Quantum Gravity

0:00 I have a Hilbert space that can describe the space of states of my computer here, and a Hilbert space describing the space of states of this projector. To get the Hilbert space for the combined system, I just take the tensor product of those two Hilbert spaces. So it corresponds to setting side by side. In n-cob, it just corresponds to setting two spaces, or spacetime, side by side. So if I have a space consisting of a circle and another space consisting of two, I can set them side by side here and get a space consisting of three circles. Here I decided to make it a little more interesting to also describe how you set the morphism side by side. You're not able to do that in all these examples, but I didn't bother talking about it in verse two. so here I actually drew a morphism and another morphism and then the result was sitting side by side and in this case the unit object is the empty manifold so the idea here is you could imagine a universe which changed topology some other universe that changed topology and then if those were solutions to some kind of theory of physics you had you could hopefully side and get another solution. Of course, that's not terribly interesting if they don't interact with each other at all. It's just two things going on in parallel. But things get interesting when you have processes where you start out with two things sitting side by side and then they change to something else. So actually people learn about these things in electrical circuit theory as well. Electrical circuit theory can compose circuits in series and So, composition of morphisms is really series, and this is more like parallel. Now, here's a big difference. The tensor product you set is Cartesian in a specific technical sense, while those in the other two categories aren't. So, of course, the product you set is called the Cartesian product. It's a set of ordered pairs. but category theorists have abstracted what's really interesting about being Cartesian and they make the general definition of what it means for any category of tensor products

2:30 to be Cartesian and this is that definition so we say that a monoidal category is Cartesian when for any particular object, x there is a morphism from that object to x tensor x it depends on what object it is by a subsequent x under this delta, but I just did keep the notation clean looking. This we think of as duplicating data. And we also have a morphism that deletes data, meaning for any object, for any object x, there's a morphism to the unit for the tensor problem. So in the world of sets, we have this. So in the deletion, is there one delta, for every x there's a specific morphism delta sub x and i was lazy and now i'm paying for my laziness i didn't write the dependency of this on x so for any object there's a specific morphism delta sub x of this sort and a specific morphism e of x of that sort yeah and in the so in the world well I'll describe what this stuff means in a minute but in the example of the category of sets what this morphism does is it just sends any element of x to the ordered pair consisting of that element comma itself so we would send the number 3, the real numbers to the pair 3 comma 3 in r squared pairs of numbers so that's what we use implicitly whenever we're duplicating information in our calculations So implicitly, when I'm writing several copies of the same number down in a calculation, I'm applying this morphism. And similarly, I could just map any set to a one-element set. There's a unique way to do that, and that accomplishes the need of throwing out any information I might have about a specific element of that set. And what we require is that certain diagrams commute. In other words, here, if I start out with a set, I map over, I'm going to illustrate with the example of set, because that's sort of easy to understand what's going on. So I'll call this a set, but this is a general concept. I map to the set times itself by this duplication, delta, then I throw out the first ordered

5:00 pair, using this deletion, E, so now I'm in the set, one element set across my unit set. Well, because that's the unit for the tensor product, that's isomorphic to the set I started with. So I can go around this way, and so what have I done? I've taken an element of this set, duplicated it, and thrown out the first copy. Well, I'm left with the second copy, which is just what I, like the thing I started with, So I could have equally as well just used the identity to take that element of my set and not do anything at all to it. So we want this to commute, meaning we want this process of going around this way to have the same result as going correctly that way. And similarly, the other way around, I could duplicate and throw up the second copy and keep the first, and actually the same as not having done anything at all. So the category of sets and many other categories have this special property, and they're called Cartesian. And this is really basic to our normal way of thinking about mathematics, and indeed life in general. So for example, every academic department has a little room consisting of a Xerox machine whose purpose is to duplicate data. And then right next to it, of course, there's always a wastebasket, which allows you to delete the data, like when you screw up in making your Xerox copy. So those ideas are very intuitive to us, and so we think that in an ideal mathematical world, an ideal world, you can take anything, put it in a Xerox machine, make two copies, and throw one out, and you have done nothing. And that's what these diagrams say. So it came as a big shock to people when they discovered that in the category of Hilbert spaces, in the monoidal category of Hilbert spaces, you cannot do this. You cannot find morphisms of this sort for every object to make these diagrams cube. And it was, that result was known as you cannot clone a quantum. So there's no Xerox machine that you can stick an electron in with a given spin and out pop two with the same spin. notion, and you've just said that. The notion, the partition notion depends on X. The deltas and the e's depend on X. Whereas in the quantum

7:30 case, the impossibility of negative theorems are for a universal machine. A universal clothing machine. It does not depend. No, no, no. X is not a state here, remember. X is a Hilbert space. In the case of Hilbert. X is an object in my category. So in the case of Hilbert spaces, I'm saying there's no operator from a Hilbert space to a tensor itself, which has the property that you could duplicate, which would act as a duplicate operator for all states in that open space. So it actually is universal in that sense. So people tend to emphasize the inability to duplicate quantum information, but it's actually worth pointing out that the parcel of this notion of a Cartesian category is this counterpart, deleting information. And it's also equally true in quantum mechanics that you can't cleanly delete quantum information. So in other words, there's no machine you can see of, even if you're allowed to make machines to implement any linear operator, not necessarily unitary operator, which does take a quantum system and erase the information about its state. Of course, people are less interested in doing that so they don't emphasize that. But you see here that in a Cartesian category, those two concepts are linked. Nor is n-cob Cartesian. So that is, if you have a general n-minus-one-dimensional manifold, there's usually not any n-dimensional space-time which starts with that one manifold and duplicates it, and ends up taking you to two copies of that manifold, which has the property that if you that there's not some way to map one of those copies down to the empty set to find a cobordism down to the empty set and let the other go on its merry way in such a way that this whole space-time is the same as one of those cylinder space-times so in the world of cobordisms it's sort of visually evident if you look at a few examples that it's not Nobody regards that as shocking or surprising, and yet we do tend to regard no cloning as shocking. But I claim that's because we tend to think of quantum theory as, by analogy, sets, category

10:00 of sets, which I'm saying we shouldn't do. Another big difference is that both NCOB and HILD have duels for objects, and this is It's part of why a process called quantum teleportation seems high. So we say that a monoidal category has duals for objects. And for every object, it's from x. There's some other object called the dual of x, x star. Together with morphisms, this e here has nothing to do with the e on the previous slide. Sorry, I didn't use the same letter. going from x star tensor x to the unit and conversely the other way satisfying a couple of identities which I'll draw in a minute called the zigzag identities. So in the world of Hilbert spaces or even vector spaces this dual is just what you call the dual Hilbert space or vector space and the dual vector space has the property that there's a linear map from a vector space tensor and its dual which applies a linear functional to a vector and becomes a complex number. And this is just sort of the same thing that turned around backwards. So that's where the notation dual comes from, by analogy with linear algebra. But in n-cob, the dual has a different meaning. It's just the same manifold, the same space, but with its orientation reversed. I didn't draw a little arrow saying which work way around the circle. It's clockwise, so if you don't see much S star, this example looks just like S. But these maps, the bordesans, in some states, E and I, what they are is they're just a result of taking a cylinder and pulling it over so that it either goes from two things down to none or from none down to two. So geometrically, when we say a monogular category has duals for objects, mean is that you can actually think of a warpism as like a pipe with an input and an output. And you can actually bend that pipe around to turn the output into the input, or turn the input into the output. Yeah?

12:30 You don't actually need orientations for this. This works in a category-oriented. Yeah, it would work without orientations. And then I shouldn't have even said with orientation reversed. They are those things that if you had an orientation reversed. If you had an orientation, putting a consistent orientation on this cylinder would make it come out, this one's going around, this circle was clockwise oriented, then the other one should be counterclockwise oriented. But that's not the essence here. So the zig-zag identities look like this. They look like the ability to scrape out some zig-zags. So there are identities which say that the I, the E, and identities can be composed and tensored in such a way to get something that's equal to an identity point. But I hope that the reason for being interested in them is geometrically obvious. If these were made out of flexible material, you could just stretch out the left-hand side and get the right-hand side. So in covert spaces, the category of covert spaces, the dual of a covert space is the usual thing. The space of linear functionals on your original covert space. As I already mentioned, E has the effect of this, evaluating a linear functional on a state of getting a number. There are different ways of thinking about I. about it is it takes each number and it's really just a turned around version of this but one way to think about it is to think of this as the space of linear operators on h and then i sends any complex number to that number times the identity operator and then a good exercise you can do if you get bored of the rest of my thoughts is just check that these satisfy the zigzag So both the category of cobordisms and the category of Hilbert spaces have duals for objects, but the category of sets doesn't. There's no such thing as a dual of a set that has these different properties. So duals seem strange to us, and in particular, Avramsky and Kierke have shown that quantum teleportation relies on the zigzag axis. On teleportation, a very sketchy description of it goes like this. You have a particle marching along. You create a pair of particles in a correlated state, a bell state.

15:00 You let one of those interact with your given particle. And then the other one will have acquired the state of the particle you started with. So this other one can be very far away from the one we started with, which is called teleportation. Of course, you actually have to move these particles next to each other to make it work, so you don't really get to communicate faster than life this way, but it's still somewhat surprising. However, it becomes a lot less surprising when you realize that the underlying mathematics that makes it work is the zigzag identity and that, in effect, what you're doing to transfer the information from the computer over here is you're letting it flow through a zigzag shaped pipe metaphorically. What happens if you try to define the dual of a set as a set of functions of this set? Functions from that set to what? To error, for instance. Well, then it just doesn't work. Then you can take one of those things and another one of your set get a real number to pop out, but R isn't the unit for the Cartesian product of sets. The one element set is the unit for the Cartesian product. So the point is that R isn't a special favorite set, just in the context of set theory. So in summary, quantum theory seems counterintuitive if you expect Hilbert's bases and operators to act like sets and functions. Actually, it's more like n-com. So that suggested some of the things that seem weird about quantum theory will seem less weird if we think of operators as more being like spacetimes. And of course that's great for people who are trying to do quantum gravity, because people trying to do quantum gravity are indeed trying to somehow marry the concepts of quantum theory and general relativity. So I think that this is an important clue to understanding quantum gravity. it's a clue that you've actually have implicitly followed. So who got this clue first? Well, perhaps Feynman was the first one to get it because he was the first one to start using diagrammatic technology to describe linear operators between Hilbert spaces and start manipulating these diagrams as if they were geometrical or topological entities when actually he was reasoning about linear operators

17:30 and these are quantum processes. And in fact, nowadays we understand, some of us actually understand Feynman diagrams in the context of this monoidal category approach that I've talked about. Penrose picked it up with his work on spin networks, which are really like simplified Feynman diagrams where all you care about is the spin of particles. And he was doing similar things. So both string theory and also spin-phone models are trying, I believe, to exploit such clues. So I think they're looking for some language to talk about quantum space-time that will blur the distinction between pieces of space-time geometry and quantum processes in a useful way. So in string theory, the fine-new diagrams now look higher dimensional, and we can reason with them as geometrical entities, are being used to describe, from one point of view, they're being used to describe operators from the Hilbert space of the two strings here, Hilbert space of the two strings. Similarly, in spin-pull models, although they're different, and the discrete flavor, the reason with diagrams like this, they describe operators. So the way I think of it now, partially because I'm so frustrated with quantum gravity, think of either of these approaches to quantum gravity as really a predictive theory at this point, but just sort of an attempt by people to try to reconcile their ideas of quantum theory and geometry, and explore the mathematical possibilities, and hope that maybe someday we'll get some theory that has predictive power. So one last thing, which I'll sort of whiz through, is to try to, I've been trying to on with this relationship, and I think it really goes back quite usefully, it's quite worthwhile under matrix mechanics. So Heisenberg, you know, invented matrices. He didn't actually know about matrices when he invented matrix mechanics. His thesis advisor told him, oh, you invented matrices again. But he just was thinking, a quantum system like an atom can be in some set of states, And then I want to describe, give a complex number which describes the amplitude for it to turn to some other state, given some particular process, and I could write those numbers in an angular array, or maybe more easily as a graph like this, not more easily, but more geometrically as a graph like this.

20:00 Well, a matrix of numbers, if you think about what that really is, in the case of a matrix of natural numbers, a very nice way to think about that is what we call a span of sets, a set mapping down to two sets, the finite sets. So the idea is these would be the indices for your row, these would be the indices for your column. If you pick those two indices, you could look at how many elements map down to both of those, and that would be a natural number. So you'd get a matrix of natural numbers this way. But you get other, more sophisticated matrices by somewhat more sophisticated spans. For example, spans of measured spaces give you general operators between Pilgrim spaces. And what you're really doing in a path integral is doing that. So when you fix, say, a position here and a position here, and you integrate over the possible paths that go from here to here, to get a number, which is now not just a natural number, but it's going to be a complex number, and that would be what you do to get a transition amplitude from a path number. On the other hand, cobordisms are similar, but they're what we call co-stands. You have this space-time here now with the two ends, the beginning and the end of your space-time, getting included into that. Now, for category theorists, this business of turning around arrows is a very natural process. So, category theorists' spans and co-spans same thing. And it turns out that there's a general theory of these spans and co-spans which you can think of as a generalization of the theory of all these things that I've talked about so far path integrals, cobordisms and matrix multiplication and so on. So I think that that's the right way to sort of find the common ground between cobordisms and linear operators. So I've been trying lately to understand scan physics from what I call a Spanish perspective to find a framework that nicely unifies the two sides, the quantum and the space-time. I think I'll stop there, so I'm ahead of time. If you require the theory to be supersymmetric,

22:30 then the limit cannot go arbitrarily far. There's a maximally symmetric theory, and I put the point field theories here, just to distinguish these theories from the extended objects, string theories. In the framework of field theory, I mentioned the maximum asymmetric theory is a theory called supergravity, and if you allow yourself to drop it, I mentioned the maximum asymmetric theory is 11 minutes of supergravity. So these theories were constructed some 25, 30 years ago already. They've been known for a long time. And at the time, people hoped that they were good enough to solve the problem of divergences. Well, of course, it's extremely difficult to prove disprove the fact that these theories are convergent or not. And the expectation was that the theories are actually not finite, despite all the cancellations that one could achieve by opposing supersymmetry. So this has led to the answers that perhaps you should give up the point field theory idea and replace and start quantizing extended objects. And, well, this has led to the development of super string theory, and then the maximum is the magic super string theory, so you think that the space-time dimension is 10. And beyond that, you can even think about quantizing two-dimensional surfaces, not just strings. There's a maximally symmetric theory called super-membrane theory. And that is also something called matrix theory. Well, string theory is a very big field. And let me just list some of the successes. Indeed, so far, no one has discovered any divergence in theory, so it seems like a good candidate for solving the problem of non-biscibility by some theory. However, if you look a little more closely, it turns out that the finiteness of a possible, possibly true to all orders, so far has only been proven vigorously up to two groups.

25:00 And for the more difficult extended object theory, super-membrane theory, there's actually, so far, no finite misresigned. One advantage of the divided string theory is that it predicts gravity sometimes. When quantizes the closed string, one of the excitations, the massless excitations, is a particle that has precisely the properties And in lowest order, string theory is able to reproduce the interactions as predicted by Einstein's theory. Of course, string theory then has much, much more. It has all different tolerant, excited states. So gravity is naturally embedded into a much bigger theory. String theory is also able to predict a, what I would call, semi-realistic spectrum of low-energy physics. It predicts gauge bosons. Gauge bosons come out of open string as open string states. It predicts chiral fermions of the type which is compatible with chiral fermions of the type as they appear in the standard model. And therefore, one can try to start building and trying to extract a standard model from string theory. Well, one of the big successes is that string theory, at least for certain types of black holes, has been able to give a microscopic interpretation to the patients by parking area law, according to which the entropy of black hole is suitable units given once this famous factor of 1 over 4, and string theory is able to derive this factor of 1 over 4 for certain types of black holes, and it's female black holes. They're not realistic black holes, but they're theoretically very interesting black holes. Well, if string theory is true, there's a real Bernardo for experimental physics in the coming years, namely we expect to see lots of new particles, electric particles, new kinds of forces, like extra bosons, and perhaps also new dimensions.

27:30 Sometimes I think about what would be the most spectacular discovery that LHC could make. I think the most spectacular thing it could discover is not extra particles, but the fact that our universe has more than just four dimensions. I think with this, we could not only also be able to persuade all kinds of funding agencies to invest more money into high-energy physics. Well, and of course, string theory has had an enormous impact on pure mathematics, as measured in fields' metals. If you look at the list of field-medalists in the last 20 years, it was probably about roughly one half of the field metals were given for work in connection with directly or indirectly related to string theory. So it seems like a big success but there are many failures or at least open questions. Well, first of all that's a projection that's always mentioned by the other side we have so far no background independent formulation of string theory. It's much like if Einstein hadn't known about Riemannian differential geometry, if he had just known that gravity is described by a master spin 2 polygon, he'd have known the cubic interaction vertex of these gravitons. And from that, to deduce the full theory without knowing Riemannian geometry is extremely difficult, if not possible. And this is the situation where we know in string theory. know the three-graviton and four-graviton vertices, but we have no idea what the geometrical, mathematical framework is for that theory. String theory at the moment also cannot say very much about this question of what does space time look like at the smallest distances. It has difficulties in reproducing the node cosmology. in particular if there's a positive cosmological constant supersymmetric theories and superstring theories are very much very much prefer a negative cosmological constant so the fact that the observed cosmological constant looks positive

30:00 is really an embarrassment there's a question of how to break supersymmetry and then something that's been much discussed about lately is the question of of making any prediction, because the Sphinx theory allows for at least this main consistent, quote, unquote, vacua, or can replace vacua by universe. So some of our colleagues have tried to turn this embarrassment into a virtue by saying, well, this is really the world we live in. It's not just our universe, but there are zillions of unobservable universes out there. I can tell you that this is hotly debated, and I, for one, am not a believer in this scheme. I would rather find this rather depressing if this was really the final answer that you cannot explain things, but you simply say, well, things are the way they are simply because only in our universe conditions would allow for our existence. So this has been termed string landscape or even a swamp. A swamp, it's this thing, swamp, I never could quite figure out how it's really defined, but I think the proper definition is swamp is a theory that's not survivable from string theory. So the question is, is string theory falsifiable? Testable. Well, before passing on, I would tell you about a very recent development, which just happened beginning of this year, and may well be the most important development in our field in the last few years. In fact, some people have actually done the calculation of N equals 8 supergravity, and found that this year is super finite at three years. I should say, I worked on N equals 8 supergravity some 25 years ago, and I never, ever thought that this calculation would be possible. What has become possible with new technology, these people even have evidence that the theory might actually be finite to all orders in perturbation theory. So although this is now back to field theory, string-inspired techniques are crucial for this derivation. And what's also crucial and new is that if this is to be true, then we need something beyond supersymmetry.

32:30 And we'd just like to mention what this something might be. This is one of my favorite subjects, exceptional symmetries, but we can discuss it over coffee. anyway, if this theory turned out to be finite to all orders, this could really change the picture completely because in part super-string theory was partly motivated at least by the expectation that this super-gravity theory is better well, the other points we can perhaps discuss later ok, now in the second, or last third of my talk, I would say a few words about which is an entirely different approach. As I said, the assumption that ultraviolet diversions might be artifacts of the perturbative treatment, and one should quantify, one should quantize Einstein's theory non-perturbatively. There are various arguments in favor of that. The perturbation theory is blind to geometry. Covariance, and general covariance is, after all, into the theory from the very beginning. Also, if you think about the metric, space-time metric, that sets scales in terms of which we measure distances and also divergences, it starts to fluctuate very much. So we may not even be able to talk about such a thing as ultraviolet divergences if we don't have a space-time metric. Well, the approach is a conventional approach, textbook-type approach. canonical, like you learn in textbooks, how to go from classical mechanics to monochamics, and simply apply this to a standstill, and see how far you get. Well, I think I should quickly pass over the more general aspects, because I understand that what we talked about, background independence, later on. So what one does, one has to do, one has to break manifest space-time covariance by making a split of space-time into space and times and time and then go work through the canonical quantization. One question one has here, by breaking this in this way and then quantizing, it's not guaranteed that space-time covariance survives in the final theory,

35:00 so we'll have to establish some analog and classical space-time covariance in the quantum theory as well. Unification in this canonical code is not the primary goal, and according to the components of this absatz, it may not even be needed. Geometrodynamics, this is the old-fashioned way of doing canonical quantization of gravity, is by doing it in terms of metric variables. explain this in too much detail, but one does. One takes the space-time, splits into time-time, time-space, and space-space components. It turns out that the canonical are the spatial components of the metric, and the associated momenta, which are just computed in the way. We know that from our own mechanics textbooks. The only difference is that this is really a field theory, not only have the labels of the space in the system, but also the space coordinates. So it's a huge collection of fields, and that's one of the complications from this. Well, there are other complications having to do with the fact that it's a constraint system, but the formalism was developed by the back in the 1950s, so it's mainly a technical problem. You have to do some hard computations, day, you have your Hamiltonian, when you simply make the usual replacement, the momentum by a differential operator, because we're talking about the field theory, this is a functional differential operator, but this is a mathematical, and so if you do this, you go through all this after a lot of work, you end up with a legion, and there's a famous Wheeler-the-Wittig legion, which is right here, There's something called, well this is the wave functional, which now depends on the spatial conformity of the metric, and this is like a Laplace operator, like in the standard linear equation, except that it's a functional differential operator. And this term here with the curvature is something like the potential term, the Schrodinger equation. But you note that rather than h psi equals h psi, this is a constraint equation which comes from h on psi equal to zero. And this function is sometimes called the wave function of the universe. That's the wave function that should contain all the information about us and the universe.

37:30 So you can think of this as the Schrödinger, in some sense, Schrödinger equation of gravity. And let me say again, it's like a stationary Schrödinger equation. Maybe it's a timeless equation. This is smart, the theory is so smart, it knows that there's no prepared time for the relativity, so at first you might think, well, that's a drawback, but in fact, some sense is an advantage. So this is wave functional, you can discuss, and one really knows how to interpret it, but it's supposed to contain full information from beginning to end as it works out. So what's the problem? So the question has been around for, the equation has been around for 40 years, but as by now well known, the interpretational problems, how should we interpret this wave function? If somebody gives you this wave function, what would you deal with it? How to interpret it, many worlds interpretation or something else? How do we get a classical world out of the quantum world? and what is the origin of time and the error of time. There are also mathematical challenges which have not been able to overcome in 20 years, although equation groups are probably innocent. It's a very singular function of the French equation. You might suspect that the ultraviolet diversity of the equations that we put on the surface in this equation is the fact that you're not able to make it all defined. So the question, is the Hilbert space of the equation of quantum mechanics so valid? In fact, in this metric form, no one has been able to define or introduce the Hilbert space of wave functions. So altogether, there's almost no progress with this in 40 years. and I once had to bring a little city next to Mr. Price de Witt. One of the inventors of his equations asked him about the equation and this is the outside box. In fact, he seemed to dissociate himself from this equation almost but somebody tells me that he then asked Price de Witt whether he would agree to rename the equation from Wheeler de Witt equation to just the Wheeler equation This was too much, then.

40:00 Well, if you're faced with such a complicated equation, you can try to simplify. We cannot solve the Schrodinger equation for the uranium atom, but you can solve it for the hydrogen atom. So you can also look at semi-classical limits, especially if I know what you think first. We can try to connect up this to classical gravity by making it kind of from the UKE ansatz, plug it back into the equation and see whether it can extract the Schrodinger type equation and then perhaps compute quantum corrections. The other simplification is mini-superspace. We simplify this metric by cutting down the infinite number of degrees of equilibrium and using some finite numbers. This is a typical answer to an integer step with one variable, the radius of the universe. And then this difficult functional-differential equation becomes an ordinary partial differential equation for a single halfway function. Well, let me also mention new development and these Ashtaka variables. I still have, what, five minutes? Oh, okay. So I'll rush through this. There's been, this deal of canonical conversation of gravity was given some impetus in 1986 when Ashdegar discovered a new set of variables, also canonical computation relations, and the main advantage of this that this is a canonical exchange simplified enormous years to the old version of Ashdegar's variables. And then, well, there's a lot of development in this field, This connection here is the basic input, but nowadays one no longer works with the connection but rather than describing it, I'll just show you the picture of what the blue column gravity person thinks space time, or space rather, looks like at the plant scale. This was actually from a poster that was made for a conference at my institute called A Loops of Life. It's a rival event to string theory, string conferences.

42:30 Spin forms, I think that John Bias will probably mention that. No? Oh, okay, well. Let me just go through, rush through, again, successes Gravity in this form can be expressed as the angle-style variables, there's background independence, there's also an attempt to reproduce the back-end side parking entry. More recently, there have been claims that the back-end times we have had to be avoided. And it's big forms, I'm not going to say anything about. So the open problems with that approach are the following. First of all, this approach has severe difficulties in connecting out to the real classical world. And one way to see it is that the metric in this formula is represented by a highly similar So, you lose sort of visual connection with what these things mean. what is the status of the Relativit equation has it really improved in comparison to the geometrodynamics approach of the old fashioned Relativit equation I think in fact there's not much improvement there but we can discuss this there's the question of quantum space time covariance the question what now finally happened to these two loop divergence that killed perturbative quantum gravity again there's no clear answer question. In contrast to super symmetric series, where at least they pose some kind of restriction on the matter coupling, here it appears like anything goes, just in agreement with the idea that you first quantize gravity and then add on matter as unique. And come back to this degeneracy problem. Strings here has a degeneracy problem with the vacuor, but there's a huge degeneracy, and at the moment no one has any idea of how to put down these different choices, and then you can define Hamiltonian, which would be accepted by everyone as the obviously correct one. So the question is again, is loop-commodality about the high-ball theory?

45:00 Prospects, so this is my last two transparencies. these two approaches loop quantum gravity emphasizes the notion of background independence at the proper quantization of geometry we really want to replace Einstein's theory by quantum Einstein's theory string theory on the other hand emphasizes the issue of ultraviolet finiteness and that happens with consistency it also has something to say about non-perturbative aspects, it has no problem reproducing classical space-time geometry as a background, the problem is rather producing the background that we see in the sky, but there's no problem of principle in reproducing classical space-time background. Well, this was the way it was set up, by starting from some background, and it is able in principle to unify matter and gravitation okay however you read sometimes that we're working on the same towards the same goal maybe through just looking at two different sides of the same coin i'm not so sure because these two approaches are based on what looks to be like contradictory For example, the question do we need or don't we need matter, the mathematically consistent conversation of gravity, where the two approaches take completely orthogonal, opposite points of view. So I think a proper summary of what this talk would be by saying that the search for the white theory is still found unfinished and you probably still have no idea what physics looks like at the Planck scale. So there has been a lot of discussion in the press in the last years. We have some quotes. More recently actually there were also books that have been hotly debated, not only in the community but also in a wider circle of interested people. So I think I would say that we're in the middle of a really exciting evolution of ideas.

47:30 And I would like to thank you for your attention. Didn't do too badly with the time, I hope. thank you what do you mean physical I mean that interpretation with this whole scheme I mean I start that you're probably not referring to the interpretation of problems related to of the universe well roughly speaking i would say that well that's my baby john can correct me because he has worked on this a lot in my view this is a way an approach where you try to discretize gravity you try to replace the continuum theory by discretized theory much like, well at least on the surface, much like what you do when you replace a conventional quantum field theory by a lattice field theory what's the mathematical interpretation what kind of, is that a lattice or what is it? well it's supposed to be a lattice but not a lattice like, well that's what I was trying to say When lattice size is a field theory, that's mainly a calculational device. And in field theory, there's agreement that this is not the correct theory, but what one has to do is one has to take the limit where the lattice spacing becomes infinitely small and thereby to recover the continuum. And this approach has been rather successful when one tries things about, say, quantum chromodynamics, where one has been able to make all kinds of also non-preservative calculations and predictions from this approach. And I think in gravity it's somewhat different because the lattice that you use in field theory,

50:00 one is given a Minkowski or Euclidean space-time background and that serves as a sort of reference for your lattice and that's also the reference with respect to which you take the limit of lattice spacing to zero. Now in gravity it's different because it's supposed to be somehow lattice independent, so the lattice itself is a much more, how should I say, it's much less rigid. And the question is, at least to me, but I haven't been able to get clear answers on that, well when it's supposed to take some refinement limit where this lattice sort of fuses with a continuum or is it such that this lattice is interpreted to be the real theory and the real real spaces the thing that really does replace a continuous space but do you have anything to I'll just say I always want it to be the real thing. That is, I want it to not be any . And another thing that's very different, this idea of the lattice is the idea of referring to this that they would fix the lattice for us here. All the different . Maybe I should go back to this picture. This is, you see, what you see here is not a fixed lattice, but you even have this, not just the propaganda picture, but you also have a propaganda movie with this, where you see this sort of fluctuate all the time, around continuously so it's not a fixed structure and I think space time well this is a space space picture because this is a Hamiltonian approach but in a spin-form type picture it probably would have a similar structure whether it's possible or not by the time.

52:30 I just want to make sure that this problem of interpretation can be very easy and easy to solve. If we don't, we don't system with instrumentation and mechanics, which is even harder than . Because in fact, the problem is one, cannot have the test between observer functions in the case of some of the And the second point is that it's quite difficult to define the object in the universe applied in that function. And the two problems are somewhat similar in this interpretation, especially if you So let me describe that point, that's a point, exactly. So the first point is solved because you don't need an option. You need to see, for instance, the way Ashrafel has described this interpretation. Just say, okay, you have a set of preparation and the mass of preparation is the system. And secondly, you have a measurement of density. So you don't even need to define the mass of this pattern. So this is important. The second point is about the Kantian or non-biblical He says, okay, you cannot define the universe of a concept, because you can't say that you replace the universe by the term or the set of terms in the sequence of experience. So, in that case, you can very well define for the Zion motion, which is supposed to give you all the possible predictions for all the possible experiments you get to. Maybe I should add to this. I'm not sure that I completely understood what you just said. But for mathematical physicists like me, the first problem is, you know, being able to calculate any kind of wave function of the universe, and this has been possible only so far as I'm aware for very extremely simplified models, mini-superspace, which in my opinion are at best the caricature of the real world, but we don't have a single example of field-theoretic solution.

55:00 And so we're really talking about something here that we've not even been able to calculate. We don't even have a single example of such a wave function. So in some sense, you know, I feel I don't have ground under my feet. If I talk about wave function of the universe, much of this discussion goes in this way, but I'm not able in a single example to exhibit the wave function of the universe. I mean it's not like in quantum mechanics if I look at the waveforms of the helium atom and I see things and I can just looking at the mathematical result I can interpret it but here you know much of this work is also say something called the midi super space is really directed or motivated by the idea if we could just compute one example of such a wave function which was not completely trivial then we would also learn something about how to interpret. So my approach is more from the mathematical. Well, I think I agree with what you say. I think everybody agrees that the reputation of this quantum cosmological wave function is clear that we somehow must define things relationally. We must be able to somehow split up this wave function of the universe into some piece that describes something like an observer and then the rest and then try to decohere the two you know i'm i'm fully aware of that i just want to say a little more about

57:30 I think that was an approach, that's where I was looking for a way function of the universe, because in some sense that describes, I was looking forward to it. No one knows that the experiments would be a certain whole purpose. And so, in these approaches I gather, the way function of the universe is sort of just like the number one universe. and the way you function has to be related to the parts of the universe. Hopefully you describe something about experiments and that's where I always have trouble with these important decisions for some of the work of scholars, collaborators, understanding, not really very much I mean, the universe may be more than any other kind of business. So you might think of any kind of business, of course, to understand the whole universe. But of course, the universe has a lot of more property to study outside the box, not very much of what's outside the box. And the gravity may be in different respects, but it may not be complete. In fact, I would say most people actively working in this field, especially in the string cycle, are very much more urgent to convene an immediate problem to mathematics.

1:00:00 What I wanted to say is that generically, theory and supernatural symmetric theories prefer much more negative cosmological constant, anti-de-sitter space rather than de-sitter space. But the reason is that supersymmetry is not compatible with positive cosmological constant, but it is with a negative cosmological constant. Now we know that supersymmetry must be broken, the world around this is not supersymmetric. But what I want to say is that it's in difficult to produce out of the equations in a natural fashion, and not a contrived fashion, a universe that behaves more or less like our own, with an initial inflationary expansion at the beginning, and then sort of different Robertson-Watt-like evolution, and there are such attempts, and there are such claims that this can be done, but if you look at them, then I sometimes get the feeling that this doesn't come out of the theory very naturally you have to force and bend and massage the theory to give it to you what you want and in the end I think this is also another criterion we should keep in mind is you shouldn't make more assumptions than the number of assumptions you make should not exceed the number of facts you want to explain and I think many of these models So it's more of a structural problem in which they give you the broken feature. It's the problem that I look at the equations, whatever equations there are, to get this out in a sort of, I don't want to have to force the equation to give me what I want. I think this is another way of putting some of the comments that started from the context of the Kantian interpretation of universal wave function. Or the original way of thinking about that is the range of possible...