Quantum General Relativity 101 — An Introductory Survey

0:00 This, as many of you know, this semester we have lots of new people in the center, so what we decided was to have a bunch of introductory talks. We already had two talks on numerical relativity by Pablo and Jorge, and then on gravity waves by Peter Salson and Sam Thin. And the first talk on quantum gravity was given by Lee, and this is the second in the series. I suddenly realized that as I was writing, somehow my sense must have known that it has something to the quantum, because general relativity is already getting fuzzy there. So the aim of this talk is the following. What I want to do is to provide a broad overview for sort of non-experts, by that I mean for people who don't really work in quantum gravity, but assuming familiarity with general relativity, basics of general relativity, and basics of quantum mechanics. This is a general overview talk, and so there are many, many subtleties which are ignored just in order to get across the main picture, the broad picture of where we are and what the issues are. I would say that this is really the thing that I will mention are based on results of lots of people, and this is only a partial list, I think. I just thought of writing, and I apologize to those people whose names are not here. But what I'm going to do is to not really refer to any author during the main talk, and I'll just insult everybody uniformly so that nobody feels particularly singled out. Because, again, the main aim here is to provide you, not really historically, but in some sense logically, a summary of the main issues and where we are. In other words, we might not have arrived at these various topics the way that I'm going to present them today. So I'm going to do, there are four top parts. First, there will be introductions. The second part is the basics of quantum geometry. Again, I will explain what quantum geometry is all about. The third part, I will talk briefly about black holes. There is going to be a talk by Kirill and two other informal teas on Friday and Monday devoted to black holes, and so my discussion will be very brief. And then I'll talk about quantum dynamics and the status of this quantum dynamics, and then I'll end by giving the general outlook for future directions.

2:30 I should emphasize that particularly in quantum dynamics, what I'm going to do is to present my personal view. is shared by a lot of people, but is not universally shared. So I'll explain to you what is, where the disagreements or controversies lie. Again, those working in gravity waves and numerical relativity, this kind of thing happens in quantum gravity because in these other topics, what is exploring consequences of a given theory, and the theory is, well, you know, it's completely given to us. It is Einstein's general relativity, and we are looking for, we are trying to work out its consequences. Of course, there can be disagreements in strategies still, but in quantum gravity, we really do not have a theory at all, and therefore, the disagreements can be deeper than typically of the type, what is the better direction to proceed, that is, intuition, about future directions. So let me begin with the introduction. and I think Lee already mentioned about a little bit about this I think he said something like as he was walking up Professor Jain asked him why do we have to quantize gravity unfortunately I don't remember what his reply was but anyway let me give you one of the standard replies it is that already since the 1930s this issue about whether we should quantize gravitational fields has really been discussed in the literature I found one of the earliest paper I think it started the earliest paper it was 1936 by Bronstein on quantum gravity and the basic reason for this I think the conceptual reason for this is the following one might imagine that one could have classical gravitational fields and for example Einstein's equation could be in the form of a classical Einstein tensor constructed from the classical metric, is equal to 8 pi times Newton's constant times the stress-energy tensor. But stress-energy tensor of math, this is matter, so these are all off-letters. So one might imagine putting an expectation value there. This might be a good approximation in some circumstances, but one might imagine making this the fundamental theory and not quantizing gravity at all. Well, if you do that, from early on, people realize that if you have a theory like that, which is a

5:00 fundamental theory, not just an approximator then there really are problems and in particular you can violate quadrality. These were problems which were already kind of noted in the 30s but in the 1970s these problems were discussed in much more detail concrete experiments were proposed particularly by I think two Princeton students and then a bit later of all places here at Penn State an experiment actually took place to show that gravity should be quantized. I don't think Lee said that. This experiment was performed by Don Pace. It's one of the thought experiments that none of us would actually perform, I think, in the sense that we would all believe. You know, you do something in the lab and does it affect the moon? That is the kind of thought experiment. It is to do with long entangled states and having one field called practical, the other field quantum, et cetera. But Don, being the third person that he is, Don Pace, he was recycling here. He actually performed the experiment, and it was actually published in physical literature. the letters. So, we do know that for very general reasons, we cannot have a hybrid theory like that, sort of, in which matter is quantum mechanical and the gravitational field is classical. Of course, quantization is a rather strange process, because what we are trying to do is to guess a more complete theory, the quantum theory, and what we are given is a less complete theory, the classical theory. And this is a very logical thing to do. The The other way around, there can be some logic to it. But this is a very logical thing to do. And so, I mean, everybody uses this word, and it sometimes looks like there is a crank that you turn in order to turn the classical theory into quantum theory. But we all know that that is not the case. But nonetheless, instead of saying all the time, we are looking for a theory which, in an appropriate classical limit, will reduce the general relativity, we just say we are quantizing the general relativity. So this doesn't mean anything more profound than that. And as one might imagine, in trying to get a more general theory, there of course is a lot of, you know a lot of techniques, so there's a lot of science in it. There's also art in it, because it's really a guesswork at some fundamental level. What is a more complete theory? It's a very educated guesswork, because we have good experience in lots and lots of systems, but nonetheless, at a fundamental level, it is really a... So, this is the problem that we know that general relativity is highly successful classical theory, and we won't write down the quantum version thereof, some quantum theory, which

7:30 in the classical limit goes down to general relativity. And then there are two viewpoints, I think, broadly, which have been there for a long time, but which are now much more sort of in focus. In other words, there are many other approaches to the subject, which in some sense have quite a different viewpoint, whereas these two viewpoints are the ones on which people have, over the years, worked on the most. So the viewpoint number one, which is sort of something that comes from biology physics, is really that the fundamental problem at the forefront is the problem of unification. We know how to deal with electroweak interactions. We kind of know how to deal with strong interactions. And the idea is now only gravity is remaining. And what we should find is a unified theory of all those interactions. In this way of looking at it then, the geometrical role of the gravitational field is secondary. The fact that gravitational field is somewhat different from other fields, in the very geometry of space-time is regarded as a secondary point the fundamental thing is that we know how to quantize my various matter fields gravity is just one of the fields there's nothing special about it really so we should treat it like any other matter field so we begin with a classical space-time background right you begin with you give yourself a classical space-time on which your matter feels that you are trying to quantize that is the basic idea usually the background is taken to be flat, flat, and then we quantize the field. But if it's not flat, then you go back to the picture in which you have to have the dynamics of the background. Yeah, so the idea is that the total metric, you want to write it somehow as the total metric GAB. You want to write it as some background, say a flat metric or some other background that it gives, plus a quantum field. very, very intuitively. So this is really a quanta seal. This is like multiplied by identity. So we're splitting the metric into sort of a kinematical part and a dynamical part. So the idea is that, in the version of string theory, the idea is that there's nothing special about gravity. It's just one of the expectations of strings. That's what we should do. And then the idea is to postpone the fact that gravity is coded

10:00 The viewpoint number two, which is coming more for general relativity circles, is that the fundamental lesson of general relativity is that gravity is correlated to geometry. And therefore, we should not have, in general relativity, when we solve any of these problems, for example, numerical relativity problems, etc., we do not have a background space-time. Space-time itself is dynamical, so the same thing should be true in quantum theory. Everything should be quantum, including the geometrical space-time. So to do that, in the beginning, we literally have no space-time at all, so we literally have to step outside the notion of space-time, so to say, try to learn to do physics without having a background space-time. We are used to it a little bit in classical theory, but in classical theory, at least the end result of whatever you are doing is a classical space-time, whereas in quantum theory, even the end result would be a state which may not be peaked around any classical space-time, which is called a mechanical state, and so at a fundamental level, really, There will be no background space than I thought. But then in this point, what is postponed is to postpone the problem of unification. In other words, we won't try to account for why all these interactions exist and why we have got all these fundamental interactions, couplings, et cetera, et cetera. We will, to begin with, we focus on the geometrical aspects. We look at what would quantum geometry mean? How would you do physics in presence of a quantum geometry rather than a classical geometry? And then we proceed from there. As it's clear, I think, from what I said, I think that both these viewpoints are incomplete. I mean, ultimately, one has to unify all these interactions. I mean, one would imagine that there is a very nice unified description. And ultimately, you cannot treat the geometry as a classical background. Ultimately, we ought to look at geometry fully non-pertrobatively without splitting the metric into a kinematical part and a dynamical part. So these viewpoints are incomplete, but they are true to be useful working strategies. And as the title of my talk suggests, I'm going to focus here on the second strategy, which is developed by General Electric. The motivation for this strategy, concentrate first on geometry, comes from the following.

12:30 I think the most exciting things that general relativity has given us are pure geometry, black hole. It's a pure geometrical object, gravity wave, that refers in pure geometry. So the idea is that these are the most exciting entities and therefore it is sort of natural to see if you have quantum geometry, what do these entities really mean. and the second thing is that second motivation comes from quantum field theory itself if we do ordinary quantum field theory then we all know that there are these ultraviolet problems there are these infinities you might have working rules in order to go around them not to worry about them in actual calculations but at a fundamental level there are these infinities and one of the possibilities could be and these are suggested by Planck actually because the possibility I'm talking about that continuum hypothesis may be wrong was already suggested by Planck in his first paper on quantum mechanics. In fact, Planck, in some sense, was more excited about the fact that with the Planck, you get a fundamental length, and that, therefore, you might get a fundamental cutoff in various kinds of physics. And so in any case, if geometry is quantum mechanical, then maybe the continuum picture we have is only an approximation, And therefore, maybe the problems that appear with the ultraviolet divergences, first of all, they appear, for example, typically in that dimension dependent. They depend on four space-type dimensions. These problems may perhaps disappear if, in fact, you do kind of feel fairly properly, but properly take into account the short-distance structure of space-time itself. It may be that what we are doing is to assume that space-time is continuing at all scales, years, as well as long ones, and maybe that is what is wrong, and if you take it into account properly, there will be, so to say, a very sophisticated version of a cut-off that will be built into this year. So these are the motivations. So now, what are the basic questions? Supposing we buy all this, we buy that we should look at this problem more emphasizing geometry, then what are the basic questions? So the kinds of questions we want to ask you are the following. If I look at this blackboard, for all practical purposes, it looks like a very smooth continuum. But we all know that if we look at it microscopically, it has atomic structure. It is not a continuum at all. It has a discrete structure.

15:00 So the question we are asking is, is the same true about space-time? Is the continuum picture that we have really a coarse-grained approximation like the blackboard that we have up here? What are the fundamental building blocks or the elementary constituents of geometry? What are the atoms of geometry? Are there, if there are atoms of geometry, fundamental entities like that, physical entities, then are there processes in which quantum geometry can interact and emit and so on quantum matter? So this will be the exciting thing that we have, for example, in the atomic physics, right? I mean, we've got quantum states of matter, we've got photons, and then there's interaction. But we all know that you can have quanta of matter, and they can provide for you, make non-trival geometry. So in this sense, called quanta of geometry. That's what happens in a black hole form. In a black hole form, there's a collapse, something collapses, you have some matter field, you have some quanta of that matter, they all disappear, and then you have an event horizon, and outside what is left is geometry, non-trival geometry. Well, is the converse true? Can we have quantum geometry being converted into quantum matter? And obviously the first hypothesis that comes to your mind is, oh, maybe that is what Hawking process is after all. It is conversion of quantum geometry into quantum matter. So does that happen? Then what are the equations of quantum dynamics of geometry? If geometry is a quantum mechanical entity, what are the fundamental equations that go on this? That of Einstein's equation. How do Einstein's equation emerge as h-bar goes to zero? Then, does the quantum nature of geometry cure the ultraviolet problems of quantum field theory? Does it cure the singularities of general relativity? Are the singularities of general relativity artifacts of our hypothesis that the continuum picture works in arbitrarily really high curvature. Maybe you have to continue to picture the very bad approximation when the curvature is say one upon a million plant length squared and then you cannot continue anymore. In other words, the continuum picture may be like a mean field approximation in magnetism. And it is true that the mean field approximation is perfectly fine. I mean, we use it all the time, right, with magnets, all the experiments we do.

17:30 And yet, near the critical temperature, it's very, very bad because you really have to worry about the blocks the splints, in that case, and is really a thing true in general relativity. So these are the basic questions. Now, many of you, of course, this is a confusing point, the next point that I'm making. Many of you must have heard that, well, the general relativity is non-renormalizable, and therefore we cannot make sense of quantum gravity, and therefore we need other things, such as supergravity, superstrings, membranes, etc., etc. The thing is that what we know about gender relativity is that it is perturbatively non-renormalizable. Nobody has any argument that I know of, I mean we fight very hard, which tells you that quantum gender relativity cannot make sense. It could be an artifact of just the perturbation of quantum. In other words, artifact of the assumption that takes time to begin with is all continuum and then you incorporate quantum effects by adding small corrections. Maybe you have to shift your vision, your arena completely from the very beginning and then maybe quantum general relativity would make sense. So that is the viewpoint that is being taken and then one wants to explore the consequences. So last thing in the introduction. Okay, so we're talking about quantum geometry. So what do we mean by quantum geometry? I mean, I talked about atoms of geometry, blackboards, and so on. But in practice, what am I supposed to do? Well, we can just go and look at what happens in ordinary quantum mechanics. We all sort of feel that for the hydrogen atom, in the classical theory, where I got the energy, the constant of motion, like the energy, the angular momentum, and the g-component of the angular momentum. take continuous values, whereas in quantum mechanics, the corresponding operators have discrete eigenvalues. And when this happens, we say that means quantization. Now we can ask the same questions of geometrical observables. In classical general relativity, if I fix a manifold and I look at a curve, or I look at a surface, or I look at a region, like this room, for example, then I can compute the length of the curve, the area of the surface, and the volume of the region.

20:00 Of course, the numbers I will get depend on the metric of space-time, and therefore these numbers are not universal. These quantities are functions on the classical space-space. So, just because these quantities are functions on the classical space-space. So, the question is, can you promote these things to operators? If so, what are their eigenvalues? discrete, and what we will see is that the answer is yes, these eigenvalues are also discrete. And so in this old-fashioned sense of the word, nothing fancy mathematical here, in the old-fashioned sense of the word, we would say that geometry is quantized. Ultimately, what emerges is that geometry has a polymer-like structure. The fundamental excitations of this geometry are really one-dimensional. that if these ones excitations, if these are densely packed, then they can reproduce, on large scales, properties of a three-dimensional continuum. But at a fundamental level, we'll see that the fundamental excitations turn out to be just one-dimensional. Now, there are many difficulties in actually carrying this over, and the typical difficulty is really that the royal role to quantum field theory has been perturbated, That is to say, normally we have a background space time at our disposal, and here we do not have a background space time at our disposal, and therefore we would like to, we need completely new methods, and this is where the difficulty was, and these methods have now largely been developed, and so we can attack these various problems. If you use a perturbative approximation, if you use a background metric like that, and then say that you have got corrections like this, it will be very difficult to see discrete structures. Because if I took, for example, a harmonic oscillator, and you do it exactly, and we all do it, and we know it has discrete eigenvalues. But for some reason, I just chose to say that I'm going to treat the harmonic oscillator as a perturbation of a free particle plus the potential from V of X up here, which is both of X squared. I'm just going to say that I'm going to treat this as a perturbation, as a perturbation of the free particle. But the energy spectrum here is, of course, continuum. It could be very, very, very hard for you. You really are going to sum the whole series and do hard work to see that the spectrum

22:30 here is going to be continuum. And exactly the same thing is happening. if you start with mincastle space, continuum background space-time, continuum geometry, it is very hard to see that you get a continuum picture. So let me now begin with the second part. And the second part is a little bit, in some sense, a longer part up here, which is quantum geometry, and then there are two other parts. And I debated about this slide, but again, this is mainly sort of directed towards the classical relative to people in the audience, so therefore I thought that they are more familiar with this particular language, so I'm going to call them geometry in this language. As numerical people know very well, you can look at general relativity as a dynamical theory of geometry you so you are here the space of all three-dimensional matrix you give yourself a three-dimensional matrix and you give yourself the momentum or the velocity the time derivative of the three-dimensional matrix then Einstein's equation tell you what the space-time is so you can think of the space-time as being a trajectory in the space of this matrix up here and you what you can do is when first one puts Einstein's equation on its own There are the so-called constraint equations in which there is no time dependence, and then there are evolution equations, that metric dot is equal to something and momentum dot is equal to something, and these are dictated in this Hamiltonian form by one of the constraints, which is the funny part of the whole theory, and this constraint looks like the constraint on a pre-relatistic particle. It looks like G alpha beta, P alpha P beta is equal to zero. If this was the constraint, it would be the constraint of a pre-relatistic particle, which, for a photon, for example, it says that its rest mass is equal to zero. In the Hamiltonian theory, if we take this constraint for a pre-particle, and then we go to quantum theory, then we just get the Klingoran equation. Because GABPAPB just goes to the wave operator operating with pi equal to zero. how we quantize our massless scale of T, for example. So here is a similar, namely that the dynamics is given by a constraint like that, where, however, there's a potential term, it's rather complicated potential term, which

25:00 is just the scalar curvature of the three matrix. So for the longest time, people have thought, well, we can go to quantum theory. And the way to go to quantum theory would be that we look at wave functions of these three matrix up here. would be like the Clangoran equation. What you do is, you get the G alpha beta, which is sort of a tensor on the space of matrix. So you take this G alpha beta, and you write this three dimensional matrix. This index alpha stands for a pair of space-time indices AB, or space indices AB. So you just convert this into H bar by R delta by delta G, H bar by R delta by delta G, and you will get this equation. And this is the famous Vila-derivit equation. The problem has been that this equation has uncontrolled infinities. You have got products of dysfunctional derivatives at the same time, same point. And this also doesn't make much sense because in quantum theory, typically the wave functions are supported in distribution and this scalar curvature term cannot be computed on the state of distribution. Now, in order to have, to control these infinities, to tackle them, try to use standard quantum field theory method. But we really cannot use them because there is no background space time. For example, you cannot normal order because you cannot take Fourier transforms on this big space up here. Point splitting is difficult because this metric G up here is indifference. There are all these problems that arise just because we do not have a background space time. So we need a functional, a non-pertensibility functional analysis on this space G up here, And this is a problem that has been opened for a very long time. So what you can do is make a transition. And this, to begin with, was just technical, and then it turned out to be important also conceptually. Instead of working with this phase G and thinking of Einstein's equations as giving you trajectories in this phase G, you can instead work in the space of some connections. These connections are like Yang-Mills theory connections, solutions of Einstein equation as giving you a nice trajectory in this space and this you can achieve by certain means of a canonical transformation so instead of looking at the solutions to Einstein's equations as being that as giving you a dynamical trajectory in the space of metrics up here you can think of it as being the dynamical trajectory in the space of certain

27:30 if you do this then all equations become simplified the important thing is really the following that it literally is true that if I tell a student who knows enough mathematics to know what a connection are and I tell them that the basic variables are going to be this connection and that can only can you conjugate momentum so like electric field in ENM this is really the vector potential, this is the electric field, this is the analog of that. If I tell you, tell this first-year student that, well, I got these SC2 connections, and I got this triage, or I got this electric field, write down the simplest equations you can, okay, absolutely the simplest equations you can, which do not depend, do not use any background metric. So all you have is this thing. So these equations have to be meaningful, that you have to be gauge invariant, write them down. And the first three equations this student will write down are the constraints of Einstein's theory. So this could have been, so to say, just knowing that you should use a variable like that, one can actually guess these equations, these constraint equations. That's absolutely the simplest one. In particular, the analog of this scalar constraint, the Willow-Devitt constraint, now has, there's a metric again on this big space up here, and again, I just get g times the momentum, Here I got momentum as P, now I got momentum as E up here. So again, the potential term has dropped out of here. So in this case, in a very precise mathematical sense, it is true that the solution to Einstein's equations, these trajectories, are the null geodistics of this particular metric that you can write down up here. Again, this is for the classical relativity audience up here. First of all, the constraints are the simplest possible expressions, as I just said. At least, the form of the evolution equation, the nonlinearities that appear up here, are also much simpler than those of theory in terms of metric variables. And the group in Cordoba, with Oscar, Ruella, and others, has shown that this system is already, I mean, the one that just results mathematically here, is already a first-order symmetric hyperbolic system, which is something that is sort of looked for in numerical relativity. here. More importantly, now there is a big similarity with gauge theory. We can look

30:00 at this as a connection of Yangnus theory and this as being the electric field in Yangnus theory. But the physical interpretation of here is that this really is a connection, which is a spin connection, which enables you to pilot transport spinners. And so the internal group of here is, if you like, is associated with spin with rotation. And this object up here are these are triads on the space sign. So these are these orthonormal triads in space, which define the metric G in space. So we can go back and forth between Yang-Mil's theory and general relativity doing this. In particular, one can exploit the similarity. And what has been done over the last five years is that by suitably generalizing techniques from Black's Gage Theory, we have developed, people have developed, a rigorous functional calculus on this space of connection. What is involved is some limits of this lattice theory to get the continuum, and this limit, however, is non-standard. It is not done as one does in gauge theories, because in gauge theories you have got a background metric, and you take the limit as the facing goes to zero, the distance goes to zero. Here you have to take another limit, which is called the projective limit. But this all has been developed, well-developed, and so you now know how to take these techniques over to develop a background independent And this now provides for us a mathematical arena to develop the quantum theory of geometry and to formulate Einstein's equations. So the idea is to work with the space of connections, exploit the similarity, take suitable limits, and that is what enables us to deal with these uncontrolled infinities we saw in the Vila-Devitt equation. So now, let me just give you the summary of the final picture, the summary of the final picture is the following. In the quantum theory, at a fundamental level, the fundamental excitations of this connection that we have got are one-dimensional. This is not very surprising to people because these are the flux lines of the electric field, if you like. The analogs of this in E and M are really the flux lines of the electric field. These are the one-dimensional sort of fundamental excitations like that. These are flux line lines. But these excitations, if you like, these are 1D excitations or nothing. There is no background space-time. There is no background matrix. There is no eta eb. So you just have this space and you have this fundamental excitations. And these excitations themselves

32:30 give rise to geometry. If these excitations are in some semi-classical state, you perceive a continuum picture around you. If these excitations are very, very condomechanical, then you don't. Look at this light right now. These excitations of the electromagnetic field up here. But this excitation happens to have lots and lots of photons, therefore I can approximate it as a classical electromagnetic field. But I could also have a state, I can guide you to another state, like in the last colloquium, photon and two photon and so on and those are extreme quantum mechanical things which cannot be visualized as classical space as classical electromagnetic fields the same thing is true here you can have expectations that are around us which are lots and lots of with we can have expectations with lots and lots of lines superimposed in certain ways then it looks like a classical space time but on the other hand you may have these expectations fundamental elementary ones, just one excitation like that, and that would not look like anything like a classical space-time. That would be a pure quantum mechanical geometry. So, here the fundamental excitation are one-dimensional things, are one-dimensional excitation or nothing, rather than what happens in this picture, perturbative picture, where the fundamental excitation are gravitons, which are wavy undulations on flat space. In this particular picture, gravitons arise only as a proximate concept. The elementary excitation, these lines that I talked about, they correspond to flux lines of area. So, for example, if I have a surface, I have, right here, for example, let me here do each for this, so here, if I have elementary excitation like that, and if I take a surface, and if it intersects this excitation, it endows the surface with an elementary quantum area, elementary quantum of area, basically some constant square root of 3 by 4 or something times Planck-Lenz squared. So that means that in this room right now, I got, look at the surface up here, and this surface we think has an area of, say, 10 square centimeters. What does that mean? That means that there are going to be flux lines of geometry, of area, passing through here. And how many are there going to be? Well, 10 square centimeters is about 10 to the 66th plank line squared.

35:00 So there are going to be 10 times that, 10 to the 67th flux lines of geometry passing through the surface, giving rise to the geometry of the surface, giving rise to the area of the surface. Please. Do you think that tapers would maximize the number of flux line areas? I could orient that tapers in order, but in the, I mean, I just asked it in a, so let me just start telling you how to come at the answer. I'll tell you the answer. The answer is no. The answer is that the flux lines are, in order to reproduce a metric around us, these flux lines are going to be, the distribution of these flux lines is going to be pretty random. They are not in any one direction. It's pretty isotropic. If this were not the case, then the space around me would not be isotropic. On the other hand, if I were in a suitable third space-time, where the space is not isotropic, then the flux lines are indeed frequently in one direction as opposed to other, and then the statement is that, yeah, if I fill some surface, then indeed the area would actually change. What are you thinking, if you have the word below us, is that somehow a non-hike problem? Yeah, I think. Okay, so you're asking the question. I was actually saying that this space is approximately flat. So you're saying that, let's take it into account. So the answer to your question is yes. The answer to your question is yes. But this can be just worked out by just taking short-term geometry. And just look at some paper, some coordinate patch up here. Okay, you fix your paper by adapting coordinates to this paper, and then the statement is that in Schwarz's space-time, if you take it into account, the gravitation will feel a blur, but you're saying it's true. So there are these semi-classical states which are weaves, which are called weaves, because the idea is that, just as my shirt is made out of, looks like a continuum, but it actually is woven because of various kinds of fabric, various kinds of threads, the idea is that the space itself is like a fabric and it is actually woven, and the threads which weave it are these fundamental excitations of geometry. So these classical states are called these. And they're obtained by densely packed elementary excitation. The continuum picture is recorded only in the semi-classical states.

37:30 And finally, there are operators L hat c, A hat c, and V hat. So these are the length operator associated with the curve, area operator associated with every surface, and the volume operator associated with any region. You can construct them rigorously, and you can show that they have a purely discrete spectrum. So in this sense, geometry is quantized, in an old-fashioned sense of the word. As a quite sociologically interesting remark up here, this work resulted through a joint effort of a lot of people and working on rather different lines of thought. But they finally converge to produce for us this consistent picture. The subject of quantum gravity brings together, so to say, sociology from particle physics or high-energy physics, and also sociology from general relativity, and sometimes there are interesting interactions between them, which at least at the beginning can look like crashes, but at the end typically look very nice for the origins of ideas. So when this was done, for example, I mean, there was black and full information coming all along, and when this was done, for example, there are many papers which are written more in the particle spirit, particle physics spirit. What do I mean by that? I mean, in particle physics, we don't have a theory, typically. You know, you're looking for a theory. Like, we didn't have theory of electroweak interactions. We're looking for it. In general relativity, we're always secure. We have this Einstein theory. So we could say, be careful. can work out its consequences. We could take our time to publish papers. We could do all those things, not an Einstein series. But in the particle physics, we don't know what a correct theory is, so the sociologist always publish quick ideas, and those ideas which survive, which are good, will survive, and then go into details with those ideas. But first, you publish lots of ideas. And in this publication, you will see the same kind of scatter, you know, sort of particle physics type scatter. There is an idea that is published. It may be slightly wrong. It may be incorrect. Then you correct it later. And then you will or more relatively circles, which are slow, which wait until the whole picture is clear, and then the papers are published up here. I just want to say one thing. I think both are very, very important, and there is a really good convergence between the two things. However, I just want to say one thing, namely, at some level, unlike in particle physics, there's a difference in particle physics. At some level, we really, really need to have a good mathematical control and really know

40:00 that our results are solid. What do I mean by that? In particle physics, even if some result is approximate, maybe you don't understand the fudge factor or something, somebody goes and makes an experiment, it works, you know the idea is right, you can go ahead and then worry about the fudge factor. Here, this is not going to happen in the near future. And because it is not going to happen in the near future, you have to be very careful. Give an example. In the beginning, people found, I was involved, one of them, but people found that, for example, for the area, we found some eigenvalues and these eigenvalues were discrete and we were very happy that eigenvalues are discrete so that shows that area is quantized but then a bunch of people were worried again including me that maybe there are some discrete eigenvalues but there's a whole continuous spectrum as well this is exactly what happens for hydrogen atom we have got bound states and we have got a continuous spectrum if this happens for for the area operator that there are there's also a continuum path, then we would not be so sure, because maybe the physical world lives in the continuum path and not in the discrete path. So the care was needed. This is really to graduate students and postdocs who are not familiar with this area. Care is needed to show that the set of eigenvalues is purely discrete, not that there are discrete eigenvalues, but the whole space is really discrete. This has been done. Now, just some details of quantum geometry, and then... So, to begin with, we have this connection, which are like the positions of particles. I mean, the analogy in particle mechanics will be like a position, there are configuration variables, and there are these electric fields of trials, which are like the momentum variables. So, heuristically, just as here you consider wave functions of x, here you could consider wave functions of the connection, and just as you have p to be h bar by i d by dx, you could consider the electric field operator operating on psi to be equal to h bar by i delta by delta x. It just turns out that for purely dimensional reasons, the points of brackets between a and e is not one, but it was a gravitational constant. And, therefore, instead of having the h bar by i, you just get h bar g upon i, delta by delta e. But h bar g is just a Planck-Line squared. I'm using c equal to one unit. So the electric field is Planck-Line squared. On the triad operator, it's Planck-Line squared upon i, then delta by delta i.

42:30 These ideas can be made precise, thanks to this functional calculus. So there really is a Hilbert special state of square integral functions in a real sense, in a mathematical sense. There is a decimorphism invariant measure on the space of connections, and you can consider square-integrable functions with respect to that. And now what happens is the following. These elementary triads that are operators corresponding to these elementary triads, that naturally they arise as two-dimensional operators, or technically two-dimensional operator value distribution. In other words, you have to smear them with some test functions on a two-dimensional surface. It's really like the flux, if you like, of this operator through a surface. This operator is completely well-defined. If you try to think of this operator by itself, it has distributions because there is flux quantization, and therefore this looks like a distribution up here. So these operators, which are smeared on two-dimensional surfaces with some smearing functions, are self-adjoint operators with purely discrete eigenvalues. And this is the basic geometrical operator, just as triads in classical Riemannian geometry are the basic objects. Once you know the triads, I can construct the metric, I can construct curvature, I can go on. So similarly here, the triads are the basic operators. Once you know them, you can compute the geometry. So let me just focus on the area operators and see what happens. So first of all, all eigenvalues of the area operator are known in a closed form. The simplest eigenstate that you can find out, which is by itself very nice, that you have got a closed form for all eigenvalues. The simplest eigenstates have eigenvalues which are given by some fundamental constant times n, where n is an integer. So these simplest eigenstates look like level spacings of a harmonic oscillator. They are uniform. These simplest eigenvalues up here. And what they correspond to is, you have the surface, we're looking at the area operator associated with the surface, and these eigenstates just correspond to flux lines, which, so to say, don't do anything funny at the intersection up here. They just go through. Because you can have flux lines with multiple intersections up here. Don't look at that. look at the simple intersections of these flux lines and each of them then gives you

45:00 this quantum of area multiplied by n and then you will get the area associated with the surface. So these are the simplest functions. However, if you look at the complete spectrum, if you look at all eigenvalues not just the simplest ones then it turns out that they do not have equal spacing That's why I'm going to do it. It looks like a technical thing. But they do not have equal spacing. But for large eigenvalues, in fact, the eigenvalues crowd, as A increases, the crowd. The level spacing goes to zero rapidly. And in such, it goes to zero as the exponential of the eigenvalue itself. So for large eigenvalues, the level spacing is going to zero very rapidly. These eigenvalues were plotted, for example. I think that's . We just put a computer program to run, to calculate them, for a hundred of them. These are all in Planck-Land units. So up to one Planck-Land squared, there are nine eigenvalues. But to go to two Planck-Land squares, you already have 99 eigenvalues. So there's really crowding, even going from one to two, thinking about the large number of eigenvalues, like 10 to the 67 that we're talking about up here. is also a plot of these eigenvalues that we made and what is plotted is really again this delta a the difference between eigenvalues multiplied by eigenvalues and you can see that there really is this level spacing is going to zero very very rapidly up here actually like an exponential the continuum limit then is up very rapidly You might wonder about the following thing. I'm telling you all these properties. I'm saying that, well, look, the level spacing up here, naively you would have thought that the level spacing is uniform. These are the eigenvalues we had first found in the very, very, very first time we looked at it. But the statement is that the real eigenvalues are not like that. They all crowd. You might say, but who cares? I mean, this is all at the level of plant plant. If they crowd very close to each other or they are separate, what difference does it make? And what we'll see is it makes a profound difference. And this will be one of the applications that I will talk about. So let me summarize. I began with the introduction. And then I told you about quantum geometry. The fundamental excitations of geometry

47:30 are one-dimensional. There are these flux lines, like the lines of the electric field. The interpretation of the flux is the area. And there are global spaces of states, and operators well-defined. And you can show that the eigenvalues of geometric operators are all discrete. This is the message facing you up to now. Now what I want to do is to turn to the third part, which is black holes, which will be applications of these ideas. And we will see, already in this case, these delicate things about how the eigenvalues behave matters quite a lot. Is that the behavior of the eigenvalues that you suggest that the areas aren't quite added? Eigen values, it's not. Any operator's attitude. Eigen states are not attitude, right? Is that correct? That if you had to join in the three-team area, then what is funny happening? Well, maybe we should talk about that. No, it is additive. I'll be much more careful than anyone said that. I mean, the statement is that these surfaces are always supposed to be without boundaries. Maybe you're not worried about that at the moment. But if you take that, then the statement is that additive in there. Okay. Okay, so now, why black holes? Well, when you see this black hole thermodynamics was developed in the 70s, and what was found when this was developed is something very nice, very beautiful, that when you look at these black holes and you study their properties, then, first of all, they bring together general relativity, quantum field theory, and statistical mechanics. But not only do they are brought together, but in fact the overall coherence of the theory, you know, for the theory to be meaningful, they had to be brought together. You don't have choice, so to say. All the three theories are needed in order to get a nice, consistent picture. For example, the entropy of the black hole is given by the area divided by Planck length squared up here. G h-bar is Planck length squared. And similarly, the temperature of a black hole is given by surface gravity, which on the surface of Earth would be g, times h-bar upon 2-bar.

50:00 See, if you did not have Planck's constant, then you could define an area, but there is no way to get a dimension less quantity out of this area, and therefore you could not construct entropy. The same thing is true with the temperature up here. Furthermore, as you can imagine, in the classical general relativity, you would think that the entropy of a black hole is infinite, because you can pump into it any amount of matter. All the information is lost inside of that black hole. Indeed, h-bar goes to zero, this goes to infinity. thing the black hole is dark which it is classically it's completely black therefore the temperature goes to zero all those things happen up here right and then with these definitions the laws of black hole mechanics are the same as laws of thermodynamics just as you have got de the change in energy is equal to TDS so that's the work done you have got your change in the mass of a black hole is equal to the temperature times delta entropy plus work done where these are the formulae that come up here. So this is a powerful hint, and a lot of people have sort of taken the point of view that one should use it. And in this particular idea, this particular program, what is in fact using it rather strongly? So first of all, there's an attractive possibility from quantum geometry, namely that perhaps we should think of Hawking radiation as conversion of quantum of area into quantum of matter. So, this idea can be checked. Is this something that can work? And this already tells us something very interesting. So, let us understand that. So, the idea appears in the following. That you have got a black hole, and the black hole is evaporating quantum mechanically. quanta of matter. As it emits this quanta of matter, the black hole shrinks. So the idea up here is really what happens in atomic transitions, namely just as here, the black hole is going to emit one quantum, it's going to shrink a little bit, lose one quantum of area, say roughly speaking, it's going to emit a little more, it's going to shrink. It's going to happen in these discrete steps like that. And in each jump like that, it's going to emit a photon and we can look at what the spectrum spectra of this radiation looks like that's what that's what we can do so now the statement is that the um what is the end point result the

52:30 end point result was that hawking did a calculation without having this quantum geometry hawking did a semi-classical calculation using a background black hole space-time and using quantum fields on that and found that indeed the spectrum, the radiation of a black hole looks like a black body and the temperature is roughly given by one upon mass of the black hole. Having this and having the idea that perhaps the simplest eigenvalues of area are the, are a surprise, assuming that the area quanta up here really behave like area eigenvalues behave like harmonic oscillator, you have got equal spacing, Beckerstein and Mukherjeev realized that there is a problem, and the problem was the following, but if in fact the area has equal spacing, so it goes like n times Planck length squared, let me put Planck length equal to 1, g equal to h bar equal to c equal to 1, so the area goes like n, you know where n is an integer, Now, in classical generativity, the area of the horizon is mass squared, and therefore, in one of these transitions, if the area changes by delta A of the horizon, then mass changes by delta A is equal to 1 upon 2m times delta A h. I'm just taking delta H equal to 2m times delta n. For this, on the other hand, if I just, the smallest, if I make a smallest transition, then delta H, the change in the area, would be one unit, so n is equal to 1. So in the smallest transition, I will have the delta m is equal to 1 upon m, and since h bar is equal to 1, this is really, this is the energy, and that will be emitted in terms of frequency. is, is in the smallest possible transition, you will get a frequency which is comparable to one upon n. That's what we are talking about up here. In other words, what we are concluding is that in this spectrum, if in fact the spectral lines were emitted, right, because of this transition, then the smallest, the first spectral line you will get is here. We would not obtain anything to the left up here. This is an exaggerated picture, but just to sort of show, the first line would be omega naught, second line would be 2 omega naught,

55:00 third line would be 3 omega naught, etc. Because these are the jumps. The harmonic oscillate, if you like, goes through one energy level, two energy levels, or three energy levels. So you will just get these kinds of spectral lines, and they look very, very different from the spectral lines that Hawking told us would result by a semi-classical calculation. So there will be a major deviation from Hawking's blackbody spectrum, even for large black holes. Now, not for microscopic black holes, but even for large black holes. But fortunately, the actual spectrum does not look like a harmonic oscillator. This level spacing, in fact, goes to zero very, very rapidly. It is bounded by exponential of minus the area. And therefore, if you put this up here, then you will find that the spectral line, the arise here but arises way out here so that time the minimum energy it can emit is 1 upon 2 and then delta H which is less than or equal to omega naught the fundamental thing the fundamental thing that we have the spectral line times extremely small factor so already the spectrum begins here and furthermore since the level spacing are also small you will get many many many many spectral lines and therefore you will get the continuum of consummation So, I want to emphasize that this is a very simple illustration of the power of the thing, in the sense that there really are checks, and you don't have that much freedom. You might say, well, at Planck scale you can do what you want because the scale is so small. No. If you do something, it can have ramifications on large scales, even for large black holes. If, in fact, my area spectra really were uniform, I would contradict Hawking radiation, and therefore semi-classical gravity and quantum field theory in curved space-time for large black holes. There is a real tie-in here, which is very, very interesting. So there are sets of quantum theory of geometry. What's wrong with quantum field theory in curved space-time? Because for large black holes, you don't expect. You expect quantum field theory in curved space-time to hold. For small black holes, where pure quantum gravity effects would be important, that there will be deviation. But for large black holes, it's like, to contradicted counterfeasal series curve space time, for large black holes, would it be like contradicting, I mean, the next step of contradicting is the neural relativity. There's no reason. There's no physical reason.

57:30 Absolutely. In this case, we have, suppose in fact we're contradicting hawking and supposing then one is going to say codophyll theory can curse this time is wrong the whole branch of thing goes out of the window I agree it's not an experimental verification but it's a very very strong statement I agree that in principle everything could be wrong but at the moment we are looking at sort of consistent things the next thing that I was going to tell you was about I don't find the appropriate oh here it is The next thing I was going to tell you was about black hole entropy, but I think Kirill is going to give a talk about black holes, and therefore I will not go into great details about it. I just want to tell you that this whole idea of quantum geometry enters black hole entropy in a very important way. The next topic is really this quantum dynamics. What I will do is the following. I will actually summarize, because it is 2 o'clock, I will actually summarize, and then, in other words, I'm just skipping one topic, but I think I would like to talk about these topics to people who are in Gaiwichi group up here. So I will summarize, I'll take some questions, we can stop, and then I'll quickly go over this topic about condom dynamics, just to tell you my viewpoints. So let me summarize. Basically, the statement is that this is an outlook and direction, and this is just a general preliminary remark, which I think I already made, that I feel that this is not a direction, well, this is kind of a direction, I feel that since we do not have pressure from experimentalists, we have the luxury of being reasonably careful, and it's not perhaps necessary to be mathematically rigorous because somebody else can do the work later on, but it's really important to be mathematically careful. For example, in the black hole entropy formula formulations, it is very important to make sure that you are talking about black holes and not about arbitrary, in other words, it's important to make sure that whatever formula you write down are not valid for situations in Minkowski's Pasta or situation in a Schwarzschild star as opposed to Schwarzschild black hole. And this kind of precision, this requires a lot of thought, a lot of precision.

1:00:00 I mean, it is much more tempting to sort of just write down the framework or theory in which gives you the results that you want, but it's sort of important to know for physical reasons, not just for making it beautiful and precise and mathematically elegant, but for physical reasons, it is very important to make sure that we are really talking about what we want to talk about, procedure is really very important again if we had experimental tests then maybe one could be sloppy one could check if something works and then worry about being precise but would not have this experimental test and that's why okay so what are the problems well this really comes from the last part of the talk that one of the one of the major directions the major directions are better formulation of the Hamiltonian constraint of general relativity and making sure that there are enough semi-classical states. Another direction, second direction, this goes from more former theoretical development to having more and more physical content in them. The next thing is looking at so-called spin forms that Lee already talked about. I think what is very interesting about spin forms is that It brings together ideas from very diverse things like regicalculus, topological condom field series, and condom gravity. And I think there is some very nice underlying picture, and it would be really nice to explore this in detail. But also, if you look at these Euclidean skin forms, I think they might actually give us a real handle on the problem of completing the condom gravity program. one would be able to take this Osterwalder-Schalder approach in which, if you know the Euclidean theory, if you know the measure in the Euclidean space, then you can actually get the Hamiltonian, the Lorentzian theory. And I think there really is a good chance that this approach can be married very well with the spin-form ideas in order to actually tell us about the dynamics of the Lorentzian theory. And again, we talked about these Minkowski spin-forms. And I personally do not see how these Minkarskian spin forms can really lead us to the quantum theory, because I do not know any analog of that in any other field theory.

1:02:30 If you have measures, if you have part integrals in the Minkarskian domain, I do not know how to make sense of it in any other theory. What we do is you write down the equilibrium part integrals, and if the measures satisfy some properties, then you get a satisfactory Lorentzian theory. However, this Minkowski spin phone, I think, would be very interesting because it would give us this wave state, these states which approximate on a large scale geometry of the space-time around us. The third topic along the same line, becoming more and more less formal in some ways, would be to look at quantum field theory on quantum space-time. So you can have appropriate weave states, either these weave states or weave states that come from the canonical quantization programs and try to do quantum theory on this quantum geometry. Just as we did quantum theory, quantum field theory, on per space time, the next thing is to look at quantum field theory on quantum geometry. Because the fundamental excitations are one-dimensional, the effective dimension of the space-time dimension And that might well cure the ultraviolet problem. The question is, does it do that? If it does that, then there will be a very nice indirect support for the microscopic structure of space-time that comes out of these approaches. There is a whole bunch of problems about black holes. Just let me take a couple of minutes to talk about them. First of all, I think we need a better understanding of Hawking radiation. There is a partial understanding of this conversion of area contact to matter contact, but there are still open problems there. they need to be solved. Then about black hole thermodynamics, I think we should understand black hole thermodynamics from the perspective of full quantum gravity now. Now we have, there's a lot of work, there's a lot of film. The framework is ready, so we should really understand this, not only classically, not only quantum fields in interspace time, but from the perspective of full quantum gravity. There is something weird about black hole thermodynamics, and there's one aspect that preserves a lot of people that I just want to tell you. So, we have a formula which says that the entropy is equal to area in some units. And this is a bit strange, because normally, entropy of a gas, or entropy of, you know, the black body radiation, or of the spin system in magnets, et cetera, that entropy is obtained by suitable coarse graining.

1:05:00 In that sense, it's a little bit fluffy, you know, exactly how you coarse graining, and so on. And furthermore, in any case, it's a function of stakes. density matrix, rho, or a probability distribution rho on the face space, and you take trace rho log rho. It's a function of the state up here. Whereas area, on the other hand, even classically in geometry, classical general relativity, area is a geometric object. It's not fluffy at all. It's very precise. It does not involve any coarse grading or anything like that. And it is really observable of the face space of general relativity. It's not a functional state, but it's really an observable in its own right, and yet the two are pretty equal to each other. In quantum theory, where an operator corresponds to this area, and on the other hand, there is no operator corresponding to entropy. It would be very strange to have an operator corresponding to entropy up here. So, this puzzle, that this is really a functional state, this on the other hand, I shouldn't say puzzle, but unease, something funny, this on the other hand is an observable in classical theory, goes over to quantum theory, in which this is a Hermitian operator, and entropy is, of course, of the operator. It could be that we are trying too hard to mimic ideas of statistical mechanics as they are, and perhaps a suitable generalization of these ideas is really needed to look at statistical mechanics from the point of view or perspective of quantum gravity. Then there is the issue of these dynamical processes. You know, what happens? Most of these black hole thermodynamics, classical and quantum mechanically, is obtained in stationary case, and one would like to understand what happens in non-stationary cases, and in particular, one would like to derive the second law in quantum gravity, area increases, and secondly, one would like to address the issue about the information loss, which has to do with dynamical process. And finally, we'd like to ask, are there some more directly observable predictions of quantum gravity? I don't think it's a healthy attitude in which one says, well, Planck-like is so small, so we don't worry about it. I think we should keep our eyes open all the time. And these could be analogs of E equal to M-square phenomena of special relativity. After all, the speed of light is very large, and therefore normally you think special relativity as being important only when the particles have extremely large velocities. And yet in nuclear reactions, you know, nothing has very large velocities. you get this profound effect like that, and it may well be that there are similar things in quantum gravity,

1:07:30 the analogs of E equal to mc squared, respect to relativity, which really don't involve very tiny scales and don't involve very large energies, and yet some funny thing happens, for which there is a consequence. For example, there are models in three space-time dimensions, in which matter is coupled to gravity, which are, there are completely, fully non-perturbating effects, which go away in perturbation theory completely, which are the following peculiar properties. Because of the specific non-unite coupling between matter and gravity, small fluctuations, quantum fluctuations in the matter, can be transmitted and become large quantum fluctuations in geometry, even in relatively same situations. happens in three dimensions, and this is very, very well understood. One of the questions is, are there analogs or something like that in the full theory? So this is the direction that I want to tell you about. And finally, I just wanted to, again for the 101 level, I just wanted to say that there are two references. There is one on our central webpage, an article called Quantum Mechanics of Riemannian Geometry, which is roughly at the, slightly more detailed, but roughly at the level that I talked about here, and then there is a paper in the Los Angeles Archives, which is also addressed to graduate students. So let me stop now, and maybe take some questions. Yeah. The entropy is not defined that way. I mean, it may happen... Close now is part two of Abhay Ashtkar's talk, which was given later to a smaller audience. further 18 minutes and 23 seconds of the second part of the talk.

1:10:00 some few words that might, more than what I did already for Pablo's question, it might not really explain things. Okay. So this is a basic question that we all solve here. And now the quantum geometry is well-suited in the sense that this is an important point that I want to make, that in ordinary general relativity, we really have differential geometry to formulate our theory. Of course, you may not believe in general relativity. You may believe in higher derivative theories, or you might believe in supergravity, but still, differential geometry gives an arena to formulate that, and this is a dual role of quantum geometry. The first role of quantum geometry was it was giving us observables, quantization of area, etc., etc., but the dual role is that it also is a substitution in quantum theory of the differential geometry, and basically it is an arena which provides us to write the quantum gravity theories non-quantumatively. It could be general relativity, but, for example, it can be supergravity. I don't want to say that you cannot write any quantum theory of gravity in this particular way, but there is a large class of theories which one can write. Now it comes to, let's go to quantum constraints. There are seven constraints, and six of them are related to three-dimensional symmetries. The diffeomorphism group and the gauge group, the U2 gauge transformation, the frame rotation transformation that you have, 3 plus 3, that gives you 6 up here. And then there is a scalar constraint, I'm awarded to call it Hamiltonian constraint, because of this temptation. Once you call it a Hamiltonian constraint, you think of it as Hamiltonian, which it is not, because it is equal to 0. The constraint says something is equal to 0. So you've got a scalar constraint, or a real-and-webitt constraint, which is 1. and the status is that this has been solved these constraints have been solved rigorously there are no infinities nowadays we see this as being a trivial thing but it is not entirely trivial there were papers in the early days worrying about infinities that would come up if you try to solve this constraint there are also papers which say that there might be anomalies just with the difumorphism constraint for example and the statement is that there are no anomalies encountered and the symmetry group that we have the semi-direct product is listed to quantum theory

1:12:30 And the very nice result up here is that the physical states up here are then closely related to the northern graph invariants, and this in particular is something that Jorge and Gambini have been working on. Now, here, for the Willow-David equation, there is a proposal for defining this quantum operator he had. In this proposal, due to Thomas Thiemann, there are incentives, which I think is, again, a technically non-dual thing, and Thiemann has also been able to introduce matter fields in this thing. So it looks like a good news, but there is however, and the however is, as Lee pointed out in his talk, if I take two scalar constraints, two Hamiltonian constraints, and if I take the commutator, you find that's weird with some lapsed fields, NO1 and L2, then you find that this is just identically equal to zero, whereas in class, from classical theory, you expect that the commutator should be a diffimorphism constraint. That's what you would expect from the classical theory. So the action of this constraint C at N on elementary states, so this is the first point, and the second point is, this is what Lee was talking about, that there are these elementary states which have to do with spin networks, and the action of this constraint on the elementary state appears to be too local. As Lee said, there is no this terminology because this is a constraint and therefore things of course don't propagate even in linearized guy which is truncated second order if you write it as a wheeler david equation truncated to second order you don't see any propagation there i mean it's the constraint equation but i do understand the point that it is too local because roughly speaking um what the constraint does at one vertex of this graph doesn't relate to what the constraint does on another vertex of the graph. So these are controls. There's no question about it. And so the summary is basically that theory in this sense is almost complete. There's a consistent theory. There are no infinities. But the problem is that it is unclear if this theory admits any semi-classical state at all. Why do you say complete or almost? Almost. It's not complete, right? Because of the... I mean, if you take T-man saying find the inner product on the space of physical state that T-Man gave us. If we have this physical inner product, then one would say that, well, this is the correct

1:15:00 inner product for following reasons, and then I got a Hilbert space, and I got this, whatever the operas it has. There's an issue about interpretation. It's the issue about getting physics out of it, but then the theory will be complete, but we don't have the inner product yet. Not on the space of the solutions to the Hamiltonian constraints. I mean, they then really belong to the dual space. There, it does not have enough power. Yeah, there's enough power before you take it to account. But this looks like an extremely interesting thing by itself in mathematical physics. But this is a very important thing. I mean, you know, we do not know if that's in the classical space, so normally people are not used to quantum mechanics at the world, and you can't have something which is completely irrelevant. I mean, that may not be the case. And here are two examples, one from elementary quantum mechanics and one from gravity itself. From elementary quantum mechanics, we begin with X and P operators, in XYZ, PXPYPZ, we construct angular momentum operators LX, LY, LZ. We compute their commutators, and then we try to find representation. And when I find the representation, I find both integer angular momentum and heart integer angular momentum. However, if you wanted to have the operators X and P well defined, in other words, if your L was X cross P, and not just L, which satisfies the commutation relation, LILJ is equal to IJ bar XL and IJK LK, if you wanted L to come from X and P, to look like the classical theory, particle moving, classical space time, if you wanted orbital angular momentum, then your integer values. However, these are there and these are very important, very interesting. These have no classical limit because if you try to construct them by quantizing a system, this system would be, the same space would be compact, it is not quotasable on anything and therefore there is no classical limit. but nonetheless, you get this. And it is quite possible that this pure quantum sector of Thomastis, the sector of Thomastis quantum in this cell. It doesn't have any classical limit. In fact, in 2% gravity, the same thing happened. In fact, in the loop representation, the first representation that was written down, first quantum theory that was written down, corresponded to precisely technically time-like factor, and this time-like factor is precisely the one which has no

1:17:30 classical limit. So the statement is that already in 2% gravity, there are sectors of the theory, which are nice and semi-classical. You get lots of states, which are, you know, But there are also states which are not semi-classical. There are states which are purely quantum, so to say. And it's possible that Thomas's thing really belongs to that sector. However, the conclusion that has been drawn sometimes, and even in print nowadays, is that this shows, this problem shows that canonical quantization fails, and we have to look for something else. And that something else could be, for example, in form or something else. And with this, I strongly disagree. What do I mean by this? Why do I strongly disagree? We'll see in a minute. Just to qualify, I do believe that we should look for other directions, particularly the spin form. But I think the major motivation for me to look at these directions is really that canonical quantization is very, very inconvenient to use in practice. It is very difficult to get physical predictions, analyze physical phenomena in canonical quantization, because it's a frozen formalism. I mean, at least you let it de-parameterize it, so now I'll try to talk about time evolution, etc., etc. It's a frozen formalism, and therefore it's going to be very difficult to use in practice. And secondly, looking at spin-form is also likely to give us new ways of defining the scalar constraints. This is really related, for example, with what Mike and Carlo have done, making suggestions for how the Hamiltonian constraint should be modified coming from a spin form. But also, it is related to this Osto-Wall-Oshada thing. I mean, maybe one can actually really do spin form and can actually show that there is a measure on the space of the spin form appropriately set. And then the statement is that there really is a Hamiltonian theory, and then one can go and look what the factor order you get there. If you want to, you can go and look there. But however, my belief is that our best chance to establish the existence of a satisfaction in non-predibility quantum theory is still in canonical colonization. It's always, you know, there's a tension here, namely something may be easy, something may be well-suited to show existence of a theory, and something else may be well-suited to extract physical predictions, develop approximation methods, etc. In particular, my belief is that spin-form methods are just so far are just in the somewhat preliminary stage,

1:20:00 and so they just haven't encountered the difficulties that we saw in canonical quantization up here. They're going to get exactly the same difficulties. There are going to be problems about defining measures, and these problems of defining measures will be really similar to the problem of getting a satisfactory regularization of the constraint. And the last thing that I see is that I want to give you my reasons why I believe what I just said. So what I believe is that, yes, I'm uneasy with the features of T-MUN construction, and I do feel that one should work in other directions, Frazier status by no means says that genomical quantization is doomed or is stuck or one has to abandon it, one has to do something else. What are the reasons? The first is that something surprising, to me it was surprising, and I never thought about it until this T-Mars thing came up. First is that the vanishing of the commutator is not necessarily an obstruction for consistency. In other words, the fact that the commutator doesn't reproduce the classically expected that the theory is inconsistent. I can construct similar situations in finite dimensional examples that are of course not very natural because I constructed them artificially in which the classical constraint algebra is enclosed but I just factor all suitably so that the quantum algebra closes and use that quantum algebra so that the quantum constraint the commutator is equal to zero. Let me start with that. I can have examples in which the classical parts of brackets of constraints not zero, but I can factor all of them in such a way that the quantum commutator is equal to zero. And yet, the theory is perfectly satisfactory, and for some very simple reasons that I can explain why this happens. Yeah, it is, everybody would agree, yeah. No, no, I should make it more. In fact, it is a theory that everybody would agree is but more importantly the same thing happens in 2 plus 1 gravity let me just take two minutes I would hope that this paper would be this small paper would be finished we're not going to write this but since this claim has been made so many times I think it's worth writing a small paper

1:22:30 I would hope that this would be actually ready by now but it is not what happens in 2 plus 1 gravity is the following we can take T-mount construction and apply it blindly step by step, completely step by step. What you find is, when you do that, there are many, many solutions to the quantum constraint T-hack operating on size equal to 0. There are lots and lots of solutions. Again, you look at the simplest ones, which would come from spin networks, for example, and you find that they are really too local. But indeed, they are actually unphysical, in the sense that in the 2% gravity, we know what the correct norm is, so you can go and check the norm, norm is actually infinite for this solution. And the physical solutions result from infinite superposition of this simplest solution. And not only physical solutions can be obtained this way, but all physical states can be obtained this way. So it really is true that I can take Thomas's prescription. Of course, I am using hindsight here. I know how to solve 2 plus 1 gravity, but I take Thomas's prescription and I can follow step by step and actually recover what we all agree and what has agreed for a long time as being the correct answer in that particular theory. What is happening here is this analogy that you might, for example, find in ordinary quantum mechanics that these very simplest solutions, if you like, which are two local. In ordinary quantum mechanics, you may actually find by by solving some differential equation, you come across plane waves. And then you say, well, these are, all I'm getting is plane waves. You know, plane waves is not physical. And I can say, well, okay, I can take one plane wave, I can take two plane waves, I can take three, I can take four. You take finite number of them, you superpose them. They're not physical. Clearly I'm doomed. My method is wrong. I should go. But we all know that is not the case. It's just that the plane wave basis has its particularity. And, for example, you can take the spherical waves, and you can expand them in terms of plane waves. So this function necessarily involves an infinite number of terms. You can't stop it at a finite distance. And that is what is happening up here. The physical state can be actually obtained by an infinite supposition of the simplest So my reasons are, first of all, the constraint algebra by itself is not an obstruction. The second is that the physical solutions in 2 plus 1 gravity, the demand description completely works and we can get the correct theory.

1:25:00 And finally, my reason is that I do believe, in any case, that there are probably better regularizations of the scalar constraint. And, for example, Rodolfo just gave us a talk on trying to give us a better regularization of this Hamiltonian constraint by going through a cilia wing there, etc., etc. So I certainly think that there are full directions to look at. So to summarize, I don't mean to say that all is well and easy and everything is fine and Thomas has solved the problem of Hamiltonian constraint and we have quantum theory of gravity which is satisfactory. No, I'm not saying that. There is a quantum theory which is almost complete, which has lots of good mathematical properties, but I am uneasy, however, and not satisfied with the status of the scalar constraint in this theory. However, there is no hard argument against canonical quantization per se that I know of. The arguments that I see some qualitative fractions are similar to the argument that we have all seen, which says that you should not do quantum general relativity because it is not renormalizable. These are two-state things that is not renormalizable, but that doesn't tell us anything about possible existence of a consistent theory. I feel we are really on a similar footing up here in terms of the canonical quantization. It is true that Thomas's construction has this peculiar constant algebra. but it is true that it is too local and the two things are related. All these things are true, makes one uneasy, but it is not a water-type argument. Of course, ultimately, however, it may turn out that people were right and that for some reason, quantum general relativity does not make sense. But it may turn out, and there's always part of it, but not for the reasons that people are given. Similarly, it may turn out that one really and really canonical quantization doesn't make sense. Of course, that can always happen, but not for the reasons that I have heard so far. That is my sort of basic view of the subject about dynamics. There's lots of interesting work to be done. Why is there not more people doing it or something? I think the real reason is because conceptually it's very important to establish that the theory exists, but if one completes a canonical quantization program, still it doesn't seem to be so easy to do some practical applications of this Finnish theory. And I think that is why there is reluctance in actually putting a lot of work in it,

1:27:30 which is sort of sociological, which is understandable, etc., etc., but that is what is happening. And the very last remark is that why is it, I mean, there's a lot of progress on the last few years, but why is it, for example, a lot more people talk about this, for example, why is it that everybody is so excited about anti-deciple space, time, and not about job matrix or something like that. And I think one important reason, there are many reasons, but one important reason is I think the mathematics one uses here is very unfamiliar. It doesn't come from standard quantum field theories, conformal field theories, etc. that people have developed up here. And that always is an obstacle, both psychological and technical, about getting into these areas. And that is why it seems like pin networks just give us a basis in a Hilbert space. And from a purist point of view, you know, basis is no big deal. I mean, there are many basis, and this is one basis. It was a big deal. But it's very interesting because once you formulate it as a full network, you need to make connections with other things. It makes the theory transparent to other people, working in topological kind of field, etc., etc., and I think that's why this is important to make this connection. Okay. If there are further questions, I'll be happy to take them. on this more at some stage and anywhere there are going to be three different stations on black holes Thank you.