String theory without space time (contd.) / subsequent discussion

0:00 There's no exhibition at all. So perhaps this is analogous to the solutions that are not of the product form, perhaps it's a bit trivial, but I certainly expect there to be solutions of the Springfield equations which have no world-applied spacetime, which has no reference to spacetime at all. Because we want things to look like space-time on the large scale. But we're only in one particular state, right? The one particular solution which describes our universe, right? The claim would be that the ground state of the theory has to correspond to one of the solutions which does have a space-time interpretation. There's no reason why there couldn't be other solutions as well. Yeah, but you'd have to have a superposition of a whole lot of them as long as it contains one. Well, moving right along, let me... As long as I'm in battle. As long as I'm in battle. Yeah, we'd like to see some of the topology on space. Okay. There is a sense in which solutions are, well, here's gauge. There are certainly sort of gauge... Right, there are solutions, right, there are gauge transformations on the space of solutions. You can ask which solutions are gauge equivalent to other solutions. As far as I know, none of the four solutions I've discussed were related to each other, but there are certainly ones you could construct that would be dangerous for them. Are any solutions going to come from one to the other? Yes, but I gave an example. If you take a radiation-dominated Robertson-Walker spacetime from here to the Planck scale, you could write down approximate solutions that would be good down to the Planck time. Okay, so let me now come to the second aspect of the question, which is... The space-time manifold, M, appeared only in the definition of the strength and script M, and the question is, is this really essential, or can we get rid of it? And to answer that, one has to look more precisely at the space of mass that one is including in the .

2:30 I would like to just very quickly remind you of some facts about subalop spaces in order to be more precise about what class of maps I'm including. So if we first consider maps into Rn, if we just do a Fourier series, we get the Fourier coefficients x sub little n. And the definition of the subalop space is one subalop space for each real number little s. This space, the total space of weight s, is defined to be simply the collection of such coefficients which are square-subtomable with an appropriate weighting factor, which is essentially n to the 2s. So for s equal to zero, this factor is not there, and you're looking at just the space of square-integrable curves, but in general you can consider it. There's a definition of what the sublux spaces are when you consider curves in Rn. Well, of course, there's a higher dimension, the higher way the sublux spaces fit in the lower one. So you will have continuous curves in the lower dimensional spaces. But if S is not equal to a half, you can always find at least one curve which is discontinuous. No, there's lots of curves which are continuous. It includes all the continuous ones plus a lot more. Suppose we want to consider loops or paths that go into a manifold. How do we define the Sobolev space of maps in that case? Well, the easiest way for our purposes is to simply embed the manifold into Rn. You can always do that for some sufficiently large n.

5:00 And then you define the Sobolev space for the manifolds to be simply a subspace of the Sobolev space that loops into Rn, where the condition is simply that the image lie on the manifold almost everywhere. So the picture is that you have maps which take this little interval into this big space of Rn, but it's actually required to lie almost everywhere on the manifold sitting inside the space Rn. I'm doing this because I'd like to understand to what extent does the manifold topology matter in the string field theory. So I originally formulated the string field theory in terms of this task space, and then I had to ask, well, how essential is that? In other words, if I pick two different manifolds, would my task space be different? Why am I not doing it intrinsically? No, you wouldn't, but that's on the next panel. So, let's see. I should say that this topology discussion was worked on in collaboration with John Rich. So, Hilbert manifolds, simply infinite dimensional manifolds, based on separable Hilbert spaces. Basically, one can use all the standard notions of manifolds. If this index S is bigger than a half, so we're sort of considering continuous loops, it's been shown that this loop space, this path space, is actually a Hilbert manifold. Now, infinite dimensional manifolds actually turn out to be much simpler in many respects than finite dimensional manifolds. One of the key theorems proved by mathematicians in about 1970 is that if two Hilbert manifolds are homotopic, then they're diffeomorphic.

7:30 That's not something which can be true in finite dimensions. Let me very quickly run through some say amusing consequences, of course you may not find them amusing, but they're amusing only in the sense that they're not really essential for my... For the argument it follows immediately that every Hilbert manifold has a unique differentiable structure. It follows that the infinite dimensional sphere, the space of all unit vectors in the Hilbert space, is actually dithymorphic to the Hilbert space. All the homotopic groups vanish. It's contractible. It's homotopic to H. It has to be dithymorphic to H. It's a rather strange conclusion, but it's true. It also follows that any vector bundle over the Hilbert space, in particular the tangent bundle of the Hilbert manifold, has to be diptyomorphic to the Hilbert manifold. It's rather counterintuitive, but that's what these infinite dimensional spaces tell you. Well, the main consequence that I wanted to point out is that if two finite dimensional manifolds are homotopic, then their associated path spaces In particular, if you consider the path space of manifolds M and the path space of the manifolds No, no, no, no, it's a Hamiltonian, which is very different than I said. Okay, so this can be thought of as a generalization of the statement that if you take a Hilbert space and you consider the Hilbert space which is H cross H, that's isomorphic to H. As a result, as I just mentioned, if you step over a Hilbert space, then it really is isomorphic. This is a generalization because I don't require that these be Hilbert spaces and can be an arbitrary finite dimensional manifold. It's still true that these associated infinite dimensional manifold are indistinguishable.

10:00 In that sense, the string space is independent of the dimensional space. There's no regularization needed in this statement in the sense that these are rigorously defined in the financial regulation theory. It wouldn't be true in terms of the financial regulation theory. No. I'm not sure I'd... It would be obviously very difficult to do. Okay, so the string space is independent of the dimension of space-time, but as I said, all of this is really in the context of F bigger than a half, so all the strings are continuous. In that sense, we have not removed all the dependents of the space-time manifold, the continuous strings, which would be the topology of M, if not the total dimensions of the base. Now, there are three reasons why... String theory should not be restricted to the continuous strings, and time is running short, so I don't know, let me just tell you the argument without going through it, and if in the questions period you want to know in more detail, I will be happy to give you the details. What is the domain of the string field anyway? Normally we think of it as coming to you from the first quantized theory. You quantize the string and you get wave functions that live on some configuration space. So it's natural to ask what strings are included in the configuration space of the first quantized string? Well, you can show by using sort of just a standard boxed string state that the continuous strings are actually a set of measure zero in the first quantizing of the product.

12:30 So that rather strongly suggests that one should not be restricting oneself to just continuous strings. There's a second argument which says that the first quantized string is like a two-dimensional field theory, with the mitigating of the distributional things that it shouldn't be. There's no distinction at this level. The problem is going to come from, again, the solutions. I will argue that in terms of the configuration space, you don't distinguish the open string from the closed string because that allows things to be discontinuous, but the problem is can you find this sort of kinetic energy operator, this sort of cube, which will reproduce the spectrum and scatter into the closed string theory? Well, if we just... If we just go ahead and assume that we're going to work with square integrable strings, except for the fact that they're not continuous, so let's just take the simplest string that we can work with, one can show that the space of square integrable strings is in fact intractable for any manifold M. We knew that this was actually a Hilbert manifold. That would tell us immediately that it's diffeomorphic to a Hilbert space. It turns out that there's technical difficulties in actually making this argument, so we can't prove it. But in any case, since it's a contractible topological space, it keeps very little of the topology of the manifold then. The next result is that one is led to adopt essentially R-infinity, the simplest infinite dimensional manifold. To be precise, what I mean by our infinity is simply the underlying manifolds of a Hilbert space, a Hilbert space without the sort of linear structure superimposed on it. We adopt this for the string-space description, and with that sort of definition, if you like, one can formulate the string-field theory in a way which is independent of the space-time manifolds and the space-time metric. Okay, well I'm running out of time. Let me make just a few comments. It's a good thing I'm running out of time because I have no more than a few great comments about what a non-perturbative quantization of this theory would look like.

15:00 Okay, I mentioned earlier that if you found about a solution which has a metric, one can perturbatively quantize it using a background metric. Field theory. What would a non-perturbative quantization look like? Well, of course, very little is known about this. The first comment is that this theory has no initial value formulation. The product involves these integrals in one of the arguments. It's not a local action on the string space. It has no initial value formulation. There's no canonical formulation. So there's really very little hope of doing something like a canonical quantization in this theory. If you just close your eyes and are very careful not to think too hard, you can try writing down an effective action which would just follow the standard procedure you would imagine introducing some sort of J integrating over all string fields now on this big incremental manifold, instruct W, vary W with respect to the source to get a field which is defined to be phi naught, and then insert W. To get the quantum effective action. And the quantum equations of motion would then be the extremization of this effective action. Now obviously there are many, many problems with this. One has all the usual problems of gauge invariance and all of that and lots of unusual problems. But maybe this is at least a rough framework that one can try to work on. Well, to summarize, let me state some conclusions. These conclusions are all modular technical difficulties, most of which I've suppressed, but go ahead anyway. The claim would be that you can arrive at a formulation of string theory without first putting in a space-time manifold or metric. Secondly, that there is a correspondence limit in the sense that solutions of Einstein's equations are approximate solutions of the Greenfield equations in 26 dimensions.

17:30 Right, so at least in this approach, all would have to be compactified. And expanding about these solutions, expanding about one solution which has been... These conclusions have some far-reaching consequences, just to stimulate the debate, let me state just a couple of them, the implications of all this. Well, there's no spacetime, which tells you that all questions like, is quantum gravity invariant under t goes to minus t, the sort of question that Roger Penrose likes to ask about time reversal invariance, even questions like, are there fluctuations in spacetime topology, only make sense semi-classically. You don't have sort of... You don't have the topology, space-time topology, in the fundamental theory. You don't have the space-time in the fundamental theory, so you don't have time. You don't know what time diversity and variance means. So the questions just don't make sense. There is, when you consider perturbative string theory about flat space-time, but I would argue that... Not that I... Well, and finally, we're all familiar with the fact that every new theory of physics introduces its own concepts and makes other concepts obsolete. And I'd like to conclude by simply, I guess, pointing out or agreeing with Carlo Rovelli in his talk yesterday that it's time for time to go. Time is a concept which has served us well. It's not something which I think is at all. It's not something which is in string theory. Look, I mean, I formulated the theory such that classically there's no time, right? There's no space-time.

20:00 It just doesn't seem to me the right way to approach the quantum theory to try to find a time and, for example, looking for time in quantum gravity is perhaps analogous to looking for absolute simultaneity in special relativity. We're having trouble finding it because it's simply not there. Now, I know half the audience is about to jump up and say, what about interpretation of the theory? What we mean by observable, what we mean by an observation at blank times is very different from what we mean today. I think we should, I'm going to adopt Brice's view, I think we should investigate the theory, see what it tells us, and let the theory suggest its own interpretation. So I'm going to stop there and let Brice answer all questions. Class space reproduces the string perturbation theory. If we were doing this for the superstring, what would then have the arguments that this would impact the affinite bottom perturbation of the spectrum with the affinite directions? Well, these conclusions have some far-reaching consequences. Just to stimulate the debate, let me state just a couple of them. The implications of all this, well, there's no space-time. Which tells you that all questions like is quantum gravity invariant under t goes to minus t, the sort of question that Roger Penrose likes to ask about time reversal invariance, even questions like are there fluctuations in space-time topology, only make sense semi-classically. You don't have sort of topology, space-time topology in the fundamental theory, you don't have space-time in the fundamental theory, so you don't have time, you don't know what time reversal invariance means, so the questions just don't make sense. There is, when you consider perturbative string theory about flat space time, but I would argue that not that I... Well, and finally, we're all familiar with the fact that every new theory of physics introduces its own concepts and makes other concepts obsolete. And I'd like to conclude by simply, I guess, pointing out or agreeing with Carlo Rebelli in his talk yesterday that...

22:30 It's time for time to go. Time is a concept which has served us well. It's not something which I think is at all... It's not something which is in string theory. Look, I mean, I've formulated the theory such that classically there's no time, right? There's no space-time. It just doesn't seem to me the right way to approach the quantum theory to try to find a time and, for example, looking for time in quantum gravity is perhaps analogous to looking for absolute time of finality in special relativity. We're having trouble finding it because it's simply not there. Now, I know half the audience is about to jump up and say, what about interpretation of the theory? What we mean by observable, what we mean by an observation at blank times is very different from what we mean today. I think we should, I'm going to adopt Brice's view, I think we should investigate the theory, see what it tells us, and let the theory suggest its own interpretation. So I'm going to stop there and let Brice answer all the questions. Thank you. Thank you very much. Looking from the back, Stanley. I would say that the theory that I've been describing is a theory of interacting open and closed-strand, which is why we're able to change the space-time metric. It is true that one has difficulty constructing a purely closed-strand field theory. That's a difficulty. I don't know whether that's, in a sense, fundamental. Well, no, that's certainly what I mean, but after the five hours of discussion yesterday, if we had had five hours on space, I would have had five hours of space to go. Five hours of space.

25:00 One that had an A that you knew or a background metric. And also, you don't know what the ground theory is yet. This may be true. No, I think all of those questions have to do with whether string theory describes the real world. Look, we all agree that we're living in a... The universe would be very well approximated by a four-dimensional space under the Rensselaer signature and all of that. If the theory is constructed non-perturbatively and has either a ground state which has no space-time interpretation, you know, we could throw it out. It's not describing our universe. But I agree. What I would like to say is that the time, I mean, the fundamental theory is not an ingredient. Why can't I do that? I don't want to do that. I don't want time for any solution. I only want time to come about solutions that describe space time. Although I need to know, certainly I need to know why is the universe the way it is. Why is sort of the apparent, you know, we see four to nine times space time. So that would be the statement that maybe there's a unique ground state. Well, meaning one of these solutions. But that's sort of a question not in principle, it's a question about, you know, relating this observation. I agree perfectly. Look, we know so little about the non-perturbative formulation that I don't want to...

27:30 This was, um... Well, there is a way of sort of classifying the solution. This was in response to an earlier question about whether all solutions could be put into this 2F non-form. And I know that you have some ideas on that, so... Let's take people, I think, in order, if that's what you want to say. I also have a quick comment and question. The comment is that synthetic formulation is not necessary to have a mutually valid formulation between synthetic geometry. Most TLC, and there are some physical reasons for it, is interesting in fact in TLC that you know of them and so on and so forth, that's not likely to be the question. It doesn't mean that there is a big difference between them. I mean, the thing that, the viewpoint that we are opposed to, and some people come up with, have similar viewpoints, and the viewpoints, and the differences between the two, we feel that while there will be many other states which will have no classical analysis, we feel that, however, there really is a macroscopic question of space and time. And so if you went and even though on the macroscopic scale today, if you did go to the macroscopic level today, it wouldn't be so simple. Whereas in here, space time always arises as a classical metric which is trained in, I mean not trained, it's like it arises out of a classical community. And therefore, the statement is that either there are these other things in which space-time is not needed at all, or the space-time as we know it today, there doesn't seem to be room for this sort of thing like what is a microscopic structure of space. Well, I don't think it's quite fair. I think that what one would say is that if you are looking, let me try out a line of argument.

30:00 Suppose you wanted to probe the structure of space-time on very short scales today. Well, we certainly know that in string theory, if you were to do that calculation of what happens when you send in particles with very high energy, you would get something very different from, you know, just in the ordinary field theory of sending in those particles, because when the particles collide, all the other modes of the string will come into play and will affect the answer. One could then interpret that as, you know, resulting from difference in space time or something like that. But I have a vague idea. If I don't have a clear idea of what it would mean to have... You know, the sort of funny structure on space-time on small scales, which I'm saluting as well. There are a bunch of, there are a bunch of nearby metrics so that the space-time looks more like some of the, some of the spaces that we can see in. That is, it doesn't seem, it doesn't seem clear to me from the presentation anyway that, I don't know how you're going to probe them. It doesn't seem clear to me that you know, that that isn't, that that's not the kind of fluctuation that you need. Once you, once you allow yourself to put in an arbitrary metric on these spaces, you lose. Well, if I understand what you're saying, you're suggesting that one sort of take a classical metric which has, on small scales, very highly curved, you know, blank-sized wormholes and things like that. And I guess that's what I'm saying I don't believe. I think that it's not that one has classical metrics on classical manifolds with very, very strong curvature. I think the whole idea of a metric and a manifold will only make sense on these scales large compared to the Planck plane. I understand what you're saying. It just looks like some of the solutions, some of the particular solutions...

32:30 I can't get a solution like that because I need to just assume that the curvature was everywhere, you know, small compared to the Planck curvature. That was just assumptions that curvature be less than one over length. If I try plugging in one of your spaces with lots of wormholes, it's not going to satisfy my field of play. I'd like to add to the question, and then you can go out there and you can make a suggestion. The question is that the structure, as you end up with, is very general. You have a function space on a space of mass which is to satisfy certain properties. And then, honestly, you could put an algebraic structure with an anti-community function and non-community function in its place. So one can imagine many other things, not only spaces of maps or groups. It's a manifold satisfying the structure of the space of maps on the space of groups, etc. So there could be many other types of solutions to this kind of algebraic structure. Right, I agree. In some sense, in the final picture, strings have no longer played such a fundamental role. I guess I would argue that if it was abstracted even more, one still has to at least... One used to think that we had to come down to experiments, and I think quantum gravity people have long since gotten rid of that restriction. But at least you have to relate... Thank you very much. But I mean, you have to relate it to, at least some of them, and what we're trying to do here is to relate it to the perturbative string theory calculations, which agree with experiments in the sense that they do correspond to gravity. Yeah, perturbative scattering of gravitons in string theory reproduces exactly what you would get if you calculated it in Einstein's theory. Spoil the story, Steve. Well hang on, we'll make it quick for you, because a lot of people are waving their hands. In fact, when we understand the structure of this form, i.e. their space of mass of solutions to some manifold, they will have the appropriate structure. They will be a solution to perhaps the classical equations, i.e. every quantum state of quantum gravity may be a solution.

35:00 The algebraic structure will come from the naturopathic structure of the solutions, and the relationship between the string theory and the quantum gravity theory will come from the relationship between the representation theories of the gravities and the geological algebraic structures. Well, of course, it would be very interesting if that was true. Well, we'd have to wonder whether there would be analog solutions in your theory, which would have fields other than gravity. In the first quantized version of the theory, the gravitational field was just one of the many excited moments. Now, in your construction, you start with a theory that doesn't have any space-time structure, and then you construct particular solutions of that theory based on the gravitational field. There is no mention in the solution about the other field. Oh, so is the gravitational field somehow prepared? Or you can construct solutions using the other fields as well. Sure. I mentioned that you could put an electromagnetic field on, for example. You can put background fields associated with other modes of the string as well. There's nothing sort of ticked out. I use gravity because that's what people are interested in. Is there then some solution which doesn't need to talk about individual fields Well, I gave one example which was sort of just this function of the ghosts. I constructed q which had, you know, just the ghosts and no x's at all. So there's no sort of way that one could construct a space time or any of the usual components of the string field about that solution. So I think it is true that you have solutions where you don't have anything like this, usually decompositional. So all that you said, the semantics, it's a minimalism? Yeah, you can just... The string field is a fundamental object. It lives on this big manifold. You can decompose it into the usual sort of field on spacetime only, you know, about particular solutions. Only in certain solutions can you decompose it into its minimum.

37:30 Can you provide a possibility on the day you need to specify solutions? Is there anything analogous you expect in this country? I don't know if there are other ways of doing it. That's what I'm going to ask you. Or, in other cases, it's more like both of you. There's some analog or representation theory that needs to be taken into account. Well, as far as I know, there's very little progress in that. I think it's a very interesting question. And because the structure looks so algebraic, I would think that the algebraic approach would be more promising than to try to sort of go back to the usual differential equations of evolution. There's something like a representation theory which would classify solutions to ask how many there are. I think it would be wonderful to talk about that. No, okay. Richard does have definite ideas on it. I hope that you'll say a few words about it. Raphael, you can try and say it. I think there are a lot of similar lines related to classical mathematics. For you, you had, for example, for any kind of mathematical platform, you had some solutions, kind of singularities, in the sense of the normal form. Let me ask you the second question first. I mean, the large thing at the tachyon only expanded about flux space time, okay? So there's no, you know, it's not necessary that this... The fundamental theory would suffer anything like that problem. There could well be solutions which don't have tachyons, and in fact, we have some examples. This T-squared solution I mentioned is in fact one that has no tachyon in its background. It has other problems, of course, but it has no tachyon. So the tachyon is not the fundamental problem for the bosonic string, I would argue, from this viewpoint. Your first question about what happens if you take a four-field-like solution...

40:00 Clearly, the curvature is going to get very big near the singularity. I don't know of any precise results. I mean, all I have is this intuition that if you somehow restrict yourself to the region of spacetime where the curvature is big, they take maps, you know, which sort of your string is never allowed to get, you know, within a blank length of the singularity or something like that, one could find an approximate solution. Of course, you have to worry about what boundary conditions you're imposing and all that sort of stuff. I don't know. Again, one of the main methods that I got from your talk, at least from the second part, is that the formulation of the theory, which depends very, very little on the structure of the target, but you were correct in the fact that it depends very little on the target space, because... The kinematical variable is functional over this object in the linear space without being abstracted into one or two objects. And then the action is just physical power. Now, at first I'd say that we could have started from everything, just the positional parameters of this power, this movement, something like that. So, if I understood what you were saying, the physical content of that is somehow coded in this linear power. So, the question is, first I would like to ask you a comment about that. And then, how much the possibility of defining this star, this product, this function, depends on the function of groups, actually, and how much it is different from the function of groups in different targets. Well, I was careful when I first set up the theory to define my string space to be simply the space of half strings, and so that my star product never required me to use sort of any extra structure on one of these half string spaces.

42:30 The more conventional picture was to define full strings, and then you have to somehow pick out a midpoint, and you have to sort of know the right half and the left half of the string, Thank you for your time, and I look forward to hearing from you in the next lecture. That measure in your functional integral for the star product, that carries a lot of, that has to have a lot of structure reflected in the topology of the math. But that's, that's the same thing I would get though. It's the same manifold I get if I had started with objects with functional integrals. So far we've got six people wanting to ask questions. I suggest we go on for another five minutes, then have coffee fairly quickly, come back and continue this discussion. It's very interesting. Well, I suggest we go on for another five minutes, have coffee, quickly, come back, continue this discussion so no one feels like they're in the cell, then go on to the rest of the discussions back to time this morning, so we don't need to rush. No, no, no, I think a lot of people wanted to ask questions, but they weren't in the cell. Ali, you had your hand up. On this point, I think, as you know, I have an argument, semi-classically, you can get a notion of extended homotopy too discontinuously. And I would like to ask you your remarks about how that affects your argument. I guess the key point is semi-classically. As I understand it, you have a path which is actually a smooth path. You look at deviations from it, which are sort of restricted in some way that they don't. And then you say, if I sort of evolve the string a little bit, I can sort of evolve the distance in this way, and then it's sort of all given by vectors as normal. I think that extra structure is what allows you to extend the sum of something which doesn't allow me to do it, because I don't have... I'm just taking all paths with this.

45:00 No, I know, but you said square-interval abilities, which leads to where it arrives in my argument, which is that you have the expanding and deviations. You're expanding and deviations too. You're 2a, 2l, plus a. You've got a background solution. No, no, no, but it's a very different thing I'm expanding on. You're expanding and actually, in the past, I'm expanding fields on that path space. It's a very different type of... There's one more question from Bill, then we'll break the copy and turn it up. Okay, I emphasize that, but there are unfortunately a few subtle things. Things like, these fields are supposed to be Hermitian, right? And you cannot take a Hermitian matrix, square it, and set it equal to zero and get anything other than zero. So there's no way that I could realize this just in terms of five dimensional matrices. Now the thing which saves you in some sense is the ghosts in the sense that these fields not only are Hermitian but they're also Grossman values. So I think it's the combination of those things which allows us to work. Now I guess I haven't thought... That's in some sense what one gets in this discretized version. So I think the answer would be, at least you've gone from fields on an infinite dimensional space to fields on a finite dimensional space. Now, whether you can actually get it down to finite dimensional matrices, I'm not so sure. But I agree that you can at least go down one step by discretizing the sigma points using this regularization that Richard and Mark Schroednicki developed. So that your fundamental string fields now live on just Rn, the value of x for n discrete sigma points. And you can then set up the same star algebra in that case. It may literally be possible to do it not in complex, but in quadrillion. Because of the non-communicativity or something. Yeah, that would be it. Alright, thank you. Let's take a short break.

47:30 Yeah, I want to get a quick count of how many people are staying for dinner. That's the evening, right? Tonight's evening meal, yes. 1, 2, 3, 4, 5. I was saying for dinner. Again, 1, 2, 3, 4, 5, 6. I'll tell them that I've done it for ten, so it's okay. People will get the 66 lappies to achieve. And who's staying for breakfast tomorrow morning? All those people? Well, presumably. Okay, so dinner for ten, breakfast for ten. Okay, thanks. Abba, you wanted to say something about the proceedings. I just wanted to... John asked me to remind you... The manuscripts are due on August 31st. They should be sent to John at BU. I think you have the address in the conference announcements. If you lose that, the physics department will still get it. And as far as the discussion is concerned, hopefully we'll get it transcribed during summer and send it to various people if they want to make any corrections on it. Can we give them the discussion before they give it to the table? If you don't submit the manuscript, the discussion will appear without your consent. It's a good threat, I think. Okay, Gary is ready for the attack to be resumed. Don, you had a question, did you want to? I just wondered, okay, you have this classical theory, you've got the action, and I just wondered if you have some fundamental way, non-perturbative ideas on how you would quantize the thing. Would you use a path integral, and if so, what are the paths in the space? I had one quick transparency where I said basically everything I know about it. I said that the sort of straightforward canonical quantization wouldn't work because we don't have anything like an initial value formulation. Abe pointed out that one could do sort of symplectic structures on the space of solutions, which would be interesting to look into.

50:00 One could imagine doing a path integral type approach to an effective action. You would then integrate overall string field configurations. I'll try to make sense of it. I don't really know much about what the non-perturbative quantization is. Lionel Mason wanted to put it. Is he here? No, he's gone. Okay. He lost his chance. Hey, it's too late. Okay. Lionel, your time has come. Do you want to get back on the bottom of the list? That's all right. I was describing the bosonic string because that theory is simpler and I'd have to suppress less in order to present it in, you know, an hour or something. It's the super string which picks out E8 cross E8. And that's still There are a number of theories that are very much around in current discussions of superstring theory. It's been extended somewhat in the following two senses. Initially, say, four years ago, when interest in superstrings began, it was believed that there was only two theories, say, in a ten-dimensional plot space time, which were consistent, the E8 cross E8 and SO32 theory. It has since been noticed that there are other theories. For example, SO16 process of 16 theory, which makes sense and is consistent in 10 dimensional flat space. It has also been discovered that there are many ways of changing the gauge group upon compactification down to four dimensions. So you can get much larger gauge groups or smaller gauge groups in lower dimensions, and as I briefly mentioned at one point in the talk, there are these new ideas about how to construct four-dimensional string theories in which you don't have to mention internal sort of classification. The theory is set up with only sort of a four-dimensional space, and those theories have lots of different possibilities for the gauge group. So if I could just, you know, my reaction to all of this is that, you know, various claims to uniqueness of string theory, I think, are very premature. So the more we understand, the more we learn about possible classical solutions and all of that.

52:30 It's still conceivable that there is one, you know, grand superstring field theory or some other non-perturbative formulation, and all of these different theories people have constructed over the past couple of years will be viewed as just different classical solutions. That's not been shown yet. And even if it was shown, one could ask, well, why am I working just with scalar fields on this string space? Why not, you know, spinner fields or vector fields or tensor fields? So I think there's lots and lots of possible ways of extending string theory. My interest in it is not so much uniqueness, but that I believe it will provide . But he didn't give the answer to your question. The reason he didn't apply the method to that theory has only caused string. The string field theory has never been applied to string theory. Well, the SO32 theory is a theory of open strings, right? There's a type of open string... Oh, there is also an SO32 open string. Sure. That's correct. But not E8, then. But not... that's... Yes, no, sorry. Spent your time. I'm slightly disturbed by the fact that... Compared to the Planck scale, right? That's the radius of curvature being the Planck line. Does that mean that when the solution is exact, it becomes the zero? I'm not able to understand from... But essentially, you said two things. The first thing you said was that I could use that because if the solution is exact, it becomes the zero. I mean, plot space is an exact solution in 26 dimensions. You can find other approximate solutions, but I think that's all we would expect. I mean, classical space time is supposed to be an approximation, so I'm not upset that it's giving only approximate solutions. If there is some condition which is a huge, awful mess to write down, which, if the curvature is very small, looks like g mean u equals zero. That's what you mean by approximate. There is some exact solution to which this is an approximation, and that exact solution has bottom zero curvature. That's right. I feel very uneasy about the apparent-disapparent structural structure in this, and so I'm looking for a place where that structure might be hidden, and a good candidate place would seem to be the choice of the space of densities of way to half that you're working with in field theory, and presumably if your choice of collections is potentially equivalent or equivalent to a choice of measure on the Silbert map, or if you're dealing with them around the other zillions of units, which they certainly are.

55:00 It seems that to make the theory, you really have to pick one out, and I can see where you can pick one out using the space-time structure that you started with, but if you're going to throw that away, I guess I'm concerned that the view is something that hasn't been specified in terms of this, or you really have to put back in the space-time structure and the equivalent structure to get out the theory. Well, I certainly agree that that's a valid worry. I've thought a little bit about how to mathematically define these functions that I need to work with, and really the only thing I know at this point is what I mentioned during the talk, this lattice regularization that Richard and Mark Trednicki proposed, and that, of course, does give us a way of defining everything precisely for finite n and arbitrarily large n and things like that. It's not satisfactory, sure, I mean I agree it's not satisfactory. Yeah, I agree one has to look into that. Let it be a manifold, but just sort of have a funny topology? The arguments I gave were showing that for any manifold, so even if it's different manifolds... So you're not changing the differential, but you want to change the topology on the inner line of the point set. Yeah, like that. Yeah, so suppose it's a small torus. Suppose an animal is flat, but it's torus about the same length. What would we get? Um, okay, you want to take a spacetime which is a flat torus at the same length, in 26 dimensions.

57:30 Yeah, that's a solution. That's a solution. That is a solution. Like there are some other people in the audience who should ask me a question. You know, what do you do in the coffee room if you don't know what to ask the speaker? One of the things that you have to make an intention, many of us for the first time are understanding something in detail, but... Could be a dangerous move, that. The one is they feel the problem. Namely, what one has done is, I mean, you had basically stated before, but what you did was, you started with this general form framework, and then you gave some solutions. You look around at some of these space-time creatures and stuff. There must be some rules and rules of other solutions, which also come from space-time. For example, suppose you have started as your candidate manifold of space-time, but with the connection. Again, one might have worked out some things, and one might have found that they used to be flat and stuff like that. So, in what sense, is that true? And secondly, another illustration of this is that one would require even further principles in terms of, you know, further something, you know, Yeah, I think it is true. In fact, various people have looked at torsion. There is an anti-symmetric second rank tensor as one of the modes of the string, and you can put that on your background space time and couple the string to it, and it acts very much like a torsion, adding a torsion to the connection. So I believe there are solutions with that non-zero. What that means is that if we are going to, you know, this is all again sort of relating it to, you know, ordinary four-dimensional relativity. Yes, I think one does need to have, need to show that not only sort of is the natural ground state or whatever for effectively four-dimensional,

1:00:00 but that the background torsion is zero. I just want to get a range of possibilities. I want to put that in as a background structure, construct a few from there, and I might say that is where I'm satisfied. These possibilities are what it is. At the moment, the horizon is wide open. So these are complex, half-flat, reaching-flat space-times. Is that what you're thinking of? They're also half-flat. I'm a little bit worried. I don't know if complex metrics would affect this argument about no calculations that I know of have looked into that. It might turn out that this argument that it is a solution if the Ricci tensor vanishes breaks down if your metric is somehow intrinsically complex. Now, continuing in our baseline, the fields which we know, semantics particularly, propagate on a space-time background. Can you construct solutions which are based on the fields without using actually the metrics? You did it for your ghost variables. Now, you make them seem similar to the features. Okay, if I understand you, are you asking the question whether I can find solutions which in some sense have a background electromagnetic field but no space time, no metric, okay. That's actually exactly the question that Bryce asked me right after my talk, okay. My answer to him was, I think that's a great question to look at, and I plan to look at it, but I've never thought of thinking of it before. Okay, so I don't know whether there are solutions like that, but I don't think it will be that hard to...