Wormholes and time in quantum gravity (last part) (contd.)

0:00 What I would like is that we start off the discussion period by continuing the, really what was happening yesterday afternoon, the discussion of time, because many of the people who spoke then, even the speakers, didn't actually know what they were saying, so we maybe let that run on for say 45 minutes or something like that. Then three other people have indicated a desire to give short 10-minute contributions. If anybody else wants to do that, could they let me know over the coffee break so I can slot it into an orderly structure? Okay, fine, let's get going then. It's a great pleasure to introduce Gary Horowitz, who's going to tell us about string theory without space-time. Well, when I was preparing this talk, I was given some advice, which I thought I should share with you. So you see, Shelley, orbifolds arise in the theory of the monster group. The moonshine module can be viewed as a toroidal orbifold of a leech lattice. Of course, Shelley, all known physics arises. What they hear, blah, blah, blah, Shelley, blah, blah, blah, blah. I'll leave it to you to decide again. I'm happy with that. I'll leave it to you to decide how well I followed that advice, and also which way I've interpreted that advice. Let me begin with just a very general remark. In any unified theory involving gravity, a fundamental four-dimensional spacetime doesn't exist, even classically. This is what we mean by a unified theory. A familiar example is given by fluid-supplying theory.

2:30 Say just n-dimensional general relativity, the solutions are n-dimensional manifolds with 3-2-flat metrics, given one, you cannot in general extract a four-dimensional space type. It's only for particular solutions, for example those of this product form, say m cross k, that one could then extract a four-dimensional metric. And as you know, in the usual procedure, if K is a compact space with symmetry, you get Yang-Mills fields arising in the four dimensions, and that's why we use this as a unification of gravity in Yang-Mills. But in the fundamental theory, you don't have four-dimensional gravity, you don't have Yang-Mills, you have just n-dimensional metrics. Three-space could be a two-dimensional field, like a sponge, which builds up in the three dimensions. I think that the point I want to make was just that unification means that you sort of don't retain the identity of your starting point, that you have combined it with some other fields and some more fundamental object. This is all classically, I emphasize. There are of course other arguments familiar to all of you about why in the quantum theory one expects our usual notions of space and time to break down. So one is led to look for something to replace space-time as a starting point of physics, something more fundamental for this unified description, and I'd like to discuss that in the context of string theory. The clue to Klein is nothing. I'm not going to push that any more than I already have. I don't think that's... Okay, so the usual picture that one sees in most string talks is that one has some fixed background space-time and one discusses the motion of strings say joining and splitting and joining in this fixed background space-time.

5:00 It doesn't look very unified, and of course it's not, right? Not only have you failed to unify the metric with all the other modes of the string, you've in fact taken a step backwards by decomposing the metric of general relativity into a kinematical background metric and the dynamical part of the gravitational field, namely the mode of the string which corresponds to the massless spin-to-field, which people interpret as the graviton, which you then compute the scattering amplitudes of, okay? So this is not a very unified picture but in fact it's not supposed to be. These pictures, and in fact all the calculations that are associated with it, describe a perturbative expansion about this background spacetime. So one uses this approach to calculate the scattering of strings perturbatively about the background. To get at the issue I mentioned a moment ago, one of course needs a non-perturbative formulation of the theory. Now, there's not yet universal agreement among string theorists as to the best, the most advantageous, non-deterbitive formulation. Various ideas have been proposed. The one which is perhaps the most developed at the present time is string field theory, a second quantized field theory of strings, and that's the approach I'll be using here. Given talks on this, I've tried to, well, I've started with the first quantized string, namely a single string moving in a fixed background flat spacetime, motivated how one would go to a second quantized string field theory, introduce an action first proposed by Ed Witten, which still retains information about the flat spacetime background, and then discuss how one can remove that background spacetime metric. I mean in the interest of time what I'd like to do today is to just reverse the order to start off by presenting the theory in its final form or at least in the form that we understand it today. I do that because I'd like to save some time to talk about some more recent work in which

7:30 One can ask whether there's dependence on the space-time topology as well as the metric. Space-time, of course, involves the metric as well as the manifold as well as the metric. And I would like to argue that one could formulate string theory in a way which is independent of both the manifold and the metric. But let me do this in two steps. So let me first assume I have a topology and see how one might define a theory. One starts off by considering a path space. So this will be open strings. So I'll consider some open interval, say 0 to pi over 2, and I'll consider the space of all maps from that interval into a manifold M. So this is, you can think of this as a space-time manifold, and as I just said, I'm going to argue later that the theory is actually independent of that. A string, see this is supposed to represent the configurations of the string. The two dimensions come from the evolution of the string in time. But for the string field theory, the domain of the string field, as the domain of any quantum theory, are the space of configurations. So you're supposed to look at sort of all possible configurations of the string. So, it is a one-dimensional... M is supposed to be... Yes, you can think of this as space-time, but not within any metric structure. I didn't say time, right? I only said time... Well, that was only because Bryce was saying two dimensions. I was... In order to define the string field theory, all I need is past things. I don't need to talk about moments in time. I don't need to do anything. I just... Here it is. What structure do I use? Is it more than one second? I don't know anything. Well, I'm using sort of a manifold structure if you want. Do you play with a manifold?

10:00 Differentiable manifold. But then you have to fix a dimension. I can't say anything. At the moment you fix everything about it. You fix its dimension, you fix its ecology and everything. But you don't care what dimension I do? But at this point, I don't care. Gary, why do you call it fast space and not consideration space? No reason. Which one of them is it? It's not a final test. No, it's not a final test. No, it's not a final test. No, it's not a final test. And finally, phi is a function on m cross m plus an extra thing called the ghost, which I'm going to not worry about too much. Now I say m cross m because I'm thinking of this path space which I defined as really the space of configurations of half of a string. You might say why do I do that? Well I do it because I can then more easily define a product in the next step. So it's just convenient to set up the theory first by thinking of a path space of essentially half a string. The string field is then a function of the product. You can think of it as say like a left half and the second factor is representing the right half configurations of the string. No. Okay. To be precise, it is actually not a function but a density of way to half because I'm going to have to integrate them in a moment and I want the whole thing to be covariant. So you should think of this as really a density of way to half. And in the full sort of classical string field theory, that would be all I need to say. And the quantum, sort of second quantized string field theory, these will be operator values. They will be the string field operators. They are, yes, they are Hermitian. There is a Hermitian condition imposed on the fields. Well, okay, that really goes into the discussion later about topology, so let me just postpone it for now.

12:30 The maps from this interval say 0 to pi, pi over 2. No, Bryce was asking about this earlier definition of script M. He was asking what class of maps I'm including in this space. It's a very good question, and it's precisely that question which will occupy me in the second half when I sort of argue how much the space depends on the topology of M. But I'd like to sort of postpone the discussion of that for now. Okay, so I have this space, a function, or it's a string field, density, weight of half if you'd like, on the product of the space plus some I define a product of this space, and the product is if I have two string fields, thigh and side, at a point x, y, where x is a coordinate of m and y is a coordinate of m. On this factor, they are points in this space of maps. They are maps, sure. But I'm going to be integrating over maps. It's like a path integral. So I'm integrating phi of x of z times psi of z and y, dz. This is very much like matrix multiplication. If you think of the first index as like the row index and the second one as a column, this is a formula you write down for matrix multiplication. Just like matrix multiplication, it's not commutative. I sort of said very glibly a moment ago that I needed my psi to be the density of white a half in order for this to be well defined. If you push me and say rigorously and mathematically what do I mean by that, I have to confess I don't know in the sort of... I'm going to show you the infinite dimensional space that I want to work in. However, there is a very nice regularization, if you'd like, of this star product, which was proposed by Richard Woodard and Mark Schroednicki. What they do is, in fact, essentially what you do in your loop space approach. They discretize a sigma space, the space of the parameter values. They can then reduce phi and psi to functions on a finite dimensional space. Then this is all completely well defined. It's independent of any background structure on space. In addition to the product, I'm going to need an integration. The integration of one of these string fields is defined to be simply taking the string field, evaluating it, sort of on the diagonal at x and at x, integrated dx. It's like taking the trace in this matrix.

15:00 I guess I'm never, this should, density of weight of half, yeah, it's density of weight of half in each factor. That's right. Okay, those are the basic ingredients in the theory. Now let me present the theory. Okay, this is work that was done in collaboration with Joe Lichten, Ryan Roman, and Andy Scrumminger a couple of years ago. The action is simply the integral of phi star phi star phi. This action is invariant under a linearized gauge invariance, which is the variation of phi is phi star epsilon minus epsilon star phi, where epsilon is an arbitrary string field. I said that this was like matrix multiplication, and although the matrix multiplication is not committed inside a trace, it is. It's surprising. And the field equation that you get from this action is simply phi star phi equals zero. That's the theory. Sorry? The integral is what sort of essentially performs a trace in this field. The integral means evaluate on the diagonal and integrate over it. So far, I've not had to introduce a space-time metric. And this formulation that one obtains is much more algebraic than geometric. And after John Satchel's talk the other day, I have a feeling Einstein would have approved.

17:30 It's not quite . But you can think of this as simply defining a non-commutative algebra. The sort of star product gives you this non-commutative algebra structure. The gauge invariants are simply commutators, and the whole thing looks very algebraic. How do you map these? How do you transform these maps? What is a coordinate transfer? Well, I'm going to view script M, which is half-string space, as an infinite dimensional manifold, and as such, any smooth map from that manifold to itself, which is invertible. It's a diffeomorphism. I want the field to transform under any such diffeomorphism of that infinite dimensional space to itself. Well, I can see the maps you start with that define, that are the points in this manifold, but I don't know quite yet what it means to take smooth maps of those maps. One particular example would be obtained by taking a natural diffeomorphism on the space time. Okay, that moves the polar space around. In particular, it moves the loops around. That then induces a gifting morphism on the loop space. Well, I want to include all of them. Gifting all the loops by a constant. I really will sort of talk about all these topological issues if I get to them in the second half of this talk. Okay, so that's the theory. And the question is, what can you do with it? The first step in analyzing a theory is to look for some solutions. So let me describe a procedure for finding solutions to this purely cubic axiom. The first step in finding a solution is to pick out a particular element, a particular string field, which is the identity under this star product.

20:00 So, phi star i is equal to i star phi, which is just phi. If you remember what the definition of star was, you can see that i is essentially a delta function, which equates sort of the left argument with the right argument. We're doing classical string theory. This is classical string theory. We'll be finding classical solutions. The second step is to find differential operators. Differential equations or eventually of course I want to get out geometry so I'll have to use some of this terminology. I suspect that one will or one could reformulate this in maybe a purely algebraic approach which might actually be going on and doing higher things might be more advantageous. But for now let me actually think of these as differential operators. Q left and Q right. Q left acts on the first argument of the string field. Q right acting on the second argument of the string field. Define Q to be simply their sum. And these operators, Q left and Q right, have to satisfy three properties. Three properties are that Q annihilates I, the identity element I just mentioned, that Q right of phi star psi minus psi star Q left of phi is equal to zero. What this is saying is that, remember, the star product basically equates the right argument of phi and the left argument of psi, that was this matrix multiplication idea, and this says that you can flip the operator, which is acting on the right argument of this first string field, into the operator acting on the left argument of the second string field. It's like an integration by parts type rule. Well, I've written here anti-commute, and that's because, as I said, these string fields do involve ghosts. I'm going to try to suppress most of the technical issues and sort of focus on just the key ideas, so I'm not going to discuss that in any detail, but the net result is that these operators Q have to be Rothman odd, and so one requires an anti-commutation relation there.

22:30 Sorry, you're absolutely right. This is wrong. I should not have interchanged phi and psi. What I meant to do was simply to move the operator over. It's like an integration by parts. So this should be phi and this one is psi. Why is it just q out here? Well, you'll see in a minute. These are the three properties I need. If I have operators satisfying those three properties... Then I claim that QL, the left Q acting on I, is in fact the solution to our field equation. Q left star I star itself is zero. That's our field equation. Furthermore, this solution has another property, which is that if I star it with another field, plus add, and now I do need to change the order, What I get is simply the total Q acting on the state phi, the string field, phi. Do you know in any sense whether this solution is general or ? I do not know whether every solution can be written in this form. I know a large class can. Is there anything checked here about it? There is. In fact Richard Woodard Has some ideas on how one might try to prove that all solutions can be written in this form. So I guess it's a possible conjecture, maybe a conjecture that Richard would want to make, that all solutions can be written in this form. I'm not sure, but it's an opening. Given that fact, we can now simply argue that if we take our string field and expand about this solution, just by writing it as this solution plus another string field, and substitute back into our action, what happens is that any time you have two or more two left i's start together and you actually get zero by the field equation.

25:00 If you have one q left i, well first of all you have three terms which come in with one q left i, but you can always combine them in pairs and replace it by the total q, you get a factor of three-halves of a star qa, and then you have the term which is just the three a's, plus a star a star a, so the action now expressed in terms of this shifted field becomes this one, q we said was some differential operator, And so you see that by shifting about a classical solution, you can rewrite the action in a more conventional form in which you've picked up a kinetic energy term, a term quadratic in the field with various differential operators. So one generates this kinetic energy term by expanding about a certain classical solution. Sure, I'm going to come to them right now. I said I didn't know if all of them could be expressed this way, but let me come to some particular ones. So here are some examples. I haven't dropped any cards. And then, in the expansion, you are supposed to get an A and so A integrations The shift of the field. Okay. So here's an example. Example one. Consider, okay, a D-dimensional flat spacetime. Okay. So here's an example. Example one. Consider, okay, a D-dimensional flat spacetime. Okay. So here's an example. Example one. Consider, okay, a D-dimensional flat spacetime. Okay. So here's an example. Example one. So this is the first time I'm going to adopt a flat spacetime. I'm going to take my operator q left to be the integral from 0 to pi over 2, something which I call the VRST current. What this is, is the current you get by taking a string, a single sort of first quantized string, moving in this flat spacetime background. Quantizing a string using the BRST procedure and construct the BRST current. For those of you who don't know what that means, let me just write it out explicitly to give you an idea of what that current looks like. The current that you get by that procedure involves one of these dose coordinates times Watt metric contracted with tube momenta.

27:30 The metric contracted with x prime twice where prime now denotes u by d sigma. Another component of the ghost multiplying p times x prime. And then there is the three-ghost code. These are, of course, the constraints of the first quantized string associated with the re-chromatization invariance on the world sheet. The DRST procedure is to center those constraints times the ghost plus the three-ghost string. So that's my object q left. You can ask, does it satisfy the three conditions that I said were needed for it to be a solution, and you do the calculation and you find out that in general it doesn't, but it does if and only if d is equal to 26. So this is how the critical dimension, which is familiar in the first quantized string theory, enters in the string field theory. In this approach, what is looking for a solution, the corresponding to the flat space time, and finds that this will be a solution only in 26 dimensions. So this is a differential operator, these P's are really a differential operator? Yeah, P's are like D by the X's. These are P's of sigma, like each of these is exactly. Um, well these are, these are nice, right. Is that a, can you take it? I mean, you need one-dimensional time for other reasons, like the dose and everything. I'm wondering whether you need it for the real potency, which is this piece here. For this particular approach, I'm not sure whether... I think the calculation of Q squared... Could you do that? If Q squared would actually be zero if you took a metric of other things? Well, we'd have to think about it. But in practice, there's other reasons why you wouldn't want that. This is, so the point I want to make is that the critical dimension is now associated with a particular solution.

30:00 What have to be careful about first quantized and second quantized. The critical dimension is a first quantized, if the quantized result in the first quantized theory, if the classical result in the second quantized theory. It's the first quantized h-bar. But isn't that just the same as changing the mass to the Compton wavelength in the Klein-Gordon plate? And then you can write the classical theory of the scalar field. Once you've done that conversion, you don't have to worry about h bar. Maybe we should set it up. It's my impression that people in this room in general would be much happier if this number was 4, not 26. I think that, well let me just make two comments about that. First of all, what I've been saying applies to the bosonic string, and as I'm sure you know, the theory which is believed to be a consistent quantum theory of gravity is the superstring, which... For which a similar analysis can be done. Everything I say has analogs. For the super string, of course, there are much more technical problems which have to be overcome, and various people are working on that. But the analogous calculation would then give you D equals 10. There have been some recent proposals over the past year or so for four-dimensional string theories into string theories now not in the string field theory sense, but in the... These are theories which sort of fundamentally exist in four dimensions. What happens is that you have extra degrees of freedom associated with the string.

32:30 There are a number of ways you can use these terms, and I'm going to talk about some internal degrees of freedom if you like, which take care of the conformal anomaly which is setting this to be such a high number, if you have those extra internal degrees of freedom, you need only four space time coordinates, and then you're naturally, you know, propagating in four dimensions. So, at least there is, in those theories, one would never have to talk about compactified internal dimensions or anything like that. The only space-time dimensions we'd be able to reconstruct would be the four-dimensional space-time. This condition is actually required for all of them. Well, one has a solution, this Q left i now, with the Q I just mentioned, and in 26 dimensions, we have a solution. We can expand our field, as I just mentioned about the solution, quantize... The fluctuations and what one finds is that the theory you would then construct doing perturbation theory in A about this background correctly reproduces first of all the spectrum of states in order to get all the masses and spins of the first quantized string and all the perturbative scattering amplitudes that one would calculate from this other approach of just considering mapping world sheets into a flat background. So this approach, which sort of started off looking like it had no connection with anything, at least is now connected up with perturbative scattering calculations and string theory. Yeah, I'm not sure. You started with a theory which didn't have any metric in it. You write down the equations, you erode them in a certain form, and then you found that if you introduce the metric, you can construct a particular solution of those equations which didn't have the metric. But that somehow doesn't say that the metric emerges from the metric-free... It only shows that there is a particular solution of those equations which is constructed from a matrix.

35:00 So a matrix is put in by hand. It's not coming necessarily from up and down. Am I right or not? Well, I wouldn't quite put it that way. I would say... There's no problem with constructing particular solutions starting with a metric, because those are the metrics that you want to get out. I'll show you later that there are other solutions which don't require a metric. There are other solutions, and they don't give you a typical formula. Well, it's presumably not the one that we want, right? We don't want a formula to come out from Las Vegas. But this is one. It's just an example. It's an example one. Do you mean any curve metric? That's example two. I'll be there in just a second. So comment one, since I've already done this comment two, is I just wanted to emphasize, and I think Carol is already alluding to, one's using a notion of time provided by the solution, the left eye, to construct the quantum theory perturbatively. Well, when one constructs the quantum theory of A, one uses the fact that we're expanding without the left eye, which has a flat metric, so we know what sort of asymptotic states are, we know what positive and negative frequencies are, and one constructs the quantum theory as a mutual. There is no tau in the string theory. Well, there is a tau only in the sense that when you calculate the string scattering in the string field theory, you want to be able to find the diagrams. And we find the diagrams right now in the two-dimensional world sheet. And so, as Raphael said, why take the curves? Why take the flat space? Example 2 is, let's try the same Q, but simply replace our flat metric with a curved metric, okay, and ask... Yes, I'm going to fix, well, I'm going to pick a G and say, when does the Q constructed by that formula with this G satisfy the three conditions I need for it to be a solution of my string field theory? And I find that in general it doesn't.

37:30 It will if, again, I have this dimension condition, 22 dimensions, if the curvature is much less than 1 over the pipeline squared, and if the Einstein equation is satisfied, then one gets an approximate solution to the field equation. Approximate in the sense that there are correction terms of the order of the pipeline squared times the curvature. So this is a statement, this is a correspondence limit, saying that for solutions of Einstein's equation where the curvature is big compared to the Planck scale, small compared to the Planck scale, you can construct approximate solutions of the string field theory. If you're worried about the fact that I got just the vacuum equations, you can put on electromagnetic fields in the space time. That requires a certain multiplication of q coupling the string to the background. And then you get that these are solutions provided these things are satisfied. And the last condition is replaced by Einstein's equation. What is your time? It's 26 minus 2 over here. Newton's constant comes from, you're doing the calculation, there's a dimension for parameter in the formula for q. Which I did not write explicitly. And that dimension full parameter is what sets the scale. And you essentially fix it to be the observed Newton's constant. You only get to do like a length. Right. You have to distinguish between A and B. Which is like what John was saying, why we have to draw in the public sense. There's one, there's basically one dimension full constant in the theory.

40:00 So that sets the scale, and everything else can be related to that. Also, there is no such thing as a cosmological constant. Um, right. I don't see... Well, in fact, the cosmological constant was fixed to zero by taking b to the 26th. But that wasn't . I thought that was . Oh, you assumed it. I don't know, you've got a solution. Actually, you're right. The way I stated it here, I said that it was required in order for it to be a solution. I'm now thinking that if I want to allow it to be a cosmological constant... That's certainly true, but that's not the issue I'm talking about here. I'm talking about the classical string field theory. I think to be precise, if you consider a cosmological constant, I could actually drop this condition. I would then get a cosmological constant which is huge, right? It would be the Planck cosmological constant multiplied by a factor of d minus 0.6. Right, then I would be violating my second assumption, so I couldn't sort of consistently do that. Well, no, no. If you don't set D equal to 26, presumably you have to have a big lamp, and then you can't have that approximation. But you haven't ruled out solutions like this. You haven't ruled out classical solutions like this. Oh, no, no. I haven't said anything about curves with a blank scale. Let me move on. So those were solutions which were, in some sense, known by first quantized methods having to do with the formal invariant sigma models and other approaches related to that. In some more recent work with the people listed here, we've pointed out that there are other solutions which try to find the conditions.

42:30 The scattering theory about it seems to have some difficulties because the ghosts don't decouple. You can take a Q to be essentially this, with P squared times a ghost. This will satisfy our three conditions and obtain a solution Q left i. The scattering theory about it seems to have some difficulties because the goes stems a couple. Scattering theory is not equivalent, but it does seem to have some interest as a possible high energy limit of the usual. There are other solutions, and let me just mention one here, which don't involve any axes at all. This is like a solution, but this is very trivial to be called analogous to. A solution includes a quantum theory which is not in the product form, but just to show you if you can do something like this, you can take an integral of just the ghost times some function, like cosine squared sigma or something, and this will satisfy our three conditions for many apps. There are some general conditions that I've got to satisfy because they're not very restrictive. These will then give you solutions which make no reference to space time. There's no x's in here at all. Perhaps this is analogous to the solutions that are not of the product form, perhaps it's a bit too trivial, but I certainly expect there to be solutions of the stream-field equations which have no well-defined space-time metric, no reference to space-time at all. Maybe you'll mention it, but we want things to look like space-time on the large scale. We're only in one particular state, right? One particular solution which describes our universe. The claim would be that the ground state of this theory has to be apart from one of the solutions which does have a space-time interpretation.

45:00 There's no reason why there couldn't be other solutions as well. I can't believe there's ever 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 that one. As long as I'm in that one. Yeah, we'd like to see some of the topology aspects. Okay.