Quantum gravity on the computer (last part) / subsequent discussions

0:00 You can actually do calculations in the large beta regime. You find that for the 4.22, well then you have a cutoff, of course, and you can calculate these things, you can estimate them, and you see right away that all the graphs of 4 external lines go like 1 over beta to be L plus 1, where L is the number, except for the 3 graph. The 3 graph has the same difference. But of course that's way up in here anyway, the large data, and you know you could get zero. So you're summing up an infinity of complicated graphs to get a result that you know is zero anyway. Well, because for large data, the field can't fluctuate very much and it doesn't feel the curvature of the atomic space, and it's essentially free. This is not on the part. Oh no, that's the sphere. We're not talking about the O-1-2. If we don't run into that problem because we're down in the chaotic regions of small data, I believe, my personal view, if you want me to, let me finish before we get back to the complex intelligence. I simply say at this moment that I don't believe the O2 or the O3 can make any sense at all in the continuum. The O1-2 is a possibility.

2:30 Well, this is just a week old, this idea, and they're presently working it out. I have one more transparency. A possible explanation for the lack of consistency is the low momentum region cannot actually be reached, even with modern supercomputers. Low momentum means p squared small compared with quant mass squared, or pi squared over f small compared to beta r over a squared, or pi squared over n plus 1 squared small compared to beta r. The beta r at the optimum, 1 over n plus 1, this implies pi square root of n plus 1 is small compared to 1, and already n plus 1 is only 11, that's our biggest lattice, and this is not small compared to 1. So it appears that we are forced to begin to deal with the hard energy regime, at least the Planck regime, which means, okay, let's try the next step in the... We've already found that if we try the mass arms out, the simple 1 over beta r, 1 over p squared plus m squared, that the mass is given very small. So let me try a different one now that is suggested as the natural thing you try. You try a p-fourth term. It's supposed to be more than perturbative. Not in the string theory, no. And of course this gives a behavior like this. And now we would have to fit not beta r and psi, but beta r and alpha. My boys are currently running that. I don't know what the results are. If you do this computation, well this is what the two-point function of the normalization I've chosen looks like. And you'll notice an independence does survive in this. And we'd already noticed an independence.

5:00 Because it stayed close to zero, there's another reason. Another reason is that if I have a mass term, I cannot write down a corresponding action, which is 012 invariant. This is the 012 invariant action that I can write corresponding to this ansatz. And you'll notice it gets quite a bit more complicated. I now have the Riemannian connection coming in. So the next step, and my boys are doing this, is to calculate the four-point function for this math and then determine what that implies for the correlation function and see if we get better agreement. I hope we get the sound sense. This is merely meant to be an approximation to the effective action. This is not, I'm still using in the computer. I never put a term like this into the computer. I don't know, I always stick with just that term with a bare mute. So this is meant to be whatever effective action this classical action leads to. Beginning of one, isn't it? I know. Alright, well, so these are the uncertainties. I don't know much about it. How do you handle the effective action in the presence of symmetry breaking? In my spare time, I'm now thinking about precisely that problem. This is the last...

7:30 No, no, no, that would be coordinate independence, yes. And I feel that what we've done here is definitely coordinate independence. I may redo this using Riemann and Orwell, but then that also, then the issue of, suppose I do an 0-1-2 transformation. I have to transform my state, which means I have to transform my boundary condition, and I'm not quite sure about the structure of the effective action under those circumstances. We have not tried anything but what I had already spoken about. Knowledge. Tried to speak to the experts on the subject. Nice talk I had. The field is wide open. We have no words of wisdom. And he doesn't believe that just because a theory is preservatively non-renormalizable that it doesn't exist. We have no evidence yet that the theory doesn't exist. Beta outgoing behavior as beta goes to zero. If that law continues, there's no reason to suspect it won't. We're now in a nonlinear region. Nothing weird could happen.

10:00 The four point function is not being consistent yet. If we can make that consistent, then I would say, well, it looks like we have a consistent nonlinear sigma model in four dimensions. All the more reason to hope that one might have a consistent quantum product. If this doesn't pan out, well then, I will say... Well, in that case, I have to appeal back to the light phone smear, and maybe that will help out. This takes two times later on in the session. Do you have enough actors to communicate? Well, we are, yes. In fact, the aerobars get kind of bad, but we're going to do that, too. George had already done that, but his four-point function, if he was using any species, was not going to be the right one. I'm not even sure we're using it right. Well, that's the way it started. You can prove that it's identical, certainly identical on the computer, to what we're doing. No, no, I didn't mean to say that. I mean, because one of them is...

12:30 The only theorems I know of so far are just simple-minded ones that data are either greater than data or smaller than data. I mean, comparing one theory to the other, I didn't deal with that over the course of time. The proof is easy to follow. In other words, put a tight potential around that one. There are a couple of remarks about the way some perturbation theories and numericals can be acted in. To the continuum pass interval in the semiprofessional unit, you have sort of done it in a few cases to get the configuration, configuration, evolution, and physics things all lined up. That can be stated through your full theory. You don't necessarily have only one answer. Yeah, that's a good point. What is it, 5-2 for this example? Well, let me apply two of the four spheres, the three spheres. Is it pi-four of s-two or pi-two of s-four? Pi-four of s-two. Are you right? Right. So that there could be another... But that's the second you have the possibility of more than one evolution that connects. Now, in addition, you've chosen mathematicians from the spatial, because they're also all lined up, which can classify as that, which gives us s-one and s-two. Now, the question arises when I go to the version where I clearly can go to the non-homotype classification in the discrete model, whether you really have seriously preserved that with your metropolis method for choosing that, even though the other homotype class might sometimes be a long ways away, and it may be a long ways over a very high barrier, which increases the statistics a number of points.

15:00 If there is a high barrier, the computer certainly isn't going to figure it out. So that you may have to put yourself in the one evolution of the trivial evolution, which is all standpoint and definition. It's good if you're comparing the perturbation theory around the trivial evolution. But if you believe in non-perturbative, fixing up a perturbation theory... And you'd better here. Well, hold on. I'm not convinced that including other topological possibilities is necessary in order to get away from perturbatives. We've already gotten away from probing. The structure, the non-perturbative structure as seen by the effect of the higher pentagonal important expansions that I have appreciated, I see those most strongly in a semi-classical approximation. Well, we already don't see perturbation theory there because we get finite. Hold on, hold on. We've got to keep order. There are too many people asking questions now.

17:30 Actually, there isn't really a metric in the gravity case, is there? There is, yeah. That's too strong a statement, but there's something like a curve in terms. Well, you see, there's also a gene you knew that multiplies the two derivatives. And that's not part of your internal space. If you take the time-time part, then all right, yeah, you know that. Well, of course, this doesn't break neatly. There's no Hamiltonian constraint. It doesn't break neatly into a kinetic part and a curvature part. So we don't have those things. I mean, Bryce, maybe with a table of analogies and this, all this is perfect to gear up. You get experience for one. With another normalizable model. Just started. All lined up. Phi 1 equals 0, Phi 2 equals 0, Phi 3 equals 1. Over the whole lattice. Just start rolling the dice and pretty soon it's way for that. And anyway, throw away the first 2,000 streets or 4,000 streets. Yeah, if Arlie can give me a map, an actual, explicit map from S4 into S2, I'll try to mimic it on the computer.

20:00 Yeah, all right, thank you. But you give me an explicit map and we'll put it on the computer. Well, typically the relaxation time is pretty fast here. Even the first 2,000 sweeps, the quantities don't differ that much from the succeeding ones. We just throw the first 2,000 away anyway. There is no critical temperature in this model. I mean, it's infinity. Beta equals zero. Yeah, because in that case... Well, if you look at the, if you look at what that psi really means in math terms, and the way things are going to zero, I didn't discuss it, but it's worse than trivial, you don't, your action is essentially, the alternative is to drive, is to drive everything to the critical point. At the critical point, you cannot have a renormalized scale parameter and a zero, because beta, beta critical is a finite number. And remember, beta critical is the product of mu squared and a squared, of mu r squared and a squared. If you try to hold mu r fixed, you can't, and you might let a go to zero. Yeah, well, no, he was actually saying you need an infinity of counterterms, and you have an infinity of renormalization group equations, an infinity of beta functions.

22:30 And maybe they could all go somehow. I don't think I want to try facing... There was never any evidence whatsoever for any of that. That was the last hope. Well, let me see if there are any more questions. Well, it may still not exist. If we cannot get consistency between the two-point and four-point ideas, then I say there's something wrong here. I'm not quite sure what. Thanks again, Bryce. I don't know why you didn't tell us. One of the outstanding obstacles to outstanding problems in constructing no boundary and providing a proper contour for doing the sum of the geometries, One, because the Euclidean action is positive definite in gravity, but also, I think, because the idea is to have any correspondence with perceived reality, we need to have a can oscillate so we can describe approximately a regime. This problem has been alluded to a number of times, and I would just like to report very briefly on a little model which Vice has been discussing, but a simplicial lattice.

25:00 You're probably all familiar with the regi calculus, that you can make simplicial geometries, and if you join Flat four simplices together, they can make a curved four-dimensional manifold, much in the same way that a flexible polar could build a curved two-dimensional surface out of flat triangles. The geometry is all contained in the specification of the edge lengths and topology. Therefore, one approach to defining sums over geometries is to approximate them by integrals over the edge lengths of a simplicial lattice. So, for example, such a construction in the case of weight and no boundary proposal for the weight... The universe would run roughly as follows. We're interested in evaluating on a three geometry which has boundary edges s, i, d. We want to integrate the Euclidean action, which is a function of all the edge lengths, both on the boundary of the manifold and on the inside, and we want to do the interval over the interior edges that aren't specified by the arguments of the wave function. There's some measure, which of course we have to specify to put in. Here we would let us, for example, take some action for general relativity, which is an issue, and we have to, as I said, specify the contour of integration, not these particular... What has then led to a series of models which are of a familiar character. They're what are called mini-superspace models. Although these are not mini-superspace models built on symmetries, that is, restricting the symmetric form of the action, but they're built on restricting the number of degrees of freedom in the... In the geometry, there's a finite number of degrees of freedom associated with books. So, for example, the wave function is a zero function of the space of all the... Now, a particularly simple model in this class can be obtained as follows. If we take the surface of a five-simplex by joining together six points in five dimensions,

27:30 much as one could define a surface of a tetrahedron. So, similarly, the surface of a five-simplex gives us a four-dimensional surface in five dimensions. All of this is made up of six flat four-simplices. It has spherical topology. If you take one of those four-simplices out, you then get a geometry with a boundary, much in the same way as if I remove one of the interiors of one of these triangles here, I would get a compact geometry, which has a boundary which is specified by the edges of the triangle, and these are the interior edges of the integration to get the functional integral. In such a model, the wave function would be a function of the 10 edges which define the 4-simplex, which has been removed from our 5-simplex. A particularly simple version of that model, further down to make it even simpler, the model of the simplicial map of the 5-simplex and all, for example, all the edges and the boundary are actually related to one another and also the interior. So one can have a, what one might call, a mini-simplicial superspace model by taking all the boundary edges equal. In this case, the wave function, the mini-mini-superspace, is a function of one boundary edge length, or one edge length which specifies one number which specifies all the edges of the boundary, and the integration is over one interior edge. In this case, it's possible to write out the action in a very straightforward form. I'm assuming here a theory in which we just have a cosmological constant. The action is then a sum essentially of a curvature term, which is proportional to the boundary edge, and a constant term, which is proportional to the boundary edge. And here is the form which Reggie action gives us. There's a term which is basically integral rdv, edge length, and then there's the standard surface term that Raphael and I, some years ago, Z1 and Z2, are just simple functions of these interior edges.

30:00 The volume, of course, is one might first think of investigating the integral, the defining functional integral, by the methods of steepest descents, which involves looking for the semi-classical, involves looking for the extrema of the action, in effect solving the field equations in this one simple dimension, which now become simply algebraic equations, just the equation of the derivative of this action should be equal to zero. When one does that, one finds familiar enough. What we would then be solving for, recall, is to prescribe the boundary edge, right, and we're looking for the interior edge, which extremizes the action, and here's what the solution looks like. This is for a cosmological constant factor equal to one. Here's the prescribed boundary edge, and here's the interior edge expressing the ratio between its value and the boundary edge. If the, a value of the boundary edge, and read across here, One finds that there are two regimes. There is a boundary that is small enough, then there is a real Euclidean geometry, basically a simplicial analogy of the force sphere, which extremizes the action, but as you increase the size of the boundary, as one would expect, because bigger and bigger, eventually it would no longer fit. Beyond that, however, you can get complex extrema, the current complex conjugate pairs. And the geometries on the insides of these objects, if you calculate the signature, for example, are Euclidean, below the critical boundary edge, and they're Lorentzian, if without worrying, right, you just sort of calculated the, use these extrema, wrote down their contributions to the value of the wave function as a function of the boundary edges. Here is the, we see this characteristic structure that, the extrema, that is the Lorentzian region.

32:30 In this fashion, when you're below, you have an inputting action, and too fast to be seen on this scale. This, if you like, is the classically allowed region corresponding to Lorentzian geometries, and this is the classically forbidden region. ...of course is the classical solutions in the continuum, right, with cosmological constant equal to three. H squared are, in Euclidean regime, a four-sphere, which contains no symmetrical three-spheres, which have a radius larger than h minus one. And in Lorentzian regimes, the solutions, of course, corresponded to Sitter space, which had no three spheres which have radius less than one over the cosmological past. The question is, I tried to take a look at the question of was there a sensible contour for the integration which would reproduce these semi-classical results in the spirit of imposing upon the choice of the contour that it should at least reproduce the notion of classical space-time. One such contour for which the integral is one convergent, it's complex, for two, for which it reproduces the semi-classical result. It's in fact the obvious one. One chooses, it's the contour which is associated with the steepest descent approximation to the functional integral, and here's what it looks like. Interestingly enough, it's closed, like this. Maybe we should start down here, at the bottom of the transparency. If you take the case when you're less than... The boundary edge is less than or less critical. Then you have one extremum on the first sheet for which the action has the correct sign.

35:00 Why is there a cut? I'll refer you back to this function. Square root singularities at C, which is the complex plane I'm plotting, which is the interior edge rescaled by the boundary edge, add a quarter, a third from the properties of the size of the interior edge. This is a fairly rich, complex structure, but basically these are all... So as you wander around, I don't know exactly how many sheets there are, but they're a finite number as you just change... Here's perhaps a better... I actually calculated the contour, and here it is for what it's worth. I have a more interesting picture. This is a gigantic lemon. This is the real axis. This is the imaginary axis. The cut runs... I didn't have time to wrap it. It runs in here, like this. We go downhill as quickly as possible. Here's what it looks like if the value of S is above the critical point of the extremum or here, lying on two separate sheets, one being above this cut and one being below the cut. The important point is that they can all be unified in one, not necessarily steepest-of-sense contour, which is one contour of integration, which is closed, and they can be distorted into each other so that the wave function, which is this curve up on the top, so that the wave function behaves smoothly. This is a contour of integration for this simple model, for which first the functional interval exists in classical space-time. We might note that it has no boundary endpoints, or that although there's no

37:30 Natural analog is a simple-to-sweet case of the really good equation, at least in the context of Jonathan's talk. There is the possibility, at least, that it's the diffeomorphism constraints of the full theory, and he and I are now looking at this question. No, that's fine as far as the speaker's concerned. Anyway, our first speaker today is Chris Isham on canonical groups and quantization of gravity. He'll tell us how to do it. Old habits die hard, and had I, in fact, given one of the review talks to which John referred in his introduction, I would have probably started by saying the following, that it seems to me, if you look at what's been happening in quantum gravity over the last 20 years, really, and also a moment, there's one major question which obviously sticks out as a basic foundational question, and that's whether you see the theory of gravity as being really a gentle movement from the classical theory to a quantum theory, or actually a fairly abrupt movement from the quantum theory to the classical theory.

40:00 Most of the work that's in fact been done in quantum relativity theory falls in the first category, and I guess as far as this meeting is concerned, the two obvious and most promising and interesting results of that are that still falls within the sort of traditional idea of quantization of a classic. On the other hand, of course, the movement from quantum to classical, the obvious current there of that particular standard is obviously string theory. In which the underlying theory, as this is actually a classical theory, it's quantized, it's not the same final classical theory, it's not VR, so I think we can fairly say that's the result of this movement. Now that's fair enough, but there's one question which fascinates me, has always fascinated me in quantum gravity, and there's a question which applies to both of these, and that's that as far as the classical concepts of the Manian geometry or the manifold or hopological structure of space-time are concerned, there are two things one has to ask oneself. First of all, what is in fact put in a priori as background? Now as a background term I mean this in two ways. There's of course the technical sense, you may actually literally have a background. There's also a conceptual sense, I mean that's where I almost literally myself some of those concepts are usually unspoken and articulate and then it's made to discover again at the end. It's a prediction. So one needs to be conscious of these. It's nice to talk to an audience of friends I must say. So that's one question one perhaps would always want to address. And the second question which is a bit like complimentary for that is because the two stages of the... These are ideas I've been fascinating about for a long time, and I want to say a little bit more about it today from one very particular... Obvious examples, of course, where we have background structure in the technical sense, all of the familiar expansion methods, weak still perturbation theory, where, of course, we have both a background metric and a background manifold implicit, and also the same marks would apply to strings in a space-time. Gary, of course, will be telling us at the end of the week about methods which avoid this.

42:30 But it's clear, if you talk about strings in a space-time at all, whether you've committed yourself at a rate fast. So these are some sort of general things one can obviously... questions one can ask in quantum gravity. And my particular approach at the moment to studying this real question, which fascinates me most of all, what is the status of these concepts at the quantum level, is to adopt a technique which we'll probably regard as being slightly retrograde. I don't think he's here. Anyway, there are some members of me. Which is to really take as seriously as you can things like... Can I draw a piece in a second? Greaves really understands, as hard as you can, what it means to take a classical theory and quantise it. Not because I actually believe, necessarily these days, this is the way to do classical physics. It's almost all on there. Which is not a very... You've seen me do this so many times. What is that? Oh, the next one. We're towards the topology. I really want to understand what type of ideas are at the top of the level. And I think these sort of investigations do not presuppose any particular philosophy of what the theories are supposed to be. It could be string theory, it could be all the positive gravities you like. I don't think one is committed in doing it for a particular kind of view. At least that's the idea I would like to hear. I was taught by a great master, Bryce. No, I think there are general issues which arise which you can discuss in this sort of framework without thinking. Now, as this is meant to be a discussion meeting, I thought I would start off by just making some general remarks anyway about the sort of a priori status of these ideas of quantization and so on, what this might mean. There are lots of questions you can ask about the whole business of putting metrics, operators as metrics and so on. But I think perhaps the best place to start is by looking at what is conventionally done with weak fields. Although I'm doing this canonically, this is not meant to be a pitch, like the old battle against the various methods, but what I'm saying is that it's just as critical to part into the other side.

45:00 But in the old weak field approaches, of course, as we know them so well, one starts with some sort of background, which doesn't actually have to be, some small perturbation. Then the quantization means, of course, quantizing the deviation from the flat space, which would quantize the matrix itself. Then you get the conventional canonical parameter. This type of expansion is manifestly incompatible with G.I.J. as only a manual metric. So the interesting questions then from this point of view are, can you in fact do a quantization without putting any sort of background metrics on at all? If you have a background metric, if you do have a background topological structure, the obvious question is what role does this topological... is the topology, if you like... Now, as I remarked earlier, to defend myself, there are similar questions applied to any probe, particularly the obvious question here, And the last point of view is, can you actually get beyond the semi-tactical?