Dionysios Anninos - 3/3 Quantum Gravity in de Sitter Space

Okay, welcome back everyone. So I'd like to pick up where we left from. Just a few remarks that are based on questions that people were asking me. So for the spinless case and similarly for the general case. If you want the expression that has the explicit uh regulator inside it it takes the following form where epsilon is a small positive number and the idea is to uh take the epsilon going to zero limit at the end and this thing will diverge but the divergences that you get are compatible with the local counter terms of quantum field theory. So I I wrote the spinless case but this pattern of of regularization is the same in general. And I'd like to make one more comment uh before uh before moving on uh to the properties that this object has in gravity which will be where things really start to distinguish themselves from black holes. So, so far I've been drawing an analogy with black holes that you might take a little bit too seriously. The thing looks amazing. >> Stand by. [laughter] Uh, so so so I'd like to break the anal. So the purpose of the first part of the lecture will be to break the analogy with the black holes a little bit and then uh in the time remaining I'd like to discuss time permitting uh an example of what might constitute a a model a simple model that's both ultraviolet uh finite and potentially infrared complete as well. Okay, I hope I have time for that. If not we can talk about it after. Now, so good. So, in gravity, so so far I gave you a picture as to why to think a potential series of reasons of why to to start considering the rather esoteric question of studying fields that live on a force sphere. And although it seems like a very abstract esoteric thing to do, I hopefully uh linked it to sufficient physical questions that it's at least a question worth pondering. And now I'd like to give you uh uh sort of a gravitational completion of the formulas that I wrote down. Now, one reason to consider, one practical reason to consider the problem uh in uh the case of not just fields on a rigid uh round sphere but but on a theory where the sphere itself is dynamical. Well, a we'd like to understand quantum effects in gravity. So, this would be quantum effects in gravity, ukidian gravity with lambda positive. But another more practical effect is that the expressions I wrote so far whether we like it or not [snorts] do have ultraviolet divergences. And you know this is a reflection of uh uh sort of the local divergent nature of quantum fields. the fact that they have contributions to things like uh uh entanglement entropy that diverge or or the splitting. And so we may want to embed this in a theory where the local divergences can be absorbed by coupling constants local coupling constants that are physical coupling constants of our fields themselves. So when we if we have a theory of gravity, yes, in addition to the couplings of our matter fields, we also have the coupling the local couplings of the gravitational theory itself. And these are couplings that we have to measure in any physical uh model. But we're equipped with bare couplings that can absorb all of the divergences of the quantum field theory calculation. So even though the quantum field theory calculation is ultraviolet divergent and you might discard it for that reason, once you couple the quantum field theory even weakly to gravity, you can absorb the divergences produced uh in this uh expressions into the bare couplings of your gravitational theory rendering the problem in a physical sense finite again. Yeah. So gravity equips us with a larger set of local coupling constants that we can use to renormalize the divergences of the quantum field here and the rigid spacetime. So that alone is of practical value. However, the real motivation uh uh to study the problem of on gravity is that all of these questions of horizons and sort of uh these questions about the entropy associated to horizons are deeply linked ultimately to gravitational physics. Yeah. So this beenstein hawking law is not a property of quantum fields in a fixed curved space but rather it's a contribution that comes from the path of the saddle point of the pattern of the gravitational field. And so here we're going to sort of follow that uh proposal uh and try to compute what appears to be the simplest uh uh object that the theory of gravity has that has the property that it's entirely diffmorphism invariant. So it's an object whose output does not in any way depend on our choice of coordinates. And in addition to that uh uh it doesn't depend uh in any choice of local field redefinitions. So when we have a theory we we talk about a field but in principle we can we can change this field by local field redefinition to a new field that has the same uh structure as our original field. And one of the benefits of something like an Smatrix compared to a correlation function is that an Smatrix takes as an input a correlation function and as an output gives you the piece of the correlation function that is insensitive to things like local field redefinitions. This is sometimes referred to as the equivalence theorem. And so it would be nice to have objects in cosmology. This is a much harder task it turns out. But objects in cosmology that are as invariant as for instance the Smatrix itself. Now there isn't a rich set of such objects but at least as a proof of concept uh this object over here is an example uh uh of so let me sort of write it down. So the proposal that has appeared in in the literature of an object that we may wish to study is to consider the path integral uh understood in a saddle point of type of methodology for reasons I'll explain in a moment uh of the ukidian uh gravitational action. So 1 / 16 pi g uh in uh root integral roo 4 g g g g g g g g g g g g g g g g g g g g g is a Romanian metric that's smooth. Uh so in principle it can have higher derivative terms as well but we'll already get interesting results at the leading order. Uh so this is our ukidian action and the task will be to compute something uh as simple as this object over here. And this is uh in a sense the cosmological counterpart of what people compute when they compute the using ukitian gravity methods. The thermodynamic formulas of black holes which I told you in the first lecture have been computed with great precision and compared to other methods for instance super symmetric methods string theoretic methods adsf methods and so on. So this is the direct parallel if it makes sense as an object in quantum gravity for lambda positive to the kind of calculations we do for black holes. Yeah. Now the proposal then is well let me sort of be more complete in principle you you may have to sum over different manifolds as well. Uh but the basic idea is that this object over here understood in a saddle point approximation is is an object that we should study to get an idea about the thermodynamics of the ditter horizon. So the gravitational this is called the gravitational sphere path integral and I'll give you a warning sign already. We've now entered the phase of the lectures where we don't have a clear uh physical interpretation of this object as we do for black holes. So this is well defined calculationally from the sense of settle point. But I will not be able to offer you uh uh uh an answer physical answer that's as sharp as the answer that's proposed in the black hole literature. Okay, just to be clear. Uh now so this is an however it's an object that has interesting mathematical properties. Yeah, it'll be a function of the cosmological constant of gene Newton. It'll be a function of the coupling constants of of the your matter fields of the higher derivative questions. So it'll be some interesting in the worst case scenario some interesting function. Okay. Now what are the mathematical issues with this object uh which are also issues with any approach to ucleian gravity. Well, one issue uh which will haunt us is called the conformal mode problem which is that very much unlike ukitian quantum field theory where the ukidian action is bounded from below in gravity the action in ukitian signature is in fact offshell unbounded from below. So there's there's offshell configurations of the metric that drive the the value of the action uh to be parametrically low. This is true in any theory of ukidian gravity. It's true for ukitian black holes. It's true from ukidian ads. It's true from ukidian flat space and it's also true for ukitian ditter. The way we deal with this problem in the context of uh of ads and black holes is that we understand this object not as a mathematically complete expression but as an expression that makes sense in a saddle point approximation where we find critical points of the integral and we expand around them to compute small corrections systematically. The basic reasoning being that this uh is presumably an object that's defined better defined in some complexified contour in the space of metrics and that uh this is a particular choice of contour that doesn't make much sense. Yeah. Yeah. >> Uh when you say that there are offshell configurations for which the action is not valid below >> then you said Macowski ads. Sorry, I I I I mean lambda 0 lambda negative theories of gravity with lambda zero theories. Yeah, because we still have to path any rate over metrics there and the offending configurations could be local pockets of the metric that vary very rapidly in their conformal factor and so it's not sensitive to to lambda. You see? Okay. So this is how I'd like to understand this object mimicking the the success of of black holes and ukidian ads. Um and so what we need to find then from the point of view of saddles is sort of configurations and then evaluate their on shell action. So I'll do it for pure GR with lambda positive but everything I say will hold more generally uh provided you have a smooth sort of field theoretic description. Uh so what we need to find are critical points of the ukidian equations of motion. Okay. So what are the ukidian uh equations of motion? So we'll have um let's see R is equal to J MU. Is this the right? Did I get it right? Am I getting it right? So the sign is important. Uh I think I got it right. Right. So so in particular we will be looking for solutions to the ukidian equations to the ukidian Einstein's equations. Yeah. That are smooth and moreover inspired by the force sphere we would that live on a closed four manifold. Yeah. to start with. Yeah, we can look for solutions of the the S4 itself. And well, we've already discussed one solution. So, by the way, if we take the trace of this, maybe I have a minus sign here. >> That's a negative energy. >> Yeah. Okay, good. So, if I take the trace, uh uh these solutions also have a constant positive reach scaler. Yeah. Okay. So, so we want solutions to the equations of motion. I feel I'm getting a s I getting a sign wrong. Yeah. Well, anyways, I'm more confident about this equation. There's something bugging me about this two. Okay, maybe someone can. So, which have positive reach scaler? Now, an obvious uh critical point is the round metric on the force sphere. Yeah. Okay. And so what we'd like to do is compute the onshell action. Yeah. So we need like I said we need the onshell action on uh the what turns out to be the dominant yeah critical point. So in order you you want to find all these critical points and there could be others. In fact, there are I'll discuss them in a moment. Uh but the one at least from the non-critical points that has the most negative action which will dominate yeah in the sense of the saddle point approximation turns out to be the raometric on on the force sphere. Yeah. And so we need to compute the onshell action but r okay taking this expression which I'm fairly confident something off. So so what is on shell the the action that we have? So r is 4 lambda and then we have a minus 2 lambda and so we have a minus 2 lambda over 16 pi g and then we have the volume of the s4. Okay, so it's the volume of the s4 uh in appropriate units. So in units where the curvature of the force sphere is four lambda. And if you do that calculation, so let me just give it to you as a homework problem. Uh if you do that calculation, uh what you find oh so here I've done it for general dimensions. Okay. So what you find is pi l^2 in four dimensions g h bar cubed where I've reinstated the the for the sake of completeness the the speed of light and the h bar and l 2 is related to lambda in the way that I've mentioned before. So the onshell action is equal to the following quantity over here. Okay. So in other words to leading order using this ukidian uh uh path in row calculation and the saddle point approximation we have that the partition function on the dominant saddle is equal to I'll set h bar and c to one now uh is equal to pi l 2 with a plus 9 over uh g or maybe let me write it as so 4 pi l 2 over 4g. Okay. So that's our first calculation. Yeah. And so this is a curious result. So let me repeat the logic. We took some theory of gravity and we path integrator of the fields and we're assuming we're in a semiclassical regime which effectively means that gton* lambda is very small. Yeah. And we're doing a saddle point approximation and studying the dominant contribution to this uh ukidian path integral. And what we find uh is the following equation where we have a formula that is quite beautiful. It combines Newton's constant, quantum mechanics constant, relativity constant and cosmology constant all in one dimensionless number. Very cute. Yeah, pretty nice that you can do that with all fundamental constants of nature. But more than that, if we go back to our static patch picture, yes, uh this quantity over here, even though we were studying physics on a four sphere, which is very abstract. So this object over here is nothing other than the area of the ditter horizon because you have a two sphere with a 4 pi, but it size is L. So it's 4 pi L 2. Yeah. So 4 pi l^2 o a over 4g. Yeah. Okay, that's nice. So out of the blue in this ukidian reasoning, we landed on a ukian path rule which is approximated by e to the a over 4g. That's nice because at least it has some geometric interpretation. And note that we didn't put in a priori a horizon anywhere in our in our sphere calculation. Nonetheless, uh we got something that uh is of interest. Yeah. So this was sort of a motivator. Uh this is in a sense a motivator because suddenly our uklidian physics not only was the one loop stuff for fields propagating on this sphere revealing some sort of interesting statistical mechanic or thermodynamic type expressions. But when you couple it to gravity, the leading piece itself seems to have this flavor reminiscent of the beenstein hawking type formula where now the area is the area of the ditter horizon. Okay, so that's sort of interesting. And to make it a little more close to home, the uh if we evaluated this quantity homework in our own for the measured observed value of the cosmological constant, this uh quantity turns out to be roughly of the order of 10 the 123. This a over 4g. Yeah, it's exactly the number that puzzles us when we think about the cosmological constant problem except in this language it's tied to something that we might uh think about from a totally different perspective which is the the the pututitive horizon entropy Beckenstein hawking entropy of the ditter horizon or the gibbons hawking entropy of the ditter horizon. Yeah. And so there's something interesting about recasting some of the puzzling numbers that we measure in nature in a language that is coming from a different angle. Yeah. So that's one point of curiosity. Yeah. So are there any questions uh at this stage? Yeah. Um I guess he he says that the motivation to go away from PFT in a rigid background was to to derive something like this where s= to a 4g but I guess it can be derived. >> So the motivate sorry no I never motivated the ukidian path just to be clear about the logic I never motivated the ukidian pattern as something that gives us a 4G. The purpose for me was to find some calculable quantity that was as invariant as possible uh that uh that I had reason to believe might be interesting because of the the fluctuations of matter fields appearing thermodynamic. But there was no reason. Of course after the fact you can reconstruct your reasons in your favorite way, but there was no reason for the four volume of a four sphere evaluated for the onshell action to give a over 4G a priori. >> Yeah. >> Yeah. It's an interesting thing that it does. We can cook coke up reasons why it does. But in fact to make the you could ask the you could give me the following complaint. Well, I still don't get it because a thermal path integ a partition a thermal partition function you'll remember from your thermodynamics classes is not the entropy it's the entropy minus the energy over the temperature. Yeah. Okay. The free energy let's say. So actually this seems to be you know a situation where there's no energy at all. And that's already a fair point except for the fact that if we go back to our Hamiltonian constraint in gravity and recall that a priori uh there's no boundary all configurations seem to have vanishing energy and so this seems to be a situation where there is no contribution to the thermodynamic free energy uh uh from anything other than entropy. It's a purely entropic uh interpretation of the black hole thermodynamic formulas which is something that happens rarely for black holes but in particular it happens when you have black holes that are naturally in a microcononical rather than canonical ensemble. It happens for what are known as BPS black holes. Okay. So that's interesting. Uh but now okay so this is perhaps a ukian motivation uh to to think to think further about this this uh ukidian gravity partition function uh that plus the loops of this the the fields looking like Harishandra characters which have an elegant thermodynamic type formula but now let's uh peak a little more delve into a little more detail yeah and so into what makes this problem different Now in the so I want I'm and I'm going to compare and contrast with the black black holes as I proceed. So the analogous problem for the black holes so differs in a variety of ways. So for black holes the manifold on which I study my ukidian path integral is not a a closed manifold. The force sphere is a closed manifold which means it has it's compact and has no boundaries. But for a black hole, we study the problem on on a four manifold whose boundary is given by S1 * S2 and we put boundary conditions on the metric on this S1 * S2. uh and the size of the S1 in units of the size of the S2 is some parameter beta such that the output of of the analogous calculation where I smoothly fill the S1 * S2 with some ukidian saddle uh uh truly yields uh a formula which is in fact uh more standard uh type of uh free energy uh formula where you interpret this beta as the hawking temperature of the black hole. Yeah. So the black hole formula has a little bit more moving pieces. There's a beta and this beta we can use. I don't know who remembers. I'm sure all of you remember that uh to get the entropy from the thermal partition function uh uh what you do is you operate on the thermodynamic free energy with this thermodynamic derivative and you get uh sure enough uh using this saddle point methodology you get a over 4g. Yeah. So here we got a over 4G but we never took a thermodynamic derivative and more than that there was no thermodynamic thermodynamic derivative to take in the first place because there's no tunable parameter analogous to beta which was a deishlay boundary condition that I had to tune in order to be able to calculate stuff. I hope the difference is clear. In other words, in the black hole case, this boundary acts like a natural reference point with a reference clock for which we can measure properties of the black hole in a robust coordinate system. Yeah, it's as if we have a measuring device that at the boundary that we measure the black hole respectively. In cosmology, there is no analog of this boundary. There's no analog of an inbuilt measuring device that we measure the cosmology or the properties of the horizon with respect to. Yeah. So no beta. That's very different. It's as if instead of having a preferred S1 inside of the force sphere along with integrating over all geometries, we also integrate over all thermal cycles which is reminiscent of what we do in thermodynamics when we go from the canonical to the microcononical ensemble. Yeah. >> Is this because observers are outside the black hole but outside the >> You may wish to interpret it in such a way and people are actively trying to do that. Yeah. Good. So now for more differences. Yeah. At one loop. So I gave you the part of the one loop partition function uh in gravity but a more complete calculation uh for gravity or or or higher spin fields in general actually uh reveals that the refined structure of the gravitational partition function. So what do I mean by one loop? Take your sphere. Yeah. And now allow the metric to vary to fluctuate slightly at the Gaussian level on top of the sphere and and path integrate over these Gaussian fluctuations. Okay, so this is a delicate uh calculation. You have to gauge fix properly, keep track of the ghosts, the ghost zero modes, the volume of the residual gauge group that you haven't fixed. Yes, you can ask Sam about details. He's a world expert on this thing. And sort of you have to sort of compute this thing. And in addition to the character expression that I wrote down at the end of last lecture which had this beautiful Harish Chandra bulk and edge terms that reproduce the heat kernel coefficients and violet you know logarithmic coefficients of gravity. There's two additional terms. One term takes the form of five log s0 sorry minus 5 f log zero where I'm going to define f0 to be a over 4g just to comply with notation in some of the literature I gave you. And so there's a logarithmic correction uh that comes from dividing properly by the volume of the residual domorphism group. Yeah. Sorry to just that to be clear. So what do I mean by that is that when I do a path integral for gauge theory, it's important to divide by the volume of the reparameterizations. But if my saddle has an isometry group, I only fix the diffomorphisms modulo the isometries because the isometries don't act yes on my background metric when I do the one loop approximation. So I still need to divide by the subgroup of diffmorphins that happen to be isometries of my saddle which in this case is the SO5 isometry group. And you need to measure you need to compute the volume of this diffumorphism group in appropriately in the appropriate units you see because this has to be derived from a local measure in order for the whole pattern itself to be local. Now those are a lot of details but the outcome of this is that the volume of uh the residual diffmorphism group that remains unfixed at the one loop level when we calculate these these uh uh one loop effects turns out to be minus 10 /2 the 10 is related to the dimension of SO5 yes 5 * 4 over2 um in units of S not which is sort of the natural coupling of the problem and so there's this additional correction to the to the entropy. Let's say if we interpret it like that. And finally there is uh one more uh correction which is uh a phase that appears due to the fact that the unbounded nature of the ukidian action requires us to calculate the Gaussian fluctuations on a complexified contour and in doing so because of zero modes that are present in the problem when you don't have a boundary compared to the problem where you do. These are zero modes that you have to integrate over that you can't fix that are not there for the black hole because you don't because of the black hole they're killed by boundary conditions that you have to retain in your calculation. And so this was a phase that was initially uncovered in a beautiful paper of Pochinsky. Uh and uh so the full answer is much more puzzling than the black hole answer. Not only because there's no beta dependence, but also because of the absence of a boundary. Uh the inclusion of these extra zero modes gives rise to an additional phase that seems to disrupt the naive interpretation of this formula as an entropy. Yeah. And there's additional contributions that that we need to take into account as well that need a you know so now this is interesting. It's challenging and interesting because it means that as it stands both because of this observation and perhaps to some degree because perhaps there were already hints of the fact that we had these edge terms that disrupted a clean trace interpretation of the one loop result. It appears that the more refined version of the proposal of Beckinstein or Gibbons and Hawking on interpreting this as an entry requires some features or some charact further characterization and this is an open problem in the literature currently but a problem that has only been taken more seriously in recent times and the type of proposals that have appeared in the literature so far it's an open problem like I said involve things like decorating uh the sphere with a ukidian sort of world line that breaks some of the zero mode symmetries uh uh that give rise to these phases or perhaps excising from the sphere uh a slightly thickened version of a world line like a little world tube where you put boundary conditions and you try to sort of manipulate the equation in such a way that the leading term is still a over 4g but that the subleading order some of these effects are disrupted and it's an open question. So an open question to see whether uh upon decorating the sphere or or or or doing something to the sphere one can get an answer that's a little more convincingly tied to an entropy. And so this is uh uh a lot of sort of ongoing research. In fact the archive today alone had three papers related to this question. uh where people are trying to to clarify in what sense the rules of ukidian gravity driven by this a over 4g uh driven by other hints in the expression might indeed be a quantum mechanical entropy or something else. So in this sense the problem distinguishes itself for the black hole and it's very much a ukidian counterpart of how cosmology distinguishes itself from a black hole. In cosmology there's a horizon that follows around an observer. Yeah, there's an observer dependent horizon and there's no preferred reference frame. In a black hole spaceime there's a boundary that we use as a preferred reference frame to measure the properties of the black hole. Yeah. So in cosmology there is no god-given temperature or clock that we can measure the horizon temperature with respect to. So there's no reason why correlator should look thermal once you allow for geometric fluctuations. In a black hole the the property of periodicity of the ukidian clock is a property of the boundary which is rigid. So you have every reason to expect that your green's functions will be thermal beyond perturbation theory or at all orders of perturbation theory. So we're starting to see the properties of cosmology in Lorenzian signature manifest themselves from a ukian perspective. I consider this a feature not a bug. Uh but ultimately a successful research if this is a successful research avenue it will have to address some of these issues. And what replaces the hypothesis of the black hole as a quantum mechanical system at finite temperature that's strongly coupled that we measure with respect to boundary. What replaces that framework here to explain all of these detailed expressions? Yes. So that's the ukidian challenge of uh ditter uh quantum gravity and uh to add richness to the problem. The other thing I mentioned in the first lecture if you remember yeah uh is that in effective field theory we don't just have gut newton and lambda we have many many many couplings that we expect to be physical and physically relevant for all higher derivative terms but we so far don't have quantities that are completely invariant that we can define them with respect to the analog of s matrix elements that define for instance the couplings of irrelevant operators in the standard model. Yeah, because you see so could it be that one practical reason to have such quantities is that they code inside of them in their detailed form all of these coupling constants. Now one way for that to happen would be that in addition to the four sphere saddle the round metric on the force sphere there were other solutions to the Einstein's equations the ukidian Einstein's equations uh that uh that also contribute yes because then we could treat the saddle point approximation on each of them would give us invariant information the sphere for instance is a conformally flat space so it won't pick up the coupling of the vile squared a higher derivative correction. But if you had some nonconformably flat space it would yeah so however sadly if you talk to mathematicians they will tell you that uh the only Einstein metric so these are known as Einstein metrics solutions to the Einstein equations the ukidian signature uh on a force sphere topology is the raw metric uh on the force sphere. Fortunately, if we don't restrict ourselves to the topology of a four sphere. Yeah. Uh it turns out that other topologies uh in for closed four manifolds uh do admit uh um CP2. So do admit um uh Einstein metrics. Now I I mentioned this. So it's an a it's an interesting mathematical fact that that we're we may have to tie the the mathematics of closed form manifolds and Einstein spaces to the physics of lambda positive spacetimes. But more practically uh it gives us more data more invariant data because now in addition to the sort of perturbative structure zero loop one loop and so forth around the force sphere which gives us some information that's invariant that's sensitive to all the couplings. I can also do it on CP2 which admits so CP2 is is a different manifold from the force sphere. It has different topological invariance and it's known to admit a metric a smooth metric on it. It's known as the Fubini study metric for those of you who are more mathematically inclined uh which is smooth and uh and uh positive curvature and we can compute loop effects on top of uh this saddle as well. There's a saddle which is given that by two products by the product of two two spheres two standard round two spheres. uh that's another solution to ukidian gravity and then there's a more abstract set of uh space times ukidian space times which is given by taking a cp2 excising k generic points so this is a mathematical operation and considering a connected sum with another cp2 never mind if if what I'm saying now is a little just assume that there's these other manifolds but for those of you who are interested in four manif manifolds. Let me just say there's these connected sums. Uh these are sometimes called delpets manifolds and the number of points that you can remove that admit uh that admit Einstein metrics on top is restricted from 1 to 8. Curiously uh if you go above eight, you can't have positive curvature. And the interesting point I want to make is that when K goes from 5 to 8, these ukidian solutions admit a non-trivial but continuous modulized space of solutions. Okay, which means that you have a continuum of distinct metrics uh on a space a modulized space. So this is the space that parameterizes all these distinct metrics uh that uh that admit different that that are priori would be different saddles to this path integral that you could use to register different coupling constants. So the point I'm making here, and my apologies for delving into a slightly more mathematical angle, but it's motivated by the things that we've been discussing so far, is that there's actually an infinite number of ukidian saddles with different topologies uh that a priori could capture in an invariant way different information of our of our of our underlying theory. Yes. And so at least from that point of view uh perhaps it's abstract but there may be an operational meaning to define uh in an invariant fashion the coupling constants uh of cosmology or at least the lambda positive cosmology because this one will register uh while this one doesn't register the coupling of v squar because it's not it's a vile flat metric uh this one will and so forth and so maybe there's an interplay between how many saddles I have and and the number of coupling constants and to make matters more interesting if I start adding other fields like a Maxwell field to my theory the number of saddles blows up even further which is in line with the fact that there's all sorts of new coupling constants like f to the 4th f to the 6 and so forth yeah and so I don't know if this is the right architecture of the quantum mechanical theory but these are invariant uh data sets that at least in principle can be used to define what we mean by a coupling constant without having to refer to a choice of coordinates or a choice of field variables. Uh that may require us to put in a measuring device or whatnot and and pay the price of having to model that and the ambiguities that that come along with it. Okay. So, let me pause for a moment and see if there's any questions because we moved into a bit of a uncharted territory over here. >> So, I think I I that's what you're saying. So, if you write your action as a sum of a bunch of operators that have different couplings, then what you're interested is like the effective action uh which has some not not so uh conservative like a coupling which would be some function of lambda. >> So the effective action is is a useful machine but in the end of the day field redefinitions will reshuffle uh different couplings of these R squar corrections. So so they're not defined in any invariant way. You could say well g mu I could do a field redefinition g mu plus alpha r mu time you know some function. So you see the problem with the effective field theory lrangeian is that these couplings are not phys yet physical quantities until you tie them to a physical observable and so you'd like quantities that are truly physically invariant. So typically what we do if we have an S matrix is we tie these coupling constants to different elements of a scattering amplitude which is truly invariant you know under all the senses of of invariance. Okay, modulo some issues perhaps with soft modes inside the S matrix that we don't know how to handle properly in gravity in four dimensions. But you see there's something to be said even if it's just sort of a proof of concept about having enough invariant data in a theory to account for what we believe is the totality of its invariant coupling constants. That's the point I'm trying to make. And this modelized space of of non-trivial sort of Einstein metrics gives us at least some operational hope that this is the case. Yeah. So this is a different motivation for the ukidian picture than the thermodynamic motivation which is the one that is currently more actively investigated. >> Yeah. >> The contribution of all these saddles which are not the spheres are they suppressed? >> They're all suppressed. So, so the the second most dominant one is CP2, but they're all suppressed. And what that means is that their action is smaller by a gap. And that means that they're very very suppressed in ukitian signature. Now, what I cannot answer to you, but is an interesting question that uh that G and I were talking about just before is what are the Lorenzian consequences of these saddles? Let me give you a sort of a point in Lorenzian signature. One thing we can do to the sphere is slice it on an S3 and use it to calculate a wave function. What is less but CP2 also admits a topological S3 slice and so perhaps that may contribute to the wave function as well and and so what remains to be answered is just because uh in uklidian space they're small effects whether that remains true if you if you study their effect at arbitrarily late times. Yeah. So I think these are effects that may let's say they may pollute the standard saddle we use to calculate a cosmological wave function and we may want to be concerned about that. Yes. So I think that's an interesting problem and alongside that let me add one more flavor that that distinguishes this problem from the ads problem flat space problem is that oh if these if these saddles are relevant to the physics you will fairly quickly note that aside from S4 which is invariant under the SO5 symmetries at least perturbatively in the sense that perturbation theor of course you're gauging the SO5 but the organization of the perturbations obeys the structure of SO5. None of these other saddles are invariant under SO5. Yeah, they all break the decider the ukidian decitter symmetries. This one has SO3 time SO3 or its double cover. This one has SU3 mod Z3 and so forth. Yeah. And so if these saddles are relevant to the physics, it they may in fact offend the ditter symmetries. And that's an important effect to understand because we have to bear in mind our assumptions as to how perfect the ditter symmetries are. Yes. In cosmology and whether there's quantum effects that pollute them. This is not something that happens in anti-deitter space. So to give you the analogy if you have ukleian ads say you have gr with negative lambda and you have a boundary which is a perfect round three sphere and you fill it in uh with a saddle of the Einstein equations with negative lambda you you can establish that the unique infilling is uh ukidian ads4 which has of course the ads isometries. If there were other infillings with negative curvature other than ADS4 on a perfectly three sphere boundary, they would not have the ads isometries and you would spontane you would have a effect that breaks the the ads symmetries which would be incompatible let's say with the assumption that this theory is dual to a CFD because the CFD doesn't spontaneously break its own conformal symmetry. Yeah. But remarkably, GR makes it much harder to find solutions with negative curvature that fill in a perfectly round three sphere uh than it does finding uh positive curvature uh Einstein spaces on closed manifolds. Yeah, I can repeat these words that I said just said for ads also for Mikowski space. So if you have a perfectly round three sphere and you want to fill it in with a piece of uh solution of Einstein equations with vanishing curvature the infilling is essentially uniquely fixed by a portion of R4 flat a flat metric on R4. Yeah. And if you had found other ones you may have wondered about the fate of the pon symmetries in gravitational physics. Okay. So these things uh may play. Okay. So then there's a question. So the other question this raises is the fate of the cider symmetries in the presence of instant tons or gravitational instant. Yeah. So that's another point that may be worth contemplating on questions. Yeah, >> I had a question that's related to the previous one about separate um about the different being suppressed. How do we do kind of the power counting there just like characteristic or something? >> Oh, so they're suppressed. You see, this isn't a topological theory. So they're suppressed because they have a different onshell action. They're suppressed just by virtue of the fact that their volume is different. However, you're thinking in two dimensions. But here these this is geometry not just topology. But to add to that they also have different topological invariance. So their oiler characteristic is different and there exists a local coupling that registers that for instance there's a gas invariant which is quadratic in R squ that you can use and crank up the coupling to isolate further uh one topology from another. Yeah. So you have both games that you can play. >> Yeah. >> When we say that these other contributions don't have the consider symmetries, is that an issue because this quantity is the analogy of like the transition from consider to something else or >> so I haven't offered you firstly I didn't use the word issue. It's a prop. It's a feature. And also I haven't interpreted for you because I don't know what the physical interpretation is. One possibility like I said is that they contribute uh to the uh quantum state of the world. Yes. So the way we construct uh well maybe just I wasn't planning to talk about this but whatever. Uh so if you go back to the first lecture I gave I told you that one way to think about states in cosmology is through wave functions. Uh uh and the wit sort of had this idea that there's a wave function of the world uh but the wit didn't tell us how to compute such a wave function but and in principle there many wave function but there was an interesting proposal of hard and hawking to produce some wave function. I don't know if it's the wave function of the universe but it's a wave function of some universe. Yeah. And and uh in fact that would have been and their proposal was following the construction of a wave function for quantum field the ground state wave function of quantum field theory or quantum mechanics where we go to ukidian time to select the ground state and we calculate uh the wave function in terms of some data on uh the the surface where we make our measurement. ments. Yeah. Hardland Hawking proposed that the way we should calculate uh the wave function of the world a world a wave function of some world uh whatever a solution to some wheeler to wit equation is by path integrating over a manifold which caps off smoothly like a portion of the sphere. Yeah. pattern your fields and restricting the data at the end of your path and row decorating the endpoint with some induced metric H. Yes. And the claim of Hardland Hawking with some evidence is that this is a solution uh uh to the um Wheeler Dit equations. Yeah. Now if you read hard hawking paper in principle they don't restrict the topology of the infilling. So the infilling most naturally and it's what we typically do is a portion of the sphere and that produces a wave function that transforms nicely with the ditter symmetries at late times and it mimics the bunch Davies wave function for small fluctuations and we all love it and we can play with it but in principle uh you know they could have had sort of other topologies that contribute to their prescription. nothing nothing in their formula restricted the topology. In fact, experience from analogous formulas in black holes tells us not to restrict to a unique topology. And this is a this is a way in which you know maybe if you fill it in with one of these other manifolds, you'll produce it a different wave function. But if you produce a different wave function, it won't be as as rigidly governed by the decider symmetries as the one that gets contributions just from this guy. And you have to prove or establish that their effects are small. And even if they're small, it would be nice to know how small they are and so forth and how they depend on time. Yeah. So this is the this is the the the the only thing I can offer you. uh and going back to the question of two dimensions what I can tell you is that in two dimensions one can actively uh ask this question and actively establish that at least for well understood models such cont such configurations contribute and more than that you could talk to Sam they dominate at late time so it's a little disturbing yeah so we h I I think we have to operate with caution because you know we don't know the rules of this game but in few the few cases where we have access to control models they seem to play a role. Those are my five cents on this problem. Yeah. Other questions? Good. If not uh I think uh we can now graduate to the last part. How am I doing for time? 35. >> 35. Amazing. Okay. So, I I should probably I'm losing my voice. Uh, so I'm going to get a glass of water. >> No, no, no, no. >> Okay. That's called scientific cooperation. Sorry, I'm losing my voice. And so let me p maybe if there's any questions at this point. uh if if not let me okay so let me then move on to the to the final topic which is uh related to the the other point I was making which is that I think in order to make progress on these thorny questions Thank you so much for what? >> Take over. >> You want to take over? >> No, I don't. >> You You want to solve the will the [laughter] >> Thank you so much. That's that's a true display of kindness and generosity. >> I went out of my way. >> Okay, now that I can talk again. Um so my so the perspective to end these lectures uh on a constructive note because you know um yeah to end these lectures on a constructive note. So my perspective is that accompanied by this general analysis and considerations which were motivated from many perspectives and so forth. I think we will have to accompany the developments by introducing controlled systematically calculable models where we can control every aspect of the theory to a degree of precision that our heart desires. The analogy to bear in mind if you will is for example when we study quantum field theories and we look for examples of quantum field theories that are completely solvable or integraable for instance uh the tuft model at large n of two-dimensional QCD or Schwinger's model of quantum electronamics in two dimensions. Yeah. or potentially unequal force superyang mills theory at least in the planer limit. So whenever we study a problem yeah uh or the SYK model for black holes which is a solvable non-trivial model of black hole dynamics. It's a toy model but it's a model that helped us understand the the the dynamics of extreme horizons to a degree that we could and I can tell you this because I lived through the phase transition to a degree that we didn't understand before. Okay. So I don't see how uh a complete a more complete understanding of cosmology will not come accompanied by analogous simple concrete calculable solvable models. And so I'd like to discuss uh an example of what such a model might look like. Yeah. and what kind of features it has and to what degree it looks like the kind of theories we'd like to control and in in what senses it's different. Okay, so that's the final point I'd like to make. I only have half an hour so I won't be able to give you the gory details of the model. So what I'll do is I'll list I'll I'll list first what is a wish I'll build a wish list of what what a model may look like and then I'll give you an example that satisfies some of these properties. Okay the most obvious item in our wishes would be a model that reproduces own known laws of physics without any divergences anywhere and solves the cosmological con. That's not going to happen. So let's let's start let's try again. Uh so I'd like the model to have propagating locally propagating degrees of freedom and I should say this this approach really was developed initially. So so there's works that inspired this approach coming from in fact Pochinsky and and Martinek and others. And from from my side, this started in a paper I wrote with uh theatric molman and uh and Theresa Bautista Solans and uh and then further work with Kara Barco. Uh um and so the so the spirit is to find a a wish list that we seem to agree upon across different perspectives. So I'd like the model to have propagating locally propagating degrees of freedom. In other words, I don't want the model to just be topology. I want the model to have geometry and probes that can measure the geometry in some sense. doesn't seem unreasonable. So I'd like the model to have a parameter analogous to G Newton lambda which as I drive it to zero the model starts to become more and more semiclassical which means that the fluctuations of the metric become smaller. Yes. that the the quantum effects can be controlled uh and that the size of the world in some ultraviolet length scale becomes parametrically large. Okay. So I'd like the model to have a semiclassical limit. uh I'd like uh the the classic so in the classical regime I'd like the model to have in addition to sort of the sitter type uh uh solutions. I'd also like it to have classically big bang and big crunch solutions which are sort of the kind of phenomena that we have sort of cosmological singularities let's say in its classical phase space and quantum mechanically I'd like the wheeler dit equation of this theory to admit uh a rich class of solutions tractable solutions uh that I can identify in the semi-class limit with which with this classical phase space. So I could sort of I can start at linking sort of cosmologies classically described with solutions of the willer the wit wave function. Yeah I I'd like one of these solutions to be the hard hawking wave function. Maybe that's I don't know why I'd like that but I'd like it because it's the kind of wave function we we study often inflation. Um, and finally, yeah, uh, I'd like this theory to have a non-trivial decider entropy. But since I haven't defined for you physically what the decider entropy means, uh what I mean by this is a systematically calculable uh uh two sphere path integral. Okay, which admits interesting loops and uh and so forth. So this is a wish list that you may have. uh for a model and the question is can I find a model the only parameter that I'm not tweaking at this stage uh is the dimensionality of spacetime yeah that's the price we're going to pay in this approach however the hope much like for black hole and quantum field theory gauge theory counterparts uh is that our wish list is sufficiently robust trust that we can derive some insight from these simplified models. Yeah. So that's the perspective. Uh and so that's the price to pay. Oh, sorry. Let me add another one. and and an analog of the conformal mode problem or the unboundedness of the ukidian action because I consider this to be feature not a bag a bug. Yeah. So that's that's a lot to ask for. So now the question is whether we can come up with models that the price to pay. Okay. Are there any questions about my wish list? So now I'd like to present you uh a model that has Yeah. >> I can't hear. the propagating degrees of freedom do you apply to gravity or >> so no the gravity on its own doesn't have prop I said I didn't specify that they're gravitational so they can be local quantum fields however their effect will induce fluctuations quantum mechanical fluctuations of the metric field itself which will in turn produce non-trivial loop effects in these kind of quantities is yeah what I will not have in this model is gravitational waves of the tensor structure of the metric field itself that I won't have yeah and so that's that's the that's the price to pay but I will have non-trivial quantum mechanical fluctuations of the conformal factor of the metric which is where the conformal mode issues appear and the one sector of the metric that appears more tied to this beenstein hawking type formulas. Yeah. Okay. So the model so one model I'm not saying this is the unique answer to this list but one model that appears to satisfy some of these properties not completely solved but it's it's is close is a model where we take a theory. So let me write it in a lrangian formulas. So we'll take the Einstein action uh we where theta will be our sort of Newton constant which is dimensionless in two dimensions and this is really the I'll write it in well let me write it in in ukitian just for the sake of clarity but you can write it in in lorenzian as well I'll write in ukitian because for the sake of time I I I'll mostly report here because there's no time then we can go outside. So the theory will have a non-trivial cosmological constant. Yeah, it'll have a coupling constant associated to uh Newton's constant which of course is not much more than registering topology in this case. And we'll have in addition some matter fields. some matter fields uh which in the simplest instance of this model but this is not a necessity I will take to be uh a two-dimensional conformal field theory whose central charge is equal to C which is a large but finite number for instance it could be 100 trillion yeah large but finite but I'll be I'll allow it to be as large as I so please um uh which has which is unitary so my matter content will be unitary uh and uh the spectrum I will assume that it's a standard compact 2DCFD with a discrete spectrum of states yeah when quantized on a spatial circle okay and so this will be uh the model and the question that we'll try to address. Well, we there will be a Lorenzian and a Ukidian uh set of questions that we'll try to understand. Yes. So, well already we see that it has propagating degrees of freedom. But now what we need to understand it's two sets of questions one ukitian one Lorenzian. Yeah. So the ukidian question is easier to formulate which will be can I compute in exact form uh the following the following partition function at least on a two sphere. Yeah. as a function of the central charge. Um so that's one ch one target for this model. And from a Lorenzian point of view, um can I establish this semiclassical limit and write down an exact expression for the Wheeler the Wit wave function and solve it. Yeah. So these are the two targets that you may have for this model. And so I'd like to report on these targets. Maybe let me give you some references so that this isn't uh completely opaque. Uh 21 06 01665 and 240615271. Okay, so let me give you a flavor of what the solution to this model uh looks like. So, first things first, because the theory I'm coupling the metric to is a conformal field theory, provided that the theory lives on a sufficiently simple topology. Yes. The way that a conformal field theory talks to the metric is governed entirely by the conformal anomaly. Yeah. And so we can already anticipate. So one of the reasons this is attractable model is that the interaction of the the matter sector with a gravity sector is fixed by an anomaly and this is a very rigid structure that doesn't require detailed knowledge of the matter fields themselves. In particular, the the conformal anomaly implies that the trace of uh the stress energy tensor of the matter fields uh is related to the Richie scalar of uh sorry of no CR 24. It's a little sensitive to your your conventions for the definition of the stress sensor with a variation of GMU new or two point but the the conformal anomaly when you place a 2D CFD on a curved space of trivial topology. So the so the violent the conformal anomaly fixes uh the the trace of the stress energy tensor uh in the in in terms of the geometry of your space. So this r is the richy scalar of the uh metric itself. However, the variation the stress energy tensor if I if I do the variation of the lranchian also receives a contribution yes from the fact that there's a cosmological term okay because I have to vary the action uh with respect to these terms as well now this term over here is a topological invariant and the variation of it yields nothing but this term over here which will contribute to the variation of the matter field. Yes. Will add an additional term to the standard conformal anomaly equation effectively yielding an equation uh of the following form. Yes. that the equations of motion yes associated to the metric sector of this theory which come by varying the full theory with respect to the matter fields and the gravitational field and the use of the universality of the conformal anomaly equation effectively fix for me uh the reachi scalar so let me write it in the following form uh that the reichi scaler of the physical metric which is fluctuating in this theory uh admits a equation a solution to the it admits an equ must satisfy an equation of motion that is given by r is equal to 48 pi over lamb 48 pi lambda over c yeah which means in particular that when c becomes very large the curvature starts to get driven down and the size of the universe starts to become very large. Yeah. So we we have an hope to identify a parameter uh in this model uh that that controls the semiclassical limit which is in fact the central it's the size it's the number of propagating degrees of freedom. That's what drives the theory into semiclassical regime. And you can systematically check uh that if you fluctuate around this saddle which in uklidian signature will be the two sphere and in lorenzian signature will be a space of positive curvature that the fluctuations are controlled by one over c effects. Yeah. So the model admits non-trivial solutions precisely because there's an interplay between the matter sector and the gravitational sector and the fact that CFDs quantized on a space uh of of on a closed space have a casemir energy. So another way to see this is that in a theory of gravity the stress tensor the total stress sensor has to vanish the Hamiltonian constraint. And so there's a balance between the positive vacuum energy that we've added and the negative casmir energy of the CF CFD fields that yields an actual uh non-trivial equation which is the simplest counterpart of the Einstein equations you can imagine writing down in two space-time dimensions. So r is equal to uh some constant. Yeah. And if we tweak lambda to be positive this theory admits decider type solutions. Now in addition to these decider type solutions the theory will we can follow the standard and working in the what I what I argued for you to be the semic-class regime we can now set up an ADM type of analysis in Lorenzian signature where we write a general two-dimensional metric in terms of a lapse function and a shift function. I I don't was this covered in the previous lecture? So so remind yourself of your previous lecture. And so we can write a general 2D metric uh in the following form. There's a lapse function. Yeah. That that registers. So the passage the the passage of the time coordinate t there's an induced metric on a constant t foli surface in the foliation and there's a shift uh uh vector that sort of tells us the relationship of one leaf with another as we build our foliation. Yes. And n and nx are associated to constraints because we have two diffomorphisms. Yes. And so these are the the the the lranch let's say at the classical level the lranch multipliers that impose the constraints and I can write down for you in this model uh the exact form of the constraint equations. So for instance the Hamiltonian constraint taking into account uh the matter fields and going to a gauge where n is equal to omega some function of space and time and nx vanishes okay so I'm going to a conformal gauge yes conformal gauge so in this gauge the metric will be equal to omega^ 2 dt ^2 + dx^ 2 and I'll further take x to have unit uh periodicity 2 pi because I I'd like my world to be on a circle. So the the ADM equations, a standard ADM analysis which I invite you to reproduce reveals the following formula. Okay, nice formula uh uh and momentum constraints. Okay. So we have in addition to our to our geometric equation over here we have the constraint equations uh of our two-dimensional world where I should say that I've assumed that my matter fields are in an energy state of uh the flat cylinder uh that is conformally equivalent to my physical metric. Yeah. So em over here minus C over 24 is the energy of the CFD matter fields as per in the sense of them perceiving this uh flat reference metric which is conformally equivalent to the physical metric and so I can talk about energy. So, em represents the exitation of the energy of the matter fields. And now I can uh further restrict on a constant time surface uh I can restrict. So let me try to make this equation more recognizable to you. So I can use my spatial diffios to go to a gauge where my conformal factor is purely a function of time. Yes. Uh I haven't assumed any anything. I'm just using a diffuse on that slide so that I'm allowed to do that because I'm not assuming that the matter fields will generally be independent of space just metric. And I'm allowed to use that tangential diffomorphism to my Koshi surface. And so then my equations become uh c over 24 omega dot squared over omega cubed plus lambda omega plus this energy difference c over 24 pi 1 / omega equals to zero. And in this choice of gauge, I satisfy trivially my momentum constraint. Sorry, the primes indicate derivatives with respect to x and the dots indicate derivatives with respect to time. Yeah. And so I've trivally satisfied this. And the equations I'm left with are equations that you may find familiar because these are nothing other than uh the FLRW equations uh where omega is the scale factor. Well, it's a slightly different scale factor because I'm not I I my metric is omega^ 2 minus dt^2 + dx² whereas in fr rw I would have the omega over here but you can do the coordinate transformation and so these are nothing other than the FLRW equations which really are the Hamiltonian constraint of GR because you have a negative kinetic term from the metric the conformal hydrometric counteracting the positive energy of the matter fields and the cosmological constant to yield a vanishing solution. Yeah. And so this model simple as it may sound uh admits in its semiclassical phase space not only an interesting geometric equation but in fact an equation which is isomorphic to the FLRW equation itself. Okay. Now this is interesting uh because FRW equation uh has nice solutions. So let's see the solutions. Uh okay. So let me report briefly. Yeah. So clearly yeah. Where did this thing go? Where's the stick? Ah where is it? where >> ah my god and we're trying to talk about quantum gravity. >> Okay. Okay. So, let me report on the solutions. And how much time do I have left? Eight and a half. >> Eight and a half. Okay. I'll report quickly. I know people are tired. Uh but it's sort of fun. These are things that you can really just plug into mathematic almost do them without. So let me call this thing this parameter epsilon. So there's three phases to this equation. Uh and remember we also have to satisfy the r is equal to 48 pi lambda / c. uh in addition. So every solution we have locally has to look like DS2 but globally DS2 can distinguish itself because it could be a different quotient of DS2 and that's the control of this parameter epsilon that gives you non-trivial solutions even though they all locally look like DS2. Of course that's at the level of the geometry. The matter fields are in a completely different state in each of these solutions. So they're physically very different. So we have three different cases. Epsilon positive, epsilon0ero and epsilon negative. Okay. And we can solve. So for epsilon negative which is the case where the kazmir energy dominates over the exitations of the field. So it's sort of in the case where the the fields are very have small energy exitation. The classical solution is a kind of uh bouncing cosmology where this parameter epsilon controls the physical size of the throat at the symmetric point and it ranges from the case where epsilon is minus c over 24 which it's the largest and as you increase epsilon it starts to make it smaller which we can view as the effect of gravitational back reaction. In this model, you pump up the energy of the the matter fields and they start to contract the size of the world. In other words, the Penrose diagram of this world becomes more vertical as epsilon is as the energy of the matter fields is driven up. At some critical point, this world acquires a pinch that happens at an infinite past. if as measured with proper time. So this is a quotient if you will of the pankare coordinates of of ds where you take x and identified. So if you do that you get a solution to Einson's equations which have uh in higher dimensions a Taurus but in two dimensions a circle which pinches to a point and here at that point of course the energies the the the the matter fields will be in a particular state. Yeah. And so you have always accompanied with your geometry the state in which the fields the matter fields are at. Okay. And finally when epsilon is negative sorry this is epsilon negative this is epsilon positive excuse me there we have a situation where we have a world where the big bang or big crunch of course these are always accompanied by their time reflection solutions since this is time reflection the equations are invariant under time reflections uh where the big bang happens at a finite proper time and remarkably all of these uh configurations correspond to different coordinate uh transformations of decitter. So this one has a flavor where the spacetime looks roughly like cos squar time some number. This one and and the energy tweak happens in this type of parameter has the form over t ^2 and this has the form over cinch over cinch squ of t. Yeah. And so all of these metrics uh all have constant positive richy scaler. Nonetheless, they're globally distinct and correspond to different physical states both of the metric and of the matter fields. Okay. And so we have a rich solution space in what seemed to be an overwhelmingly simple theory. Yeah. And so this is my I don't know where my list is at this point. H I erased it probably but this would be the fact that we have big bang and big crunch type solutions in addition to decided solutions in our phase space. Yeah. And so already this is sort of looking interesting and I probably run out of time so I'll just say it in words in the last two minutes. So I'll just say it in words. One can now quantize this theory ala willer dit and in the conformal gauge that I'm working in the quantum equations for the constraints uh become equations that govern a very specific type of conformal field theory which is known as timelike leavville theory plus the matter CFD. So you can map the problem of the constraint algebra and the constraint equations to conditions on the operators of a quantum field theory coupled to the matter theory known as timelike leville quantum field theory which is a conformal field theory which is not completely understood. It's a variant of a much better understood theory known as space-like leville theory. Uh however it's believed to exist. It's believed to be a conformal field theory. we have a lrangian formulation of it. Yes. Uh with a wrong sign kinetic term which has to do with this timelike uh pro feature and the fact that there's a conformal mode problem. And so one can map effectively to make a long story short to you the question of whether or not we can axiomatically solve this quantum cosmology with all of these features to whether or not we can aimatically solve perhaps via conformal old school conformal bootstrap methods um uh the problem of timelike leville quantum field theory. And so at least in this simple context, the question of what is what is quantum cosmology uh which is an a question that we'd like to ask maps maps to a remarkably sharp question uh of what is the conform what is the definition of the conformal field theory uh of time like quantum leville field theory which is a remarkably sharp potentially sharp answer to the question of what is cosmology? technology at least in this simplified world that obeys all of the conditions of my wish list. Yes. So the point then to take home from this is that accompanied with our general considerations of horizons and wheeler equations uh will be qu questions of this form and this may be an instance where we can really set the stage for a complete answer to a toy model of what is a cosmology that has the features I just displayed to you. And if alternatively if there's an obstruction to the existence of this theory for whatever reason this may teach us about what obstructions there are to the seemingly innocent assumptions we made about a theory that has local fields a constant positive lambda unitarity and so forth. Okay. And so that's that's an example of what a model might look like uh which people are actively investigating. So, I've run out of time, but uh thanks for listening and we can talk more outside. Yeah. [applause] All right. Great. Uh maybe we have time for a question or two. >> Okay. Good. It is indeed late in the day, but come down, ask your question. >> Yeah, come down. >> Let's get coffee. We'll come back at uh 4:30. >> All right. >> Thank you again. Thank you. [applause]