Loop algebras & loop-based representations in quantum gravity; subsequent discussion (last part) (contd.)

0:00 The question about physical problems, where you won't get completely mired into really hard and foreign mathematical questions, is this very problem? I mean, I suggest that I or someone... That's right, terms of state, terms of function of the root state. That's right. And you say, gee, doesn't that mean that we've shown that we've got a solution? And my response is, no, that doesn't. And the reason why that doesn't mean that we've got a solution is that that thing itself, say we did go ahead and put an inner product on its largest basic state, and we don't want to say we did, and we took its inner product, that state itself might diverge. So what you're going to suggest is that the... That is, presumably, when you say the difference between the power of time and the state, that's just the leading term. There's some very complicated function of delta. It's really equal to... What I propose to you is that it's very possible, and exclusive examples are known all over, every single anomaly that exists in human theory is of this type, where when you do an inner product in a given set of states, the things that you thought were subdominant terms wind up diverging on you in such a way that they cancel out those apparent things that are going to zero. Um, okay. It would only help, I think, if this was looked into and these things were carried out.

2:30 So, first, I think it's a serious issue. Second, there are really two other things. One of them, this delta comes from the regulator in the operator algebra. All of these terms act with a constraint. It doesn't know about this regulator. This regulator is in the operator algebra. And for every value of the regulator, there is an algebra of operators which act on the space of functionals. So if you're going to give me an inner product on the space of functionals before I apply the constraints, that inner product If it's going to be, it seems to me, I don't see how that inner product is going to know about the value of the regulator, because it's got to be that for every value of the regulator, there is this operator algebra. Well, there's got to be, we've demonstrated. So, I think it could be looked into, but where the inner product gets the inverse powers, the additional inverse powers of delta, is mysterious to me. I don't see it. Sure, of course. And you might say, oh gee, it's obvious that that's finite, and maybe it is obvious that that's finite, considered as a functional of this path, but we're going to take an inner product of it that's involved in some kinds of integrals, maybe not over a path, but over something else. And that's going to turn it all into a C number so that the only thing left at the end is delta and kappa or whatever other dimensions or constants there are in theory. It's not at all clear that doing that inner product doesn't result in... There are a number of things that come in like one over delta, and in fact every single anomaly that exists can be set up to have just exactly that form that you want, that is naive, that you've written down there, that is naively, it's equal to something that's going to zero, definitely take an inner product, the inner product diverges badly enough so that the whole thing is some finite constant or constant. No. Okay. Let me make my second remark. And then, okay. Yeah. Perhaps this should be your kind of final message, right? That it looks like it's not very...

5:00 So I just, I mean, the example that you talked about, there are the criteria, so which is a product for you from this perspective? I mean, are you saying that we just consume, or is that expected? We just consume all that? No matter which in the product I include, no matter which in the product I include on the space A of E, it will always be the case that when you take the limit, you get an amount. And so the whole question is a different question now, and I just don't know where to begin now. Right, but that is because one of the criteria, and here I don't see any criteria. That's why, I mean, I agree with you, but I don't know how to even begin talking about it. One way to begin is to examine it. I agree with that, but I don't know how to transfer that music to your text. If you want to learn more about us, you can go to our website. Why not apply this to mathematical prices based on the values used in the model for mathematics and physics? It's a good story. Why not apply it to the laws of physics? It's like, you know, there's a couple of things that you can do to have some basic data that you can still do on paper, but we've got some sort of studies to show something there, which are assigned to be used in various kinds of programs. And remember, there's a field of pediment, so there's a lot of money. Oh, I see.

7:30 Yeah, I think we're really running over our time now. John, do you want to stop the next one two minutes late? Yeah. About ten after five? Yeah, that's up early. So I'd like to conclude. Is this yours, Lee? We begin by apologizing for what undoubtedly will be a rather elementary talk. In which I will not certainly not attempt to solve any problems in quantum gravity but I promise also not to pose any new questions. Certainly no new questions will be posed in this. Actually my purpose is twofold. First of all I'd like to just simply describe in the matter of a progress report some work that Carl Kukas and I are doing which Use some rather old ideas about quantum gravity or how one might approach the quantization of gravity in conjunction with some new techniques that have been developed recently by Carl in applying them in a very simple situation, namely the quantization of a single relativistic string. This leads me to my second purpose, which is simply to persuade you that the relativistic string in its first quantified setting is an excellent simple model, an excellent paradigm. By which one can use it as a proving ground or a testing ground for many conceptual and technical issues that one might like to study in a much more formal context of quantum general relativity. So, if you'll pardon the pun, I guess I would like to suggest a dual role for the string in quantum gravity theory. First of all, of course, we know that it's providing us with a framework which has the potential to define in some sense what quantum gravity really is, as Gary Horowitz undoubtedly will convince us of. In the near future. But also, as again, much in the same fashion as the simple relativistic point particle has served as a nice paradigm for many ideas, I think it's time now to start applying some of these ideas to the string itself, where one still can do some calculations that have some rigorous meaning, but now we have something that's slightly more non-trivial, a slightly more non-trivial, generally covariant system. To see why this is so, let me just, and also to set up some notation, let me simply just give you a 30 second sketch of

10:00 The Hamiltonian structure of the string. It's very much just a theory, the classical theory is very much a theory of general relativity coupled to scalar fields in two dimensions. We have an action that describes the string, which has a two-dimensional Lorentzian signature metric on some spacetime, which for a closed string is the simplest case for this procedure, is just a cylinder topologically. We have some scalar fields, which geometrically are the embeddings of this two-dimensional manifold into a string, or into a Nicosi space or target space. The reason why this is very much like a general relativistic theory is because to get the field equations, we simply have to vary both the metric and the scalar field with the string embeddings. That is to say, the metric is varied, it's not just some fixed background. Consequently, the action does have an invariance under the usual action of diptyomorphisms on these tensorial objects. So, if we go to the Hamiltonian description, we unavoidably end up with something like the Hamiltonian and momentum constraints in general relativity. Here's the Hamiltonian constraint. It's quadratic in momenta. It has a potential term. We also have a momentum constraint, linear in momenta. And these constraints have their usual geometrical interpretation in terms of deformations of Cauchy surfaces. One nice feature of two dimensions is that the surface deformation algebra, in fact, is a truly algebra, unlike in higher dimensions. In fact, just to be specific, the Lie algebra is two copies of the Lie algebra of the diphthyomorphism group of the circle, which is identifiable with the Lie algebra of the conformal group of isometries of the cylinder. So this is the basic structure, and as you can see, it's very much like a simple model for general relativity, and so in any attempt to study the quantization, as people often do in string theory, many, many of the familiar issues and programs arise. And before I get into all the detailed nastiness, let me simply try to amuse you by giving at least a partial list of some of these programs and issues. So, we have, of course, a canonical Dirac-type quantization, a constraint quantization. The constraint there is usually called a covariate quantization. It's not quite what you might have imagined for a conventional covariant quantization of a general relativistic theory, mainly because we do have this conformal grouping giving a projective representation, so things get a little funny, as you probably know. We also have what I'll call an ADM quantization. This is a quantization where one chooses a time in a space variable and deparameterizes. This is usually called the light cone gauge.

12:30 We have what is the battle and fratkin' Bilkovitsky formalism that Jonathan described for us, where one works with an extended phase space, even classically, including these ghosts. This is usually called BRST quantization. Of course, we have Euclidean functional integrals. We have something that's very geometric, like a two-dimensional quantum cosmology, if you like. And so you might think of this as a model for the Harvard-Hawking program. If you put a blend of many of these ideas together, you eventually recover something that is akin to Tyler Warren's proposal for quantum cosmology using proper time quantum mechanics or some generalization. I've even seen papers that have twistor quantization of the string, although I certainly don't understand that. Okay, so these are some of the programs I encourage you to add more. And some of the issues and features, of course, that arise in any such programs are something of this nature. This is just a partial list. Of course, it's a geometric theory, so we have some notion of dynamical geometry. We even have, of course, for other reasons than what one might try to do on gravity, but one certainly has non-trivial topologies being considered and quantizations on space times with non-trivial topologies, both at the pathometer level and at the operator level, which is very interesting. Of course, there's an interesting issue that the Euclidean action is not positive because of the Lorentzian character of the target space, so one might find some use of that issue in understanding the same issue in quantum gravity. Of course, then we have a whole other set of issues that one can try to explore, like the relation between Euclidean and Lorentzian quantizations, as we both well studied in the string case. You have Dirac versus ADM, which also there's an interesting comparison available there. We have what I would call a parabolic versus a hyperbolic. Yes? Yes, that's right. I mean, traditionally, really, what people do is they rotate everything, so the thing really is positive when anybody does a calculation of string theory. But in an algae, what happens is general relativity, where one just rotates the space-time and ignores what the eternal time is doing, in some sense. There is no rotation. Do you mean by rotating the time variable in the target space? I quite agree. No, I quite agree. The time direction of the equations identifies the external and the internal.

15:00 I quite agree. No, I agree with that geometric interpretation. You can embed it anywhere, any place you want, but I quite agree with physically what should happen is you should use Lorentzian, all Lorentzian, and all Euclidean. I'm just saying as a model, this is something like what happens in general relativity where we don't have this target space to give us any physical intuition about it. People who lecture on string theory, they'll spend the first few lectures with the book. Why don't you start looking at these things and doing a quick rotation and ignoring all these? Well, in some sense, this is my Euclidean versus Lorentzian dichotomy, which perhaps could be studied. I mean, I think it's not clear how the Euclidean and Lorentzian, at least at the operative level, it's not clear at all how these things relate. An important issue, and these, in fact, the issues in red are things that I hope that Carl's work and mine will perhaps be able to shed some light on in this very simple context, is parabolic versus hyperbolic. The parabolic approach is basically where one has some preferred time, maybe it's a many-fingered time, but the basic dynamics are controlled by functional Schrodinger type equations, which is something we're all familiar with. Hyperbolic approaches, I simply mean things like the Wheeler-DeWitt type quantization or Fine-Gordon type quantization. These are two ways that one might try to quantize any general covariant theory, and it's interesting to see if we can compare the two. In this latter approach, the Wheeler-DeWitt or hyperbolic type quantization, one can... The string has a lot of symmetries, classically, and one can try to see if these symmetries tell us anything about how one might construct a single universe, a single universe quantization. Symmetries are thought to have been important to do that, and of course ordinary general relativity has no symmetries as it's conventionally formulated. Because of the relative, now this is a relative term, simplicity of the classical system at least and its quantization, one might be able to investigate the role of ghosts in the BFB formalism in the construction of physical space, maintenance of Hilbert space, structure, etc., etc. Also, and an important feature of what I want to talk about is to, we also can have relative simplicity of the theory allows us to study, to some extent at least, the status of the diffeomorphism group, or actually it's the algebra in the quantum theory.

17:30 What role it plays. Now I'm not talking about diffe as one. I'm talking about full two-dimensional diptychomorphism group, or it's the algebra. Okay. Let me just sort of outline briefly what we're trying to do. And as I say, it's based on some techniques that Carl has developed for quantizing parameterized scalar field theories. And it's a form of direct quantization. In fact, it's a parabolic approach. And the idea is to quantize via some form of operator representations of the algebra, the two-dimensional diptychomorphism group. Classically, one constructs classical co-moments of that leality. That is, functions on the phase space would serve to provide a homomorphism from the leality of the dimorphism group into the phase space. And this actually relies heavily on some techniques that Karl and Prezaisi developed. Having gotten a handle on what these things should look like classically, one can try to turn them into operators. So we'd like to represent these as well-defined operators on some linear space which may not have an inner product on the totality of the space. Interproducts are very important in this formalism, but inasmuch as at some point we're going to actually identify some of these variables with time and space quantities, one has to be a bit careful about what one is using to define an interproduct. If one can represent this linear algebra in the quantum theory, holomorphically, then one can try to implement dynamics by calling your physical states states which are effectively diffeomorphism singlets, so to speak, that is states which are annihilated by these diffeomorphism Hamiltonians that generate the action of the diffeomorphism group. Once we've done that, these are effectively functional Schrodinger equations, one then can take the space of solutions of this and try to impose some sort of Hilbert space structure on them. So, of course, if these are functional Schrodinger equations, then we're basically playing familiar games in the functional context. And that's the basic strategy. Now, there's roughly two ways we can implement this strategy, corresponding to two different ways that one might try to represent the real algebra of the two-dimensional diffeomorphism group on the phase space of the string. It is, again, applying some techniques of Haisham and Koukos. The strategy is to extend the original plane space, this is classically speaking now, by the addition of embeddings. And these embeddings are the locations of the Cauchy slices, which are just circles, and these circles in the space-time itself. These embeddings and their conjugate momentum, which I call pi, can't be introduced ad hoc.

20:00 One actually has to introduce them in a way in which they're closely locked in to the space-time, which is two-dimensional space-time metric, through some sort of game-fixing conditions. These gauge-fixing conditions, convenient ones for the string, are conformal gauges and harmonic gauges, and they also have some generalization to general relativity. This generalizes quite nicely to the Gaussian coordinate conditions, and this generalizes in general relativity to the harmonic gauge again, or the Dodger gauge condition. That's a technical detail, anyway. We introduce these embeddings, and having enlarged the facelift, it's quite straightforward to... Based on this work that's already been done, it's quite straightforward to represent the two-dimensional Li-algebra and the diffeomorphism group on the phase space. To do that, we simply take a vector field, which is to represent the element of the Li-algebra. We restrict it to the embedding, Z, which remember Z now is an element of the phase space. We use that function on S1 to smear some functions here, H-alpha. These H-alphas have the form of a term which is simply the embedding moment itself. Plus some combinations of the original Hamiltonian and momentum constraints, and these combinations are functions only of the embeddings, which are kinematical type variables, not dynamical variables. Or if you like, it's a function which is just the embedding momenta plus some combination of what you might call the energy and momentum densities of the sphere. When Fein's-Wohling constructs these objects, these are well-defined functions on the phase space, that they do indeed polymorphically represent the, the algebra of the two-dimensional, but they're more presently. And it's these objects that we want to try to see if we can realize them as operators. But first of all, I should point out that these diphtheromorphism Hamiltonians serve to evolve initial data along the flow lines of these vector fields and give us the correct dynamical evolution. And to get physical solutions, we of course have to impose some constraints. One of these constraints at the top here is simply the vanishing of the diphtheromorphism Hamiltonian itself. And we also, of course, should impose the original Hamiltonian momentum constraints. That initial data, when evolved by the action of these Diphyomorphism Hamiltonians, evolves to give us conventional classical string solutions. Now, this could all have been thought of as heuristic, just motivating the following construction. We'd like to represent the Diphyomorphism Hamiltonians as operators on some vector space. And as I say, this vector space need not have an inner product. We basically make a heuristic substitution, because this, as I say, is just heuristic, the classical aspect, but we basically let the momentum operators conjugate to the embeddings act by variational derivatives, a single variational derivative acting on functionals of the embeddings, so part of the wave functions or part of the would-be wave functions are functionals of the embeddings, and we have this variational derivative. We then must make the energy momentum fluxes, or energy momentum densities, well-defined operators, and what we do is we basically use a Fox-based description.

22:30 We represent the string dynamical variables as you would represent variables on a fox face. Now, as you might well imagine, we have to perform some normal ordering. That's one way to make, at least in this simple context, that's suitable to make these, this operator well-defined, ultimately. However, it turns out that if we want to represent the two-dimensional E-algebra and the diptychomorphism group in this way, that this normal ordering is not good enough. It renders the operator well-defined, but it's not good enough to get the homomorphism. In fact, what one has to do is perform an additional reordering of these operators, as Carl was showing in the parameterized case, and this reordering amounts to performing a reordering that depends on the type of embedding or the type of Cauchy slice you're on. So a given observer or a family of observers, if you take an instant of time, that's some slice, and depending on what kind of slice you have, that determines the ordering that you must put here. And the effect of that is to add on a potential term, and that potential term is simply a function of the embeddings only. I probably won't come back to that, but I will say yes and no. I'm not saying that I can quantize the string in a manner which is anomaly-free. No, that's certainly not true. In fact, what I'd like to show you is this quantization scheme reproduces conventional string results. I would say that I do have an anomaly-free representation in this sense of the two-dimensional diphtheromorphism theory, or Lie algebra, thereof. Well, there may be an interpretation that implies that, but I don't see it happening. I don't see it happening. The ghosts haven't been included at all. In some sense, it may be true, but I don't know how it would work. Psi, the wave function, is a functional disease, just an ordinary functional disease, nothing fancy, and also an element of the Fox space, based on the string variables themselves.

25:00 Well, that's something I actually wanted to ask people while I was here, because this formula actually looks very similar to some string field theoretic things people have done. I'd be very interested in talking to you about that if you have an idea. At any rate, since we have done this, we can impose dynamics by imposing the functional Schrodinger equations, which amount to saying these diptyomorphism Hamiltonians annihilate physical states. And this is just a conventional functional Schrodinger equation, but we have this quantum correction here. I should impose the rest of my constraints. I should impose the Hamiltonian momentum constraints themselves. I really haven't done anything yet, except impose an elegant way of imposing dynamics. Having formulated these diphtheromorphism Hamiltonians, the Hamiltonian momentum constraints themselves serve as generators of asymmetries within them. Not as really dynamical generators, but as symmetry generators. And this symmetry is the conformal group. Or at least these things generate the Lie algebra. As we all know, when we try to represent a conformal group on a Fox space, we end up with a projective representation, and we can't avoid that here either. So in fact, if we demand that all the states are annihilated by all the Hamiltonian momentum constraints, we end up with the usual statement that there is no such solution. So at best, what we end up with here in this formulation is really, it may be, an elegant rewriting of conventional string quantization. As I say, I've seen this formulism actually in a proposal for string field theory that I... I don't even know what they do right let alone how it really relates. This is due to Rubin and Das, I think. They do something like this. This is sort of a more covariant treatment of the same thing. So this leads me to method two. So we're getting more than halfway over here. Now the idea is to really get at the heart of the matter, which is that just like general relativity, or we think just like general relativity, the string is an already parameterized system. Okay, the time and space variables are already there in the theory. So we try to extract them explicitly from the phase space and use that to define the diphthia, morphism, Hamiltonian. So let me move on a little bit more quickly. The idea is, the essential thing to do is to find a canonical transformation on the original phase space with actually a slight extension for technical reasons which doesn't really play an important role, I don't think. We find a canonical transformation that partitions the original variables into kinematical variables, embeddings in their conjugates, along with some dynamical variables.

27:30 And these two sets of variables, of course, can be treated quite differently. Right. Then we, now we have, again, the same structure we had before. We can play the game with the diffeomorphism Hamiltonians and take the same form, but now these pi's, of course, are rather complicated, not too complicated, but non-trivial functions of the original phase-space variables and, in fact, they mix coordinates and momentum together in a non-trivial way. Physical initial data, classically, that are evolved into physical solutions by the diffeomorphism Hamiltonians must satisfy the fact that the diffeomorphism Hamiltonians vanish. This, in fact, is equivalent to the vanishing of the Hamiltonian momentum constraints themselves. So, in fact, what we've done is found a way of explicitly representing the diffeomorphism, two-dimensional diffeomorphism, in terms of the original Hamiltonian momentum constraints. There's also an extra constraint which has to do with this technical enlargement that was just a technique. Right. So now we can go ahead and quantize exactly as before. The embedding momenta are variational derivatives. That's all they do. They just take functional derivatives in a classical way, really. We have to choose some ordering for these factors, and we do a normal ordering again. And in fact, this normal ordering can be thought of as using a particular conformal vacuum in this two-dimensional theory. We find that, again, that is not good enough. One cannot... To get a homomorphism using that ordering, one has to perform again an embedding-dependent reordering in a very non-trivial way, and one ends up with a potential term that represents the effect of it, yes. In terms of these variables, I can't recall because it's very messy. It's fairly non-trivial. It's probably non-polynomial. If I remember correctly, it's something like a spatial derivative. Probably correctly if I'm wrong. X1-alpha again is a fairly simple function. X1-alpha is a simple functional of the original embeddings. K is the extrinsic curvature for the slice, and that's a functional again of the embeddings, because you can use the embeddings to define these things. So this is really a scale of what would be a space-time index. That's right. This is the part that corresponds to the non-trivial part of the model. That's right, that's right. If you go to the privilege foliation, everything just becomes ordinary quantum field theory. It's a completely invariant scalar field propagating in a two-dimensional space zone, so nothing fancy.

30:00 Yes, in fact, this compensates for the original anomaly, but it just amounts to reordering this in a fashion that depends on what slice you happen to be on. So, I'm almost finished here. So, oh yes, so now we can impose the constraint here. We get the functional shortener equation. Everything goes through as before. A set of wave functions which we know how to interpret more or less. If you look at the status of the conformal symmetry group, which now is represented as functions of the root of now what I call the dynamical variables, one finds again you get the projected representation of the conformal group. But now because I've actually imposed all of the constraints I need to impose, this doesn't play a role in the construction for the quantization. So since I'm out of time, let me just get to the end. Well, a direct many-finger time quantization is possible in this very simple model. Well, alternatively... We've found a way of representing, not in a very tight sense, but we've found operator representations of the algebra to the eventual lithium morphism group for the string. To do it, we had to do some things that people had always thought would probably have to be done anyway in the form of gravity, and this is how it works. You definitely need to choose time and space. You need to have a canonical transformation which does that separation. This canonical transformation mixes the phase space, the original phase space, non-trivially, which is up to variance. So configuration space alone is not good enough. We really need full phase space. Loosely speaking, you could say this is a non-standard polarization that's being employed. Loosely because I'm really not using geometric polarization. There's no inner product being used here. Also, even having preferred time and space variables, even in a many-fingered form, was not good enough to make the theory respective to the amorphous invariance, if you like, in the manner in which I've shown you. One had to perform a rather non-trivial factor ordering. Obviously, there are other conclusions one might try to draw. Well, in this second version, those variables are functions of the original string phase space variables. So, in fact, this is very much tied into light cone gauge quantization. I didn't want to say we didn't just take up too much time. Those variables that are serving as the many-finger time are just the light cone variables of the string.

32:30 And the momentum mixed up in a very special way. Anyway, there are many questions you can ask about this, but two familiar questions that immediately come to mind are the following. Of course, we've had to choose a time, and the choice of time is not unique. And this is a sufficiently simple formalism that one might get a handle on how this theory behaves as we change our choice of time. In particular, the formalism goes through completely unaltered if we simply redefine our time by the action of a conformal isometry on the normal. Because it's basically a conformal time, nothing changes. But a much more interesting transformation arises as follows. The simplicity of our choice of time in this model is simply that it's related to the original phase space by what is effectively a linear canonical transformation. It's a very simple transformation. It's not quite linear because you're rescaling, but it's fairly harmless. So it's basically the very simplest canonical transformation you can perform. However, even in that restricted class, there is some freedom, and that freedom corresponds to the symmetries that we have classically of performing Lorentz transformations in the target space. These Lorentz transformations have the character of rotating what we originally thought was kinematic into what is dynamical, and vice versa. So the symmetry group that we have classically, if we could implement it quantum mechanically, would somehow below this distinction that I've created between kinematics and dynamics. So it's an interesting question, and one we don't have an answer for, and which we are working on, is to understand what is the status of the Lorentz group as a group of symmetries in the quantum field. Secondly, and a related question, yeah? It's not clear. It may not be. It may be that we end up with, it is a critical dimension, or it may turn out because this embedding potential, which is partially, at least this time, is a highly non-trivial functional of a non-Lorentz invariant combination of the string phase states. Very non-trivial. And so it may be that certain generators of the Lorentz group will not even be symmetrically conserved, not because it's in motion, it's not symmetrical. I thought it did. I think I, I know I can answer that. I don't know what the answer will be. I don't know. I will be able to tell you if Lorentz was implemented under what conditions. That's an item you knew. Yes, I mean, it's quite possible. No, I haven't tried it. I haven't tried it. This is right where we are. So it's a matter of just, as I said, it's just a progress report. What state is it?

35:00 Well, the problem is that to solve this equation in the Heisenberg picture is one thing. You can do it. No, no, no, but this is, this is, these, it's a transverse. It's the box space of the light cone gauge. That is the physical space in the Heisenberg picture. Yeah, okay, so the only other question which is related to this Lorentz invariance question is simply how does this hyperparabolic approach relate to a Winnie-Dewitt type approach to manifest Lorentz invariance as present. And let me just conclude by saying these are questions that... The main result is that these are questions that can be answered in a very precise way and hopefully illustrate in a simple context some of the ideas that we would like to apply in a more formal context to quantum gravity.