of questions i happen to be interested in and the branches of mathematics that i happen to be interested in happen to be ones which were directly applicable to mathematical physics so i got really interested in areas like you know differential geometry functional analysis and kind of the the mathematical foundations of of general relativity and so then when it was when it came time to kind of pick a research area i could have done something that was like super pure and abstract but it kind of seemed more interesting to do something that was related to the foundations of gr and that kind of that was like the gateway drug into mathematical physics i see so a man of relativity relativity is really interesting huh it is yeah once you get a hold of that einstein you know that einstein version of relativity you still you kind of like get hooked on it i think but yeah that's really cool man um could you maybe explain some of the basics so you know with wolfram's project we start with these structures called hypergraphs right so they're nodes basically that have connections and with those nodes that connect and if you guys are unfamiliar make sure you check out the first podcast we kind of give a summary of these things and then you know you can also see what jonathan clarifies in this podcast but going back so we have the nodes that are just basically points you can think of them as that are connected by these um i don't know what you call them edges or connections maybe edges for two or more but yeah so they're basically these nodes connected to lines or connections that formulate a space a hyper graph space could you explain maybe um how you can connect that hypergraph space to the physical space that we're actually used to in let's say reality right right okay that's a very important first question how do we connect the model to reality um so uh what i would say is that okay so we know or we we've suspected for a long time um in like broadly in the field of fundamental physics that this approximation or this idea that we have that space and space-time are smooth and continuous is really only an approximation that it kind of has to break down at some point and the reason for that is we know that there's this particular length scale the so-called planck length or planck time scale at which sort of relativistic effects and quantum mechanical effects start to become comparable right so relativistic effects happen at sort of high energies quantum mechanical effects happen at low distance scales and there's a point where these two scales cross over and that's sort of the planck scale and so to describe the structure of space and time at that scale or below you really need a theory of quantum gravity and you know we don't have one of those yet um and so everything above that scale we know is well approximated by thinking of space and time as being smooth and continuous but below that scale people have started to think it's interesting there's been a kind of convergent evolution of these ideas where people working in many different fields of sort of quantum gravity have ended up converging on the same idea that maybe space time is fundamentally discrete so like in string theory uh you know there are these notions of entanglement networks that people like uh mark van ramstump and sean carroll have been investigating there are ideas of spin networks in loop quantum gravity there are uh you know corresponding ideas in other fields like causal set theory and causal dynamical triangulation so lots of different approaches to quantum gravity have kind of ended up with this view that space time must somehow be fundamentally discrete but all of these different approaches they start by essentially considering dynamics in a continuous space or continuous space time and then discretizing it so you know you you you solve your equations of motion or something for it for a smooth continuous space time and then you construct a discrete sort of mesh approximation to it or something that's how something like causal dynamical triangulation would work and it's kind of obvious why people have done that because we know how to define dynamics over these smooth spaces we can just write down some some equations write down some partial differential equations and solve them uh you know solve the equations of motion and you you get the dynamics what no one has really done is start with the assumption or start with a discrete structure as the fundamental object and build up to the continuum and the reason no one's done that one of the reasons no one has done that is because it's really hard to define dynamics on a discrete space because you can't you know all the standard approaches to doing physics that involve you know equations of motion and lagrangians and you know least action stuff they don't work you can't define pdes over over a network in any kind of reasonably straightforward way you can't define equations of motion for a network in any reasonable straightforward way because they're discrete so what you need is a radically new way of thinking about how to get dynamics and that's really what the wolfram model is so it's it's starting from this assumption that is becoming less controversial over time that maybe spaces is inherently discrete but rather than trying to obtain the discrete approximation from a continuous one it's assuming the discrete structure kind of from the very beginning if that makes sense and then defining dynamics in terms of you know substitution rules and things which we'll talk about later but that's the that's the basic idea that's really interesting so um yeah i like that um that's that's that's interesting though because if you if you start from discrete space um how do you go about actually let me give an example of how you actually figure out how to get around some of these issues with um starting with a discrete space rather than starting continuous and then making it discretized yeah absolutely i mean so um i think a good example is uh say molecular dynamics versus fluid mechanics so we know that um in reality you know fluids are not um they're not really smooth they're not really continuous i mean they they behave like they like they are and for like practical fluid dynamics or engineering problems or whatever we treat them as continuous we solve you know partial differential equations like the navier-stokes equations or the euler equations that describe the fluid but we know deep down that actually if we look with a powerful enough microscope it's just a bunch of discrete molecules bouncing around and then there's a kind of complicated problem of kinetic theory that tells you how you go from the discrete molecular dynamics in order to obtain the continuous fluid mechanics and there's this whole complicated mathematical derivation called the chapman enzog expansion that lets you essentially make that transition you start from the boltzmann equation that describes the discrete molecules and by doing this complicated expansion you end up with the with the navier-stokes equations of fluid dynamics and that's how the navier-stokes equations are kind of derived in in in the context of statistical physics so in a sense what we're trying to do is the analogous thing but for space-time so we start from something that's a bit like molecular dynamics so just like in molecular dynamics you have rules that say you know i have some cluster of molecules and then they interact in some way and i get a cluster of molecules in a different configuration we have exactly the same kind of thing so we have a we have a hypergraph as you say that represents our space and then we have rules that just say i have a piece of the hypergraph that looks like this and i replace it with a piece of hypergraph that looks like that and that's kind of a very minimal model for dynamics of these kinds of systems and then in a certain class of cases you know you can start to define things like the boltzmann function but for you know hypergraph nodes rather than for molecules and that means that in a restricted class of cases you can do something that's very much like the derivation of fluid mechanics from the molecular dynamics you can derive continuous equations and continuous laws of motion just from these discrete hypergraph transformation rules and the really phenomenal thing is that in at least a large class of cases those continuous uh sort of like the analog of the fluid equations but for these hypergraphs turn out to be the einstein field equations for general relativity and that's kind of that that realization was one of the key things that made us realize that this this approach might actually be a fruitful way of thinking about quantum gravity so that actually brings me to an interesting point about um the wolfram model here jonathan is that you know i see all these real cool things that kind of that you guys will say will come out of this theory but as a physicist who kind of is more um the model is very abstract for me so i try to keep it very close to um you know the reality of the reality of things and i can't really sometimes make the connection with the abstraction that it seems the model has to you know the reality of this situation so it's like um i guess for example would be just even with space so if we're going back to space again you know i see these real intricate hyper graphs you know these these like membrane kind of structures that look like you know these neuronal you know connections in the brain let's say and then i try to think how does that actually connect to actual space because you know when we see space you know we think of it as space time i know the wolfram treats time in a separate sense kind of but um you know we think of space as a three-dimensional you know i don't even know you can maybe think of it as even a bubble or something if we want let's say that a bubble of a bubble of space so then i try to think how does that actually how does the wolfram picture picture match up with that i don't really see it because it's like you know in the space time that we have we have you know metrics let's say for example that can tell us tell us how space time curves and whatnot you know when there's mass involved in the wolfram model let's say if we have the space hypergraph i don't even really understand how the particles fit in there or or mass or just like what what is a connection there can you maybe bring it down to earth what that connection is yes no okay excellent question but it has there are many parts to it um and so uh yeah everything i've just i've said so far has been very abstract and you're right you know if we're if we want to talk about physics rather than just you know pure mathematics we have to make these these connections so um yeah okay so so in the in the formal okay it's used the kind of technical vocabulary the idea is that these hypergraphs are representing approximations to romanian manifolds or space-like hypersurfaces which is just those are just the mathematical structures we use to describe ordinary physical space as you say we conventionally think of that as being three-dimensional those of us who aren't string theorists or whatever um and so as you say you know you normally have in both space and in space time you have a notion of a metric and the significance of the metric is that it lets you if you have a pair of points you can the metric lets you you can project this metric tensor along that across those pair of points and you can work out the length of the path and so uh and there are a bunch of these different kind of differential geometric constructions that you can do in space and in space time that let you determine as you say things like path distances things like curvatures um things like uh torsion all these different kinds of quantities and so you ask a very important question which is how can they map into the case of hypergraphs um and the answer is that for each one of these kind of concepts in differential geometry there is a direct translation um and once you learn the translation it it all becomes somewhat intuitive so like the analog of a metric okay so if you have a network uh you can solve what is often called the shortest paths problem for that network so you can if you if you have a pair of points a pair of nodes uh you can find the shortest path between those nodes which just means you know a a continuous path between those two points which traverses the fewest number of edges sort of as an intermediate thing and so uh that gives you a notion of something that limits to a to a continuum distance at least for some networks so as the network gets very very large the distance quantity you would get by solving the shortest paths problem converges to the same distance that you would get by computing it using a romanian metric tensor which is kind of the standard way you would do it in continuous geometry so actually defining distances is not so hard defining curvatures is a bit more complicated and it's a bit more complicated because um essentially it with an ordinary space with an ordinary manifold the notion of dimension is always fixed and this is kind of a postulate of differential geometry in fact uh dimension is what's called a local invariant it means that you every connected piece of your space has to have the same dimension with a network that isn't necessarily true so um in ordinary physical space okay suppose you wanted to ask uh how do we in how do we know we're in three dimensional space rather than four dimension or two dimensional space one way we could do it is by you could take a sort of a bunch of gas molecules and you could just diffuse them out and and you you grow out some kind of sphere of diffused gas and you could ask okay how does the volume of the sphere of diffused gas scale proportionate to the to the radius of the of the sphere and so if you were doing this in two dimensions your your sphere of gas would really be a circle so it would grow like pi r squared if you do it in three dimensions it's going to grow like the volume of a sphere four thirds pi r cubed and in general if you do it in d dimensions it'll grow like r to the d so by looking at the volumes of a kind of a diffused ball if you like of stuff uh and looking at the exponent d in in that expansion you can work out the dimension of the space this is what's called the hausdorff dimension of the space and you and you can do exactly the same thing in a graph so you can you can look at a point in a graph a node in a graph and you can look at all the nodes that are adjacent to it and all the nodes that are adjacent to those nodes and the nodes adjacent to those and so on you grow out some ball of radius r and you just ask how many nodes are there inside that ball and again if the if the network is approximate or the hypograph is approximating a d dimensional space it will grow like r to the d the bit where it gets interesting is that in differential geometry there is a correction term to this expansion so so these these volumes of of balls are given by essentially a power series expansion for the metric tensor and that power series expansion it its leading order is um is r to the power d uh but it has sub-leading terms it has a subleading term to the power of proportional to r to the d plus two and the coefficient of that term is proportional to the curvature of the of the manifold um and specifically in the romanian geometry case it's proportional to what's called the richie scalar curvature which is an important quantity in relativity and other things and so that means we can use that same construction to define an analog of reachy curvature for hypergraphs for networks and hypergraphs we just we do the same expansion we look at a discrete ball of radius r and we look at the sub leading correction and so so long as you know the continuum dimension that you're supposed to be approximating you can therefore work out the curvature at any point and so and there's you can get even more sophisticated information so like there's you know there are notions of reachy curvature tensors which are sort of directional versions of curvature they tell you how curved the manifold is not just as a scalar but actually projected in some direction and you can do the same okay in the continuous case you can infer that by looking at the growth rate of a not of a ball but actually of a tube that's projected in some direction and again just like just like that in the in the network you can you can kind of grow out a continuous you can grow at a discrete tube and you can you can measure how it you know how it's volume scales um so a large part of this project has actually been taking these concepts that exist in ordinary physics or in ordinary geometry continuous geometry and kind of producing intuitive hypograph translations of them but the nice thing is that those translations can be produced and as the hypergraph grows really big they become they essentially converge to their continuous analogs just like when you uh you know if you have like a finite i don't know finite difference approximation to a partial differential equation and you make the you know the the different stencils really really small it converges to the same value as the you know as the continuous integral or whatever you're approximating it's exactly the same thing very very interesting man um you're convincing me the more you talk about it because uh it makes me think um i need to um maybe the connection is harder to see because of you need to know more about network theory or what would you call a graph theory yeah graph theory specific well graph theory in general and and also there's a similarly named field which is network science which is a bit different so so graph theory is essentially is you know it's a field of math and computer science um and so that's where a lot of these concepts come from but we've also borrowed some techniques from from network science which is about how you analyze networks or graphs that are really really big um and so that that's obviously something that we care about because we want to we care about these questions about what happens when these when you basically take the limit as these things goes to go to infinity and that's really hard to do in general and then there's also this other field which is also connected which is discrete differential geometry which is which again we've made heavy use of which is essentially all about this question of how do you translate the ideas of differential geometry to uh spaces that aren't continuous that actually are fundamentally discrete so people have done that for um these quite abstract topological things called simplicial complexes uh but but we are doing it on a much more down to earth scale you know basically defining differential geometry on graphs and networks very interesting so you're getting a lot of these similar mathematical equations let's say that are arising out of this wolfram theory that you're seeing parallels some of the things that already exist in physics or you're finding things like the curvature and you know uh dimensionality um and actually that that makes me think of another interesting point that i had did you question one no no okay sorry um yeah that makes me think of another thing that was interesting um that i saw in your theory that i wanted maybe you'd experience more although you talked about it some um you you mentioned the dimensions the way you find the dimensions by actually you know taking that ball of gas let's say and then allowing it to spread out in space so the interesting thing that wolfram's theory um also predicts is fractional dimensions so a fractional dimension i mean uh we mean juan talked about this on the first podcast what does that even mean right right okay excellent question um so that this uh this is a concept that's been investigated in lots of different contexts but i mean most people will have encountered it maybe if they've encountered it they will they will encounter these concepts of fractals so um these are sort of geometrical structures that have i think entered at least part of the popular consciousness just because they form quite pretty pictures so a lot of people have seen pictures of like the mandelbrot set or julia sets or things things like that and and uh or you know even these kind of more boring ones which are like you know coke curves or these space-filling curves that which have like self-similarity patterns to them um and the reason they're called fractals or the the um the etymology of the term fractal comes from fractional dimension so the the the um the unifying characteristic that all these fractals have is that their dimension is a bit funky so if you take the mandelbrot set or something like it uh its dimension will actually vary depending on which part of it you're looking at and it's not necessarily an integer and what that means is that um okay i briefly mentioned this concept of hausdorff dimensionality but maybe it's worth just saying a little bit more about what that is so household dimensionality is the fancy word for exactly what i was talking about before this idea you you measure dimension by just looking at the volume of you know how the volume of something grows so like when we say a line is one-dimensional it's because its volume grows linearly as you scale it but with a square it grows quadratically as you scale it and so we say that a square is two-dimensional with a cube it grows cubically so we say that it's three-dimensional and so on um but because these are just exponents of some variable and because you can define it you don't necessarily need to define it you you we have a notion of non-integer exponents uh it's possible for you to define geometrical constructions that don't actually have an integer scaling law and these are these form what are called fractals and so this is um this notion of household dimension is one notion of dimension the notion of dimension that i guess people are more maybe more directly familiar with is often what's called topological dimension which just means the num basically it means the number of um independent vectors or or orthogonal vectors that you need to span a space right so uh you know when we say that this that our physical space seems to be three-dimensional it means that that just means that there are three directions right you could that way that way that way and as far as we know there isn't you know everything else is just a linear combination of those um and so clearly by definition topological dimension has to be an integer so that's i think partly why people get a bit confused because they think well how you know how can you possibly get a non-integer number of dimensions because that you have to have an integer number of these these vectors but these fractals have this interesting property which is that in fact this is in fact the formal mathematical definition of a fractal is this is a space where the hausdorff dimension is not the same as the topological dimension in fact the household dimension in a fractal is strictly larger than the top so it in most like well-behaved nice geometrical spaces the household dimension the topological dimension are the same right in ordinary euclidean space that we you know that we generally think about they are identical that you know the square is is both two dimension has both two topological dimensions and two household dimensions but in these fractals it's not as well behaved the household dimension is is different it can vary it can be it cannot be an integer but the topological dimension is always fixed and an integer so when we talk about non-integer dimensions we're talking about non-integer housed off dimensions and so you get non-integer dimensions for exactly the same reason that you would in a fractal it's just that the the scaling relation is such that we we get a power law or we sorry we get an exponential law but where the exponent is not it just happens not to be an integer um and the reason for that is just that these hypergraphs are much less constrained than ordinary manifolds right so ordinary manifolds ordinary uh continuous spaces have very very strong mathematical axioms that essentially enforce that the dimensions have to be you know nice and well behaved and integer and so on uh with a hypograph it's kind of like the wild west of geometry there there's no there's no such you know a strong set of axioms so some hypographs will correspond to nice manifold-like structures but many more of them will correspond to sort of more fractally things with weird house stuff to mention and uh so this this sort of uh prediction such as such as it is is that given that fractals and how and you know non-integer household mentioned seem to be quite common in hypergraphs it's not unreasonable to assume that if this is a good model for physics what's probably happening is that we have a universe that is a very good approximation to three spatial dimensions you know four space-time dimensions but whether just like in the mandelbrot set there can be these small kind of non-integer uh disturbances so it may be the case that you know in some regions of space it's actually not exactly three-dimensional it's maybe 2.999 dimensional and in some other places it's 3.0001 dimension which just means that there's a slight discrepancy in the in how these volumes would you know how a volume of a ball might scale uh and that's kind of interesting because that you know we think that we've started to um compute what the kind of potential observational consequences of those kind of pockets of different dimension might be and it seems to be a fairly robust and potentially testable aspect of the theory this idea that the dimensions are not don't have to just be integers everywhere that you can actually have small corrections to the dimension in different regions of space that sorry that that's my that's the first attempt at the answer yeah that's a really good um good explanation yeah it leads me to a couple questions because um i'm trying to think why would you even um want to invoke hausdorff in the first place because you know if we live in a topological universe or top uh universe topological dimension um it's like why would you not want to just stick with that but i guess maybe is it that house dwarf is the only way to be able to measure these things in or just another technique i guess maybe or is there some kind of also is there some kind of relation between house dwarf and topological dimensions that can be translated right right okay great question so so in again in well-behaved spaces yes they can be translated so so in in ordinary manifold structures and so on they are in fact they are the same notion of dimension uh you know manifolds are well behaved precisely because all these different notions of dimension that you could define all happen to coincide um so as for why we use household dimension instead of topological well a because it's more directly relevant for curvature calculations so you know in that calculation i gave i mentioned before where you you compute the curvature in terms of the subleading term of this expansion uh the dimension that appears in that equation is the house dot dimension not the topological dimension um which is important if you try and apply geometrical techniques to to fractals and define curvatures of fractals and things the other reason it is which i guess you kind of alluded to uh is that it's just in in networks and hypergraphs it happens to be way easier to measure topological dimensions so sorry it happens to be way easier to measure household dimension measuring topological dimension is kind of hard because you need to have a notion you need basically to import ideas from linear algebra and you have to essentially treat the you know the hypograph of the network as being like a vector space so you can then define you know notions of orthogonality or linear independence and you can talk about spanning sets and bases and things and it's not even immediately clear how you would define a vector right in a in a hypergraph right you at some level you could say okay a vector is just a path but then it's like okay well now i have to define the notion of a dot product right i have to find inner products between vectors and it turns out for subtle reasons there is no there isn't actually a unique way to do that um there are spaces in which you can there are so there are particular hypergraphs which are well approximated by vector spaces but there are many more that aren't and so even defining a consistent notion of a dot product is really really difficult and you kind of you need that in order to define topological dimension whereas household dimension as i say all you need is just the notion of a volume of a ball and the notion of a radius and that's very easy to get in a network so uh so household to mention just happens to be the easiest thing to compute and also happens to be the thing which is relevant for uh for deriving general relativity for you know relevant for space-time curvature and things very interesting so i guess one more last question about the hausdorff thing so we can move on but i just want to just finish it off with this so with that you're saying that the hausdorff dimensionality if it matches with the topological one in good in well-behaved spaces you you mentioned that there could be pockets of different dimensionality or different dimensional uh house dwarf dimensionality even in our space but is it preferable for you to look for a hypergraph structure let's say that has a topological uh dimension that actually matches the house dwarf dimensionality so we can call that actual space or actual that's the one that corresponds to our universe well it's i mean it's obviously preferable for exactly that reason right because uh you know all of our current observational evidence suggests that you know we do we don't live in a fractal you know we the the topological dimension and the household dimension of our universe are the same and so if if it is the case that they're not identical these corrections must presumably be quite small so we'd be talking about you know we're not talking about you know you have one region of space that's seven dimensional another another region that's two dimensional we're talking as i say about you know one region that's three point zero zero zero one dimensional or something and uh the reason is because because of how this expansion works um okay let me i won't get too deep into the math but the the uh locally uh you can you can approximate an exponential with a quadratic right if you look closely enough it's quite hard to distinguish an exponential and a quadratic right which means that when you think about these this expansion which has as i say this exponential contribution from the dimension and this subleading quadratic contribution from the curvature when you look at very small scales it's actually quite hard to distinguish between the two or in other words a large change in curvature could equally well be called the the the effects of a large change in curvature are indistinguishable from the effects of a small change in dimension at least if you look very locally because you're basically making a large change in a quadratic and a small change in an exponential and so uh because we don't know off the bat we don't know a priori what the dimension of our spaces is we have to be open to the possibility you know ordinary ordinarily in general relativity you make the assumption that dimension is fixed and curvature varies whereas in these models you cut that you can't make that assumption you have to at least allow the possibility that both quantities are varying simultaneously and so so the the most physically plausible scenario is that dimension is almost fixed it it's very very close to being fixed but it has these small put up you know there are small disturbances in in in various places and that we may be able to measure and so the kinds of hypergraphs that we're looking at or the kinds of dynamics that we're looking at are or looking for are ones that produce those kinds of spaces ones that you know spaces that are well approximated by something which has integer dimensions but where there are small perturbations and they are they do exist and and they're they're reasonably common but not incredibly common and so it's it's not um so it may be that in that sense our universe may end up being quite special uh in in in the sense that it's somewhat distinguished in this space of all possible universes but we don't we don't really know how that story will play out yet gotcha very very interesting well yeah jonathan i i like for me i have this one particular line in the project kind of had me uh shocked really um because the team lays out what does you know like when you say well like what does quantum mechanics and gr being similar say about physics because you all make this statement that g like gr and quantum mechanics are effectively the same and that was like a little bit shocking to read as a physicist because you're just like what how how do you justify that yeah yeah uh so i wanted you to elaborate on that at least for the listeners too like that hearing that was shocking to me could you elaborate on that yeah so it's a pretty bold statement right so um yeah okay there's there's a there's a bit of um there's quite a lot of sort of stuff underlying that statement for sure um and it's it's it's a little bit easier to make it quite nasty it is because i'm sorry it is only because i do have more questions about yeah yes but yeah so yeah yeah sure sure so okay here's the here's the basic rundown so um okay let me give a quick explanation of how you derive very roughly how you derive gr sr and gr in these models so when you have a hypergraph you're applying these rules to it you're replacing bits of hypergraph with other bits of hypergraph so sometimes those rules you can sometimes those rules will have causal dependencies between them so sometimes rule b could only have been applied if rule a had previously been applied and that might happen for instance if rule a produces an edge that's then used in the input of rule b so in other words the input and the output have some have some you know they they overlap in some way so that uh you know because the input for b used to hype used an edge it was produced by the output of a b couldn't have been applied unless they had previously been applied so in that case you can say that there's a causal connection between the between event a and event b or rewrite a and rewrite b and so you can construct what we call a causal graph that shows essentially all of these causal connections each each node is now an event each a rewrite and the causal graph shows the causal dependencies between all those rewrites and that gives you a directed graph that um in a so if the hypergraphs are like approximations to space the causal graphs are like approximations to space time or to be a bit more precise they are approximations to what's called the conformal structure of space time so space time has a causal structure it tells you if you look at the light cones in space-time they tell you which events can influence which other events and so on and that's exactly what the causal graph tells you it tells you which which of our rewrite events influence which other ones so it's kind of a minimal model for space time and then if you and then you can it using a sort of using the the sort of process i defined before uh i mentioned before which is just you know transitioning from this kind of discrete molecular dynamics like stuff to the continuum equations uh you can derive a continuum theory of these causal graphs under some assumptions so if you assume certain things about the rewriting dynamics specifically if you assume uh well if you assume a that the hypergraph converges to something that's finite dimensional and you assume that uh that the rewrites are random in the sense that or okay to be more precise that they are weakly ergodic which essentially just means that you know there's no kind of there's no bias in how the rewriting occurs yeah um and that's the same that's the same as saying that uh in molecular dynamics you need to have molecular chaos you can't just you you can't have all the molecules like drifting in one direction or something um and you make this additional assumption which is what we call causal invariance that is essentially a translation of lorenzion variance or general covariance but to the discrete setting and that just says that the causal graph the causal structure that you get is the same regardless of the order in which you apply the events so sometimes you might apply the events and you get you might you might apply the events in a different order and you get a completely different sequence of hypergraphs with a different causal structure and those we know cannot those kinds of rules can't be compatible with special relativity so to get special relativity we have to enforce this constraint that the causal structure is always preserved when you change the rewriting order and that's like saying that the conformal structure of space-time is preserved under changes of reference frame which is exactly what special relativity says right right um and so from those three assumptions it turns out that's sufficient to define uh to derive general relativity to define the act to derive the einstein field equations and the way that that happens is you um well you essentially you you do it a lot like with the continuum theory you you write down essentially a sum over curvatures over reachy scalars um and that then you take the limit as this causal graph becomes infinite and then that sum over these ritchie scalars converges to an integral because you're because of this ergodicity assumption you can switch out this discrete sum for an integral once you take the infinite limit and that integral is exactly the einstein-hilbert action of general relativity which is the action integral from which you get the einstein field equations and that's that's the basic rundown of the of the derivation there's a lot of technical detail which we could show if you're interested but that's that's the idea okay so then you've got quantum mechanics so this is this is where it gets a little so slightly more abstract but hopefully not too much more abstract so um i already mentioned that you can have a situation where applying the rules in a slightly different order leads to a slightly different hyper graph so in reality you don't just have one possible history it's not like there's a linear trajectory from one hypergraph to the next to the next because there are many different rewriting orders and there's no canonical rewriting order just like there's no preferred reference frame in our universe as far as we know in general you don't get a single evolution history you get a tree of possible histories or actually to be a bit more precise it's it's more complicated than a tree because not only can these branches of history diverge they can also merge again because the hypergraphs can just accidentally become the same so you actually get this very complicated sort of bush-like structure that we call a multi-way system or a multi-way evolution graph and when you're first confronted so at first we thought this was a bug of the formalism that it's like okay we don't get a unique history evolution history we get all these kind of different ones we tried to figure out ways to make this problem go away but we later realized that it may have been a feature that that is um in quantum mechanics you also have this notion of you know alternative histories right so in in the at least in the in the mathematical framework of quantum mechanics what it means when you have anti-commuting variables is that essentially any observations that take place of those variables must take place on on different branches of history and so from taking that idea to its logical conclusion this is where you get the so-called many worlds interpretation of quantum mechanics that you ever had kind of pioneered and so this notion of a multi-way system of many possible evolution histories for the for the hypograph seems very many like right so it seems like this may be giving us something that's a bit quantum mechanical so the question is can you make that a bit more precise can you make that correspondence more more formal and the answer turns out to be yes so in quantum mechanics when you're doing you know when you're doing when you're dealing with the mathematical structure of quantum mechanics you are doing everything in relation to these hilberts what are called hibbert spaces these abstract spaces of quantum states and there are certain algebraic operations you can do on hilbert spaces like you can you can take the tensor product of two hilbert spaces which just means that you're sort of combining two quantum systems together to make a composite system or you can uh you can take what's called the adjoint of a hilbert space which essentially just means you're playing time in the other direction or you can take the dual and things like this there's a whole list of these things you can do and uh our multi-way systems also have an algebraic structure to them uh you can take two multi-way systems and you can take their tensor product or you can take the multi-way system and you can reverse all the arrows so the time runs in the other in the other way and the question is are those algebraic operations really the same as the algebraic operations that you do for in hilbert spaces for quantum mechanics and what we were able to do using some fancy categorical methods was to show that actually the answer is yes the the axioms satisfied by the algebra of multi-wave systems are the same as the axiom satisfied by the algebra of hibble spaces which means that actually there is a direct like one-to-one correspondence between concepts and quantum mechanics and concepts in multi-way systems okay that's already kind of interesting so then the obvious question is well if we know that the continuum theory of these causal graphs as i say under these assumptions that you have finite dimensions you have random essential you know randomness in the rules and you have uh causal and variance we get this continuous we get these continuous equations which are the einstein field equations the the obvious question is what are the continuous equations for the multi-way system and okay this derivation isn't everything i've talked about so far is stuff that we've actually been able to prove what i'm about to say has not yet been made mathematically rigorous but it seems to be true on the basis of computer experiments so i just want to i want to make that clear that this is not the the statement i'm about to make is not proven it just seems to be correct empirically but it seems to be sorry right exactly right so it seems that the the continuum equations or the okay the same argument that gives you the einstein hilbert action in the causal graph if you apply the same arguments to the multi-way system appears to give you the path integral of quantum mechanics um so so in in that sense there are these two action principles that we know that we know in fundamental physics right there's einstein hilbert that gives you gr and there's the path integral that gives you qm uh or in the relativistic case that gives you qft and this what we mean when we say that it seems gr and qm are really the same theory is that it appears the same mathematical argument that gives you the einstein hilbert action for the causal graph gives you the path integral for free in the multi-way system um and and therefore these two action integrals are just they are representing the same idea just applied to two different applied at two different levels of the model one at the core level the causal structure and one of the level of the multi-way structure um now actually presenting a rigorous derivation okay quantum field theory doesn't even have a rigorous derivation of the path integral yet so ordinary and and that's been going for like nearly 100 years so uh we it's suffice it to say we don't have a rigorous derivation of the path integral either um but it's something that we're trying to work towards and and but as i say initial numerical data sequences seems to suggest it is ultimately just the same action principle that gives you both qm and gr sorry that was a very long answer for that oh that's a very good answer no it's very good because um when you mentioned the mult uh multi-way space um you do bring it up in in in the topic of discussing quantum mechanics in this uh technical uh work how i mean i was kind of shocked to see that it gives rise to the uncertainty principle uh with it has to do with community or commutation rules and and how you y'all map it to a certain space and stuff it it it was all sort of rattling um for me but that was a very nice uh explanation though i mean damn bro that's big if true that's all i gotta say yeah for me it was it was how could you say something so controversial yet so brave i was i was really floored by it but um the other thing that i kind of wanted to uh ask you was um like is the abstraction to to just this level of information structures or like information constructs seemingly more important than the actual physics you think wow that's an interesting that's an interesting question um uh some i mean okay it's gonna sound slightly heretical to say this sure i think the answer is i think you're right i think the answer is yes i think um in many ways the most important okay um i i i wanna i don't want to say anything that sounds too strong you've already described that just say this is my opinion yeah that's all you got it's fine we'll shield off the haters um yeah i i don't want to be too controversial i mean so i don't yet okay it is not yet clear whether this is going to yield um sort of whether this general approach is going to yield significant advances in fundamental physics uh i think it i think it is and i think i think stuff is looking really promising uh but obviously we can't know that for certain until we actually have an experimental prediction and it gets validated or falsified or whatever um so for the time being the best we can say is that what we've got is a really interesting and very rich and very beautiful mathematical structure that appears to have very very bizarre uh formal analogies with fundamental physics right so the very it's okay it's a bit like what may be happening okay the worst case scenario is that it may be a bit like uh what is called colossal klein theory in physics which you guys may well have heard of right so collusion calling theory this kind of predecessor have you guys done a podcast on it or no we should show our list actually very very brief like mentioned we mentioned it yeah right right so so yeah so as you guys will know and listen to the podcast right it was it was a kind of string theory predecessor where you just um you know the you you say okay gravity we get gravity from curvature in four dimensional space time maybe we get other forces from curvatures in higher dimensions so you go to like five dimensions and you see oh you get electromagnetism and it's like well that's amazing that's a really interesting analogy formal analogy between gr and electromagnetism turns out that it's not physical because you you end up massively mispredicting like certain properties of the electron but still it's an interesting mathematical connection that then led to a bunch of interesting physics in the worst case that may be what's happening with this model that that is it may be that this is not quite the right way to think about the foundations of physics it may be a bit like color decline theory uh that you know we will get a lot of stuff right and then somewhere we'll mispredict the mass of the electron or something and the theory will come crumbling down but just like kalutsukaine theory got absorbed into this other program and turned out it was a kind of fruitful way of thinking about things that just kind of had to it needed some tweaking uh the same thing may be happening here that we we've got something that has surprising formal connections to the foundations of physics uh but it ends up being absorbed as part of some larger program that may happen um i don't want to take bets on what the probabilities are but that's that's at least one possibility so the question you ask is a really very interesting one therefore what is the most interesting thing is it the detailed physics connections or is it somehow this more this more philosophical angle that is it is fruitful to think about physics in these purely informational or computational terms and if i'm being brutally honest with you i think the answer is it's it's the latter right the the more interesting more interesting than any one detailed physics connection is the fact that what i think we're really learning by discovering by you know investigating this formalism is that uh thinking about physics in terms of computations in terms of ideas like algorithmic complexity or you know graph theory or uh you know computability or you know computational complexity those kinds of ideas they actually do have a role to play in fundamental physics that is a really important point to make and um again just like with discrete space other approaches to quantum gravity are also reaching the same conclusion right so there's this whole uh i mean you guys are probably aware there's this there's a sort of um a program in fundamental physics right now called the it from qubit kind of idea which is this like this notion that somehow there's a deep connection between uh qubits that is quantum information theory and you know things like the structure of space time so this is an idea that kind of has originated in string theory and in ideas like the like gauge gravity duality cft correspondence where suddenly people are now realizing that you can compute uh things about black holes things about gravity by looking at essentially quantum information theory on the boundary of the black hole and then things like that so you can you can do like entanglement comp entropy calculations which are really a computational idea and infer things about gravity which seems really crazy and i think basically i don't think physics has really absorbed that idea completely yet the idea that these purely computational information theoretic ideas actually have physical consequences and this model is kind of the most extreme version of that in that if this model is actually correct it means that all of physics basically is a result of information theory and computation and i think yeah um that's a really excellent point that you make that this is in some sense that is the single most interesting thing about this no for sure because i do see you you see branches of physics reaching sort of like statistical mechanics has has been i mean famously uh reached this point uh or or sort of played with this point of information theory uh and everything and so i'm like okay quantum mechanics is like the shannon information theory okay and uh and how in the app like remove the physical properties from the from i guess the the situation system or something but the system and you you and just leave it to the abstraction um you get something that i feel is more fundamental or seems more fundamental and no and i mean like the looking at the hyper graphs and these multi-way systems uh it's just i i do it makes it made me ruminate on that and so yeah i thought i'd ask you about it but yeah great answer i it it was just interesting to to see yeah i thought that was a very nice answer and also um i guess jonathan i wanna i guess jonathan i wanted to just say um to put a petition out there i think you should change the uh project to the wolfram gerrard project just saying uh so i it's i should clarify it's you know there are there are more people than just myself and stephen working on this so this is um you know okay the okay it's perhaps worth mentioning just a little bit about the history so um you know as you guys know and i think you mentioned on the on the last podcast right now stephen in another era was a kind of was a theoretical particle physicist and uh a you know a reasonably kind of high-flying one but eventually he kind of he bailed on academia to go and start from research and build mathematica and things which was probably a better use of his time than continuing doing particle physics um and but but he's always kind of maintained this interest in physics and an interesting kind of science and computational method more generally and so uh in 2002 he sort of proposed as part of this book that he published called the new kind of science he um proposed that this might be a fruitful approach to doing physics this idea of thinking of space in terms of networks and hypographs and things like that and thinking of dynamics in terms of their substitution rules and he kind of laid out in rough terms how that might work and he kind of left it there and i think he's i think he's sort of expected that people would take this up and run with it but nobody really did and so for you know like for nearly 20 years it was kind of it just stayed there you know in that book and and and that idea never really got off the ground um but then yeah so then what what was it two years ago uh myself uh and another sort of young physicist guy max piskanov and steven kind of we decided that if anyone was gonna do this project it kind of had to be us and it basically had to be now and so uh that was when we sort of started working on this in earnest and after we announced the project back in april uh since then loads of other people have kind of joined forces and helped contribute both to programming and doing like crowdsourced computing and to helping us you know do research and write papers and things so like um okay i mean to mention a few specific people like uh the these these quantum mechanical derivations i was talking about before you know like connecting the algebra of multi-way systems to the algebra of of finite dimensional hilbert spaces that wasn't just me that was that was a like it was a collaboration between myself uh monogamy and namadori xxx asawala and some other people um there's uh folks like hatha marshallawi who's also been helping out with kind of formalizing a bunch of the connections to topology and homophobia theory there's now you know we have about i think on the order of about 50 actually uh affiliates kind of and students working on on various aspects so thank you for your you know thank you for the very kind suggestion but it's it's important to note that it's this is not just myself it's not just steven it's it's there's a large team of people uh you know doing these things well jonathan i gotta say i think you guys are doing really good work man and i think you are um very good at explaining um what's actually going on with the um wolfram project and i mean i would love to hear more maybe even you know if we can have you for another podcast at some point um we would definitely love it i think this was a good a good a good thing to um relay to the audience here yeah for sure but since we're coming on time i got a little scared here because uh john i just saw a low memory usage here and i don't yes we want to make sure we save this uh yeah this podcast but it was great having you on jonathan and this helped clarify a lot of things i think yeah both me and terence so yes yes you for sure have turned me to to the wolfram team a little bit more today for sure okay this is some victory switch to yeah yeah yeah and like if if the listeners i'm pretty sure the listeners will want to see more and i think uh they'll be excited to hear more maybe we could flesh out a little bit more of the project because i i barely scratched the surface i didn't even ask all the questions i wanted to ask so yeah i've got like 50 more so any you know hopefully we we won't be bugging you too much for another uh for another episode like this but at some point we'll get up we'll get him again hopefully this was a pleasure no thank you it would be it'd be wonderful to do a follow-up sometime and uh but yeah i mean thanks again for inviting me the questions were great and this was yeah this was a lot of fun so thanks thanks again cool all right thank you jonathan and guys remember to like share comment and subscribe if you haven't already uh check out the websites once again guy guys eigenbros.com buying rose on instagram igambrose on twitter and i can browse two on tick tock and then of course guys follow uh or join the patreon if you can it's patreon.com eigenbros if you support your boys and uh we thank you so much jonathan it's great talking with you man thanks a lot cool you

Channel: Eigenbros

Views: 5,279

Rating: 4.9477124 out of 5

Keywords: eigenbros, physics, science, Wolfram, Wolfram Physics, computation, Jonathan Gorard, Stephen Wolfram, hypergraph, causal invariance, podcast, mathematics

Id: AKDD1AfDJBM

Length: 55min 1sec (3301 seconds)

Published: Fri May 21 2021