The Biggest Ideas in the Universe | 21. Emergence

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll today's big idea is a somewhat tricky one it's one that has received both attention and also a little bit of controversy within physics as well as in philosophy and that is the idea of emergence emergence in the sense of what we might call a higher level theory emerging from a lower level one and I'm gonna use a lot of words like what we might call in this video because some of these ideas are not agreed on or at least the way to talk about them or is not agreed on so I'm gonna try mostly to stick to things that are indisputably true rather than diving into the most controversial aspects of the subject but be aware that out there there are people who use the words in slightly different meanings and disagree with each other nevertheless this is a topic of emergence that deserves its place in the biggest ideas in the universe because it's part of this program we're doing in the most recent couple videos of mapping matching the micro physics the fundamental physics of quantum field theory or whatever the ultimate theory of everything turns out to be to the observed world that we see you know remember we talked about probability and things like that the fact that we're not Laplace is diamond so now we can get a little bit more deeply into what can we do given that we are not Laplace is diamond can we nevertheless talk usefully about the universe and that's where emergence really is crucial it's not just optional right it's really really crucial so we absolutely have to face up to this idea so let me just do a little bit of explaining what where the idea of emergence comes from in science the idea again these are gonna be the ideas that i'm gonna assume are true for the rest of this video you may disagree that's okay you may always disagree there is only one reality and this might seem like a strange thing for someone who believes in the many-worlds interpretation of quantum mechanics to say but for many worlds are for never ready in the the reality is the multiverse it is all the worlds all at once okay so this is not talking about branches of the wavefunction this is talking about the global wave function of the universe so where's assuming there is some thing called reality there's not that many people who think that there's not anything called reality but there are plenty of people who think that the job of science is not to describe reality they distinguish between what may or may not exist as reality and our observational outcomes when we look at the world and they say the job of science is just to describe what we see when we look not to describe what is really there so I am NOT in that camp so I don't even know how to talk that language so I'm going to assume there's something called reality that we're trying to describe but there are multiple many many many theories that can describe reality all at once so the multiple theories describing reality in different useful ways so the idea is that you can sort of capture different in fact let's say different accurate ways it's not just useful useful as a little bit weaker than I want to say you can capture different aspects of reality using different theories of physics so just to give away the game right away the classic example is the one we already talked about last time with atoms and molecules and kinetic theory being some kind of attempt at a fundamental theory in 19th century physics and then the emergent theory is a theory of gases or fluids or whatever thermodynamics okay thermodynamics fluid mechanics etc so you don't need to know all the details about all the molecules to nevertheless have a good theory of the world so the idea here is what we're saying here is that the multiple theories we have in mind would be the kinetic theory of the atoms molecules and also the fluid dynamics theory all by itself so you can derive fluid dynamics from atoms and molecules but you knew fluid dynamics first right or let's put it this way I don't forget about whatever that historically actually happened you don't need to know about atoms and molecules to have fluid mechanics okay it's an independent autonomous way of talking about the world that is accurate in a certain regime and not in others okay and that's what I want to say here these theories can be different theories that can be different already said that the theories should agree in overlapping domains so the idea here is that there is something called the domain of applicability of the theory when you have the fluid dynamics description if you know that there is also a more comprehensive atomic kinetic theory description you don't expect the fluid dynamics description to tell you anything in the regime when you only have one molecule in your system right it's becomes true in the limit when you have many many many molecules so there are different domains one molecule many many many molecules in some domains the theories are either going to be completely inapplicable or they will disagree but when they are accurately describing reality there is a regime in which they overlap and then they agree with each other so that's the basis you know that's based on this fact that we think that there is a reality so where the theories overlap and are supposed to be valid they both better or they all better be describing reality okay and to be honest usually you can talk to a scientists a lot and they will very rarely use the word reality they're not against it but they know better they know that they don't know what reality is they assume it's out there most of them but well we know what we have our models and what we have our experimental data and we have some interplay between those we like to think that our models get better better at capturing reality but to a large extent when we talk about emergence in this way and we talk about the relationship of some higher-level theory to some lower-level theory we're talking about relationships between theories themselves rather than relationships between theories and reality because who knows what that is I mean the relationship is we capture some of it but who knows how much of it who knows what captures the rest etcetera so the basic picture that we have to do to say the same thing we just said but to say it in pictures emergence the usual way it's used will be a little bit more careful later on but the notion of emergence is a and again the usual way higher level and that usually is taken to mean macroscopic large scale many moving pieces going on underneath theory emerges emerging from a lower-level microscopic theory and you could also say fundamental instead of microscopic that annoys people but none of these words are exactly right to mean fundamental makes it sound like it's more important which is just false okay just because a theory describes what goes on at a more comprehensive way doesn't mean it's more important right if you want to get from here to the moon Newtonian gravity is more important than quantum gravity okay or even more important than general relativity it helps get you there in an efficient way so fundamental is doesn't it have exactly the right connotations microscopic isn't right either because the theory is supposed to be true at every level it's just that it becomes important at tinier levels I once tried to have a campaign to replace the phrase fundamental physics by the phrase elementary physics elementary in the sense of basic like you know the the stuff out of which you could potentially derive everything else but also unlike fundamental elementary sounds a little deflationary right it sounds like oh it's just a simple stuff it's not the you know the big hard stuff no one agreed with my proposal though so I didn't really catch on so higher level and lower level or another way of saying the same thing you just got to remember which is higher and which is lower so the way we do that is we say you know here is reality again this is something that I'll mention but most scientists can get away without talking about and then there's some microscopic theory by which we mean the theory that purports to be comprehensive as comprehensive as it can be describing reality and it might not do the whole thing so let me let me see if I can do this I'm going to take this ellipse I am gonna duplicate it so I get exactly the same ellipse they'll put it down there and the idea is that this region of reality is described by this theory okay but then there's some smaller region of reality which is like a subset you then again this is just the usual picture we'll try to be more careful later but usually when you have the gas versus the you know the fluid dynamics description versus the atoms description there is a smaller domain of applicability for the gas description in principle you could always describe what's going on in a box of gas just in terms of the atoms and molecules might be very impractical but that's what Laplace is deeming would do so there's some strictly smaller theory the macro theory that describes the macroscopic world do the same trick here put that down there and then say here is the region of reality oops I should do this down here also should my region of reality what did I do oh my goodness I know where that came from I should just delete this video I'll leave it in for the video people like it when I do silly things in the real time in the video there we go okay so there is some macro theory that is a subset of the micro theory and therefore describes a smaller part of reality and hopefully ideally all these are going to be well understood all these relationships and so this is the emergent theory in this case that's roughly speaking what we what we mean now the reason why I have to be so weaselly worried about this and keep you know saying not everyone agrees is because again to preview a little bit there are people who think that there is an important difference between a theory that is microscopic and a theory that is comprehensive so what is implicit in a lot of these discussions of emergence is that the microscopic theory literally describes tiny little cases that somehow come together to form big pieces in the macroscopic theory that's kind of what happens when you go from atoms to a box of gas but it's not what happens in quantum mechanics for example where things are much more subtle and maybe it's not what happens in things like consciousness you could talk about whether there's a theory of consciousness that emerges from a theory of neurons in the brain and there people are less happy some people are less willing to think that a microscopic theory of neurons could in principle describe macroscopic consciousness that's a more contentious topic anyway but anyway this is this is the basic picture that we have in mind let's get into some details let's give you some examples so the classic example that we already mentioned is atoms or particles emerging into gases and to look up whether it's just 1s or 2's in the middle gases it's 1s I've typed that word a million times in my life but there you go so the more microscopic theory is the theory of atoms or molecules or particles or whatever and you have some point in gamma space or you have endpoints in Moose Pass and the language that we used last time so you have some phase space variables X I P i okay the position and momentum of every single particle in the gas in the box of gas or whatever and then you have some theory up here of the gas or the fluid whatever it's gonna be and there the language you speak is a completely different language and that's the important point here you talk about the density the pressure the temperature the velocity field of the gas and so forth so in this case there's a very very special case because here you can actually derive the higher level macroscopic rules there's literally a map you can construct from the microscopic situation to the macroscopic situation so I'm not gonna do that in any detail but you know basically how it works the density I know that you might not be familiar with the Greek letter Rho being used for density but that's what we used the density comes from the number density of particles of atoms times the mass per particle so the density here we really mean the mass density and these are both things you can work out if you know where you are in phase space for all the particles you've got how many particles there are in a region and what their masses are right you know that the temperature comes from the average kinetic energy of the particles comes from means it's proportional to and the proportionality depends on details about you know how much they spin how many dimensions you're in things like that the pressure comes from you know the force across an imaginary barrier so we mean there is imagine the in your box of gas you put down a little wall and then you said well I have a bunch of particles here they're gonna bump into the wall and then careen off and I can figure out the total force there if that wall were there that's what the pressure is the pressure is how much force would be exerted on you if you put your hand in there okay so you can calculate all those from these quantities that you're given in the microscopic theory but that's unusual that is not necessary in any sense it is far more common that you discover the macroscopic theory completely independently of knowing the microscopic theory and that by the way is number one amazing the fact that that's how nature works and number two a perfectly good reason why you need things other than fundamental physics in your life okay this is the fundamental lesson of a very famous article written by Philip Anderson Nobel Prize winning physicist who passed away recently he wrote an article in the early 70s for physics today I think it was called more is different and the point of the article was it's when you have a collection of many things atoms or electrons or whatever and you consider their collective behavior there are things that happen that in practice you never would have guessed knowing what the individual constituents were there is collective behavior that gives rise to new kinds of phenomena whether it's super conductivity or phase transitions or whatever in principle maybe you could have taken the microscopic picture put it on a computer or giving it to Laplace's daemon and figure out exactly what happened but there's a way of talking about what happens purely at the higher level that gives you some amount of insight that is irrelevant that is that is independent I should say of what's going on at the lower level that's why I Anderson says it more is different now there's an ongoing controversy between different camps of people about whether or not you could in principle put the microscopic theory on a computer and simulate it or whether or not it's so different once you get to the macroscopic level that you couldn't even do that and we'll talk a little bit about that later on what I really wanted to say so the two things I want to say about this picture right here are number one in principle there's this relationship between the micro theory and the macro theory it might be impossible or difficult to derive but in principle it's there but number two you notice in this case that the relationship between the micro theory and macro Theory changed the very nature of what we said exists okay in the micro theory you said I what is this thing what is this box of what you have a box of what what is it and you say well it's a collection of particles they have certain masses certain positions certain velocities but if you were speaking the vocabulary of the emergent higher-level theory someone says a box of what you say oh it's a gas with a certain temperature and pressure it's a completely different set of words and concepts that you're using and that's a kind of the magic of emergence there you're describing the same thing in two very different ways and we're gonna get pretty deep into exactly how different that can be but so this is a very common example but let me give you another example which is equally true and relevant but kind of sneaks under the radar a little bit so let's call this the sneaky example of emergence and that is center of mass motion by which we mean when you talk about you know we already mentioned going to the moon in a spacecraft talk about celestial mechanics the earth going around the Sun going around the earth whatever if you want to solve that if you're you know people do this you want to solve for the motion of the earth through the solar system over millions of years okay so you need to account not only for the moon and the earth but also all the other planets and even like the shape of the of the earth a little bit but to a very very very good first approximation you just need to know as far as the earth is concerned it's center of mass and the velocity of that center of mass so you have the whole earth here earth for example and of course the earth is made of many many particles roughly ten to the fiftieth particles in the earth depending on how you count I'm not sure why I think that's true there's some number embedded in my brain in my memory but in a recent video I mixed up lithium and beryllium so you shouldn't really trust what's embedded to my brain but there's a center of mass for all those particles and you can figure out where that is and then there is a velocity or a center of mass momentum also so you have X center of mass and P center of mass and one of the very crucial things that Isaac Newton himself figured out was that in terms of Newtonian gravity that's all you need if the objects are spherical you don't need all of the information about the individual particles you can boil the information you need down to where the center of mass is and then from there you can get the orbit of the earth to very very high approximation not perfectly because you know there are tidal forces the fact that the moon orbits the earth which you know is true and you can get again a good approximation to that both from the center of masses of both of them but the moon also gradually moves away from the earth and that's because of tidal friction between the moving the rotating earth and moon and to understand that you need to go beyond the center of mass approximation but this is a really really good approximation and it's an example of emergence because the theory that you're using to describe the earth going around the Sun is a theory of a point a point mass with three coordinates for its position and three momentum right that's not what the earth actually is but it's a really good emergent description of it so in both of these cases we have a situation where there is a many to one map when you do that emergent description from micro to macro well I mean by that is for the gases up here if you're and this is you be familiar with this from the entropy that we talked about in the last video given the density as a function of space and the temperature as a function of space and the pressure etc there are many microscopic points in phase space that would look like that macroscopically up to some you know tolerance or precision or error bar likewise given the center of mass of the earth and the center of mass momentum of the earth there are many arrangements of the individual atoms and molecules that make up the earth that would give you that so there's a many to one map and that is the phenomenon we call coarse graining so both of these examples of emergent phenomena are ones where coarse graining plays a role but that's actually not it's not necessary that I'll talk a little bit about what that means what I want to what I want to emphasize here is that when we talked in the last video at entropy coarse graining played an important role in Boltzmann's definition of entropy right the entropy was the logarithm of the number of microstates in a coarse grain macro state but that was just descriptive of the configuration that we were looking at right so when we talked about the evolution of the state we sort of talked about it microstate by microstate we said that it's probable that most microstates will do this or that this emergence thing is a deeper claim because this is saying that not only can we coarse grain but when we coarse grain so when we map all of the microstates of the earth into is just a center of mass that leaves us enough information to predict what's going to happen next there is an autonomous theory of the coarse-grained States themselves we didn't need to make that claim entropy we just said well there's some macroscopic observables and that's what we're gonna keep track of here we're making the much stronger claim that that coarse graining that comes from macroscopic observables or whatever or give us enough information to make a predictive theory about what's going to happen next that's kind of an amazing fact and that's why merchants is so awesome also by the way I forget how much to say this now yeah I'll see you later there's another amazing fact that I will say later good so let me contrast this both of these cases where there's a many to one map there are also cases where just a one to one map okay so there's there are also cases of limits rather than coarse graining and this is does this count as emergence when you take a limit of a theory and get another theory I don't know who cares I don't I don't really care what the official definition is this is a related concept and you should be clear when one concept is relevant versus another one so for example what I mean by this I mean by this well let's just do the example and I'll be clear so consider general relativity you're all experts in general relativity because you've seen the video called gravity where we talk about which n relativity says those of you who might not have those of you living a thousand years from now going through the archival footage first look at the video on curvature on geometry and topology then you can do the video on general relativity and general tivity theory of gravity and curved space-time and it's sort of a generalization of special relativity in some sense because special relativity is a set of rules that forces there to be no gravity okay and the rules of special relativity gravity is not there space-time is completely flat so we can think of special relativity as a limit of general relativity where we turn gravity off you can do this formally you can send Newton's constant to zero the Newton's constant of gravity capital G right if it were zero then gravity wouldn't exist and you wouldn't have any curvature of space-time so you can you can take Jenna Relativity as you move this up here a little bit and take a limit where you turn gravity off and you can get special relativity there's a different limit you can take because sometimes you have like in the solar system what if you wanted to calculate the precession of mercury okay like Einstein actually did you don't need the full-blown apparatus of all of general relativity you don't need where about gravitational waves or something like that so there's an approximation you know even further than that what if you just wanted to calculate the motion of a rocket going from here to the moon what do you really need is Newtonian gravity right that's all you need you could do it in general relativity but it's a whole bunch of formalism and math and Greek indices you don't really that are not helpful or necessary so what you want to say is that in a certain regime Newtonian gravity is just as good as general relativity and that regime is formalized by taking a limit the limit is gravity exists to get Newtonian gravity can't get rid of gravity entirely but gravity is weak okay so when gravity is weak you can't have things like black holes or you can't have things like the Big Bang or anything like that but this is not enough because even when gravity is weak there is things that Newton's theory cannot handle namely anything moving near the speed of light like a gravitational wave for example likewise if the planets in the solar system we're moving close to the speed of light relativistic effects would be important in Newtonian gravity would not be enough so the Newtonian gravity limit of general relativity is both when gravity is weak and velocities are much much less than the speed of light when particles are moving slowly and gravity is weak you can show you can by my text book in general relativity space-time and geometry and we will derive Newtonian gravity from general relativity in this limit and then you can sort of combine these limits you can take special relativity and take the limit where velocities are much much less than the speed of light or you could take Newtonian gravity and turn gravity off entirely and in both cases what you would just get is Newtonian mechanics non-gravitational Newtonian mechanics so this is a nice easy to understand case where certain comprehensive theories have limiting cases and it's exactly the how they fit together in exactly this way right the comprehensive theory general relativity has a wider domain of applicability than special relativity or Newtonian physics but the difference is in the examples I gave earlier there was coarse graining right we were forgetting information whereas these are fine-grained limits these are one-to-one maps one-to-one between situation and general relativity a situation general relativity uniquely defines a situation in Tonie gravity for example okay it's not that there are many many different little microscopic pieces moving around in general relativity that all look the same to Newtonian gravity it's just that there are regimes where generality works and Newtonian gravity doesn't and works it's important here because Newtonian gravity still exists in those regimes right like you can take as much energy density or mass density as you want and squeeze it into a ball and say what would the description of that be in Newtonian gravity if the general relativistic description was a black hole that means that Newtonian gravity doesn't work there but it still says something ray still exists so the theory itself if I guess what I mean is that oftentimes the macro theory exists or says things in a wide region they're just not true things they're just not things that actually describe what is happening in the world so you get both of these kinds of behaviors you can get emerging theories both from a many-to-one kind of coarse graining map or just one-to-one fine-grained maps in the case of limits so if we want to classify all the different ways that theory can emerge at higher levels versus lower levels in this case by the way in these limiting cases it's not even clear people would use a higher level versus lower level language but sometimes they would and again I don't care what the language is that they would use so basically there are two questions that you want to ask about maps between theories that will help you sort of understand what's going on one is do they have the same structure or ontology as the philosophers would say ontology is just the set of things right the objects the notions the ideas out of which the theory is constructed so if the structures are the same then we can call them homeo structural same structure and if they're not then they would be hetero structural and we saw examples of that right the center of mass motion was the same kind of theory it was just Newtonian mechanics of positions and velocities just in a much smaller number of variables so the structure was the same whereas when we went from the atoms to the gas the structure was something completely different and the other question which we all just which we've just said are the maps one to one or many to one we draw this as an arrow and if it's one to one then we say that the maps are fine-grained and if it's many to one then they're coarse-grained and there are examples of all of these right so you get two questions you have two answers each is a little 2x2 grid and we can do that we can we can draw the grid and we can fill it in so here is homeo structural same kind of structures same ontology or hetero structural and then there are fine-grained maps and then there are coarse grained maps okay so what is it an example of a fine-grained map that is homeo structural well what you're saying here is that one state in a theory Maps on to one state of another theory and the kinds of states are the same kind of thing so in other words nothing really changed right but maybe you have a good theory and a bad theory or maybe you have a theory that is accurate in a wide variety of circumstances and another theory that just claims to be a simplification in some sense so basically what you have in this in this box up here are theories that are just simplifications of bigger theories not really this is where the word emergence would rarely be used I think in this particular case but for example you know any of the oldest Fira kakao examples that we talked about you know frictionless planes a ball rolling down a frictionless plane that's described within Newtonian mechanics with the same ontology of points in phase space and forces and F equals MA and so forth you've just decided to ignore friction right so you haven't changed the structure and you haven't changed the number of different states in the theory or anything like that you just simplified your life so that's that's the thing you can do but it's not very highbrow in terms of what do you mean by emergence for the homeo structural same structure but coarse-grained well then we have center of mass motion right that's a very good example for different structures fine grained well general relativity mapping on to Newtonian gravity the Newtonian limit of GR [Music] it's a different kind of structure because general relativity says what the world is is a curved Varenne Sein space-time manifold that's what it says it is right and Newtonian gravity doesn't say that Antonian gravity says that space and time are absolute and there's a gravitational potential field on top of that which obey certain equations the inverse square law for example so the language that you speak has become different even though it's describing the same set of things in the domain where Newtonian gravity works and the biggest sort of you know most interesting box here is when you coarse grain and have a different structures so for example when atoms are coarse graining to give you a guess or fluid description so all of these actually happen which is a good thing to keep in mind when you're debating different examples or aspects of emergence and so with that little classification scheme in mind let me mention a couple of features this thing I mention two quick features that you keep in mind one is the thing that I said I was going to tell you which is an amazing fact appear with you know the earth being describable in terms of its center of mass not quite sure exactly how to phrase this best what I wrote down here was emergence is precious so what I mean by this is when you do this example of taking the earth with 10 to the 15th particles and describing in terms of its center of mass you know you're saying every particle lives in a six dimensional phase space right so I didn't you know get did not only getting the order of magnitude right here but the phase space for the earth is something like 10 to the fifty or 1758 one dimensional and you're saying I can describe its motion to a good approximation by appointing a six dimensional phase space so what I'm doing by this coarse graining procedure is I'm throwing away a lot of information I could write down the positions and velocities of every particle in the earth in the center-of-mass frame of the earth right so I could give you what the center of mass is and I could give you the positions and velocities of all the particles relative to that center of mass and what I'm telling you here is that none of those relative positions in velocities matter at all I can ignore them I can throw them away once I've told you what the center of mass is doing that's all I need to know and that's what's extremely precious that's what I'm trying to get at here because the idea that I can take some system described by an enormous number of variables an enormous number of degrees of freedom and not tell you what the overwhelming majority of degrees of freedom are doing is amazing it's not generic it's not robust it doesn't usually work just imagine I told you you know so what I'm what I'm saying is for that example we've thrown away you know all but one in ten to the 50th of the pieces of data okay but there are simple examples where I throw away half the data so I'm throwing away much much less and I get nothing what if I've told you the position of every particle in the earth so that's way more information but I haven't told you the velocity of any of the particles in the earth okay tell me how the earth moves you can't do it at all you know nothing about how it's moving even though I giving you enormous ly more information if you get right down to it if I give you the position of every particle in the earth and the velocity of every particle except for one what can you tell me about what the Earth's gonna do next nothing literally nothing and you say well no I think the Earth's probably gonna move over here but it's always possible that that one particle has a momentum in the other direction bigger than the comfort total momentum of all the other particles right strictly speaking if I didn't give you that momentum I can tell you nothing so the idea that you can throw away information an enormous amount of information all but a tiny little sliver of information and still get a theory that predicted that predicts accurately what's gonna happen next is enormous ly special and non-generic and precious okay it reflects something real and important about the nature of reality that such descriptions exist at all there's no theorem that says when I have a theory described by ten to the fiftieth degrees of freedom there exists a description while we need to keep track of a handful of them and I can figure out what's going to happen next that is very that's very weird in some sense right I don't know why it's true and I mean maybe you can argue for if I don't think it's anything we understand at a very deep level so this relates to what Daniel Dennett calls real patterns I did a podcast with Dan Dennett if you want to talk if you want to see his viewpoints on this the point being that sometimes when people talk about these coarse-grained high-level descriptions they will give you the impression that what they really are conveniences right it's easier to deal with this coarse-grained approximate description and often they want to do that because they want to question or undermine the idea that the higher level things are real right tables and chairs aren't real just atoms are real or something like that but Dennis says no the fact that there is this pattern the fact that you can ignore almost all the data and still find out what's going to happen next in the life of this macroscopic system is a real objective fact about the system so these higher level things deserve to be called real they they're they have an ontology of their own there really is the center of mass of the earth there really is a table and a chair even though those are higher-level emergent phenomena okay the other thing I wanted to say is there's a phenomenon called universality or it's called different things in different contexts in a sort of computation or consciousness context it would be called substrate independence and the idea here is that if you go back up to the gas okay you got some atoms you can have this gas you can derive the properties of gas from the atoms but as I said you could have found the laws of fluid mechanics without knowing anything about atoms in fact you can have two gases whose you know important qualities are the same their densities their pressures and the equations governing their motions are the same but they're made of different atoms right so you can have the same microscopic theory you're sorry you can have two different microscopic theories giving you the same macroscopic behavior the same emergent behavior in other words knowing what happens at the emergent level is not enough to fix what happens at the microscopic level you were familiar with this actually from when we talked about the renormalization way back when we talked about the fact that you know compare this to the idea of ultraviolet completions what we said was you know in renormalization theory we admit that we don't know what's going on at very small length scales very high energies so we say but that's okay we can find a theory that only applies to the low way to the low energies and long wavelengths the infrared theory and there can be more than one possible ultraviolet completion that gives us the same infrared behavior so this is a good news bad news situation as is so often the case I mean by the way in the in the computation or consciousness world the substrate independence is the idea that consciousness or thought should be thought of as a process and that process can be instantiated in neurons but it could equally well be instantiated in silicon you know in a computer or something like that and again now when you put it that way people might be more reluctant to sign on some people are all for it and some people are more skeptical but the good news is we can say useful things about the macroscopic world without knowing the theory of everything right we can fly a rocket to the moon without knowing the complete theory of quantum gravity because we can get an emergent theory that is very very accurate without knowing the microscopic one the bad news is we can know the mic the emergent theory and still not know the microscopic theory so this is why as we mentioned in the renormalization video it's hard to get data at something like the Large Hadron Collider that is relevant to quantum gravity because that's in a regime which is just not described like there's many things that could potentially be going on the Planck scale where quantum gravity is important that would lead to the same behavior at the Large Hadron Collider or at observable everyday energies so this the potential existence of universality many microscopic theories giving the same macroscopic behavior that's a problem in some sense for physicists it gets it gets in the way sometimes but it's there it's real so we got to deal with it there you go okay so for the rest of this video I'm not done yet this is now we're gonna shift gears a little bit because much of the discussion of emergence in the emergence literature uses examples like either boxes of gas or human beings either you know consciousness and human beings or the relationship between sociology and psychology and stuff like that what I'm most interested in is quantum mechanics and I think the quantum mechanics is a crucially important example to look at carefully here how the classical world emerges from the quantum world and I think that most people don't do a very good job of it so let's you know you've already become experts on quantum mechanics now you're an expert on emergence let's put them together and dig a little bit more deeply than you will usually get told so let's look at the relationship of quantum to classical mechanics which is a relationship of emergence the classical world emerges from the quantum world that's try to fit it in to the framework that we've already given and there is a usual story so let me tell you the usual story and how it falls short a little bit okay the usual story relies on you know when I say the usual story I mean when you ever say as story as usual it depends on what community of people you're talking about I mean this is what you usually find in textbooks there are experts who have a more sophisticated story than this but the usual story invokes something called air infests theorem paul ehrenfest creates statistical and quantum physicists early 20th century friend with Einstein and Bohr and things like that I once gave the ehrenfest lecture at the University of Leiden that doesn't make me so special many people have given the lecture but it's a fun thing to do and he has a theorem that talks that is very very important for the relationship of the quantum world to the classical world so he says consider a particle moving in a potential so here is position X and here's V of X okay so you have a particle with a wave function moving in this potential and he says classically what are we trying to do here we're trying to relate the quantum description to the classical description okay so let's first remember the classical description well the time derivative of the position is given by the momentum over the mass and the time derivative of the momentum is given by minus the slope of the potential the derivative the potential with respect to X okay so you have some classical particle it's gonna roll down the slope in this case is going that way that's the direction in which so the slope in this case is negative so it pushes the particle to the right okay good and we would like to know well okay what happens quantum mechanically that we can derive this from we would like to see the submerged we'd like to start with the Schrodinger equation and derive this classical behavior in some limit or something right some situation anyway so what do we do well there's already a problem there's a huge problem let me rearrange this to make it a little bit more convenient the huge problem is the ontology is completely different as we've been emphasizing over and over again in classical mechanics a particle is a point in phase space it has a position and it has a momentum those are things that exist in quantum mechanics the particle described by a wave function right so in QM you have sigh of X and T so for one thing P the momentum doesn't even appear there as we talked about way back when momentum has to do with the wavelength of a wave you can sort of convert from position space to momentum space so I could say sigh of X and T or what is sometimes called sight tilde of p.m. T two equivalent ways you can go back and forth between these one fixes the other one exactly and X and P in this in this quantum world are not coordinates for the particle they are observables so this is what we tried to emphasize that there's no answer to the question in quantum mechanics what is the position what is the momentum and that's why the uncertainty principle is not a statement about what you can measure it's a statement about what the actual observables how they relate to the wavefunction there is no wave function that is perfectly definitely defined as a delta function in both position and momentum so this is already kind of like a conceptual problem how do we relate these observables that are not coordinates in a world where the ontology is a vector in Hilbert space two phase space to this X and P right that are gonna have to have some behavior that we would hope that would look classical so what we can do ehrenfest says or anyone says is we can define expectation values so this is a fancy way of saying if we were to measure these observables the position or the momentum if we were to start with the same wave function and measure position over and over again so when you actually measure the position wavefunction collapses so you have to imagine that there is some way of restarting the wave function exactly where it was and you measure the position again and again and again and you find out what is the average okay that's the expectation value so you define X bar as a function of time so you're not this is a little little bit tricky to say you're not actually measuring it you're saying hypothetically were you to measure it over and over again you can predict what the outcome would be and find the average of that that's what this is so you're not actually disturbing it you're letting the wave function smoothly evolve and talking about how its expectation value of all's so X of T can be written as the integral of x times the wave function over all possible X's and likewise P of T for reasons that are kind of complicated we're not going to get into it's minus I times the derivative of sine with respect to X DX okay then that kind of makes some sense right because the momentum has to do with how quickly the wavefunction is changing but anyway the point is that there are equations right given the wave function you can figure out the average value and so now you have numbers right before you just had a wave function a vector in hilbert space by defining these expectation values now you have numbers it would be nice if these numbers the expectation value of x and p just obeyed these classical equations wouldn't that be nice as the Beach Boys once said well they don't but they come very close and in the right circumstances they kind of do so that's how ehrenfest theorem works so what ehrenfest says what he proves is that the derivative of the expectation value of x with respect to time rember all we're trying to do is read arrive these classical equations right here Newton's laws turns out to be exactly what you would want the expectation value P divided by m that one worked but what about the derivative of P bard the expectation value of the momentum well it turns out to be - take the derivative of the potential with respect to X and take its expectation value okay so this thing here the thing in the bar is the integral of DV DX times sine of X DX so that looks pretty close like what we wanted was so we had classically DP DT is minus DV DX it would have been nice to get DV d DP DT equals minus DV DX bar right so in other words we were hoping for this equals minus DV evaluated at the average value of X DX so in other words the difference between these two things this and this is this thing at the bottom which is what we want says go to the point which is the expectation value of x the most likely place to see the particle and ask what the slope of the potential is that's what we want that's what we want to be pushing around the particle this thing at the top says take all the different points of the wavefunction sample them by asking what is the slope with the potential there okay and that's what gives you the change of the expectation value of the momentum but that's a quantity that has no so if I have let's draw a picture because this otherwise can be impossible to understand so here's V of X okay here's X here's the potential that we drew and let me draw a wave function let's imagine we have a wave function that you know has sort of different little parts to it so this thing right here is saying you know there's some slope here there's some slope there there's some slope there at all the different points where the wave function has support there's some slope to the potential average all of them that's what ehrenfest theorem actually says is relevant but you can see like that's a completely irrelevant quantity in Newtonian mechanics Newtonian mechanics once you'd actually have the slope of the potential where the particle is and what you're coming across is the fact that in quantum mechanics there's no such thing as where the particle is so it's a little bit more difficult to interpret what's going on I tend to dwell on this and be very persnickety about exciting that people get it wrong very easily but it's clear what to do so what to do is consider cases where the wave function is localized near some particular location okay so this kind of wave function just makes no sense that's not going to work for ehrenfest theorem which lock then it's not going to give you classical behavior according to ehrenfest theorem but what if we were lucky enough to have a situation where potential looks like this and the wave function looks like this okay then roughly speaking all that matters is the value of the slope of the potential near where the wavefunction is peaked so we call this a wave packet a wave packet something where it's nearly localized in some particular place and there's a formal mathematical way of saying this it's the the idea is that does the width of the wave packet has to be small compared to the rate at which the slope of the potential is changing okay the second derivative of the potential so there's a very mathematically precise way of saying your wave packet is localized enough to pretend that it has one single location and then you can show that - dv/dx expectation value turns into - DV and X bar DX and then you get the classical limit back so that's then just a force on the particle okay so what it's saying here is that ehrenfest theorem gives you classical motion for a quantum wave function in certain circumstances certain kinds of quantum wave functions can behave classically and some can't okay this kind of wave function has a chance of behaving classically this is in the classical limit this kind of wave function up here just doesn't behave classically at all so you have to be a little bit careful about it so everyone understands that you know sometimes they they gloss over you know there's a question why is the wave function ever localized you have to answer that but that that those can be answered there's another thing though that is much more subtle and interesting which is I mean you know that when I localize a wave function that means that I know - pretty good precision what I will measure the position to be which means that by the uncertainty principle I don't know what I'll measure the momentum to be now if the thing that I'm talking about is a big thing is the earth for example okay I can know the position to pretty good accuracy and the momentum to good accuracy so this is where bigness comes in you notice that no we're here my discussion of ehrenfest theorem that I say that the object was big wha because ehrenfest theorem cares about the wave packets being localized and then you can say that the instantaneous equation applies and gives you the classical limit but if it spreads if though if the wave packet spreads very quickly then it no longer applies and so ehrenfest theorem only gives you classical behavior for a minute for a second for a nanosecond but if the particle is really really big and heavy the spread in both position and momentum can be small or position and velocity can be small and in that case the classical limit remains true but what about the other case you know there are cases where the momentum spreads where the velocity a small variation in velocity leads to a big difference like for a single particle so like for a single particle like an electron moving through a bubble chamber when you see that single track that looks pretty classical mechanics the Arum an electron is not at all heavy electrons light it should spread out all over the place something else is going on we know what's going on we know the electron is being observed it's wave function is being measured repeatedly by the bubble chamber and that leads to decoherence so that's a clue as to what is going on so not just electrons there exist big classical systems like the earth the earth is not an example but like the earth that are actually the earth is an example let me explain that that are chaotic in their classical motions ok what what chaos means sadly it's not a whole big idea of video by itself but the chaos phenomenon is the situation where a small deviation in initial conditions leads to a very large typically exponential deviation in the final place where the thing goes ok so in many cases that's not true if you drop two balls off a building and watch them fall if you drop them as slightly different times they will still land at slightly different times chaotic behavior happens when slight deviations lead to huge deviations in near future so I said the earth was not that but it actually is but the time scale for the chaotic behavior to show up is really really long so the Earth's orbit around the Sun is chaotic if you give it a long enough time we can't predict the position of all the planets in the solar system to very high precision a billion years in the future because of this chaotic behavior but there are systems that are much more chaotic you know chaotic on a much shorter timescale so there are my favorite example is Saturn's moon Hyperion Hyperion is a little lumpy thing it's a very tiny moons it's a captured asteroid that is a moon of Saturn and it looks kind of like a potato potato shaped and so this was worked out a while ago the fact that it's a lumpy thing so I don't know I'm not gonna be able to draw it very well Hyperion moon of Saturn okay it has different moments of inertia you know it's pulled and pushed by the tidal forces in Saturn and the other moons of Saturn and therefore it tumbles okay so it I'm just gonna write tumbles what I'm gonna do is I'll do this I'm going to try to find a video of Hyperion and I'm gonna try to put it into the file that I published in this YouTube video so we'll see if we can embed a video a level of technological sophistication I've never actually done before but what you see when you look at the video even if it's only in your mind's eye is that Hyperion tumbles and it tumbles chaotically because it's being pushed and pulled by all these other things in Saturn's orbit Hyperion just sort of wobbles in an unpredictable way so if you think that Hyperion is really a quantum system it's big you know it's it's heavier than you are that's for sure but a small deviation in its orientation if you didn't know exactly how it was oriented classically the classical trajectories is quickly diverged you can't predict what attitude what orientation Hyperion is going to be in if you translate that to the behavior of the wave function for Hyperion what it means is if you start Hyperion this tumbling potato shaped moon of Saturn in a wave function that is peaked around some particular orientation and rotational velocity that wavefunction will quickly spread out over all these different possibilities so if all you thought about was the wavefunction of Hyperion and he made a prediction for what it is you know a year later it should look like a blob like a spherical cloud of probability it should not look like a picture that you can take we've taken pictures from NASA the video I'm gonna show you is just a simulation it's not really photographs but it's based on photographs it looks like there is a planet there with a real a moon an asteroid of former asteroid that is now a moon with an actual position and location like classical things should have why does it look classical to us ehrenfest theorem does not give you the answer the wave packet of the orientation of Hyperion is chaotic and should spread out over a short time scale the answer is D coherence of course is the same answers for the electron the point is that Hyperion is in the solar system it is constantly being bombarded by photons for example as well as by other things and they're essentially branching the wave function of the universe or if you like they are measuring the wave function of Hyperion all the time and so what that means is that there is an effective emergent description of what Hyperion is doing that is classical but not deterministically classical so what you get is a stochastic classical dynamics so this combination of quantum mechanics and chaos theory is saying that not only if you didn't measure the state of Hyperion exactly would it be unpredictable it's saying it is unpredictable measure it all you want because you can't put its wavefunction into a state where it won't spread out and then be observed and be branched in different ways by the environment around it ok so there's still classical behavior in a quantumly chaotic system but it's a little bit more subtle you got to think a little bit more about how it gets there and you can't just look at the system itself you have to include the environment and decoherence and so forth ok I see an hour has gone by two more things I want to tell you one is kind of kind of a stretch but that's ok you're very much smarter and well informed now than you than where you were 20 videos ago or whatever it was when we started this whole series I want to tell you about the fact that because now that we understand emergence and now we understand how you know classical physics relates to quantum physics so we've gone both ways that's let's pause and say that when we first did quantum mechanics we emphasized that the typical way that you made a quantum mechanical theory was to start with a classical theory and quantize it and we said look nature probably doesn't do that I mean nature probably is quanta from the start but nevertheless that's the way we do it and what we've just done is to remind you that you can go the other way as well you can start with the quantum mechanical theory and get a classical behavior from it in the right circumstances so the domain of applicability of that classical theory will be smaller than the domain of the quantum theory but that's ok what I want to do now is mention the fact that this going back and forth is not unique ok there is no rule that says that given a quantum theory there is only one classical theory you can get from it or the other way around given a classical theory you can quantize it and that is non uniqueness both ways given a classical theory you can quantize it in different ways this is a well-known thing operator operator ordering ambiguities people talk about not that interesting to be honest more interesting is there are two different classical theories that you can quantize and get the same quantum theory two very different classical theories and this is just so mine stretchy mind-boggling and weird that I even though it's beyond what we're normally doing here and the greatest ideas I have to explain it to you so let me give you an example let me say say what I'm giving you an example of one quantum theory can arise by quantizing two or maybe more different come on spell different classical theories just blew my mind anyway when I first was shown this example in grad school so here's the example again not something you will normally be told and pop physics discussions but that's okay this is something called the sign Gordon theory the klein-gordon theory you may remember it's just a theory of a scalar field just you know doing whatever it wants a scalar field with some potential V of I so scalar field Phi potential V of Phi and Sidney Coleman my old quantum field theory professor from grad school wrote a paper about a particular theory klein-gordon theory where you took the potential to be looking like the sine of Phi so a potential that went up and down so you have Phi V of Phi and then the potential looks like this okay so as a joke he called it the sine Gordon theory and sadly everyone loved that name and he was very chagrin he was slightly embarrassed that people love that name they kept it but that's what we call it now the sine Gordon theory okay and in particular so I went so this is just a theory of a scalar field with this kind of potential that's that's what it is and in particular let's look at it in 1 plus 1 dimensions so 1 plus 1 to still true but we're talking about we say one plus one clean one dimension of space one dimension of time so we have a line that evolves with time and there's a scalar field moving around wiggling in this particular potential so usually it wants to wiggle near the bottom of the potential but sometimes maybe it'll hop over the top you know depending on what's happening okay so basically there are two things that I'll just say what I just said again there are two things that can happen there can be like small oscillations near the bottom and in the quantum theory those would just show up as particles right scalar bosons spin zero particles but then there are also configurations that go from one minimum to another right and we might remember we talked a little bit about that idea and we talked about topology because this is a topological defect known as a domain wall if you have two different minimum points of the vacuum two different vacua two different minimum points of the potential that the field is in one of them over here one of them over here then in between there needs to be energy when the field climbs over the potential to get there so in one plus one dimensions you can plot that right so here's X now this is um here in this plot Phi was the horizontal axis now Phi is the vertical axis and X is the vertical axis X is the horizontal axis so I can imagine a situation where Phi is let's say if I continued this down right so here's a minimum there's a minimum there's a situation where Phi is in 1 minimum and then as sort of quickly goes up to the next minimum so if this is call it minus V and plus V this is minus V and plus V okay and this is for unfortunate historical reasons called the kink solution get all of your kinky jokes out of the way right now but it you can see it's sort of a kink in the field that goes up and it goes down so there you go so this is a theory that has both particles sitting zlatan the potential and it has these domain walls kinks Solutions okay I mean generally you would have both here is the here is the miracle these small oscillations as we said they're just scalar bosons they're pretty typical if you look at the bottom of a sign potential or a cosine potential or whatever I think we even mentioned the fact that it looks kind of like a parabola so it kind of looks like a simple harmonic oscillator so it kind of looks like ordinary things we've already quantized and talked about ordinary scalar bosons with different masses as long as the oscillations are very tiny compared to the variation of potential but there's also these kinks and these kinks themselves are kind of particle like you know if we were in three dimensions than a domain wall is a big thing that stretches across the universe but if you're on a line right if you only have one dimension of space then a particle is just something is located at one point in space and this kink is also kind of located at one point in space right I put it here at zero but I could have put it anywhere in fact it can even you know I could expand this a little bit you take this picture over here I could let space go on a bit here and I could imagine a configuration that stayed in this vacuum and then it went back down to the other vacuum right so this would be if this is a kink over here this is an anti kink going the other way so you could have a series of kinks and anti kinks and then go up and down again get all the jokes out of your system and when a king can tie kink hit each other they annihilate right they go away because you can just flop over so this whole situation over time can evolve into something with just has some vibrations then goes like that right the kink and the anti kink can just annihilate so all those words the kink solution looks like a particle especially if you know the parameters the potential are such that it's very thin it kind of looks like a particle it has a position as a velocity and a kink has anti kinks and they can annihilate make some sound like particles in fact makes them sound like fermions doesn't it makes them sound kind of like electrons you know there's particles and antiparticles and they could annihilate right so what's Coleman showed in his paper is that there is another theory of fermions that is a classical Lagrangian you can write down with the feature that when you quantize it you get exactly the same theory as when you quantize the sign Gordon theory so that fermion theory is called the massive Turing model I'm not gonna go into details here but it's a specific model of fermions I've got a letter mister Turing's name it's a specific theory of fermions and anti fermions in 1 dimensional space okay that's what it is you can write down it has a Lagrangian etc and so then you can compare the sign Gordon theory to the massive Turing theory and in the sign gourd there you have scalar bosons and you have domain walls or kinks in the masseter in theory you have fermions because that's what you put in the fundamental degrees of freedom or fermions but you know if you take an electron and a proton two fermions put them together to make a hydrogen atom the hydrogen atom has spin zero or spin one depending on how the spins align right so the hydrogen atom is a boson it's made of two spin 1/2 particles and you get either a spin 0 or spin 1 combination depending on whether they're aligned or anti aligned so you can and that's a generic fact you can take two spin 1/2 fermions and if they bind together you could make a spin zero or other spin boson bound States of fermions or bosons so you get boson ik bound States and what Coleman was able to do was show that these theories are exactly the same at the quantum mechanical level the scalar bosons in the sign Gordon Theory map on to the bound states of Fermi on anti Fermi on in the mass of Turing model and the domain walls and the sign Gordon Theory map on to the fundamental fermions in the massive Turing Theory domain walls the son Gordon theory upon the fundamental fermions the message during theory so I'm not gonna go to the details here obviously you can look up papers there's this whole idea this is a broader idea called boson ization in 1 plus 1 dimensional field theories it very often is the case that you can describe things using fermions or they bind up into bosons so again the claim I'll just repeat the claim I can write down all the gironjin in 1 plus 1 dimensions for a scalar field with a sinusoidal potential the sign Gordon theory I can write down a completely different-looking lagrangian for a Fermi on in 1 plus 1 dimension that interacts with itself in certain ways as an anti particle the massive Turing model and then I can do quantum mechanics I can quantize both lagrangians and I can show that both theories are secretly exactly the same quantum theory so two different classical theories giving rise to the same quantum mechanical theory and not just you know some different parameters or some different labels one is a theory of bosons the other there's a theory of fermions so the point of this is if you were to say given this quantum mechanical theory what is it really given the quantum mechanical theory which is just some Hamiltonian operator acting on Hilbert space given that is this really a theory of bosons or is this really a theory of fermions the answer is there's no answer there's no answer to that question this description that you made up in terms of bosons and fermions there's a classical language and then you quantize the theory to get the real true theory which is the quantum theory and there's more than one classical theory that you can quantize to get this quantum theory so the answer is what is it really what is the true fundamental stuff that we're talking about here is not a question that has an answer all this talk about bosons and fermions which is the theory really made of and which are just the emergent ones that's not an answerable question in in this particular model good if you like that you're gonna love the next example of the same thing which is something you maybe have heard of which is the ad SC ft correspondence so my point in showing you the sign Gordon terney model correspondence was to say we can quantize two theories one is fermions one is bosons show they're really the same theory okay so you don't need to start from the same starting point to make a quantum theory that is really the same that the classical starting points can be very different the ad se of T correspondence is an example where you start with two different theories in two different dimensional space times and you get the same quantum theory at least we think you do or at least if you do in certain regimes and where the details are a little bit much less clear here because gravity is involved here but that's why it's exciting as well as why it's incompletely understood so ad s is anti de sitter space which is a solution to Einstein's equations with a cosmological constant lambda the vacuum energy the energy of empty space less than zero a negative value for the cosmos real constant so this is a cosmological solution with nothing else in it so no particles no radiation just empty space with nothing but negative vacuum energy de sitter space is the solution you get with nothing but positive vacuum energy so this is anti de sitter space and CFT means conformal field theory so in conformal field theory is just an example of quantum field theory conformal though means that it is completely scale free there are no parameters in the theory that pick out a certain length scale so if you for example have any particles that have a mass then you do not have a conformal field theory because a mass for a particle picks out a Compton wavelength and that's a length scale conformal field theory looks exactly the same at every length scale if you want to you know QED quantum electromagnetism at very very very low energies well below the energy of the electrons you're really nothing but photons interacting with each other that's almost a conformal field theory it's not quite because of details because the electrons are kind of still there in some sense but it could form what the field theory is really exactly the same at all scales so it's kind of a weird kind of field theory but the conformal field theory does not have gravity so the conformal field theory we're talking about here is just a quantum field theory it's perfectly well defined you can write down the Lagrangian you can quantize it you can do all the quantum field theory you want the anti-de sitter theory we're talking about here is gravity in fact super gravity in fact string theory on anti-de sitter space so that's much less well-defined you know we have some ideas about what that is like but this is a true the idea is the claim the hope the aspiration is that this is a true theory of quantum gravity in certain background so what we know how to do is look at small deviations look at particles or even black holes in that background but it is not you know some people complain because it's not a completely background independent theory and that's a fair complaint this is a theory where at least the boundary conditions are those of anti-de sitter space so what is the relationship between these two theories the relationship is the following remember we were drawing pictures and we talked about gravity we do different pictures of the Kruskal diagram for the black hole Kruskal diagram has this nice property that light cones are always at 45 degrees so that's a step along the way to a more elaborate thing called a Penrose diagram or a Carter Penrose diagram Brandon Carter and Roger Penrose invented a way to draw a whole space times even if they're infinitely big draw them in a finally piece of paper so here's the Penrose diagram for anti-de sitter space it looks like a cylinder and it has the property I said finite piece of paper I know but in this picture time goes up eternally and space is the interior here and space is finite sorry space is infinite anti-de sitter space is an open universe that goes on forever but I've drawn it in such a way such that it fits into this disk here okay I've contracted it like that so this interior of the cylinder is anti-de sitter space okay and it has a boundary has a boundary that looks like a cylinder like the edge of a cylinder right and so the this was first done by Juan maldacena in 1996 maybe something like that 97 98 around there you know I was a postdoc at the time and what Malda saying it was studying was abs in let's say 4 plus 1 dimensions don't ask why but there's good string theory reasons why that was a logical thing to study so the boundary of abs is this cylinder it looks like it's a circle times time but the circle is infinitely big so it's really just flat ok if you take a circle or a sphere and make it bigger and bigger it looks locally flatter and flatter so this is literally an infinitely big sphere which is the equivalent to flat space-time and the boundary then is 3 plus 1 dimensional and you can define a conformal field theory on it so in other words you have two different space times you have the space-time inside which is a five dimensional anti-de sitter space and you can define gravity on that you can have gravity and other forces of nature strings and other things that would appear in super gravity so you can do quantum gravity and five dimensional anti-de sitter space then you can also completely separately have a theory a quantum field theory a conformal field theory without gravity on the boundary okay which is one dimension lower in this case is three plus one dimensional and what mala Sina says is that these two theories are the same these two classical languages the language of conformal fields moving in flat space-time of four dimensions total and this other classical language of gravity in five dimensional space-time with anti-de sitter boundary conditions they are two different ways of talking about the same quantum mechanical theory this is called a duality between these two descriptions the sign Gordon necessary models also call the duality you can think of a duality is two different classical versions of the same quantum mechanical theory and I can't do much I can't do much about the details of anti-de sitter space and the conformal field theory but the point is I just try to drive home how true it is when I say that nature works as a quantum mechanical theory and every time we talk about a classical theory that's just convenient to us okay this example was two different classical theories one bosons one fermions but they're just secretly two different ways of talking about the same quantum theory this is two different theories one with gravity in five dimensions and one without gravity in four dimensions and again the same theory you might say well there must be more degrees of freedom in five dimensions than four dimensions but that's not true because in both cases there is an infinite number of degrees of freedom and infinity is still infinity infinity times infinity is infinity and this is you know the the most most obvious beloved manifestation of the holographic principle the holographic principle being the idea that in a black hole or other places where you have a horizon or gravity is important you can encode all the information about space-time in one lower dimension on the horizon of a black hole or something like that here you're encoding all the information about the five dimensional anti-de sitter space in the four-dimensional boundary you can you can there's a dictionary it's not perfect again you know we don't know as much about this as we want but we tested to the conjecture and it's very very strongly supported so we believe that any question you can ask in that five dimensional anti-de sitter space can be answered in terms of a question asked from the four-dimensional theory and these days there's a very say the words I'm not gonna explain anything at all it's a very exciting idea called entanglement wedge reconstruction dispute about buzzwords here and basically this is an approach to thinking about black holes in anti-de sitter space evaporating so remember I told you about the black hole information loss problem here's an example in this anti de sitter space and for in five dimensions well I could make a black hole there because gravity exists in this theory and I could let it evaporate and I just promised you that any question I asked in anti-de sitter space quantum gravity should be answerable in the boundary conformal field theory where gravity does not exist and all of the obstacles to understanding quantum gravity are irrelevant so I should be able to make a black hole watch it evaporate and ask what happened from the point of view of the boundary so people have been trying to do that now for over 20 years and the problem is yes you can do that but actually seeing how the information gets out has eluded us so far so this approach of entanglement wedge reconstruction seems to be a set of clues about how the information can get out of black holes in this case not something to talk about right here but you know maybe a future podcast I'll try to get into this anyway we yet again another way of saying quantum mechanics is what the world is made of classical physics can emerge in different ways the CFT and the 80s theory are two different emergent descriptions of the same underlying quantum theory okay final topic which is completely different now for something completely different about emergence is weak versus strong emergence all right this is nothing to do with quantum mechanics even fundamental physics really but maybe it does and I alluded to this idea earlier this idea of when the macroscopic theory emerges is there a sense in which in principle and Laplace is demon type fashion you could figure out what was going on from the point of a microscopic theory so that's weak emergence if you believe that that is possible so we mean we don't mean like it's less strong or powerful we mean it's a it's a weaker notion it's claiming less okay weak emergence strong emergence is something a more radical claim that's what it means to be strong in this case so in weak emergence the macro theory is let's call it autonomous which means to say within the macro theory the theory of gases or whatever I can ask questions I can answer them without knowing what is going on in the micro Theory that's the sense in which it is autonomous but it can be simulated exactly by the micro theory so in principle you're saying I have some box of gas I'm going to predict what it's gonna do next but you're also saying that if I knew exactly what the phase phase point was for all the atoms and particles in the box and I knew all the microscopic laws of physics I could put that on a computer figure out what would happen next and I could also derive what would happen to the box of gas okay so in other words in weak emergence you imagine that there is some micro theory and the micro theory has a lot of different states right there's a lot of different this is not space I'm drawing here this is the space of states in the microscopic theory whatever that might be may be wave functions or whatever and then there's a map from that micro theory to the macro theory and let's imagine this is a sort of coarse graining situation so there are fewer states in the macro theory and the claim of weak emergence is that given some particular state in the micro theory I can literally map it to a state to the macro theory I can also evolve it in time right to a different state I don't know what it does it goes up here and it many many many states once again and the laws of physics in the micro world give me what microstate it goes to and I can play the same game in the macro theory I can show that this goes to some other macro state and the point is that there still is a map from the micro theory to the macro theory and this diagram commutes as we say in the math literature in other words starting from this microstate to drift different colors you can follow it you can either go to the macro theory and then evolve it or you could evolve it and then go to the macro theory and you get the same answer right that's what it means to have weak emergence you can evolve the macro theory but you could also equally well evolve in the micro theory strong emergence which let me just tell you I don't think exists but people talk about it so I feel like I should mention it strong emergence says no you can't in other words strong emergence is the idea that there is a micro theory and there is a macro theory but the macro theory even if it looks like the macro theory is describing a large collection of micro things the rules of the game are fundamentally different in the macro theory they are not reducible they are not simulated on a computer there's something that is truly new that it's coming out in the macro realm in fact in many social sciences that's the way this particular word emergence is usually used if you if you google the British emergentist s-- you will find a whole bunch of people who say exactly this that you know you can't reduce what happens in large groups of people to small numbers of people so the collective behavior thereof there's some fundamentally new non simulated thing and that's what strong emergence would be about so I don't believe that actually happens in the world but I do want to say that it could you know I want to be open to the possibility so what would that mean in the language that we previously talked about what it means is something like this like here's reality what it means is basically so here's why I don't believe it because let's say that you're thinking like a physicist and your micro theory is the standard model of particle physics or the core theory so we can include gravity there's a Lagrangian right there's an equation for the standard model plus gravity and like it or not for any given wavefunction you plug it in in that Lagrangian or in the Schrodinger equation you convert to a Hamiltonian it will tell you how that wave function evolves uniquely there's no ambiguity there so if you make the strong emergence claim if you say that in principle the behavior of a human being let's say cannot be reduced to the behavior of the atoms and molecules and forces that make up that human being you're saying the lagrangian of the standard model must be wrong you can't say it's completely right and never violated but it's insufficient to predict what is going to happen for a human being those two statements are incompatible with each other and it's very very hard to imagine the ways in which you would violate the lagrangian for the standard model macroscopically but not microscopically okay I mean you can say the words but no one has ever come close to writing down equations or actually giving a principled suggestion for what the new laws of physics would be that's why it's very hard and in some sense the reason why it's very hard is this idea that we have of locality right I mean it's built into the framework of the standard model that the world is a quantum field theory and quantum fields exist at different points quantum wave function has entanglement between different points but the fields that you're entangling exist at various points and they interact at the same point in space we talked about this many videos ago in the force and action and energy video locality is really the underlying thing that makes quantum field theory go and what you need to do if you want to believe in strong emergence is to give up on that locality in some fundamental way you need to give up on the idea that not only are the laws different but to give up on the idea of sorry not only that was the same but there's it's even correct to think about a human being as just a collection of electrons and protons and neutrons so the way I would do it in terms of the pictures I drew before is you have reality you have our micro Theory which might be the core theory the standard model and then you have some macro theory which is your favorite theory of I don't know consciousness or whatever and the claim is that the macro theory is not a subset of the micro theory so there is a domain of applicability for the micro theory and there's a domain vac ability for the macro theory that looks like this right so there's a whole part of reality which is described by the macro theory but is not described by the micro theory that's when strong emergence is conceivable so you're saying that sure if I look at what happens to one electron bumping into some small number of other electrons then the standard model describes it perfectly but once it's surrounded by Avogadro's number of electrons then the standard model fails to describe it that is outside the realm of validity of applicability of the theory there's no reason I think in fundamental physics to suspect that anything like that actually happens the only reason is you know I think that the the advocates for this would say well they want to understand consciousness and they don't think the consciousness is explicable in terms of what happens in the standard model of particle physics I think it is your opinion may differ but I'm offering this as a peace offering to people who actually believe in strong emergence I think they should be more clear about what it could possibly mean I think this is what it could possibly mean if you could try to make it work quantitatively in terms of equations and so forth good for you that would be a huge progress I don't think it's the way to go but you know emergence is murky there's probably a lot that we have yet to learn

Channel: Sean Carroll

Views: 84,972

Rating: 4.8883495 out of 5

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Id: 0_PdLja-eGQ

Length: 93min 41sec (5621 seconds)

Published: Tue Aug 11 2020

Reddit Comments

[u/eigenman]

Definitely one of my favorite topics.