An answer in search of a question - "proofs" of tridimensionality of space / Why 3 dimensions of space aren't enough

0:00 I'll start. So I guess it's, hopefully everybody knows this is a session on the dimensions of spacetime. It's going to be four particular talks, and the way we want them to do it is three talks that take about 25 minutes, and then about five minutes of discussion time, then straight into the next. And we're going to start with Craig's panel, and then we're going to move on to the next. Okay, thank you. This paper, this presentation corresponds to this long 14,000 word paper that's coming out in studies in March. I don't know what's going to be more depressing than if I can squeeze in all 14,000 words in 25 minutes. Anyway, I thought I'd start off with, you know, since I was going first, start off with a kind of justification for the symposium by setting the context a little bit and, you know, talking about why might philosophers of science be interested in dimensionality or why might they be interested in dimensionality again. And obviously, you know, I guess the context where physics now is looking at, you know, all sorts of extra-dimensional theories. So I thought I'd, you know, go back to the Greeks or a little bit afterwards. You know, higher dimensional objects were well known mathematically for nearly forever. And so people would know about the dynamo, the dynamo-cubus, where you multiply by a cube. And then people like Pappus would say, you know, don't work with these things, you know, don't think about them, they're impossible. You'll get a, you know, you'll hurt yourself. And you get the similar sorts of things through the Middle Ages, through the 19th century. Even today, if you look in that philosophy of left and right books, you'll see stuff about, you know, Cadena. Greater than four dimensions, metaphysically impossible or conceivable by philosophers. Today though, if you think about it, a lot of philosophers and natural philosophers and scientists throughout history have thought about dimensionality. Almost everyone had a kind of line on dimensionality. Kant, Hegel, Leibniz. You know, just going forward all the way through. And, uh, Russell, um...

2:30 And then it all sort of died. Philosophy and physics became obsessed with relativity and quantum mechanics because they're more sexy. So it all kind of died, but when you think about all the advances that happened regarding dimensions, it seems like it's a place to look at again. The definition of dimension by Ericsson, Menger, covering dimensions of A, that's one place where there's been a lot of change in the 20th century. In physics, obviously, there's been a lot of change. The five-dimensional collusion-flying theory and the generalization to superspring theory have lots of different dimensions. Do you want to read that? Is that not a joke? I agree too. In philosophy, you know, Poincare and other conventionalists, you know, also had no problem with higher dimensions. So physicists do not have no problem with higher dimensions. Mathematicians have no problem with higher dimensions. Philosophers, you know, if you had Poincare, then you ask him how many dimensions there are, he would then reply... How many do you want? And come up with some recipe for this. And let us not forget the discoverers of Dimension X. So in Kant's first published work in 1746, Thoughts on the True Estimation of Living Forces, It's speculated that the dimensionality of space follows from gravity's inverse square law, and this idea, you know, I think it was revolutionary at this time. I don't know of anyone else who had thought about dimensionality from a physical point of view rather than a kind of mathematical conceptual point of view. This then generated this huge industry. So you can see that, you know, Kant had this idea of getting the dimensionality from the physics. Paley then did Hegel, Reichenbach, Ehrenfest, Barrow, Berkel, Bergbacher, now 1999, Carnap. It just goes on and on, the list just goes on and on.

5:00 One of the most famous papers of this literature is the one by Ehrenfest in 1917. In which he says he's going to give five different demonstrations of where physics picks out three dimensions as singular, he says. And so these are supposed to be five, he says these are five answers to the wide space three-dimensional problem. In the paper, in the beginning, he says, I don't really know whether this whole question, why is space three-dimensional, he says, I don't even know whether it really makes sense when you think about it. He says, well, that's for other people to think about. So, whether the question is justified is for other people to think about. Whether it is or not, here are the answers. That's the title of the book. If we want to just sort of spell out the Kantian argument in a little bit of detail, I mean he doesn't give the argument, it's intuitive enough, but it's useful for what I'm going to do later to just mention the connection with Gauss's law. And so let's do that. So what I want to do in the talk is first talk about the connection with dimensionality, Gauss, and gravity. There's some more contemporary stability arguments for the dimensionality of space. And I didn't give one or two criticisms of that. Okay, so everyone knows Gauss's law, and you can have Gauss's law for gravity. Instead of thinking about the room, the density, the fluid of the thing, you could have... g, the gravitational field strength, dA is going to be some little area, s, the surface, so there we have the flux, there we have our Newtonian force, and then Gauss's Law, which is a statement that the total flux through a curved surface is the same shape, but mass m is equal to the change in that mass, so then we're going to have that surface multiplied by constant. So the connection is really, you know, transparent, short and sweet. If you restrict to a simple case where you just have, you know, a source and then we're assuming that the force is coming out radially and the surfaces are spherical, then we can go from here to here and then...

7:30 The integration is just F go to minus Gs. That's all based on the assumption that G is radial and S spherical. Now we just plug in G and S, put them over there, plug them in, and then we get our Gauss's law back. As you know. And then you can see how the dimensionality come in here. Well obviously the r's are only going to cancel it. They're both r squared. So the force has to be coming out and getting weaker as it's going out there. It has to be getting weaker a certain way. One of the r squared way. And you can see that this whole argument wouldn't work if space wasn't three-dimensional. If space was, you know, if we were to think about higher dimensions and look at the s for the next dimension, you know, then that's going to be a portion of r. Then it wouldn't work. So you can see that you need the dimensionality to be... So you can see there the sort of connection between dimensionality and Gauss's law and the force law. Obviously this kind of thing is going to be a problem when you go through a non-Euclidean context. The space looks like that and all hell is going to break loose for this sort of thing. So if you don't assume that space is flat, isotropic, etc., then there's going to be trouble. So if you were to try to run this kind of Kant argument or suggestion, maybe not an argument really, but a suggestion that dimensionality comes from the inverse square law, but now you want to do it from... Well, then, you know, that's going to be really hard. You can, uh, there will be, you know, probably there will be some constraints and, uh, but my suspicion is that it will be heavily solution-dependent, um, as I hear. Maybe, uh, one can find some generality by restricting to highly symmetric space-time equations for matter that contain only differentials of second order or less.

10:00 Or whatever other type of restrictions you can think of. Alternatively, this is the way to separate people who are not thinking that these proofs are going to disprove string theory or something. You can just abandon all the aspirations to view these arguments as relevant in the weak field approximations or in Newtonian theory holds. So when I turn later to all these different proofs of the three-dimensionality of space, How can these people be saying this when, you know, there's all these superstring theorists and stuff saying there's all these higher dimensions? Are they saying that they're, you know, a priori false or something? But the answer is no, they're just retreating to, you know, the space has a look, you know, three-dimensional at a certain energy level. Okay. It's not so set up. Let's now look at the argument I really want to consider. Which is one particular species of a whole general class of theories where you try to take some particular aspect, something that happens, and then use that thing to show that three dimensions are picked out as special. So the most famous one, I think, is the idea that you only get stable orbits in three dimensions. So Paley in 1802 has this kind of argument, and then there are further refinements of it later. But this kind of argument, in terms of its general strategy, is the same as, you see the same thing for hydrogen atoms not being stable. So there are lots of papers on whether hydrogen atoms are stable in the outside three dimensions. There's all sorts of papers, you know, trying to show that the three dimensions are special for reason X, and sort of pointing to some phenomena. There's some which are kind of cool, outside the purview of this, where that reason X might be some completely mathematical, non-empirical feature. Those are kind of interesting. So, and the idea is, at least in the contemporary literature, is definitely that this argument is supposed to explain why space is three-dimensional.

12:30 So I'm not just giving it as some sort of piece of the total evidence which makes us think that space is three-dimensional, but it's supposed to be, you know, the explanation of why space is three-dimensional. So I want to look at one of the, just a sketch, I mean, all the details are on the paper. You can go to the paper if you like. But just to give a kind of schema of how all these arguments work. Here I'm mimicking one from Birkel, who begins with the Poisson-Laplace equation for the gravitational potential, and now he's assuming it's having this form in all dimensions. Note also that that's just the differential form of Gauss's law. And note that that's the same thing in EM, so you can see the arguments that you can do in the gravitational case and what you can do in the electromagnetic case. So then there's the solution, constant over r to the n-2, and then the force, where it's r over r to the n-1. Now there'll be a definition of, so this is sort of an assumption. Now they'll come in with a certain definition of stability. So here you have different ones. So Ehrenfest has one, Verkel has one, Barrow has one, they all have different ones. And they'll come up with some kind of stability condition. So his is that the... There's a certain, you know, so when you're just thinking about a sort of Keplerian case where you have one going around the other, and you're stable for him if r is between a minimum and a maximum value, and then that's it. So it's not a kind of fancy conceptual stability that you get later, the analysis is there later. And so you get some sort of condition like this. This will then rule out n is greater than or equal to 4. What about n equals 2? What about n equals 1? Well then all sorts of, frankly, kind of PSA type arguments come in. Because no one really cares about n equals 2, and clearly no one cares about 4 and n equals 1.

15:00 If n equals 2, you see some things which are kind of interesting, like Hawking saying that if n equals 2, then your a-order will slice you in half. Or then they all of a sudden jump up on the classical context and then go, ah, but in DR, you know, there's no gravity in the lower dimension. But that's cheating, right? Anyway, there's tons of them. I wouldn't say tons, sorry, three. There's objections to this. So Bertrand Russell has one, Beth Branson has one, which is thinking of Russell's, and the paper BJPS in the 50s by Edward Minkow, which has another one with what, why does gravity get to be the one who decides how many dimensions? Anyway, I don't want to talk about those. In the forthcoming paper, I have some specific objections to this specific version of the argument. So the stability conception is too weak, I say, but also it's even worse than that. It's too strong, too. It's getting in both ways, really. And general theory is going to cause trouble, some trouble, not too much trouble in some ways. As you can run, actually, Tanglieri ran a... If you do a short-style solution, you can do a stability argument for that. What I want to do is I want to talk about two general objections. So, two objections. One, the first one is, is the three-dimensionality actually explained in here? You want to make, I mean, it was very natural to make a kind of distinction between question one here, which is, how many spatial dimensions are there, versus two, why are there that many spatial dimensions? Another commentator, in the modern era anyway, thanks to their ancient two, I'm happy to go along and say, you know, these folks proved that they were, if they were good. So supposing that one of these stability arguments actually could be made with a decent concept of stability, actually works, shows a kind of real constraint between the dimensionality and certain phenomena like stability or whatever.

17:30 And I would want to say that, you know, they're part of the total evidence package for whatever the three-dimensionality is, and so I'm happy to say that, you know, question one is answered, but I have problems with question number two. Here's a way, uh, uh, I think very, uh, ten, ten-year-old, uh, I wouldn't say, but that's what I said the first time. Um, yeah, here's the way he puts it. He says, take the claim that there shall exist stable orbits, make it a principle of indimensional dynamics in gravitation theory, and then show that the proposition that space is three-dimensional becomes a theorem rather than an axiom of such physics. Reichenbach, if you read them, he clearly distinguishes these two questions, and surprisingly goes on and wants to answer this one as well. Let's suppose the argument worked. Then you have some kind of argument like this, where space has n dimensions, the general law that the attraction between masses varies inversely with the n-1 power of distance. If so, then plans are stable only if n equals 3. Plans are stable. Therefore, if n equals 3, the case that n equals 3. Presumably, if you really want explanation, explanation, then you want the must-be reading in 5. Stability would give a compelling confirmation of the hypothesis that space is 3D, but wouldn't explain the dimensionality any more than finding a raven. The blank explains why all ravens are black. Then you get, it's not surprising here, then you get all those sort of anthropic arguments. I hate those so much I won't even talk about them. But one way you can think about the argument would be from a Lewisian best system point of view. Could you take the claim that the planets are stable and add this to the best system? Would it be a competitor against other best systematizations in the world? And I think it clearly wouldn't be. So I don't really have a problem with this kind of argument except that this just won't have the status. People want it to have. It's not going to be very strong or simple. It's not going to be that strong compared to the F equals MAs and Dirac equations of the world.

20:00 And it's deceptively simple. It's not going to be like this because not anything can get into orbits. And so you're going to have to start putting in stuff about charges and whatnot. And that's going to give you a pretty messy... Anyway, I don't think this objection would bother Reichenbach. You know, he's going to say that he's got a correlation between two phenomena, a reduction of two phenomena to one, so I don't think he'd really be bothered. So I think this is a more pressing problem. You know, when he, in just a sentence or so, he said, you know, what's meant by the physics of R4, R7? Guess what? I mean, it should really resonate with you. I think you've read that your Poincare, you know, where Poincare is going to let you, you know, he can get you all your observations consistent with any number of dimensions because he's going to let any physics vary in any different dimension. But, you know, you can't do that if some features are to be special. So you have people like Barrow and Kippler expressing a principle of similarity, so they say that the assumption that the laws of physics and the rules of other dimensionality are similar to ours is possible. But this is problematic, and to show why, I thought I'd go to a contemporary argument, and I don't have much time, but let me try to go through this. It's called the BLM argument and it's in the Journal of Mathematical Physics in 1999. It's a very interesting paper with a lot of cool things in it. But one of the things, I mean it's not the main thing, but one of the main things is that they show that there are stable hydrogen atoms in higher dimensions, contrary to one of their best results. And then, as they note, the same reasoning can then be used to show stable gravitational orbits, contrary to this huge industry I mentioned. How? Well, they, you know, do the classic, you know, turn a modus tollens into a modus tollens, or modus tollens into a modus tollens, or the other way around. So they hold fast stable orbits in an inverse R potential and then work backwards, as it were, and find that they're forced to modify the Poisson-Laplace equation for the Newtonian potential in the same way that they had to modify that equation for the electrostatic potential when they were doing the hydrogen.

22:30 And so you can just rerun the same reasoning they did for the hydrogen atom, rerun it here in my simple case with a gravitational case. And so they grab it, and so when I had that Gauss's law, the differentiable form of Gauss's law, V and F, instead of working to find V and F, they were stipulating V and then working to find Gauss's law. Not surprisingly, they didn't have to modify the Gauss's law. So now the question is, so if you assume, so here you have the quantum him problem in action. Assume there's stable motion in n greater than 3 described by one of our potentials, then you can modify Gauss's law. Or go the Ehrenfest way, assume Gauss's law and then show that there's no stable motion in n greater than 3. Which way to go? Well, you know, maybe you know, but I don't know. The principle of similarity doesn't help me much. From the force law, potentially, we can get Gauss. I mean, especially if you think about, you know, how easy it is to derive the force law from the Gauss law and the Gauss law from the force law. The assumptions are so mild in each way that, you know, which one is sort of metaphysically the right way to go, you know, good luck to you. The main problem, I think, is just that there's simply no background theoretical framework here for these other dimensions. And so you're importing some physics from when you're talking about these other worlds with other dimensions, but there's no sort of guidelines for which one, for which bits of physics. So both their interests in BLM are extending some physics into higher dimensions and not some other physics. Absence of developed physical theory that takes dimensionality as contingent and offers principles of physical constraints on what can happen in different dimensions, there seem to be no standards for knowing which laws hold what dimension. Let me just conclude by saying So I think these sorts of proofs and things are really very interesting, but they're interesting primarily in answering question one, not two. And it's interesting to think about this question from how it changes, the question why is space three-dimensional, from how it changes given the background assumptions. So if you go back to 1746 and go back to Kant, remember that early Kant was a relationist.

25:00 If you're a relationist, then the question seems to make sense. Given a relation as basis, whatever, whatever it is, how do I get the spatial temporal properties? Well, you know, that seems like a good question. And so, you know, dimensionality is one of these spatial temporal properties, so, you know, it makes sense, given the base, how do I get that? In fact, the inverse square law in the Kant table is not a bad answer, right, for that. Or, you know, fast forward a couple, you know, many hundreds of years to, you know, Kaluza-Kulani and that. I remember that Calusa in 1921, the first paper, he doesn't have a compactification mechanism or anything. There they are, just the five naked dimensions with nothing to squeeze out anything smaller. So, you know, it would be a natural question to say, hey, Calusa, you know, why does space seem to be given that it's not? And so, here again, you can see that question making sense. That's it. With respect to this, the problem that arises when you go to general relativity, I mean, have people, I don't know the answer to this question, but have people, instead of trying to run the argument in the small, so to speak, of general relativity, so, because then you don't have to worry, I mean, it's a manifold, right, so you get small enough and it's going to start to look like the context in which you did the argument originally. That's right. Yeah, I don't know of anyone who's done that. But yeah, in the very small, I suppose, it probably could be wrong. But then you think of these sort of arguments as a kind of approximation, not telling you that space has to be three-dimensional, but locally it does. But it has been done for the Schwarzschild solution, which would be the natural one to use for the Kepleric case. So then it gets staked a little bit. So they get like Newtonian and Keplerian objects. I'm translating everything from their hydrogen atom stuff into the gravitational case, but that's where I know the stability of it. But they have the same force law. I think so. I mean, but they have to modify the Gaussian law, and so it works out in general, too.

27:30 If you have any other questions, then we're going to thank Craig. That's right. I'm going to disprove everything that's been said. It'll come clear enough, quickly enough, that I'm not a philosopher, I'm actually a physicist. I gave thought to trying to talk like a philosopher, but then I gave up on it. So the entire talk is going to be just as if I am a physicist, but you don't want to hear me talk about philosophy anymore. I'll make you talk about it. I'm going to talk about these ideas that Craig alluded to, from modern physics including string theory, three dimensions might not be enough, and so the argument is basically first that there could be extra dimensions, you might not think that's so obvious, then I'm going to talk about quantum gravity as the motivation for considering these things. The idea of extra dimensions of space predates the idea that quantum gravity is an important problem, but today, the biggest motivation is trying to understand quantum gravity. Our best theory of quantum gravity is string theory, and if you believe in string theory, then you believe in extra dimensions, and that's basically the entire argument that I have. The good news is that we can go look for them. It is possible, and we get very, very lucky, that these extra dimensions could show up in some experimental way. I will go through very quickly because you are all experts. Anyone who is not an expert can raise it. Oh look, no one is on there. So general relativity is simply this one sentence statement that gravitation is the curvature of space-time. So implicit in that statement is that space-time has a curvature, can be curved, it's a dynamical energy that can respond to the existence of energy momentum in the background. And therefore, you could imagine, just knowing that could happen, unlike incoming gravity where the space-time is fixed as a background, in general relativity, some of the dimensions of space could be bigger than others. Some could curl up and be hidden from us. And this is the idea that was taken advantage of by Lutz and Klein a long time ago. Again, as Craig alluded to, the idea is simply that there are extra dimensions of space, but they're too small for us to see.

30:00 So like a tightrope walker walking on a tightrope, if you're much bigger than the rope, you can successfully model it as being one-dimensional. Or if you have a straw that you look at from very, very far away, it might as well be one-dimensional. If you look at it more closely, you see it's actually a two-dimensional straw. If you look at it more closely, you see it's really a three-dimensional straw, because even then, the straw material has a thickness. So the idea of loose decline theory is that there is some manifold, some compact manifold of spatial dimensions. There are too small for us to see. At every point in space in this room, there are extra dimensions we cannot have access to. In physics speak, small spatial light scales correspond to high energies. We often hear physicists say, we don't have high enough energy particles to access the extrications of these extra dimensions. The original idea of Lutz and Klein is really good, but it doesn't work. It does not actually describe nature, and here's why. The fact that there are extra dimensions that are really small and come up into little balls doesn't leave Leave an imprint on physics in the big three spatial dimensions. Namely, all of the features that tell you what kind of little manifold you have show up as fields in our three dimensions that can vary from place to place. So consider the simplest possibility of three big dimensions, one small dimension curled up into a circle, an S1. Then there are two things you need to know about this circle that exists everywhere in space. You need to know how big it is and how it's oriented compared to the neighboring circles. Those two facts show up in our macroscopic world as two sets of fields. The orientation of the circle shows up as something that looks just like the electromagnetic field. That is good, because we observe an electromagnetic field, maybe it comes from the orientation of a little circle. But the size of the circle, the volume of the one extra dimension, shows up as a scalar field, which if you quantize it will be a spinless particle which has zero mass. We would have detected such a particle long ago, and we haven't. That particle doesn't exist. And therefore, the Klein theory in this particular manifestation is wrong. So it's an interesting lesson, though. You can throw off extra dimensions, but you can't do anything you like. You can't just say, well, they're too small to see, therefore we could do anything with those extra dimensions. They will show up in some way in the physics of the 3 plus 1 dimensional space model. So, after it was realized that stuff like that was going on, the Euclid-Klein idea was basically abandoned as a group to work on,

32:30 although people remembered that it was there. It didn't disappear from our consciousness. Instead, we worked on particle physics, and between the 1920s and the 1970s, we figured out particle physics. The standard model of particle physics, as far as any laboratory experiment ever done, is completely consistent with all the data that we have. And it describes the set of quarks and leptons and force carrying particles that is phenomenally successful. The loophole, the thing it doesn't quite yet fit, is that it is based on quantum mechanics. The standard model is a quantum field theory. Without quantum mechanics, it wouldn't make sense. We knew there was quantum mechanics already. This framework, in which we understand all of particle physics, three of the four forces of nature, dramatically relies on the success of quantum mechanics. Meanwhile, gravity is left out. We do not know how to take Einstein's theory of general relativity and make it consistent with quantum mechanics. Therefore, we have two theories, both of which are completely successful in their respective domains. Quantum mechanical standard model and classical general relativity but they're usually inconsistent with each other. So if you were really a pragmatist you could just say well I don't care. I will sometimes do gravity, I will sometimes do particle physics. Here is a reason, in principle, that you should want to reconcile gravity with quantum mechanics, namely that you would like to answer the question, what is the sun? So the sun, which you're familiar with, is that big ball, very obvious. It serves different purposes to us here on Earth. It is a source of light and heat. It's very useful. And the fact that it's a source of light and heat relies on the fact that the particles in the sun obey the rules of quantum mechanics. You can have nuclear fusion, it gives off energy, and it's that energy that comes to us. So if you're trying to describe the sun as a source of light and heat, you describe it as a set of all particles that have wave functions that obey the laws of quantum mechanics. It also is a source of gravity. The Earth goes around the Sun. And therefore, you might want to describe how the Sun creates gravity, and in the context of general relativity, you need to describe it as an energy momentum tensor for the different constituents in it. The fact is that we have no conceptual framework in which to simultaneously describe the Sun as a source of gravity and a source of light energy. Well, I think that we should want to do that, even though it is just a question of principle. There is also questions of practice.

35:00 So here is a map of the temperature and isotope facing the cosmic microwave background, left over radiation from the Big Bang. Our best model right now, although it would be certainly very hard to prove that it's true, for the origin of these tiny changes from place to place in the temperature of the cosmic microwave background, these perturbations that grew into stars and planets and galaxies, is that this is a snapshot of quantum mechanical fluctuations of some scalar fields in the early universe during a period of inflationary expansion in the early universe. In other words, the curvature of space-time as described by general relativity What was going on in the early universe. On top of that, there were quantum mechanical fluctuations, and the combination of these two phenomena at the same time leads to this picture. If you want to describe this picture, you need to be able to talk about quantum gravity, and you can't do it yet. So it's not just a question of physics. So, you already know quantum mechanics. Let me give you it in my words. And here's where my lack of philosophical vocabulary will be painfully obvious. I'm going to talk about it the way I think about it. This is what quantum mechanics is. Remember, there's a one-sentence version of general relativity, gravity is the curvature of space-time. There's a one-sentence version of quantum mechanics. What we can observe is derived from, but not the same as, not in one-to-one correspondence with, what actually exists. That's the great conceptual break when you go from classical mechanics to quantum mechanics. What you see is not what there is. So in classical mechanics, you have an object, a system, that has a state. The state can be described by the position and the velocity of the particle. What can I observe about the particle? Well, you can observe the entire state as accurately as you like. You can observe its position and its velocity. You can find it oscillating back and forth in the simple harmonic oscillator potential. In quantum mechanics, you can observe either the position or the velocity of the particle, if you like, to arbitrary accuracy, but not at the same time. And what you can't observe is what actually exists, because what exists is a wave function, a function of space, that gives you an answer to the question, how likely is it to observe a certain velocity or a certain position when you look? So there's a disconnect between what actually describes what's going on and what you're allowed to look at. So here's what that turns into when you apply that notion of quantum mechanics to particle physics, to the interactions of elementary particles described by the standard model. In classical mechanics, you could have an electron absorb a photon. Just like that. That would be what Maxwell's theory of electromagnetism could tell you what would happen. But this picture, this Feynman diagram, is very classical. This is an electron with a point-like object running into a point-like photon and turning into an electron.

37:30 In the Chicago Tribune Causal Puzzle today, one of the answers was, electrons. And I guessed photon, so I was wrong. What really happens is not a point like electron or a point like classical photon. What really happens is a little wave function of electrons interacts non-linearly with a little wave function of photons. And Feynman's great insight was to figure out how to calculate what happens. We already knew what happened, but we need to figure out a few ways to draw it and then to calculate it. Namely, the fact that these are not points but really little wave functions can be taken into account by taking this diagram and adding to it every other diagram that you can draw, consistent with a few simple rules. So not only does the electron come in and bump into the photon, the electron comes in, emits a photon, bumps into another photon, and then reabsorbs its original photon and then goes on its merry way. Or, the photon comes and emits an electron positron pair, reabsorbs them, and then bumps into the electron, and every single possible diagram has to draw. That way of drawing a whole bunch of classical diagrams takes into account all the quantum mechanics that is going on in the world. And these diagrams are pretty, but they also are calculationally useful. To each diagram, you associate a number, and that number is the contribution of the diagram to the likelihood that this process happens. Not every electron runs into every photon. There's a chance that they go by and they will run into each other. This set of contributions tells you what that chance is, what the scattering probability would actually be. Once you have these loops, see this is a simply connected diagram, this one has a little loop, that one has a little loop, every other diagram except the first will have loops in it. If you're not careful, when you calculate the numbers corresponding to diagrams with loops, the answers are infinity, which is bad, and it causes a lot of angst and a lot of philosophy and stuff like that. You can renormalize these things. There's a generational gap. The younger you are, the less renormalization bothers you. In an attempt to remain young, renormalization doesn't bother me even a little bit. But you need to be able to do it if you're going to claim that the theory you're working with is truly fundamental at some level. Maybe your theory is not fundamental. Maybe you have the Fermi theories of weak interactions.

40:00 It's a wonderful, wonderful theory, not fundamental, useful in some circumstances. But if you're going to ask whether or not gravity, described by tenor relativity, is fundamental, it better be renormalizable. But it's not. And despite many, many person hours, a very small physicist, no one has figured out why you take general relativity and turn it into a renormalizable quantum field. That is the technical obstacle to constructing a quantum theory of gravity. The reason why is because when the loops become really, really tiny, that's a short distance, that's a high energy, and these high energy processes are divergent. They give you bigger and bigger contributions. So a conceptual leap you might make, if not at all rhetorically what happened, or after the fact explanation, is you might smooth out these diagrams with little tiny loops by taking the original point particles and replacing them with little strings. So if you imagine that this electron is really a loop of string, the photon is really a loop of string, and this interaction where one bumps into each other is described by this pair of pants diagram, where one string touches another one and turns into a third little string, then the combination of all the Feynman diagrams becomes a combination of all these string diagrams of all possible topologies. Why in the world would anyone have that idea? It's a whole long historical story that is worth telling, but I'm not going to get a chance to tell. The idea is that each individual particle, whether it's an electron or a photon, is the same kind of string, just vibrating in a different way, displayed here as being blue or red. So the idea of string theory, then, is simply this. Every theory in physics has a one-sentence statement. It's a good science for people with short attention span. The idea is that every particle, every elementary particle, if you look closely at it enough, resolves into a little one-dimensional loop of string. That is relative to this, because it's the only other thing you have. So a proton, for example, is a combination of three quarks, a characteristic size of about one pheromene, and minus 14 centimeters. I had to say Fermi's name a lot because I'm at the University of Chicago. If you zoom in on one of the quarks that makes up this proton, you would see that it is a little loop of string. You have to zoom in a lot. The strings are reportedly at the Planck scale, and at minus 33 centimeters, a proton which we only think of as small is 10 to minus 14 centimeters.

42:30 So you take that idea, you take the statement, every little particle is really a loop of string, quantifies it, make it into a theory, ask what happened. That's all that you need. There's no extra input other than that. One of the things that happens is you get gravity. The original people who were doing string theory didn't want gravity. They wanted to describe a strong interaction. They kept trying to take these strings and quantizing them, and they kept predicting gravity and getting annoyed. So one of them said, you know, gravity exists. Maybe we're not in a theory of everything. Then their heads blew up to huge sizes. So after this conceptual leap was made, we started thinking of string theory as a quantum theory of gravity. It's a quantum theory unlike every other attempt to quantize gravity, I'm going to tell you why string theory predicts gravity. This is the highlight of the talk right here. Why does string theory predict gravity? Well, first it came out of gravity. So gravity is complicated through the curvature of space lines and so forth. One of the predictions of general relativity that didn't exist in Newtonian gravity is gravitational radiation. Just like if I take an electron and I shake it, according to Maxwell's equation, it gives off an electromagnetic wave. If I take a mass, or two orbiting masses, like two black holes, and I shake them, or I let them warp each other very, very quickly, the disturbance in space-time caused by that propagates out at the speed of light as a gravitational wave, an oscillating gravitational field. And right now experiments are going on here on Earth to try to detect gravitational waves correctly. LIGO experiments, Planck experiments, and satellites in outer space. So how do you detect it? How do you know if a gravitational wave is passing by? Well, you set up a bunch of test particles, a bunch of objects that are freely floating, no forces acting on them. Let's say you arrange them in a circle, just to be convenient. And then a gravitational wave passes by. What happens? Well, you might think, well, the gravitational wave moves all of them up and down, but since everything is relative, you would never know that that was happening because you would be moved up and down as well. Gravity affects everything equally. Instead, what you detect is the tidal force, the stretching of space-time relative to what it was. So this initial circle of particles gets deformed into an ellipse, first upwards and then sideways and back and forth in a plus sign. There's another polarization that oscillates back and forth in other times.

45:00 But that's how you would detect a passing gravitational wave that would disturb your initial circle into an oscillating plus and minus sign. That is classical gravity according to Einstein's equation. That is the effect of passing gravitational waves. And then you say, well, what are the fundamental modes of a vibrating string? And this is one of them. One of the fundamental modes of a vibrating string, you take a little loop of string and it does that. This is not a coincidence. This is why string theory predicts gravity. The simple fact that a loop of string can go like that means that if you take these little strings and think of them as making up particles and you quantize the theory, you will predict gravity whether you like it or not. String oscillations look like gravitational waves. So there's many different ways a string can vibrate. Remember, all the vibrations look different. This is one unescapable mode of vibration, and it looks like the spin-to-graviton that you would need to make generalities. So string theory is good. It is the only theory we have, or at least the most promising theory we have right now. Congratulations to all else, gravity plus the standard model of particle physics in the context of quantum mechanics. But there are other things that you are forced to get. So you start with the idea of a simple vibrating string, you quantify it, and you are forced to get gravity. Another thing you are forced to get is extra dimension. This is the low point. The low point is, why is it that when you quantize a string, you are forced to get extra dimensions? And I did not figure out a way to make sense of why that is. The long story made short is that when you quantize things, it often is the case that there is some symmetry in your classical theory that is broken by the fact of quantization. Sometimes that is actually true. These are called anomalies, and you can observe the anomalies. Other times, the symmetries that you were starting with in your classical theory are really precious to you. You don't want to break them. You think if you break them, you made a mistake. It turns out, certain symmetries in string theory that you really, really want to preserve can only be preserved if you find mechanical anomalies in higher dimensions. Four dimensions of spacetime, three of space, one of time, will never do the job. So how many do you actually need? Well, it turns out there are different versions of string theory that require different numbers of dimensions. The simplest string theory is bosonic string theory. Bosons, you might recall, are the particles that have integer-valued spins. Like spin zero, it takes bosons. Spin one, the photons, the gluons. Spin two, the gravitons, etc. You can make string theories that have nothing but bosons in them. They naturally live in 26 dimensions.

47:30 Those are not thought to be realistic places to start describing the real world. Number one, because there are half integer spin particles in the world. There's the electron, which is a fermion. Particles that spin one-half or three-halves are fermions that don't exist in this theory. Also, this theory is not stable. There is no vacuum state of this theory that can just stay there all by itself. That's what it means when you say there's a tachyon in theory. Fields start growing exponentially even in any spin. But then there is the superstring. Super because you have a symmetry relating bosons to fermions in this theory, and that is, of course, supersymmetry. It was first invented in the context of string theory. Supersprings have both bosons and fermions. That's good. They are stable in the vacuum. That is good. And they predict that space-time has nine spatial dimensions, one time dimension for a total of ten dimensions. So if you stop there, which we were stopped at until about ten years ago, you would say that string theory predicts there should be six extra dimensions of space. We normally think there are three. Superstrings live naturally in nine. However, about ten years ago, we promoted string theory to a bigger theory called M-theory. And M-theory predicts that ten-dimensional superstrings are one version of the theory, but there is another version as well. This is kind of cool. This is the second high point. M-theory includes another theory which is 11-dimensional. And sometimes it's called 11-dimensional supergravity, sometimes it's given different names. The point is that this is an 11-dimensional theory, 11 space-time dimensions, so 10 dimensions of space, that comes not with strings but with two-dimensional extended objects called brains, short for membranes. So these two-dimensional extended objects naturally live in 11 dimensions. And you can say, what is the relationship to string theory? Well, to get from 11 to 10 dimensions, do what Lutz and Klein taught us to do. Take one of those dimensions and compactify them on a circle. Then what happens? If you say, let's say this is a circle, a circle can be described as two points, a line segment with opposite points identified. These two planes are identified, and here is a two-dimensional brain living in this ten-dimensional space with an extra dimension of time. Compactify that dimension right there, if you compactify the ten-dimensional space down to nine dimensions, the two-dimensional membrane becomes a one-dimensional string. You can even go backwards, which is the beautiful part of it. You can start with a ten-dimensional string theory, crank up the coupling and the energy, and show that another dimension arises where you didn't think it was before.

50:00 So, the best thing to say is that string theory lives in either ten or eleven dimensions of space. If you really wanted to be picking one, you would say it lives in eleven dimensions of space-time. Seven total extra dimensions needed to make string theory. However, people got a little carried away with this notion of brains. They started playing with the notion that there are not just brains, but higher dimensional objects. And you can't do anything. Again, it's a very constrained theory. Some kinds of brains are okay and some are not. One of the nice features of certain kinds of brains, called D-frames, is that fields can be confined to them. In other words, you can have a brain embedded in a bigger space, but there are theories that live only on the brain. If you were made of the stuff that couldn't get off the brain, you wouldn't know that you were living in extra dimensions. So here is a new way to hide extra dimensions of space. Maybe we live on a brain, a three-dimensional brain, embedded in some larger dimensional space puzzle. If we were made of the kinds of particles that couldn't leave the brain, we would never know. It would be exactly like the little people in Flatland embedded in some bigger space, but you're not allowed to leave your lower dimensional subman. The nice thing about this idea is that it is testable, because there's one thing that you can never confine to the brain, and that is gravity. When you combine all the particles of the standard model of particle physics, the electron, the photon, all the other puzzle answers, can be confined to the brain, but gravity will always escape into the book. And therefore, you can say, well, Craig just gave us a talk saying that we know that space is three-dimensional because the gravitational force goes as one over r squared. How well do we really know that? If you have an extra dimension of space that is, let's say, that big, literally that big. Had we tested Newton's law of gravity that says that the force is 1 over r squared on scales of 1 millimeter? We tested it in the solar system, but how well did we test it? Well, when this theory was predicted, we tested it only down to 1 millimeter. The prediction is that on scales r bigger than the size of the extra dimensions, you have n extra dimensions, all of size l. On bigger scales than that, you can ignore the extra dimension, and Newton's law of gravity goes to 1 over r squared. But on scales smaller, you begin to see the extra dimensions, and therefore the force due to gravity can go as 1 over r to the 2 plus n if you had n extra dimensions. Everything actually works very nicely if you have two dimensions at the size of a millimeter.

52:30 This can induce people to do experiments, taking metal objects and moving them very, very close to each other, measuring the force of gravity between them, and we've managed to improve the experimental bounds by one order of magnitude. So now we know there are no extra dimensions as big as a tenth of a millimeter, a hundredth of a centimeter, and we're getting better. Still, remember the size of a proton is 10 to the minus 14 centimeters, so we have a long way to go in terms of freedom to have large extra dimensions. Another way we're trying to find them is to make gravitons. Gravitons are these particles, these spin-through excitations of the gravitational field, generally thought to be so incredibly quickly coupled that conserving a single graviton is impossible. But if you have extra dimensions, the coupling to the gravitons becomes much bigger. So you build a particle accelerator outside Chicago, this is Fermilab, I think this is photoshopped in there, this is Skyline. You collide protons together and they make gravitons. Here's a spray of gravitons coming out of a particle accelerator. You don't see them, because they're very hard to see, the gravitons. But what you see is that the total energy of the stuff that went into the collision is bigger than the total energy of the stuff that went out. There's missing energy in your detector. It has a certain signature in terms of the angular distribution of the stuff that comes out. And you can actually constrain the existence of gravitons leaking into your extra dimension. When I teach class, the rule is that if any cell phone rings, I get the answer. Very effective. This is going on at Fermilab right now. At CERN, the Large Hadron Collider will turn on in 2007. If all goes well, there will be better experiments for it. Finally, here is the ongoing controversy. Here is one of the things that we don't know. If you take these seven extra dimensions and curl them up into a little ball, what is their shape, volume, topology, and geometry? Well, there are certain discrete ways that you can do this, as it turns out, and their hope was that one would be right. There will be a unique way to go from 10-dimensional string theory to 11-dimensional M-theory down to the 4-dimensional real world. You would find it, and you'd be finished, and you'd all become philosophers. There's no danger that that will happen. It turns out that if you can do it one way, you can do it something like 10 to the 500 different ways. Taking these extra dimensions and compactifying them into some stable 7-dimensional geometry. This is just one slice of some perspective 7-dimensional geometry. So what you want to do is you want to get your graduate students to work through these one by one.

55:00 And you think that the universe is doing the same thing. The universe in fact has already done it. It is possible that the reason why we live in the vacuum we do is because every vacuum exists somewhere in the universe. Somewhere further away that we can't see. And we live in a particular vacuum that are hospitable to the existence of life. So you know what this is. This is what Craig doesn't like me to talk about because it gives him indigestion. It's the anthropic principle. But there is one problem that it might solve, namely the cosmological constant problem. You may have heard of this, the idea that there's an energy density in empty space. To predict what that energy density is, you can compare it with the observed value, to see the observed energy density in empty space. The theory and the prediction differ by a factor of 10 to the 120, the largest discrepancy in string theory terminology. Even by cosmology standards, this is a large structure. But if every single possible compactification existed somewhere, maybe large values of the cosmological content exist somewhere else. Because the physical features within each vacuum depend on the way you compactify. Maybe we live in the region. Where the cosmological constant is small because if it weren't, if the cosmological constant were large, it would grip up the stress. The energy against the empty space would give space-time such a curvature that it couldn't be in here cutting a constant. So this is a picture that that might actually happen. We don't know whether it actually happens or not. So in conclusion, there are more things on the list of things we don't know than on the list of things we know. The things we know are not facts, but imperatives. We need quantized gravity. It's just not acceptable to deal with the classical theory of gravity and the quantum mechanical theory of everything else. String theory is a leading candidate. It might not be right. It's not the only candidate out there, but it's by far the leader right now. And they imply extra dimensions. What we don't know is what are the extra dimensions doing? How are they stabilized? We still have the old flutes-inclined problem that all of these theories have the size to take a dimension that predicts a massless scalar field, but we don't observe. We need to stabilize that size somehow and give a mass to the scalar field. We live on a brain or in what we call a ball. Are there any dimensions that are big and perhaps detectable? It's possible, but by no means necessary, is that theory. Are there other domains elsewhere with different compact associations? And if there are, just the fact that we live in a nice hospitable environment actually counts as an explanation for anything in a philosophical sense. So this is what makes it exciting. We have a lot of data coming in, a lot of theoretical possibilities going on there.

57:30 I would definitely vote there are more than three dimensions. Thank you. I wanted to ask a bit more about gravity coming out of the string theory. I take it it's general relativity, not just gravity. It is a modified version of general relativity. Okay, let me ask. I saw the first step. You essentially have quadrupole radiation, so you've got a secondary tensor somewhere in there. Now there are two more steps. Now you want to get the Einstein equation, and then you want to make a connection to mathematical geometry. Now let me observe. There is a result. It just says... If you take a second rank pencil field, which carries energy, maybe its own source, you bootstrap up, you get Einstein's equation. That's right. That's what we're doing. So that's an old result. That's one way to do it. Okay. All right. And then the second question, how do you go from that field to the interval in space and time? Well, part of that result, to which you referred, is that you start with some other background geometry, put this field on, and let it couple to everything. Then the original background geometry disappears from any of the equations. The only thing that ever appears in an observable quantity is the effective geometry that comes from adding up all the effects of these interacting spin-to-pin equations. So, at that point, it is really a philosophy question. The observable thing is this effective curved space-time metric that solves Langstein's equation. Whether you want to call it the metric of space-time or not is up to you, but it's what we can do. You've now described the well-known result, string theory. That's right. That's one, I mean there's, like any good result, there are many ways to derive it. This is one way to use string theory to derive general relativity. There are other ways that involve words like vanishing of the beta function and stuff like that. But you can do it more than enough ways that you're convinced that it happens. How much time do I have for questions? About three minutes. When you quantize a string theory, you want to preserve symmetries, and you don't like these anomalies. Is the only reason to preserve symmetries so you can get re-anomalization, or is there some other independent reason you want? Yeah, you want a theory to make sense, and you really want a definition to make sense. Renormalization is part of it. It's better than that. I'm not quite sure I can put it into words right now. I wouldn't say, no, it's simply renormalization, but you want a theory that you know you don't need an infinite number of input parameters to answer any questions. You don't want to violate unitarity. The probability for anything happening should never be greater than one and things like that. I couldn't give you one realist, but it's a little bit stronger.

1:00:00 Let's see, I have a few questions, but maybe I'll just follow up on what you were just commenting on, and that is, as far as, you seem to be equating renormalizability with making sense, and this is common part of it, and in fact, super string theory, we need renormalizability because otherwise it wouldn't be consistent, and... As a philosopher, I've often been troubled by this vocabulary. In particular, it's not inconsistent to have a prediction that is infinity. It's just false. And so I'm just wondering if there is, if you would be happy with that characterization of the requirement that we have renormalizability. Well, we want a useful theory. We want to have a theory that is true or might be true about the world. And if what we're getting at is... If what we're giving out are infinities, then of course then we don't have that. So I'm wondering whether you would be comfortable with characterizing it that way instead of it making sense. Well, I think there are two statements.