Subsequent discussion (last part) / The role of topology in quantum gravity (contd.)

0:00 He briefly mourns the fact that you can't seem to construct anything that has half-individual spins geometrically, and then says, but nevertheless, there is a natural way to incorporate spinners within the context of general relativity. You have a covariant, meaning that you can give locally for spinner fields. Well, Finkelstein and Misner suggested in 1959 in this paper that Non-Euclidean topology might have half-integral spin geons, and although the particular mechanism, they didn't suggest a specific mechanism, but they suggested this is a possibility and speculated about what topologies might allow it. So the particular topologies they suggested were not the It just wasn't the right class, but it is true that in this context that we've been talking about, the diffeo associated with the 2 pi rotation, that is, if you look at the n-rotation numerator exponentiated and look at the diffeo associated with the 2 pi rotation, that's a diffeo that's not in the component of the identity for almost all pre-manifolds. And so what is it? Well, the image of a path under this diffeo R2 pi, what does it look like? At infinity, it just looks like a 2 pi rotation. Now, the meaning of the diffeo is insensitive to what it looks like on the inside. And in general, you can't take this vector field and make it look like a rotation all the way on the inside if the topology is not. But you can make it anything, and it still has the meaning. So, canonically, the way to do it is just to make it go to zero at some finite region on the inside, and now you have for all asymptotically flat manifolds, let's do that, and then that says that, that says that, say, a geodesic for a particular metric that goes in like this and comes out over there gets mapped into this geodesic, it goes in, At infinity, you rotate by 2 pi. As you go in, you rotate by angles that are successively smaller between 2 pi and 0 until when you get close to the geon, you're not rotating it at all.

2:30 So this is the 2 pi twist. And this is called a rotation parallel to a sphere by topologists. And it turns out to have been, in the last 10 years, an interesting question for topologists to decide whether this was... In general, it's not. So Witt went through this and I think the statement is only for handles and lens spaces is this deformable to the identity and nothing else. What's a handle? A handle is what you're, is what probably what everybody here is familiar with. You cut out two balls and identify the surfaces so it's the three-dimensional generalization of the two-dimensional half. So, wormhole, wormhole. A wormhole that stays in our own units. So here's an example in, so now let's ask more generally what this full group of Diffio's is. So in addition to this, to the two pi twist and the exchange that we looked at, there's, so now we come to your question about why does one talk about these three manifolds in terms of the symmetry groups that you use to construct, you could just talk about identifying. I don't want to make faces of a polyhedron not worry about the symmetry groups, but the statement is that the group of inequivalent diffeos, so here's a cube with opposite faces identified, and the group of inequivalent diffeos, here is the symmetry group, the symmetry group of the cube itself, so the symmetry group of the space-time,

5:00 With the particular metric, the group of isometries ends up being the same as the group of diffeons, or there's this close relationship. So here the pi zero of the diff is the covering group in SU . For the group of isometries, you just take the cube and make any isometry of the cube, and now you put it back down and you've got a diffeo that's not in the component of the identity of this space, so I can show you what happens to a path that goes to a homotopically distinct path, so obviously the diffeo is not in the component of the identity, and really all you have to do… So, it's an immediate proof that O star, this together with the fact that the true pi rotation is not in the component of the identity, shows you immediately that O star is in pi zero of the dip. And all you have to show is that pi zero of the dip is no larger than that to decide what the full group is. The elements you know how to physically interpret here are geon exchange and two-pi rotation, and you can look in higher dimensional theories. So, Palooza-Klein here means just higher dimensional gravity with the metric that makes it look asymptotically like a Palooza-Klein geometry. And then ask in that context, let's ask in that context whether you have half integral spin. Well, you know, the groups themselves haven't been examined, but certainly you get happen to go spin from, well, from the same mechanism, but from what looks like a broader class of examples, that is, they have a different character in some sense, because you're holding the metric. You don't have to have the topology of the higher dimensional space fixed as you go out to infinity. You don't have to have a change in the topology of the higher dimensional space that takes it away. That is, Carlip in particular looks at a manifold with internal space S2 cross S1, but the whole spacetime is just the manifold cross S2 cross S1.

7:30 The topology is just R5, but it's the, so here it's the metric, it's the metric itself that gives you, and the asymptotic restriction on the metric that's giving you the half integral spin. Well, let's see, can I take five minutes here, or what do we got? Alright, so let me... Let me finish. So that's the comments on kinematics. And now I want to finish just by talking briefly about the dynamics, of which we know much less. I think what's most useful here is to just suggest why one thinks in a Lorentzian framework that you can get topology change. So you imagine topology change arising from a path integral where you start with an initial three geometry, go to some final three geometry with a different topology, and interpolate a smooth four manifold between them. So there's this theorem of Reinhardt, which Gerhardt says Mizner proved independently. That says, if S1 and S2 are any closed pre-manifolds, then there is a compact core geometry whose boundary is the disjoint union of S1 and S2, and such that S1 and S2 are spacelike, no field-like, just as smooth spacetime with Lorentz metric interpolating between them. It's just saying they're co-bordant. Yeah, no, they're co-bordant and there's a Lorentz metric. Oh, Lorentz metric. Okay. If that's true, you have to violate causality, so Garoult shows that if S1 is not diffeomorphic to S2, then the manifold has closed-timelike curves.

10:00 There's a simple corollary that's given in the appendix of this paper, a recent paper by myself, Parker, Papastamatiu, and Zhang Y. There's a term that says if you have an asymptotically flat for geometry, so now this is for compact, and now it's simple to extend it for asymptotically flat, for the asymptotically flat case, so you want to require that at infinity the vector field just points straight up and that it's transverse, points in at the initial surface out at the final surface, and it's simple to, it's a simple corollary to this term. Manifold is non-singular, but not the field it was. Well, you could always, you could always take, so what's the, let me just say briefly what the criteria, the idea is that you have a time-orientable Lorentz metric if and only if you have a non-vanishing vector field which you can take to be time-like at each point. So if you've got a Lorentz metric, then you just, it's time-orientable, you take a vector field pointing to the future in each light cone. Conversely, if you have a non-vanishing vector field, you can always find the Euclidean metric, normalize the vector field so it has norm one with respect to the Euclidean metric, and then a Lorentzian metric is given by the Euclidean metric minus two T A T B, where T A is the non-vanishing normalized vector field. A closed manifold allows a non-vanishing vector field only if the other characteristic vanishes, and it's easy to see why, but I think I'll skip it. Let me give you some examples and then finish with a minute worth of objections. So here's the only example in two dimensions. You start off with a topology chain. So you start off with these red lines. You're going to look at RP2, what we started out looking at over here. So you've got, so this is going to be a universe that has no past boundary. So we identify diametrically opposite points of this past circle.

12:30 And now it has this future boundary over here. So it's only got one boundary. And now it has in this case, yeah, well, I mean, down here, I've identified, it's just RP2. Let me, the whole manifold is RP2. And now I claim it's simply the, this drawing over here is really just the Mobius strip. So here's what would be the pass boundary, but I identify diametrically opposite points. So now it's just our P2 with the disc removed in the future. And now if we just draw the curve, yes, so I'm going to put on a vector field that just is vertical everywhere. And now I come down perpendicular to this boundary, and I smoothly go up over here. So this is a smooth vector field, really a smooth vector field up to sine, because now here I'm going down and here I'm coming back up, so it's time-non-orientable. There are closed time-like curves, as it turns out, and it's smooth. Thank you for watching. Well, this is the same as a Mobius strip. You've just got a single blue boundary. That's the blue boundary of the Mobius strip over here. The identified circle down here is the same brown circle down here, and this vector field is just the transverse vector field on the Mobius strip, and the transverse vector field down here.

15:00 Oh, well, I don't, yeah, I don't, I don't, I don't think, let me comment on it at the end. Let me give another example. Here's a child universe branching off. And over here you can't, over here you can put on a Lorentz metric as long as you attach, as long as you attach a copy of CP2 to make the Euler character vanish on this metric. And finally, I don't have time to show you the vector field at this point, but I just want to point out that if you look at the manifold that represents pair creation of two identical or mirror image prime factors, just by taking the ordinary manifold representing a prime factor for all time and bending it around, That you always can put a vector field, a non-vanishing vector field on this that gives you a Lorentz metric making it look like a pair creation manifold, where the vector field points to the future up here, to the future down here, and is non-zero in the intervening region. So I can show it to you if you want. Always close timeline curves. Okay. Well, that was the theorem of Yerox that I quoted when I first started. Yes. Right. There are always, any time you have topology change with a Lorentz metric, with a smooth Lorentz metric, you always have closed-time lectures. I'm not sure whether or not you like closed-time lectures or don't like them. All right. So, well, that brings me to the last transparency over here, objections to topology change, and now I'm going to comment on that. So let me bring back the next quote. As the first objection, I'll finish the quote by Weill that I started with. If you assume that space-time is a manifold, then you're disregarding the possibility that a microscope of ever greater magnification would reveal ever new topological complications.

17:30 Little prime factors on top of little prime factors. Little handles on top of little handles. And I think you mentioned this yourself. Ad infinitum. So that suggests you're not really looking at a manifold at all. Let me give it to you. I have it written down. It's philosophy of natural science. Self-similarity. Self-similar message. Apological message. Well, if you're going to strictly require it to be a manifold. Well, Bryce suggested that if you're going to have different four manifolds that you sum over, then there seems to be an apparently free phase in summing over the four manifolds. How are you going to resolve that ambiguity? And then he and Anderson looked at a field theory on a background space-time with the changing Well, they looked at it with the changing one topology, skipping down, but this is what they looked at, and found that such a field theory, well, in this case, it may not exist. I see. The reason I said it here is because the speculation is that on a spacetime with changing three topology, the field theory may not exist. The fact is that if you just look at changing one topology, it doesn't exist. Question is, does it exist if you really have three dimensions? And there the question is, so there the question is, are there, so now let me ask the following question and say why I think it's the relevant question. You can ask whether on space times with closed time like curves for which you have a well- For which you have some space-like surface, you can ask, is there a Cauchy-like problem that's well defined? That is, can you find a unique solution, for example, to box phi equals zero for generic data on such a space-time? I think that's a relevant question because if you think of doing a path, if what you want to do is to change topology by doing a path integral, then you can organize the path integral by first

20:00 Summing over the matter fields and later summing over the geometries so in particular for so for one geometry you'll be summing over you'll be summing over the matter fields on the fixed geometry and then you'll be constructed and you construct in that way so for so in so you construct in that way A solution to the quantum version of this equation and if you're going to have a quantum field theory then you for the matter field you'd expect to have a classical theory as well or if you just wanted if you try to construct the classical theory simply by propagating that you ask what happens when you propagate positive frequency solutions from the future to the past. It's hard to answer that question if you don't have any propagation from the future to the past. So you want to know, are you going to find any spacetimes that allow you to do this for generic initial data? And you might hope that there are a set of spacetimes that this is not uncommon, that you can do this, that you can find the unique solution. But certainly it's not always true. So you certainly can have situations where you have closed time-like curves and where there is, where it's certainly not the case that generic initial data, certainly not the case for arbitrary solutions with closed time-like curves is you can involve generic initial data. Now you might think that often you develop singularities for generic data, so that means when you do the path integral, you'll find that the action blows up and these will be ruled out. It might be that there are situations where that doesn't happen, where you can do the path integral, and those would then be allowed. So I don't know the answer, but I think the question is. Alright, thank you.

22:30 So that essentially because the topology changes, you're having a base method by going to a close period quantized theory that allows topology to be part of space and at a perturbative level it's considered to be flat and so you don't pick up the difference. All the other factors you can get are derived from the theory from other examples. The second remark is using this decomposition where I kept the manifolds open, using polygons. If I represent all the pathologies, the point right now is theory of the boundaries. So one thing I kind of want is theory of the polygons. I don't know. Mr. Rock, are you here? I'm sure you're here. I have a question. There's another topic I wanted to ask you about. If I were to say, in my day and age, it's not likely to involve four or five members, but in my day and age, it's something that's going on, because, you know, if you've got a four or five members, it's a very mild form of singularity, so it's a much better term at one time. Otherwise, it just seems to me that it's going to be on the list. This seems to me that it could be unlikely to make a difference. Yeah, okay so let me let me just I think I think that's right and in in part what you're going to use is going seems to me philosophically to depend on whether you think that

25:00 At the same scale where the topology changes, or something close to that scale, the idea of the manifold is going to break down as well, or the idea of the smooth metric is going to break down as well. So I'm really taking a conservative point of view over here and asking, what can you do if you just suppose that you still have a smooth metric, a smooth Lorentz metric? And a manifold, and that these things make sense at sizes on the order of the Planck size. Now, I don't think that's outlandish. It seems to me it's not impossible that the Planck size is being picked out for us as a size where the, well, it's picked out by g, c, and h-bar. Local-looking Lorentzian spacetime is breaking down, but I don't think that that has to coincide with the place where you're going to get a full unification, say, of the... Where you need to get a full unification of the fields, that is, that could happen at a smaller scale. So you could imagine different scales, one where you have to worry about non-Euclidean topology and you retain a well-defined metric, and then at a scale smaller than that, you have to worry about more fundamental issues. So that's the philosophy here, but I certainly don't. I'm not adverse to looking at the, to supposing that you get at very small regions compared to this, just abandon the metric, let it go, let it become degenerate. Yeah? I think we're going to have to give time for the next lecture. Okay, let me make one small propaganda remark. A few years ago, a couple of people and I wrote a two-dimensional graph, and it's a very nice style of work, but I suggest that you not be able to understand it. The topology, geometry, and the use of arbitrage in the part of the West Coast, the south of the equation, and the use of quantification in the part of the West Coast, the South of the equation, and the use of quantification in the part of the West Coast, the South of the equation, and the use of quantification in the part of the West Coast, the South of the equation.

27:30 This is a very broad offer, and this is something that's required, because you know that you have formulas here, and you have classes to come up with, so it's easy to get out of the building, and you can come up with something by some kind of time, and you don't have to go hard enough to provide possibilities, and you can be very, very careful about it. Okay, thank you. Now, can we call after five minutes for the report on some of these works that are all made?

30:00 The work I want to mention complements actually several of the points John made originally, and is not work by me, but work by mathematician Lewis Crane. It arose from some physical speculations about what kind of manifolds might dominate the Euclidean, excuse me, the Euclidean path of the globe. We know that we'd like them to be nominated by finite action configurations, and that's been pretty well studied by Hawking and his collaborators, but if we want to think about the Minkowski effect and as well, one wonders what kind of configurations might occur here if we allow strange topologies, but we don't want the field equations to be satisfied. Now as John said, so I'll come back to that again in 30 seconds at the end, the basic idea of Lewis is that while classifying Euclidean four topologies is complicated, well, can't be done, there's no classification, perhaps one could classify Platt-Muntauskian topologies And so he defines a structure of what is interested in four manifolds, which are flat, menkaustian, and causal, and complete only in the past or the future, future or past.

32:30 Please. Flat? I'm sorry. Lorenzo. Identifying. Yeah. And the picture is the following. This is familiar for people from string theory, where you take either spheres or hyperbolic spaces and identify it by symmetry groups of the metric, just as John was talking about the three manifolds, and this leads to the classification of higher genus Riemann surfaces. As John mentioned, a very similar construction leads to the classification of prime factors for the three manifolds, and it's interesting to note that the group SLTC All of this comes in both places and one can draw a picture to understand why, if one looks in metastatic space-time, then one has a sphere, which is a sphere of the null direction to the point, and we all know how the Lorentz group acts on it. If one takes discrete segments of the Lorentz group, identifying that sphere using that, one gets... The Euclidean group construction of hydrogen and steam on surfaces, but the Lorentz group is also acting on hyperbolic steam manifolds, which we're all familiar with the orbits of the Lorentz group, and the generic steam manifold topology comes from taking hyperbolic steam manifolds, like the generic means there are many more from the hyperbolic identifications than from the others, because there's much more room in hyperbolic space. So, the same groups which classify the higher genus Riemann surfaces classify the three manifold topologies that you can get on a hyperbolic space, and in fact, when one does this, sometimes these three manifolds have boundaries, and then their boundaries are sets of Riemann surfaces, which are the Riemann surfaces connected with the same group. So that's the relationship between Lorentz group back in here and Lorentz group back in other key manifold topologies.

35:00 The idea of Lorentz is that the same Lorentz group can be used to make identifications on a region in Kowski's space-time inside the future null cone of a point. And so if one continues the process, one has some four manifolds here, which has a singularity in the past. His features are complete. His causal is flat, obviously, and his cross-sections are the three topologies you get by identifying the hyperbolic spaces with the climbing roots. Now, there's a theorem here and a conjecture, which Lewis thinks is about to be a theorem. The theorem is that any... Of these three manifolds, which come from identifying the hyperbolic space, the Euclidean space, or I see, I'm sorry, H3, Euclidean 3, or H2 cross R, is in fact the cross section of one of these things. And he gives these things, by the way, a name, which will mean simply that the name of these things is a tardis. Of any three-manifold gotten in this way is a cross-section of a TARDIS, which is a flat Lorentzian causal of some. So you give a particular three-manifold, then you give what you call a particular TARDIS. Right. And the conjecture is that all TARDISes are classified and exhausted by classifying their spatial sections like this. Can you take the unit cell and draw a line from the origin? I'm sorry, the unit cell and the unit axis? And the hyperboloid, the unit cell for the axis, I know, and look at the...

37:30 Well, the part... The group seems really to take that cone to itself. I'm not saying... The group seems really just to give you pathogens. Okay, so you just look at the whole family of hyperbolas. Each of them are identified in exactly the same way. Right, right, right. So that's this. This is very simple. The hard thing is, well it's not, I don't know if it's hard, I don't, but going back what you're saying, we define this class of manifolds which we might be interested in if we're doing a metastatic path integral. The fact that although we can't classify the things that would come in the Euclidean past integral, we can simply classify the things that come in the Tencent past integral. I mean, you haven't even given an argument to the Euclidean past integral. These are the flips. Yeah, these are the flip cases. The conjecture is that all the flat ones come by the Euclidean past integral. Right, right. Whether or not, whether that's allowing ourselves to be the Euclidean past integral. Right, right. This is just the minimum cosmology that's improved by the education. Yeah, yeah. Yeah, no, that's right. Okay, well, are there any other questions for this session? Okay, well, it appears that we've ended right on time, but I should encourage you to come back in time because this next session on time is really going to be packed, and time is supposed to prevent everything from happening all at once, but if we don't get back on time, everything will happen all at once.