Some Half-Baked Thoughts about de Sitter Space - Leonard Susskind

we're delighted to have lanny suskind who's going to be telling us some thoughts about holography and to sitter space i think the official title will soon have fake thoughts well at least not your liberation all right let me explain why the title what i meant when i said have faked in fact let me just go on a rant a uh an old man's rant um let's begin with a night thought in fact let's imagine a group of very smart theorists who were all about quantum mechanics and relativity but had never heard about modern astronomy or particle physics if they were anything like us they would eventually discover black holes quantum field theory string theory supersymmetry the holographic principle large end matrix theory adscft in other words from the pure theory point of view they would be about where we are now what would they take away from all of this here's what i think they would say i think they would say first of all if you want to put gravity and quantum mechanics together you better have a frozen time-like boundary to anchor the theory and to define observables ads is great in this respect and maybe flat space is okay to make the string scale much smaller than the cosmic radius in this case the cosmic radius would be the ads scale the boundary theory better be super strongly coupled but knowing as much as we do they might believe that the only theories that can be pushed to super strong coupling are supersymmetric that would also fit what they know with what they know about matrix theory where without supersymmetry you can't even separate two particles asymptotic boundaries super strong coupling supersymmetry these would be their touchstones what would happen then is some crazy man showed up and told him that he had seen the data and could assure them that space has no boundary that space time is more like the city space than an anti-disorder space that there's no super symmetry and despite that locality works down to ultra microscopic scales i'll tell you what i think they would say i think they would say baloney your model is in the swamp land they say this not because of some made up swampland bound on the inficon but because the model violates the entire foundation of their mathematical understanding okay what should we take away from this parable the answer is i don't really know i don't think we should stop thinking about abs and related things it's obvious that there's still much more to learn we should continue but constantly keep in mind that we're missing some very big piece of the puzzle and not to be afraid to discuss the limits of what we've learned personally i've been thinking about the sitter space but to tell you the truth i don't really believe in the project i have real doubts that eternal descent sorry that eternal stable this inner space absolutely stable the city space exists mathematically but even if it does it's not a good and reasonable starting point for cosmology it's a theory of boltzmann fluctuations and i'll talk about that a little bit and as i explained years ago that's not a good place to start for cosmology personally think something like eternal inflation a vast landscape including terminal vacua and something that i once called a fractal flow was much closer to being the right thing than eternal deceiter space but that's a very very big piece to try to bite off all at once so for today we're just going to talk about just a little bit about the sitter space and some aspects that just a couple of aspects that i've been personally interested in uh because at least at one time i thought they might really be a no-go situation for the sitter space okay let me let me share my screen you won't see me once i square and i share my screen because my ipad doesn't permit it so here we go screen sharing screen start broadcast no i seem to what happened here start projects can you see my screen yes can you see me yes all right so what the question i really want to address is does the holographic principle apply to the sitter space hold on let me move my uh let me move my apparatus here a little bit so it's easier for me to deal with yeah does the holographic principle apply to the suit of space and if so in were formed uh here are a few papers that that address some of the questions that i'll talk about but the first thing i want to talk about is what i mean by the holographic principle and what i mean is essentially what the holographic principle first meant you have a region of space uh region of space means space across time and it has a boundary there's a bulk inside it a time-like boundary in fact this is a bulk inside it it's described by all the various things that we describe bulks in terms of gravity metric action einstein hilbert action newtonian gravity it contains contains black holes hawking radiation all the things that we think should exist in the bulk on the other side of the divide is the holographic theory or the hologram we can call it again time-like spatial boundary and on that time-like spatial boundary there's a theory of some sort which is purely quantum mechanical it has a hilbert space it has unitarity it has a hamiltonian it has things like entanglement complexity but it does not have gravity at least not in its formulation it doesn't contain gravity assumption uh the holographic principle is that the right side can describe everything that's on the left side some examples they're very familiar to us uh matrix mechanics matrix theory which is a pure theory of the quantum mechanics of matrices on the right hand side is 11-dimensional conformal field theory ads syk jt gravity and so forth i wish i could say there were a lot more but i don't know very many more uh and notice the divide is a sharp divide between what is on one side gravitational bulk physics on the other side or pure quantum mechanics okay let me before i uh get into what i actually want to talk about let me just say in one particular situation one particular type of theory which is called dscft i want to explain which side of the divide this falls on is it a holographic theory is it a bulk theory so let me just remind you very very quickly uh at least this is a description that juan gave us we have the center space and we make some cut off at a very late time well it doesn't have to be a very late time a late time and we describe the wheeler dewitt wave function on that cutoff surface by a conformal field theory the partition function of a conformal field theory involving the set of operators a those could be we think of those as matter fields a metric on the slice not a bulk metric but a metric on the slice that we slice off and we construct parametrically in terms of the coupler in terms of the metric gij we construct a wheel or do it wave function if we want to calculate anything from that wheel of the wood wave function we have to square it and insert an operator of some sort and that inevitably means in the end of the day we'll be doing an integral over g over the metric the whole thing will have the look of a euclidean quantum gravity problem with matter on a space like d minus one sphere i want to say i'm going to say this is not a holographic theory not in the sense in which i said it may be very powerful extremely useful certainly in the hands of one it was very useful in studying uh features of inflation but it is not what i mean it's at best a hybrid of some kind of boundary theory and some kind of bulk theory so i will not be speaking about uh about this kind of um this kind of theory what is our best bet for a truly holographic theory of the sitter space and the only alternative that i know is to have a holographic theory associated with each static patch or maybe it's each pair of static patches so the first question i want to address uh one question is everybody still there i've had this habit of losing audiences and going on and on myself okay yes at least some people i don't mind if everybody leaves their mic open but at least some people leave their mic open and every so often when i become insecure i will say okay did you hear that please answer me okay first question is where is the hologram now when i say where is the hologram i mean in the same sense as we say the hologram for ads is on the boundary next what does the theory compute in other words what is the theory about what do we know about the properties of a holographic description of a static patch and finally i want to talk about implementing the symmetries now i may not get to all of this i'll get to as much as i can the symmetries that i'm talking about incidentally are very special symmetries for the sitter space they take one static patch into another here i have this is a picture of two this is um not a true penrose diagram it's it's a doubled kenrose diagram but that's not important here's the static patch associated with a particular i don't know we can call them a observer and here's another static patch and there must be transformations between these transformations between different static patches and also transformations which preserve the static patches are the symmetry of the sitter space we would recognize the sitter space by implementing these particular symmetries these symmetries are very difficult to implement no it's worse than that they are impossible to implement but we'll come to it that's one of the things i want to talk about okay all static patches are equivalent that's at least in classical to sitter space they have a metric here's the metric f r is i understand this thing called f of r is called an m blackening factor i didn't know that the m blackening factor uh is universal the parameter r is the hubble radius the cosmological radius of the space and uh if we are in a static patch then what we see is first of all the center of the static patch that's little r equals zero i'll give that a name i'll call it the pode why i call it the pode well i just like the word code the horizon at r equals capital r that's the boundary out here and anybody inside uh the static patch looks out and sees the horizon a kind of inside out version of a black hole okay now let's go to a penrose diagram a true penrose diagram of the same situation what we find when we draw a parallels diagram is that static patches come in pairs they come in pairs which if our experience with ads is any guide at all those two pairs would be expected to be entangled uh in entangled pairs of static patches one of them i've called the pode and of course the other one is the antipode and they just mean the opposite ends of the uh the boundaries of the the um the boundary the poet and the anti-boundary of the diagram are not true are not true boundaries of the sitter space they're points we'll come back to in a moment okay so what am i going to assume i'm going to assume or i'm going to try to assume try to explore the possibility that the holographic description of the static patch or the pair of static patches involves a hilbert space which is a left cross right hilbert space the left patch and the right patch that there's a hamiltonian the hamiltonian that i'm going to think about is the boost hamiltonian the hamiltonian which slides which not doesn't push everybody upward but pushes us upward let us say on the antipode side and downward on the code side and it takes the form of some h right minus h left as i said none of this can i prove all of this is something i wish to explore from the look of the diagram you might expect that the state of the system is the thermal field double as it would be in the sitter space as in that in the eternal black hole of the center space and the temperature and this is a peculiar feature of the sitter space it's an uncomfortable feature i think that if it did have such a holographic description in terms of a hamiltonian there is only one allowable temperature i think what that means is there's only one allowable temperature which in which the symmetries of the sitter space would be manifest or would be true okay the temperature is one over two pi r r being the hubble radius and the hamiltonian the boost hamiltonian annihilates the thermo field double state these are all as i said assumptions everybody still with me yes yes okay now let's look at the specific let's look at this uh penrose diagram for a moment just looking at the penrose diagram you couldn't tell is this the ads eternal black hole or is this the the sitter space that we're after the penrose diagrams are the same however the geometries are very different in particular a slice through t equals zero here would look completely different the ads the two-sided abs geometry would look like a wormhole which expanded out to infinite boundaries on either side whereas the sitter space the same penrose diagram where the same slice would look like a sphere with the code at one end and the antipode at the other end so they're quite different and to um just to represent that difference let's add some structure to the uh to the penrose diagram structure that russo introduced which are these wedges first of all we can label the sides of the diagram by the radii by the radii of the spheres at those points and we see that ads and ds are essentially exactly the opposite from each other i'll let you stare at that for a moment you all know a great deal about it okay one of these wedges these wedges for our purposes now the wedges just tell us whether when we move in the direction [Music] outward or move from the direction of the point here that tells us that the local two spheres are expanding as we move out and as we move along the tail feathers here of the little arrows the radial the the radii of the two spheres decrease that's all we need to know really about uh about the buso wedges here and notice that they are exactly the opposite in the ads two-sided black hole and into sitter space okay now i want to answer the question where is the hologram where are the degrees of freedom where shall we locate them at least nominally so let's come first to ads and let's take a pair of what buso would call screens i'm not sure we call them screens but two surfaces one over here and one over here we could work with just one of them and according to busso's rule the entropy found on a segment of a slice here this pink segment of the slice there that entropy is bounded by the areas of these um of the spheres at the tips of these buso wedges well that means in this case here that the that the number of degrees of freedom the maximum entropy on this slice here if these two points are very close to the horizon is just barely big enough to describe the entropy of the black hole itself it is not big enough to describe all the things that can happen out beyond the horizon what we do of course in ads is move outward we go outward toward the boundary put our screens there and according to buso's rule the entropy on the entire slice or almost the entire slice is bounded by the areas of the spheres way out here if we take those points out to the boundary we have enough entropy or enough degrees of freedom to describe everything on a space-like surface and that's why we put the abs hologram at the boundary but now let's look at the sitter space imagine we put our screen over here and then according to the rule the rule is that the entropy on this little piece over here this little pink piece over here is no bigger than the area of the sphere at that point but the sphere at that point has a small area it's over here right over here the area at that point is small and shrinks as we go to the pode or the antipode and so by putting a putting our screen out near the pod in the anticode that's nowhere near big enough to describe all the physics that would happen in the entire um static patch instead we're required to put the screen oh if i keep doing that the screen very close to the horizon if we put the screen very close to the horizon then according to busso's rule there's enough degrees of freedom near the horizon here to account for everything that can happen on the entire uh static patch so the lesson that the sitter hologram or at least the hologram describing the static patch of the city space is at the horizon as opposed to anywheres else uh we can probably useful to think of it as a stretched horizon slightly away from the actual mathematical horizon and uh for the usual reasons okay everybody with me any questions are you gonna discuss back reaction that reaction of what of the excitations of the matter excitations and the space time i'm just referring to the fact that in ads you know near the boundary that becomes arbitrarily small versus sitter yes yes that's that's a very big difference the um the boundary of abs is what i would call cold it's cold because the energy scales too excited are so high whereas the horizon is not cold uh so this is a big difference and how we are to how what would to make out of that difference well i'll talk about it a little bit but i don't claim to fully understand it in any case what kind of degrees of freedom what kind of theory might we place on the horizon should it be a quantum field theory should it be a quantum field theory whose degrees of freedom are located on the horizon and the answer i think is no the reason is that the deceiter space horizon or the deceiter space itself is a fast scrambler i'll remind you what a fast scrambler means is that if you drop an object into the uh toward the horizon that its degrees of freedom mix up with the entire horizon in a time called the scrambling time uh a fast scrambler is one which scrambles fast fast means logarithmically fast all right so you can do this you can do this problem uh in a baby form in the sitter space for example you can drop in a charged particle this was the first attempt this was long ago the first attempts at understanding scrambling were in terms of dropping in charged particles and seeing how their charge spreads over the horizon that was for black holes we can do exactly the same thing for the sitter space we can drop a charged particle in we can solve its field and uh calculate how the it's really the normal component of the electric field which is a substitute for the electric charge how the electric charge spreads over the horizon and one finds very much like in a schwarzschild black hole that the time to spread over the horizon is logarithmic in the entropy or logarithmic in the radius of the horizon it's a fast scrambler so the sitter space like a black hole is a fast scrambler and it does not scramble ballistically as it would in a local quantum field theory that would suggest well let me say yeah that would suggest uh that the degrees of freedom are all to all k local degrees of freedom coupled in a manner let's say similar to syk or something like that uh we're gonna i'm gonna we're gonna have to modify that but this is let let me not say that suggests all to all couplings let me just say that all to all couplings are the kind of things which do lead to uh to fast scrambling or one of the kinds of things that lead to fast scrambling so the uh the horizon degrees of freedom should definitely not be thought of as a local quantum field theory but more like a highly non-local all to all coupled system all right now just mainly as a foil for talking about things later i want to introduce a toy model the toy model isn't a good model it does not describe decently any kind of anti-sitter space but it's it's a useful foil for uh for discussing the sort of space later first of all the pode is a point of instability what i mean by that is that if you had a particle or an object a galaxy or whatever it happens to be which was slightly displaced from the code and you allowed it to evolve it would eventually fall through the horizon that means it would move away from the pode and fall through the horizon at a distance r or capital r away from the pole that's any the most yeah in that statement how do you define where the pole is is that just a chord that's stable that's where i am yes that's absolutely it's well um once i pick a code once i pick a code i can then look at the motion of an object relative to that poet i pick a poet meaning what does that mean that means really what it means is i pick a point in the remote future and in the remote past and think of the causal patch i didn't really fully define the causal patch uh raphael what is it what is the death what is what is the causal patch is it the domain of uh influence of the uh it's what you just said it's the uni it's the intersection of the past yeah the future of these two right yeah so you pick a pair of points one in the past one in the future you construct the static patch associated with them having uh having uh constructed it then there's a natural code which sits at the middle of it and uh and anything slightly away from that code following geodesic trajectories will fall through the horizon all right so it's it's natural again yeah okay can i ask go ahead i'm just so mathematically i think this is a well-defined statement um but non-perturbatively i don't think he would expect that we can make sense of an infinitely long world line in this inner space yeah and so it's not clear that there's any operational meaning to that definition no i agree with that i i think what we're trying out now is a set of ideas which may which we may find in the end simply don't work and uh that that would i would like to know if we can form a holographic theory of this type uh i would not be terribly disappointed if in the end we find out that there are limitations and that this can at best be an approximation which i have yeah i had a small comment this picture you've drawn about the instability isn't enough way to say that the sitter is a fast scrambler because in a scrambling time what purple particle disappears behind the horizon right no no absolutely that it's a scrambler of any kind yeah but no but it the the the motion of that particle relative to the code is an exponential growth of the it's a typical exponential growth associated with an instability and the the non-relativistic model which is most like this would simply be a an upside down potential the pose sitting at the top now let me go back a step if i were to build a simple model of the sitters of anti-disinterspace anti-disinterspace a model that's simple enough that i could explain it to a small child i would say it's a sort of container with a potential that pulls stuff toward the center a gravitational field that pulls stuff toward the center anti-dis anti-disorder space is the opposite of the sitter space so the situ space is the opposite of anti-disorder space in this respect it looks like a cavity with a repulsive center and an instability for objects located at the top of a potential ear anything at the top of the potential if you displace it a little bit will fall and will fall down to the minimum i also want to imagine that there are in non-relativistic particles and these particles have been put in with some energy some temperature and that they sit down at the bottom here and form a kind of thermal gas with an entropy of order the number of particles this region is the horizon this is the pode anything you drop in will fall off and eventually be in the horizon the horizon degrees of freedom just exist there in boring eternal equilibrium very boring but with occasional interesting boltzmann fluctuations by a boltzmann fluctuation i mean a large-scale fluctuation in which a significant number of degrees of freedom might find themselves up near the pode or someplace else it doesn't have to be at the pode but a poet is a complete as a convenient place to think about that's about all that can happen well that's not all that can happen but this is one of the things that in principle can happen and uh of course it happens very infrequently but this is all that happens in a pure dissider space or in a pure model like this occasional fluctuations and it is my belief that what a ideal decision space is is it's or a theory of an ideal visitor space is it's just the theory of these fluctuations but the fluctuations can be interesting they're very intermittent they don't take place very often but they can be interesting uh sometimes people call them boltzmann brains okay the questions that i want to address is how do you calculate the probability for these fluctuations how do you calculate the probability for these fluctuations and i'm going to give you three formulas for the probability of a fluctuation one is you calculate the entropy of the system constrained to have the baltimore fluctuation present i'm calling the boltzmann fluctuation theta theta or oh i can't remember theta uh constrained subject to the condition that the boltzmann fluctuation is present calculate the entropy of the system the difference between that entropy and the entropy of the pure the sinner space or the pure sorry the entropy without the boltzmann fluctuation that is called delta s and the probability for the fluctuation is just e to the minus delta s that's one of boltzmann's formulas another formula is that it's e to the minus beta times the energy of the fluctuation those two are consistent for a uh for for an isolated system and the last formula is let's suppose that we can quantum mechanically construct a projection operator that projects out states in which this theta object is present then the trace of rho times pi of theta is the probability that theta is present all three of these are the same basically the same formula where rho is of course just the thermal ensemble all right now there is much this is one thing that you could calculate how likely is it that uh that such a fluctuation exists there's many other things you could calculate of a more dynamical character for example the transition probability that if you start with theta you end up with theta prime up on the top beta prime could exist could be a slightly different state or it could have an extra particle falling down in or whatever can we calculate that transition probability well in principle yes if we knew what these projection operators were this is pi of theta this is pi of theta prime uh we can time if we know the hamiltonian we can time translate pi of theta prime and we can calculate the trace of rho times the correlation function of pi of theta with pi of theta prime and that's basically the transition probability to go from here to here so the point is thermal equilibrium if you could calculate everything that thermal equilibrium entails everything possible in thermal equilibrium there's a lot of stuff there there's an entire uh description of evolutions of galaxies evolution of uh and um it's not totally boring but the time interval between interesting things happening is very long all right so calculating delta s this is a challenge how to calculate delta s in the toy model here you can estimate delta s uh it's basically just the entropy of s itself of theta itself is the way to think about it if a bunch of degrees of freedom separate themselves from the mess down at the horizon here then the horizon has fewer degrees of freedom instead of having n it has n minus little n and the entropy of the horizon is diminished by a factor n minus little n uh let's uh let's keep this simple let's just suppose that the object up here also has entropy of order little n but much smaller than the coefficient n here that that is something that you'd expect all right then you'll find that the uh that the delta s for creating a fluctuation up here is just of order in the number of particles that come up here or the number or just the entropy of the object theta up on the top and it would say the probability of forming such a fluctuation would just be of order e to the minus some constant times the entropy of the system up here little s stands for the entropy of the system up here so keep that in mind that for simple non-relativistic models delta s would just be proportional roughly of order the entropy of the thing that appeared that were uh that was searching for up on the top of the of the code all right now this is not the before question did you say little c is much smaller than one [Music] it is typically smaller than one where um no not in this formula sorry not in this formula um in comparing the n here and the n here i would say this c is less than one and the reason is simple the particles that exist up here if they were found down at the bottom they would have a fairly large volume to move around in if they're constrained to be up on the top they have a smaller volume to move around in and so the entropy of a group of particles stuck up near the top here would be likely less than the entropy if they were allowed to fall back down into the mush okay this is this is not terribly important because this is not a good model lenny yeah i'm a little uh confused by the statement about the entropy of the matter that you're making so yeah the the horizon entropy will will decrease by an amount that's basically just given by the area decrease of the of the horizontal now we're doing the toy model this is the toy model it doesn't uh doesn't know anything about general relativity i thought that the the big n minus small n was supposed to capture that i i thought that that was supposed to just some sort of shrinking of the horizon is that not that is true that's what we're going to do next okay accurately it doesn't count it doesn't capture it correctly okay then i'll hold off asking anything until you're yeah good by the way i don't know if i mentioned in the beginning i think i did but uh a lot of what i'm talking about now is very closely connected to uh concepts that tom and his collaborators have discussed and um so good okay so let me move on yes no but what uh what raphael is saying is is correct but it's jumping ahead of the game here now i want to generate that jump and say let's calculate let's see if we can calculate using general relativity what delta s is here okay so let's calculate using general relativity what delta s is and to do that well first of all we start with the metric of the decidure space by itself and calculate its uh its horizon is r equals big r its entropy is pi big r squared over g in four dimensions and uh and that's the initial entropy now we consider a system theta and let it be a black hole let's think of it as a black hole for simplicity let's just take it to be a black hole a black hole sitting at the center at the code what is the entropy of the system no i don't mean the entropy of the black hole i mean the entropy of the whole system including the black hole and whatever else happens to the rest of the system and for that all we need to do is take the metric of the deciduous space including with its black hole this is this is all you have to do to get the entropy of the black hole into sort of space is to add a schwarzschild 2mg over r inside the blackening factor then recalculate sorry hold on recalculate the area we can't recalculate the area of the horizon the horizon is defined by setting in blackening factor equal to zero we find epsilon here the decrease in the radius is just mg and where we find we find that the entropy of the system that now contains theta this is not the entropy of theta it's the entropy of the system which contains theta and other stuff is just pi g times r minus mg squared and the deviation of the entropy from the original descender entropy is just 2 pi m r that's just this product pi over g r times mg that's just 2 pi m times r the g's cancel and that's it so this is the probability of nucleating or whatever it is or whatever we call nucleating a black hole of mass m big r here is not the schwarzschild radius of the if this was a schwarzschild radius of the black hole this whole thing would just be the entropy of the black hole this is not the schwarzschild radius of the black hole this is the cosmic radius of the horizon okay so we have succeeded in calculating it and of course we can use that to calculate the probability now boltzmann tells us something else boltzmann also tells us that the probability for theta is e to the minus m divided by the temperature this is just the boltzmann factor for an object of mass m and so thermodynamics basically tells us that m over t must be two pi m r we know this anyway in fact but uh but let me just say it this way that the consistency of thermodynamics with fluctuations tells us that m over t must be two pi m r and it tells us the temperature of the dissider space i emphasize this because it's a way of thinking about the sitter space thermodynamics which doesn't require us for example to change the cosmological constant which doesn't require us to uh to think about variations of things that we may not be able to vary instead of thinking about how things respond the changes in cosmological constants and things like that you think about how energy is partitioned between fluctuations and the rest of the system and that gives you another form of thermodynamics which in this case just tells you that the temperature is one over two pi r and that's correct sorry can i ask a question here it's a naive question it looked like you dropped the term proportional to m squared yes i did i did because it's much more does that mean that there should be a correction to the formula for t and t yeah yeah yeah yeah yeah plus order m squared g squared uh but but uh you know that's happening then the m's don't cancel when you said the two things even so that's very well that's extremely small by comparison with this big thing here how big is the quadratic term is the entropy of the black hole itself m squared g squared pi over g that is equal to let me write it another way it is equal all right it's equal to the same thing that i've written here except substituting the short shield radius of the black hole so the schwarzschild ladies of the black hole is a kilometer the radius here that goes into this formula here is uh 10 to the 10th light years so the m squared g squared term is entirely negative negligible and that's the whole point here the whole point here is that something has given us an answer which involves a much bigger exponent here than you would have gotten by just using the entropy of the black hole itself i don't know if that answers the question or not um sure well why couldn't you make the black hole big oh you can no no you can you can you can i should have said that i'm going to think about a small black a small fluctuation i'm working to first order in the uh yeah you can think of a black hole whose entropy is as big as a third the entropy of the um i i think it's worth pointing out here that our usual way of understanding black the temperature of deceiter space is by thinking of small quantum field theory fluctuations around the dissider space so by definition we only really know that this interspace has an exactly thermal formula for small fluctuations around the vacuum that may be um that that's all we know that's from the gibbons talking derivation that's all we know yeah okay i think another important point though is that well you're focusing on the production of small black holes by fluctuations but for small for small enough m those are not the dominant things that will be fluctuating um and and the correction to the boltzmann formula will be more important than than m squared for for a more extensive objects like you could fluctuate some radiation but it's going to definitely be small compared to this well this is a kind of beckenstein bound which uh which yeah it'll it's it's not going to exceed this but it could saturate it for example a small black dot no no radiation but if you but if you just if you yeah if you radiate one hawking particle of a typical of the typical uh desired temperature that that has of order this entropy namely one it has an entropy of order one yeah yeah right right which is the same as right that's delta s m yeah in that case m is one over r if if the fluctuation is small enough then its own entropy competes with this formula that is correct let's say that again now i didn't understand that this is a big number if m goes like 1 over r then this is oh delta x if m goes like 1 over r yeah oh that's really small yes indeed and and it's something that can't be localized really on the code i see i know i unders i understand your point okay but m could be 1000 over r right so there's just a big regime where you're not making black holes but you are fluctuating stuff and yeah yeah yeah yeah no that that's correct but uh but those things all have wavelengths which are it's not cosmological at least astronomical i understand what you're saying so let's uh so let's just restrict ourselves to black holes of um of um mass much bigger than the inverse of r in other words black holes of ordinary black holes okay so um good now i want to come to another formula a distinct formula for delta s for the probability for um for nucleating such a thing let's start with delta s equals 2 pi m r okay and let's let's calculate r first of all pi r squared over g that's this formula over here is equal to the s naught means the entropy of the decisive space north simply means deciding space pi r squared over g is equal to s naught so from that i can come from that i can compute r on the other same type of formula over here so the entropy of the black hole it's 4 pi m squared g and that will tell me what m is over here i multiply those two together and here is what you get you get that the fl that the um that the delta s the thing that goes into the boltzmann formula and the exponent is the square root of s naught that's the decider entropy and the black hole entropy a curious formula and i claim that that kind of formula has all kinds of hints in it about what the fundamental what the holographic degrees of freedom are uh tom knows this very well and i'll just repeat the story here i want to compare this formula here which contains only entropies of the various uh constituents the the horizon s naught is the original entropy of the horizon s little s is the entropy of the black hole let's compare that with the toy model first of all the toy model says that the fluctuation or that the delta s in the toy model is just about a little s the delta s that we compute is much much bigger than that so it means that compared with the toy model it's much much harder to fluctuate a object of mass m than it would be as much harder in at the sitter space much less probable and one question is what is the what is the reason why it's so much less probable so i'm going to describe now a model which uh was discussed uh by banksville and do i have this right tom tom are you there yeah that's right yeah i also discussed it somewhat later tom and i discussed it i can't remember how long ago in ancient history's i remember sitting in a cafe someplace in some foreign country i don't know where it was discussing these kind of things maybe even before i no it wasn't before it was not before this but it was before this banks and fishler had a follow-up paper on it and i'm going to discuss their understanding and my understanding of uh of this uh of this formula over here okay i'm so sorry can i ask one more time i don't understand the significance of of or the legitimacy i guess of why you're restricting to black holes and the fluctuations that you consider so even if i don't allow you a mass that's less than if even if we restrict to m being greater than the planck mass which i think physically there's no particular reason to but let's do that then still for for small enough m and sufficiently small cosmological constant the fluctuations are not going to be dominated by the production of small black holes there's just more entropy there's more phase space uh in the production of of lighter particles gases of lighter particles and and the in the statement the key statement that you're making about the relation between these entropies is not the k you know doesn't hold uh in that regime this this geometric mean formula okay so where where okay so where does it where does it hold it it doesn't hold for exactly the same reason that it wouldn't hold forth where does it begin to hold how how massive do you have to be when it starts to hold i think you really have to restrict to black holes for that formula to be correct that's what i was doing yeah that's why i'm asking what is the significance of that i mean it seems to me that black hole since they don't dominate the the the thermal ensemble at uh at low energies they do there's some energies they do isn't that right yeah that rafael if you take any ordinary particle mass and put it into that formula not a massless particle with with uh frequency that's i mean wavelength that's you know a a few uh decades less than the than the uh cosmological horizon that this formula is still correct if you've done this which is correct but the limit that i i would like to discuss for a moment is one where we take the mass to be fixed it could be bigger than the blank mask and we take the cosmological constant to zero in that limit you would not be dominated by small black holes and this formula would not be correct and so i'm trying to understand why that's not the right limit to think about well this formula for the entropy is true even if the object is not a black hole because the the field the gravitational field of a massive object far from the object always looks like the schwarzschild field the formula i'm referring to is the one in which the entropy of the object that's being fluctuated and the little s is the entropy of the black hole right yeah that's correct if i take that to be the entropy of the object that's being fluctuated more generally then this formula is not true i i think oh i think that's correct no no i'm sure that's correct this is this is specific for black holes absolutely no the formula that is correct though is square root of s naught times the mass of the object um something like that well that can't be dimensionally correct but the yeah the mass in plank humidity the massive plankton that's that's right yeah yeah yeah yeah but uh permit me the luxury of just thinking about black holes for the moment and uh we can come back to that okay so how does uh how do these uh models work i would call these toy models with strings attached instead of just having n non-relativistic particles we have n i guess we could call them b zero brains uh but they're objects which strings can attach to another way of saying is we're talking about matrix models mij is the string field operator the creation and annihilation operators that create and annihilate strings connecting i to j these are familiar things from matrix theory and uh other other situations and uh we have a degree of freedom like this for every pair of in constituent or in uh in these zero brains i'm just representing that by drawing a collection the black here are the zero brains and the blue are not necessarily strings but blue represent the modes that represent the matrices mij they may or may not be occupied by having strings excited typically in thermal equilibrium at a temperature the kind of temperatures we want to think about the strings would be excited and so in the in the horizon degrees of freedom and so we would expect a uh a big soup of strings in the zero brains with on the average some number of strings connecting every pair of these zero brains that's a model that's a uh that's a model uh for the physics here and the degrees of freedom are these m i js and very similar to what we do in matrix theory how many mijs are there there are order n squared of them and so we would expect that this system in thermal equilibrium above some very low temperature would have an entropy of order n squared in this case not n but n squared uh what would we mean by a boltzmann fluctuation well in the model the idea is that a boltzmann fluctuation is what happens when a group of particles disconnects itself completely from the rest of the horizon degrees of freedom i won't try to justify that it seems like a reasonable and certainly it's the kind of thing one expects from matrix theory when this disentangles itself and uh and disentangling itself first of all may say something about entanglement but i also just mean that the string modes which connect the two lumps here are in their ground state that's the way we might describe a uh a boltzmann fluctuation and that boltzmann fluctuation will have an entropy of order little n squared little n squared uh degrees of freedom okay so what's the what's the difference between the original entropy of the starting horizon and the horizon plus little fluctuation well it contains a term n minus little n that's the remaining number of the zero brains and the horizon degrees of freedom squared plus order little n squared in other words it will be proportional to big n times little n but big n times little n is nothing but the square root of s naught times s this formula with the s naught and square root of s naught times s that's special to four dimensions right you betcha that's correct that is correct and we're going to talk about that okay we're going to talk about that and i don't have a i don't have a wonderful story to tell about that but i do have a little bit of a story to tell about it which is kind of interesting so um all right so another way of thinking about it then is that delta s is simply the cost of freezing the off by off diagonal i mean the strings freezing out or saying there are no strings connecting theta with the horizon in other words it's the cost if you like of just eliminate cutting them kind of cutting them is not the right way constraining them to not be there the strings not be there i'll call it the string freezing factor this this this type of factor and i'll refer to this effect as the string freezing effect um and what it's saying is if you want to reproduce the the kind of um entropy for fluctuations in the sitter space there's something going on which requires you to cut the connection between the degrees of freedom of the black hole and the the horizon degrees of freedom okay now somebody i believe it was juan asked me about higher dimensions so let's see what we can say about higher dimensions it would be lovely and nice if the story just persisted for higher dimensions as we'll see it does not okay to go to higher dimensions we do the same thing we take the blackening factor now the blackening factor is the same for pure decent space but if there's a mass this is not m this is mu over here where mu is equal to this object this is a general formula in little d dimensions don't worry about it don't try to memorize it in the next 10 seconds you'll fail well i don't know maybe you won't and the horizon of course is just a place where the blackening factor is zero if you work it out again the first order in mu or also the first order in the mass of the uh of the little black hole you'll again get something which is big r minus something proportional to the mass and now contains in various ways big r okay you can use this to calculate the change in the area due to putting in a mass and again if you work it out you'll find it's exactly equal to m over t this much doesn't change it's exactly the boltzmann factor in every dimension that had better be otherwise we'd be making some bad mistake uh delta s the uh of a fluctuate of the is equal to the mass over t again but now i want to express this in terms of the entropies of the two objects again and so we have to go through a rigmarole it's not too hard it's pretty easy just use some of the equations uh that i've already written down and here's the formula that you get delta s the uh the string freezing factor contains a numerical factor depending on the dimension here which is not not important for us here and it contains powers of the of the entropy of the black hole and the entropy of the um of the deceiter space for d equals four if i haven't made a mistake it goes back to the same formula that we had before but for large d as d gets larger and larger it tends towards something which only depends on small s notice the the exponent for large s goes as one over d and so largest disappears and what this means is somehow it's much easier and high dimension to disconnect the um the fluctuating degree of freedom from the horizon degrees of freedom and the answer is very similar to what we had in this early toy model just something proportional to small s for large dimensions in other words another way to say it is the string freezing effect is much weaker for a large dimension in fact as we get up to very large dimension it looks like the string freezing factor goes away altogether and uh that's a bit puzzling for the following reason well let's see how we can say this um it looks like the degrees of freedom which connect the fluctuating system to the horizon have become totally unimportant one might say that my own might worry about that one might worry that the all to all coupling required fast scrambling uh disappears at large d that the uh that the degrees of freedom become not so thoroughly interconnected and you might worry what happens to fast scrambling under that circumstance well the answer will turn out to be that you don't really need all to all coupling to have scrap ass scrambling you need something that i'll call enough to enough to make fast scrambling and how much is enough and is it consistent uh with this kind of formula and i'm going to construct now a very crazy model which shows that these things are consistent that you can that you can let's let's say we can reproduce this formula here with a system for which we have good reason continues to be a fast scrambler i'll call it the sparse matrix model and here's how it's defined again you have a collection of the zero brains the left diagram here represents all of the possible modes that were there in the original model namely every v0 brain can be connected to every other d0 brain by any number of strings the mig mijs are there for all possible pairs of the zero brains the sparse model is defined as follows uh hold on a second yeah you go through this interaction diagram and you go to each degree of freedom and you either eliminate it or you don't eliminate it probabilistically the probability being given in a moment so you start eliminating and probabilistically are you going to end up with you're going to end up with a sparse version of the of the interaction diagram i will take the probability to not delete the probability to not delete is some constant divided by n the number of the zero brains to a power p this means that as n gets large you eliminate a larger and larger fraction of the um [Music] of the modes you eliminate them as possible excitations the number of surviving modes in the end is c times n squared y n squared that's the original number that we started with divided by n to the p that's the probability to not eliminate is c over n to the p that's the number of surviving modes that are left after this decimation of the system what is the entropy the entropy of the uh pure deciduous space that's just capital n to the two minus p yeah this this is n to the two minus p the number of surviving modes is n to the two minus p we might expect that that's equal to the entropy uh of these degrees of freedom what about the theta what about the small black hole well we do the same thing that we did before we take we remove n degrees of freedom little n degrees of freedom from the horizon soup and we calculate the remaining entropy and that will be little n minus little big n minus little n again to the two minus p and we can work that out and we find that the delta s the entropy deficit it's capital n to the 2 minus p minus 1 times little n which we can rewrite in terms of the entropy here are the entropies the entropies are n's raised to powers and we can rewrite it as s naught to the two minus p minus one well you can read it i don't need to read it out loud it looks sort of similar to the formula that we're trying to reproduce is the formula we're trying to reproduce again a little s to a power a big s to a power and we can equate these powers for example we can equate the power of little s here that's s to the d minus 3 over d minus 2 to s to the 1 over 2 minus p and that gives us 2 minus p is equal to d minus 2 over d minus 3. we can do the same thing with big s we get another formula well as it turns out these two formulas are identical they're not two different formulas they are the same formula and they give you the answer that p this power p in the uh in the in the statistical elimination of modes it turns out to just be d minus four over d minus three not a particularly uh special number notice that it goes to zero at d equals four that means that you didn't uh that you didn't decimate at all you didn't cut out any of the degrees of freedom and it goes to 1 as d goes to infinity as d goes to infinity this goes to 1. and so we can ask now for different d's in particular for d equals infinity where my biggest concern is where it looks like you've decoupled everything what does the interaction graph look like in other words what are you left over with in terms of a graph when you've uh when you've eliminated uh string modes according to the probabilistic rule involving this d here you know how about d equals three many uh d equals three is not a good example there are no black holes in three dimensions there are no deceit black holes in three dimensions anyway i think there's anti-descent of black holes but there's no disputed black holes there's no flat space black holes i thought there were three then i should have just hit the brown clothes huh small black holes well it can't be because if there were small deceit space black holes and there would be small flat space black holes so i don't think so um no they're not yeah okay okay so what are the number of surviving edges after you do this this decimation well the number was 2 to n to the 2 minus p it was n squared when we didn't do any decimation n to the two minus p as d as v goes to infinity that just goes to a constant times in that's the number of remaining modes and numbers notice that the number of remaining modes did not go to zero as d went to infinity it went to n that's proportional to the number of vertices so that means the interaction diagram has about as many uh um edges as it does vertices somewhat bigger but only by a numerical constant the constant c here is called the degree of the graph and it's simply the number of edges coming out of each vertex on the average so we have a random graph of average degree 2c that's that's the way the graph theory works out here's a non-random graph of degree 3. the non-random graph of degree 3 is just a lattice in this case excuse me i just lost my picture hold on uh it's just a lattice and if you were to calculate you know you you keep going until you run out of the zero brains you keep building it until you run out of zero brains this is a very non-random graph and it has a diameter which grows as n to the one-half diameter means the largest distance between any pair of points the one-half is just because it's in two dimensions it's a power law in n the way the diameter of the graph grows random graph of fixed degree or a fixed average degree is a different kind of graph they're called expander graphs so i'll tell you just a little bit about expand the grass here's a picture that looks like a cabbage here's a picture of an expanded graph the reason i drew it with wiggly lines is it turns out the eye much prefers the wiggly lines when looking at these things i tried drawing with straight lines and you couldn't decipher anything the wiggly line somehow made it more visible the nodes are the zero brains the links tell you which modes you've kept in the system and these expander graphs i'll tell you what they look like they're first of all statistically homogeneous they're everywhere pretty much the same they're not perfectly homogeneous because you did this by a random selection but they're statistically homogeneous and if you start any place at any point and work outward whoops excuse me what you will see is a tree graph start any point and work outward and on the average you will see a tree graph a growing tree graph which grows exponentially that's true starting anywheres although it's only easy to draw starting at one point and that tells you that the number of nodes increases exponentially yes the number of nodes increases exponentially with the distance away from a given node and it tells you that the diameter of the until that of course stops you only have a finite number in [Music] i think it stops when you've covered about half the nodes when you've covered about half the nodes it stops growing the diameter of the graph meaning the largest separation between any pair of points grows like logarithm of n that logarithm of n is important first of all it would tell you that the diffusion time on this graph you start something happening somewhere and it diffuses outward takes a time of order to log in not into a power as would be the case for a non-random degree 3 graph diffusion times log n it's also true that quantum systems on expander graphs while the expanded graphs are vastly sparser than all to all couplings they are much sparser they are still fast scramblers i refer to a paper by uh su suskin and swingle studying fast scramblers on these kind of graphs and furthermore expanders are the spiciest fast scramblers if you make the graph any sparser in other words if you make the probability any larger to eliminate edges you will destroy the fast scrambling property so that's the situation now you look at things you have d equals four d equals five equals six up to d equals infinity the equals infinity is where we're getting these expander graphs d equals four we have all to all couplings and everywhere's in between they're all fast scramblers if we went past p equals one the fast scrambling would disappear so it seems this is by way of an existence proof if you like that the weakening connectivity implied by the diminishing string freezing effect is consistent with fast scrambling i was worried about that i thought i thought the fast scrambling would be destroyed okay i uh let's see um [Music] i've gone on a long time longer than i thought i would the next topic i was going to take up all right ed what what is the time uh right um you can go on a bit um not too much more no i i i'm getting quieter myself okay yeah for the hardcore will reconvene for discussion when you can explain whatever you haven't had time for but at least give us a couple highlights of what this would be all right the yeah um so supposing i give you a uh a fast scrambling quantum system it could be then it has some some manifest symmetry some obvious symmetries um but nothing very sophisticated how can i tell that excuse me how can i tell if that quantum system represents the sitter space and not something else i mean the systems i've described they're very similar to the same systems that i might try to use to describe black holes how can i tell if it's the situ space or something else and i think the answer is the symmetry of the system the full symmetry of the system in particular the symmetries which relate different static patches here again is two-dimensional the sitter space it has a pair of static patches green and pink there are symmetry transformations of this decider space which move the static patches and take them to new static patches those symmetries are a non-compact [Music] od1 group of some sort and um we might ask is that symmetry realized by the system by the system that i've constructed if it is then it has a chance of being the center space if it's not well then it's something else so the question that i want to raise then is what do we know about the possibility of being able to implement the symmetries which take us between one static patch and another static patch or between one pair of static patches and another pair of static patches notice that the end point of this static patch over here where is it over here is behind is in the um is behind the horizon so to speak of the original pair of static patches so we're moving things around taking things from inside the horizon to outside the horizon and these are symmetries these are not symmetries that we would expect for any kind of black hole system they're new symmetries that the sitter space has to have so i would say we recognize the sitter space if the system uh implements the symmetry group well um all right i'm gonna i'm gonna i'm gonna stop here what i'm gonna tell you is number one that it is impossible to to uh to represent the symmetry group uh there's a no-go theorem about it the no-go theorem is an old nogo theorem for many years ago but that old nogo theorem also applies to ads2 and it applies equally well to a system that we've been studying for a number of years now namely let's see where is it in syk jtads2 system the same kind of none uh and there we've learned what we what we've learned is that the symmetry can at best be approximate but i think it's also true that that symmetry can be i don't know if it can be exact one can the sl2 symmetry of jt gravity be exact on in the ensemble average sense um i'm not sure how to think about the symmetries in the ensemble average uh way i mean they're acting on some hilbert space and your old argument applies yeah yeah yeah um but there are certain things that can be true for example some of okay yeah i mean of course yeah well okay what i was going to say what i will say if we have a little more time later is uh the two are the two situations are very parallel what happens in the sitter space and what happens to the um anomalous symmetries in the anti-deceiture space of of the syk model seem to be the comment is that the symmetries can be exact in the n to infinity element for example syk yes yes and also right i think right right and i think they can also be exact in the n equals infinity limit or the uh uh of the sitter space if such a limit exists and makes any sense but what i think i would suggest is that perhaps the logic of the breaking of the symmetries of the pseudospace is similar to the logic of the breaking of the symmetries of uh the ads2 space and that the upshot it's just a suggestion is that the sitter space is really realized as an ensemble average rather than uh for individual exact versions of it but uh i think i've gone on long enough now i've certainly gone on long enough for me i've reached an age now where i can't go for more than an hour without getting tired well thank you frank thanks very much for listening okay um shortly shortly we'll adjourn for 10 minutes and those who want can join for a discussion uh on the other zoom link but maybe before we do that i'll ask if there's any quick questions anyone wants to ask no well just just one remark i'm hoping that what i meant by saying something was um half baked is only that it needs some more time in the oven before it becomes fully baked as opposed to the other idea of half baked okay so are there any questions yeah lenny i have a i have a question that the um constraints in the in the d equals four case i can think of those as just uh a constraint on a fixed hilbert space there's a uh you know you you have some operators you have to set those operators equal to zero on the on the constrained states i i didn't understand whether your probabilistic constraints had had a similar interpretation right the probabilistic constraints are not probabilistic in that sense they're they're probabilistic in defining what modes exist in donut don't exist i'm making up a system this is a highly artificial thing to do okay it's only a purpose it's only purpose is to convince myself of an existence uh proof so let me just go through i'm not saying statistically that there are no strings what i'm saying is statistically there's only a certain subset of the modes allowed that that can exist okay so i should think of the model as a model on this expander graph period just that yeah okay that that now i understand yeah and i'm not advocating i'm not advocating the expanded graph picture of the sitter space i'm simply i simply used it to convince myself that there was no conflict between um this formula here not the you know which formula i mean yeah i know exactly right here and uh and the fast scrambling character right i actually have a comment which is that finitely connected graph which for you corresponding to the equals infinity seems like the most naturals fast scrambler because in that case the logarithmic time scale exists geometrically whereas in the other cases you have much faster scrambling and to get lower that's right when you address the coupling yeah you have much faster scrambling but then you readjust it by readjusting the coupling j yeah by dividing by some power of n i think that's what you're saying yes so in the other cases you adjust the coupling to get the answer you want but yeah in the sparse case in the extreme space of finite connectivity it's more than that though the answer automatically for geometrical reasons yeah that's that's correct so definitely uh but yeah if that was all we knew i definitely would have said that was the most natural fast grounder i i i know what you mean i know what you mean and then matrices wouldn't uh matrices would not fall into that class but uh but even the all to all couplings still do satisfy the fast scrambling rule that it that the scrambling time is beta times log s because when you rescale the js you also rescale the the scale of the temperature yes so if they make black holes or at least they're believed to yeah yeah

Channel: Institute for Advanced Study

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Id: FynK-B6X1HY

Length: 91min 49sec (5509 seconds)

Published: Tue Mar 30 2021