Remarks (fragments) / Time in quantum gravity / subsequent discussion / Wormholes and time in quantum gravity (contd.)

0:00 What we had originally in L0 is practically of no interest to us since most or all of it could be absorbed depending on how many baby universes there are, kinds of baby universes. Most or all of it can be reabsorbed in an additive redefinition of the alphas. The fundamental theory of everything turns out to end up telling us nothing. Because all predictive power is going down the wormholes. And there we are. This is a position I was in at the end of my first paper on the subject, which you might have seen, called Black Holes as Red Herrings, and it's also the position Hawking came to in his paper with Leflamme that came out around the same time. Yes? No, I don't think it would, because they would be someplace. If you said there were a quantum configuration that made a knot. Okay, and all of a sudden you said, gee, I'm that tiny region, I have some space-time over here that's not nice and smooth and Minkowskian, but it's got a little Klein bottle down there, I see. Okay, that would be quantum tunneling and Klein bottle, or Klein bottle, anti-Klein bottle pairs would be materializing through quantum tunneling, but it wouldn't be the same thing as a baby universe. It would be somewhere. And that makes all the difference in the world. No, it's not simple. It is simple, but it takes around 20 minutes. Okay, you'll find it in my paper. Yeah, Richard. Yeah, absolutely. That's the normal algorithm term up here. Those are there in the drawing. They're all there. Is there any assumption that we have this one parent universe? No, no, no, no, no, no, no. I deliberately do a thing where the parent manifold M has two components that could have 22 components. As I say, this, but I am assuming the loop gas technology, that is to say, I am assuming that there's a probability, in this sum, there is a negligible probability for one wormhole ending on top of another and things like that.

2:30 The SI is the action of the wormhole, which is assumed to be an instanton-like solution of the classical field of gravity. KI is a function of the ambient fields in which you put the wormhole in. It's the sort of thing you get by doing a functional determinant. Think of that Hoechst calculation of the induced proton decay due to electroweak instantons. Electroweak instanton action is like SI. The thing that multiplies the effective action, that combination of quark fields and lepton fields, I believe six altogether, is like KI. Where is the location? There in KI. KI is a function of the ambient fields on the manifold and therefore That's the, but of course you integrate over all of those, the x. Yeah, ki is a function of x. No, but it means the x. That's right. Yep. That's, that's the summation of the integration on one whole m. Yes. I asserted it. Yeah, I could demonstrate, as I say, it takes around 20 minutes of pressure, pressure of time. It is totally straightforward combinatorics. It was in fact, I should give credit where credit was due, it was derived in collaboration with Steve Giddings and Andy Strominger and it appears in my paper, a proof appears in my first paper on this subject, Black Holes as Red Herrings, and also in Giddings and Strominger's second paper on black holes. No that turns out to be totally illusory as we will see. The initial and final here is just... I haven't thought about the topic. Let me get to point two. At least I want to get to the vanishing of the cosmological consciousness before my allotted time vanishes. Euclidean gravity, as I said, is a crackless swamp, and all we will need of it is very little information.

5:00 The only formula I will need in my analysis is the formula for the expectation value of the scalar field. I will accept the Hartle-Hawking formula, where you sum over compact manifolds without boundaries, and I will say the expectation value of the scalar field is given by this expression, where S is the Euclidean action for your theory. This certainly follows from the, if you accept everything in the original Hartle-Hawking paper, from the expectations I have of the scalar field in the Hartle-Hawking ground state, and it so closely resembles the conventional formulas of ordinary field theories that I am perhaps more inclined to trust it than I am the rest of the structure. It will turn out to be all we need for diagnostic purposes. Notice no question, since I'm just doing a local field at a single point, I don't have to worry about what time slice I'm on or anything like that. For the naturalness discussion, which will come up in around seven minutes, I will need a generalization. This is going to give me the desired delta function peaking, no doubt about it. But I want to consider variants that don't give me such a delta function, and that would correspond to considering manifolds with boundaries. This is diffeomorphism in theory. It's independent events, and then it's different bio-events, and it's not the same. Well, yes and no, its expectation value is certainly independent of x. I don't have to know where x is once I sum over all three geometries and everything else to result for any x is the same as for any other x.

7:30 In this case, you don't have to tell me that this is ill-founded. Let's follow it and see where it goes, and then if it goes somewhere interesting, let's go back and see what can be done about it. I guess it doesn't matter whether you can assume that those m's are all connected? No, well, it's quite irrelevant in this thing whether you sum over connected m's or disconnected m's, because disconnected m's just give you the same factor in the numerator and denominator. In the theory with wormholes, it will look as if it's awfully important as if there's something over a disconnected end, but that will be because we've already used the wormhole summation formula, and the things we sum over will not be really disconnected. They will be connected by wormholes, which we've moved at an earlier stage to the complement part. Now, accepting this formula just as a diagnostic tool, okay? I can now give, in two rather short transparencies with nothing suppressed, really, the argument for the vanishing of the cosmological constant. To keep my equations from getting too clumsy, I'll denote all of the alphas, and remember there's one for every wormhole type, by just the single number alpha, and the alpha will be the product of all these. Now the first step is to imagine we're working with some theory cut off at some scale just below the wormhole scale with some effective action in which the alphas appear just as numerical parameters. And I want to compute the expectation value of psi using the Hardow-Walking boundary conditions in such a state. Of course, I just write the sum, the same one I had before. I define the denominator to be the partition function z and I make explicit reference. To its dependence on alpha, and thus I see a formula that will be useful to me two lines below, e to the minus s effective of alpha phi of x is the Harle-Welking expectation value with parameters alpha times the z of alpha. There's nothing here but definitions, definitions Bryce would tell me of an ill-defined entity, but by gentleman's agreement, we're not going to fight about that until I'm done with the argument. Now I'm going to do the same thing, but I'm going to do my scale a little further up. I'm going to now consider manifolds that have wormholes on them and all of that.

10:00 I'm still going to use the Hartle-Hawking boundary conditions, which means there are no baby universes. In terms of the alpha basis, that's of course a Gaussian. The way I've normalized things is minus alpha squared over 4. Alpha squared is here, of course, the sum of all the alphas. Therefore, I use exactly the same wormhole summation formula I had earlier. Indeed, I might actually, if I can find it, pull it out for you for a moment to look at. There it is. This tells you how to do it in an alpha basis. So I'm just going to take a linear superposition of alpha bases with a Gaussian. Well, nothing is easier. Both the numerator and the denominator are to be done with this formula. I have an e to the minus alpha squared over 4 in the beginning and the end, so I get an e to the minus alpha squared over 2. Then I have exactly the sum for every fixed value of alpha. The Cardo-Hawking expectation value times Z of alpha, the denominator exactly the same thing without the phi. Staring at this, assuming the expectation value of phi varies from one value of alpha to another for a sufficiently large set of phi's. Don't forget if this is arbitrary phi, it could be of my original scalar field was 2 to the 4th to the square of this 42nd derivative. Dotted into components of the Riemann tensor, what you will. This should be able to build enough diagnostics to tell one value of alpha from another, and if one can do that, one sees the probability distribution of alphas in this Heidel-Hawking state, whatever it may be, is either minus alpha squared over two, z of alpha, d alpha, where z of alpha is defined here. I have run out of time, Mr. Chairman. What should I do? You're too indulgent, sir. Maybe I'll skip naturalness and fix it with the other constants.

12:30 Just continue. Okay, in response to widespread popular desire, I will just continue. Now, the third part of the argument is to study z of alpha. I will demonstrate that z of alpha has the stated delta function peak in it. Z of alpha, you may recall, was just there one transparency ago, was the sum over closed manifolds of e to the minus s. A closed manifold can be disconnected. Remember, at this stage, considering disconnected manifolds is not an arbitrary operation because they really were connected by wormholes, which we used at an earlier stage of our combinatoric analysis. However, since the action is additive, the situation is precise like that normal field theory where the sum of all vacuum-to-vacuum filing graphs is the exponential of the sum of the connected graphs. By the same combinatorics, this is the exponential of the sum over close-connected manifolds of the same thing. Now, I propose to study the sum over close-connected manifolds by the usual background field effective action technique. I will introduce an effective background field for the gravitational field alone. I'll just integrate over everything else. And write this as either minus gamma, an effective action, which of course depends on alpha, and depends on the background for Euclidean 4 geometry, G. Then the usual rule is we'll evaluate this at the stationary point of G, and if there are several stationary points, the one of minimum action is the one that counts. In addition to all the other horrors of quantum gravity, there is a technical obstacle here. The background field method rests on writing the quantum field as a background field plus a deviation. Now, I can't think of any, since we have the possibility of different topologies for these connected manifolds. I can't think of any way of writing a metric on a four-torch, for example, as some background on a four-sphere plus a deviation. So I will just do this separately for each topological class and have a separate effect for that.

15:00 This may seem to be... When I was a boy, there was a parody paper that was circulated that was supposed to mock people who just did formal things in field theory. You'll remember it, Murray. It said, according to Schwinger, blank. No, no, in the spirit of. The main thing is that it's got to have in the spirit of. Schwinger? Somebody. Yes, blank. Hence blank, whence blank, thus blank. Okay, fill in the paper and send it. Well, this may seem to you to be exactly that kind of vacuous formalism because I have succeeded in writing a functional interval which I cannot perform in terms of the sum over an infinite set of functions which I cannot evaluate. However, This is not the case because there is one set of manifolds for which we know the effective action exactly. These are very large smooth manifolds. If I take a very large smooth manifold, I know the leading term in the effective action is simply the volume of the manifold times the cosmological constant. And that is the bottom line cosmological person, including all radiative corrections of everything in the absolute ground state, no metastable impostors need imply, for exactly the same reason that in ordinary quantum mechanics trace e to the minus ht is dominated for a large T by the absolute ground state energy, including all corrections from everything. All of them. All of them. And I've integrated out all gravitational fluctuations. I've just got this background here. This is what we usually mean when we say we do classical physics, you know. These are how we measure the parameters in classical physics. The next term is also known. The term that grows not as a volume but as a square root of the volume, that's the Einstein action, which is the real G. Of course, the real lambda and the real G in this context depend on the alphas. Then there are higher order terms, which I will return in a moment, but for the calculation on this transparency, they can be neglected. Now, the stationary points of this thing are, of course, Einstein spaces, and in Einstein spaces, as you well know, this term is twice as large as this one and oppositely signed.

17:30 So the action gamma is minus lambda times the volume of the space. At this point onwards, my argument parallels one Stephen Hawking gave in 1984, and I probably won't have, I should say that, but there's a difference between them. Therefore, for positive lambda, we must find the Einstein space of maximum volume, for negative lambda, that of minimum volume. For positive lambda, the answer is known. It's a four sphere. You do the computation, you get something like this. Plugging that in, we see log z grows like the exponential of one over lambda as well. That's going to be important later. For negative z, the space of maximum volume is not known. At least it wasn't known to Yao when I asked him a few weeks ago. But whatever the answer is, it gives a positive action and therefore it goes to zero. That's why it goes to zero. Now this is it. This infrared blow-up is the thing that makes the cosmological constant vanish. If I imagine putting in an infrared cut-off in the crudest possible way, say by saying my sum over manifolds would be cut off at very, very large manifolds, Then instead of having something that went to infinity as lambda goes to zero, I would have something that would go to some very large, but non-zero value. I'm going to get these things in a fourth period. Then if I normalize my probability distribution and then let the infrared cutoff go to infinity, I would obtain a delta function restricted to the sub-manifold of the alpha space for which the cosmological... Now I can make a couple of points here. One is that this development of a delta function at the infrared limit is nothing novel in field theory.

20:00 If we consider the ordinary theory of a free massive scalar field in a box, and look at the probability distribution for the spatial average value of that field, that is a Gaussian with variance inversely proportional to the volume of the box. As the volume of the box goes to infinity, that becomes... And as anyone who has studied the Higgs phenomena and spontaneous symmetry breakdown knows, that can be a quantity that determines the constants of nature as measured on a low energy scale. So there's a very close parallelism. The final result, although the intermediate step was quite different, between this and normal phenomena of ordinary field theories. Secondly, I should say I am tacitly assuming that it is possible to adjust the alpha parameters throughout the cosmological constant vectors. Certainly, since the leading term in the expansion of this effective action is just a bunch of constants times the alphas, if the cosmological constant is sufficiently small, there's no problem doing this. On the other hand, if the cosmological constant is gigantic, it may need a very large alpha, and then you have to worry about whether second-order effects in the alpha or third-order effects will cancel out the first-order ones. So there I really can't say what's going on and what's not going on. I think it's fair to say that it looks plausible that if you have a cosmological concept that is small on the wormhole scale itself, which is what determines the natural size that goes into the alphas, then you can certainly cancel out by this thing. If it's larger than that, it's an interesting thing. Maybe you can cancel out or maybe you can't. I can't say anything about it. Well, I'm assuming there's some way of making the theory make sense. This is not a mini-superspace computation. This is a real effective action. You've done something. God knows what. If it doesn't exist, I'm a dead man. But you've done something, and I think in this thing you really want to look for the real minimum. You've already done all of your Euclidean analytic... Absolutely not. Well, no, I don't think that's a bad thing. I know lots of models where I've introduced auxiliary fields and things like that, like large M computations, where the stationary points of gamma are in fact where the effective action ends up not being bounded below. And there's nothing wrong with those theories. Okay, now maybe about...

22:30 Well, we observe that z is the function that multiplies the probability distribution with e to the minus alpha squared d alpha. Okay? If I observe here that z is growing enormously rapidly as lambda goes to zero, like e to the e to the one over lambda. Setting all constants equal to one. Even the e to the one over lambda, for lambda positive, positive. So one-sided approximation to a delta function, but that's still a delta function. Murray, what, I mean, if I put in an infrared cutoff, this one here, so that I only go up to a four sphere of a given size, and then I stop. No, no, no, no, no, no, no, no, no, no, no. Just a minute. I will get it. So that means that if I put in an infrared cutoff in this computation, then instead of z going to infinity, we'll just go to some very large value and then go off.

25:00 This is from the right. Of course, from the left, it'll just go down to zero. Now, this, if I take this thing and normalize it and let the infrared cutoff go away, I get an output from it. And if you don't believe me for this, you should bloody believe me for a free field theory in a box. A free massive-scaled field in a box. No, my expectation of phi is still restricted on those manifolds for which alpha is adjusted so that... Now, I mean, if including quantum gravity is so sick that there's no way of making sense of it, and if I computed the expectation value of phi even with fixed alpha, I obtain nonsensical and divergent results, okay, then this whole scheme falls apart. But let us for the moment assume that if I obtain, if you keep the expectation value of 5, when I'm here with an ultraviolet petal throughout the world of planet mass and fixed outwards, I obtain some finite result that is dependent on alpha, then the whole thing goes to earth. I've explained it. I think we're getting at the critical question, and perhaps it's better that I finish this, and then you can all fall on me. You can have a piece of my flesh, whatever you want. Now, let me turn to the question of naturalness. As I said, Hawking found this very same exponential factor in a semi-classical mini-superspace computation in 1984, and he wrote in a little paper called The Cosmological Consonance Probably Zero, but if you read the paper, you discover he's uncharacteristically cautious about what this thing means, and the reason is... He first considered a theory in which the cosmological concept was an adjustable parameter, let us say, because the theory contained a theta angle, as in axionless QCD.

27:30 So you change it around, you change the vector of energy. Then he computed the Hartle-Hawking wave function, obtained by his punctual input description, and found that, true enough, it had an exponential factor and it would rule out. When the theta angle was adjusted so that the cosmological constant was zero, just the exponential, not the exponential of the exponential. And he said, but we don't know quite what to do with this. Is it something for real or is it just part of normalization conventions? For example, if I were to consider a particle moving in three dimensions in a spherical, symmetric, square-web potential, And I normalize my wave function here to a size like sine kr. Then I would find, of course, for energies lower than the barrier height, the wave function has an exponentially large factor in it, the famous tunneling factor. Above the barrier height, it does not. Anyone who tried to draw conclusions from this about the a priori probability of finding a particle out here with a given energy would be making an ass of themselves. Now, the problem with Stephen's original factor was if it was of that kind, if he could have obtained a solution of the wheel-in-the-wheel equations of a hard-of-working boundary condition if he has multiplied a string by any function of theta. Now, this is not true in my case. That e to the minus alpha squared over 2, called 4 by Gell-Mann, represents the Hartle-Hawking boundary condition, that there are no boundaries and therefore there are no baby universes. So that's a real difference. On the other hand, the distinction that makes this link is an important difference. I could, you know, Hartle and Hawking are nice, terrific guys, but still... Still, their words do not have divine authority, and maybe you want to do the same sort of integral with some other boundary theory.

30:00 If I had done integrals with different fixed boundaries, and made appropriate linear combinations, as explained in my third article, Transparency, I could have replaced that e to the minus alpha squared over two by an arbitrary function, f of alpha. But so what? Then I could have picked this f of alpha to have an essential zero that cancels the essential singularity here. Or even maybe to be a delta function, flipped on one particular value of alpha. Now what happens to that? Well I would like to argue to you that all such f of alphas are in fact unnatural. And the argument goes as follows. It was important in doing the wormhole summation formula That I cut my wormholes exactly in two to make baby universes. If I had taken them and cut them somewhere else, or not used the exact cross-section, used some other field configuration that closely resembled it, the amplitude for creating or annihilating a baby universe would not have been the square root of the amplitude for a wormhole going from one place to another on the same manifold, and that wouldn't have worked out right. Nothing wrong with that. This is something that happens all the time in ordinary field theory. If you don't normalize your fields correctly, you don't get scattering amplitudes that obey unitarity. You have to put an explicit wave function in the normalization factors to correct for it. If I don't have just the right boundary conditions for the baby universes, my amplitudes won't come out right. They won't obey the equivalent of unitarity, and I'll have to put an explicit wave function in them. Those racial factors will have to multiply by some number to power. That's the number of incoming and outgoing days in universes to make things come out right. Now naturalness has to do with what happens to physics at a lower energy scale if we make a small arbitrary change in physics at a higher energy scale. So let's imagine making a small change in plant scale physics. I know hardcore string theorists say it can't be changed, it is what it is. I will address that question later if requested. Let's imagine we make a small chain. If we make a small chain, the infinite times that represent the normals will change by a small amount.

32:30 Our baby universes will make a small mismatch. In the n language, it will be exactly as if strings were multiplied by some small power, e to the minus small number, e to the minus epsilon m. Consider the case of positive epsilon. I can understand why small changes in flat-scale physics change things like the action. But that actually changes the shape. The minimum radius may go up or down. Right, but I thought those n's were just the number operators. That's right. It means if you use the same boundary conditions to compute the amplitude, you won't actually get the amplitude for nn. You'll get that amplitude times a factor of u minus epsilon n1 plus m2. I just put it on the state rather than the operator. Divided by the sum. No, no, no, no, no. If you use improperly normalized fields in field theory, you compute a nonunitary S-matrix, which you do not correct for just by dividing something from the denominator. You use differently normalized fields. Remember renormalization group people. Change the physics on a small scale. You change the effective coupling constant on a larger scale. You also change the normalization of the fields. That's why you have the gamma term in your renormalization group equation. This is it. Now, what does this do to f of alpha? Well, the alphas are, of course, dual variables to the n. But we know exactly what that is. n is a i joined a, and a is a differential operator on the alphas. And it's not hard to work this out. Indeed, I didn't work it out. I looked it up in a paper on the harmonic oscillator. This is what it does to s as alpha. It's a horrible expression, but I want to focus on one. If s of alpha prime is so much a square integral or even so much a temporal distribution, s as alpha is analytic. It turns a delta function into a Gaussian in general temperature distribution.

35:00 So even if you carefully picked your boundary condition, say I'm going to have 42 baby universes with this amplitude and 42 with that, 47 with that, et cetera, et cetera, et cetera, et cetera. I'm going to evaluate the expectations by the state defined by that in just such a way as to cancel out the physical physics, that boundary condition which you have carefully We'll no longer do the job. God, too much overtime. Quick, quick, quick, then I can ask lots of questions and you can go eat. Last transparency. No, no, remember this thing, you get a delta function because z of alpha has an essential singularity. It doesn't do anything. No, no, no, it's an essential singularity. No analytic function times a function with an essential singularity has an essential singularity. Zero times, it's true if it were just one over z multiplying it by z would be enough to kill it. But if it's e to the one over z multiplying it by z or even by z to the 50th is not enough to kill it. And in fact it's worth an even one over z. But if you were a delta of z times any power of z. That's right. But if you claim you're what? I claimed after I normalized it, this would not be equal. After I normalized it, it was a delta function. You could put in the central zero here, but then this procedure would cancel out again. The central zero would kill it. A delta function localized someplace else would kill it, but that's unnatural. As soon as I change the Planck scale of physics, the boundary conditions that lead to a delta function or an essential zero disappear. Yeah, I know, it could be negative. That's why it could be positive. Well, here I'm assuming for simplicity that there's only one kind of baby universe.

37:30 For many, this could be a matrix, but it would still have the same generic quality. Let me fix all the other constants of nature. I ran out of time. This is disgusting. In order to discuss how you fix the other constants of nature, it's very easy. It's the same computation. Just let me change variables to alpha naught, which is the combination that I'm fixing here, alpha naught equals zero, and all the others which I call alpha half. That's a nonlinear chain. Gamma I've computed is minus one over alpha naught, but that was because I neglected all those other terms in the effective action that involved the coordinate system. If I include them, I would of course have terms that are... Higher powers in the radius of the four sphere, and therefore could be expanded in the power series in terms of alpha naught, which is essentially connected to the radius squared of the force. I'm assuming here an ordinary power series, just for simplicity. It could be that fractional powers or logarithms or something like that appear, that's not going to affect it. Now the fact that I know that this is not just a simple exponential, but the exponential of an exponential becomes... I previously showed you that everything was pushed onto the manifold alpha naught equals zero. Let's look at two different points on that manifold characterized by two different values of the alpha hats. If I look at the ratio of the z, that's the fact that I'm doing it at these two points, since it's log z that's equal to e to the gamma, e to the minus gamma, this is e to the 1 over alpha naught times this thing minus this thing. Please notice that for different values of alpha hat things are blowing up at very different rates. It's not pushed uniformly onto the manifold alpha naught equals zero. In fact, it is pushed onto the sub-manifold at which gamma naught is minimum. And that's not the end of the story. If I now expand things further out, it is pushed down onto the sub-manifold of that manifold, where gamma 1 is minimal.

40:00 You can use the property of e to the 1 over x. I'm sorry about that. It's an elementary function, but it does this rather to your property. And furthermore, things are concentrated as you remove the infrared cut-off. It goes on forever, onto the sub-manifold where the next parabola is streaming, and the next one, and the next one, and the next one. Until you run out of alphas. If there's an infinite number of alphas, maybe you'll fix everything or maybe not. So, in fact, all of the ambiguities that we thought appeared at the very beginning when I was talking about the wormhole summation an hour and ten minutes ago are, in fact, possibly removed by this new physics that takes place on the wormhole scale. It's just the situation we thought we were in after the theory of everything. We have no small distance physics. There is a vague hope, which I will say do no more but raises it with Scott, that possibly these plunders could be. This is not a well-organized talk. It's been a shambles. I've run all over time. But I hope I've explained the essential ideas of this view to you. And now I will let you attack me, please. I certainly feel that the world comes from a community that is half neutral and many of which are extraordinarily self-reliant. On the other hand, the use of all that technology is very minimal. So let me try to replace... Now he's now giving me more hands than a typical member of the Zimbabwean club. Let me suggest that the mechanism... For the elimination of technological concepts, there's perhaps more general than where it comes from. All of the students that you meet in those performance calculations, there's some theory which let me suggest goes something like this. We have quantum field theory, which we understand...

42:30 We have, from something else, some non-morphality. We have some operators which don't live in our manifolds. We can have coupled some state space somewhere else because it seems that the only characteristic that you really use is the fact that you don't carry momentum and energy. And so you have some state-space, some states which are not local and disconnected from the life of physics, and they're happening on such a scale, for some reason you don't understand, to do some coupling between the work of physics, the things you like to think about, and this non-local physics. And if you don't know this non-local physics, you'll be plastered with this stuff, and then you run out of it, and you'll never have to fill one hole, you'll be putting it on gravity. Well, I'm a little nervous about that, because you do need it at the back of the state. It's true that if you had a mechanism of that sort, you'd get this sort of exponential, but then you'd say, what does it mean? Let's think of this example. If you get exponential growth out here, what does it mean? The fact that we have this geometrical picture of it in terms of baby universes means we can answer what it means in some sense. Excuse me, but it seems to me that the crucial ingredient to all of this was the fact that when you got the effective action, it was streamlined by this huge force field. What? It has a negative Euclidean action. That's the key ingredient in getting this delta particle to 0. If you never use that Euclidean... I don't have, I don't have, I don't think there's any problem with, with using Euclidean quantum integrals to study the periodic scale. No matter what the gravitation theory is. I think that there's a... So maybe the universe has gotten out of the game at this stage. This is... Listen... The place where we would worry about just sitting in front of gravity is just the ultraviolet. I don't think... No matter what the theory of gravity is, whether it's canonical, classical, or string theory, general relativity, I think very general arguments come up for the community to sit in front of things like the study of nuclear artifacts in the vacuum. And that's really, I think, why we're sitting here today. So I think that... It's possible.

45:00 Yeah, but the argument... I mean, you have to do something crazy at some stage, but... No, but you need a drawing like this. When we... Even when our universe is hot and... Well, if I were doing Harold Hoffman, talking about the species, I would have the boundary there, of course, bound to a small, hot universe. Even when I have a small, hot universe on the boundary here... Very early on in our time, that can be connected by a wormhole to this enormous cold object, another part of the same video manifold. Anyone who has slept for at least one string theory talk in the last year knows that an insertion of a small boundary looks like an addition to a ground density. And this guy at the other end of here is making a cosmological costume. It's important that there be a disconnected universe governed by the same, otherwise disconnected universe governed by the same laws of physics as govern ours. You've got to replace pre-arrangement with precognition. That's the only alternative I can think of. Why is it that, after all these things we're telling you at this course, there's actually these ones with all of those spikes and all of those branches going off to a house full of them? Well, they're important. I mean, I didn't say they were important. I assumed they were the only important thing between the clock scale and what I call the minimal scale. I assumed they were the separation of the order of magnitude. Is that believable? Well, that's the question of whether these exist at all in the sense of gravitational instantons and whether there are other gravitational instantons or exotic topological characters that exist because they are not quite so symmetric that might be making even more important contributions. I can say nothing about that. Once I have done that, though, once I've gotten to the other side of the one-hole summation formula,

47:30 Then the rest of the thing goes. The four-sphere is important on the other side because the four-spheres beyond time and space are the minimum actions that could possibly come to life at this moment. I didn't say it was a four-sphere, but it turned out to be some sort of four-donut. And it would have been a four-donut on the other side. It's some awful, awful space of infinite units beyond Donaldson's wildest nightmare. That would have been there. It happens at the school.