Discussions

0:00 Good afternoon. Good afternoon to Kinga Marco-Palula and her husband, O. Stryer. She unfortunately had a miscarriage this week and still recovers now, and I assured her that you would understand that Olaf, her husband, stayed with her, and she was very thankful for that. So this afternoon we will just have the one paper by Sean, and then we'll go immediately into the roundtable discussion. To introduce our two speakers. Today's session will be on a rather different note than yesterday's, which more or less started from the point of view of pure mathematics and stayed within that noble realm. Today we're more or less starting from the point of view of physics and the role of space, more generally space-time structures, within physical theories. From my own point of view, we never deal directly with reality. We always deal with reality through theoretical models. The remarkable thing about general relativity is how it revolutionized our approach to space-type structures. Every previous theory involving space-type structures, and outside of thermodynamics, I guess all of physics directly involves space-type structures, every previous theory was based on fixed background space-type structures. We could call Galilean-Newtonian spacetime for the so-called non-relativistic theories and Minkowski spacetime for the relativistic theories. So in these theories, there was kinematics based on the invariance group of the background spacetime, which formed the basis for the dynamics. Dynamics had to be in accord with the kinematics. General relativity, for the first time, we have a theory in which all spacetime structures, This is represented by a metric and the inertial gravitational field which is represented by a connection which would be compatible in the case of general relativity. Both of these structures themselves are dynamical. So my slogan is in general relativity, no kinematics without dynamics.

2:30 On the other hand, the other great theoretical advance of 20th century physics, quantum mechanics, non-relativistic quantum mechanics. Based on the Galilean-Newtonian space-time and relativistic quantum field theory based on the Minkowski space-time are theories with fixed background structures. And the role of these background structures plays a great role both in the formalism and the interpretation of the formalism in terms of measurement. If you want to measure amplitude for a certain process, you have to know where to place your instrument and when to turn it on and where to look. I mean, the source generating the field, or what a structure is, where it is and when to turn it on, and you also have to know where to put your detecting instrument and when to detect. As I said, the background space-time structure plays a great role both in the formalism and its physical interpretation. The formal point of gravity, at least as I see it, is how to reconcile. The great advance of general relativity and background independent theory, in which all space labs are dynamical, and the accomplishments of quantum field theory based on them. In the course of our discussions leading up to this conference, I happened to be reading a paper by our first speaker, Louis Crane, Categories of Geometry and the Mathematical Foundation of Quantum General Relativity. We explored the possibility of replacing point set topology by higher category theory and topos theory as foundations for quantum general relativity. We discussed the Barrett Crane model and problems with interpretation and connective construction of causal sites. It immediately occurred to me that Dean Crane was the best speaker to represent the point of view of the implications of category theory and topos theory. The concept of space-time structures, and that's what he's going to tell us about today. Professor Crane was a math degree at the University of Chicago, but as he said to me, he always considered himself in between math and physics, which is not an easy or angular position to be in, at least in the American academic world. Nevertheless, he spent some time at Princeton at the Institute for Advanced Study, at Yale, and he was a visiting professor.

5:00 He's a scholar in Portugal at the ISP, in Canada at the University of Western Ontario, and in France at the Montpellier. He's currently, and for a number of years, has been at the University of Kansas. And today he's going to speak to us about, I love the title, the title of your talk is... Do you remember the passage from Lewis Carroll, what do you call the title of the talk is? Yeah, okay. So, thank you for that very kind introduction. Nevertheless, I decided to call my introductory thing me because I want to express a point of view since it is a philosophic gathering. So I want to talk something about the underlying philosophy, in fact it won't be very long. I think the point of yoga is that mathematics and physics should not be used as separate sciences. You can separate them temporarily, but sooner or later they depend on one another, and in their development they depend on one another. I think all of our real knowledge comes from our interaction with nature, and when we abstract these things in the mind, they proceed for a while, but they eventually sort of die out and we have to go back to nature, and I suspect that there are some people who sense out of their point of view is not completely opposed to that. So I always thought, let's think more concretely, well what is geometry? Is it mathematics or physics? Well, to go back to Euclid, I don't know much about the 18th and 19th centuries, so I'll go back to the elements. He's actually thinking about real intervals, which he's thinking about moving and bringing together. Some of the books tell you how to do things in a strategic compass. So, geometry, the three words mean measuring the Earth.

7:30 So, it begins with an interaction with the entire natural environment, and of course we walk around the Newark Peirce Center for dimensional continuum all the time, and on mathematical development of geometry in all its senses. We begin with that and then generalize it. There was a lot of history, discussion of history, so you could mention one, a point, I was told that this was the foundation of all mathematics. I was told when I was eight that you have to think of it as a collection of infinitely small things called points. And I thought a minute and I said, do you mean atoms? And he said, no, no, smaller than that, points. And I said, I don't believe that. The person who was telling me this said, well, you can't do math if you don't believe that it's the foundation of everything. So it was at that point that I became a rebel. But interestingly, this definition here, I always thought, well, that's just a stupid old that doesn't make any sense. But actually, as we now understand, if you're a set theory with topos theory as the foundation of mathematics, About which topos you pick, definition two makes perfect sense after all, because there are some points that actually have directions to them as good as any other point. So I think that's interesting. So the advance from point sets of topology to category theory is a lot to understand definition two. Now we've got to work our way back to definition one. And of course in the topos theory, the notion of a point complicated thing.

10:00 It's not that there are no points at all, but they are sort of relative. One man's point is another man's mason. That's right, that was a joke. So, those are sort of the ideas, but it's been very difficult pursuing such a career, because particularly in America, which is dominated by pragmatic philosophy, there's an enormous gap between the two subjects. I want to argue that the two subjects have to be developed together, and in fact, the critical issues that face both subjects end up pointing to the same place, although this is not very well known to these people. And in fact, it will be resolved in all probability together. I'll make a historical example since it's seduced into history. You know that when Newton invented calculus, he did it by inventing these infinitely small numbers called fluxions. And Berkeley immediately passed them and said that there's no such number. So, nevertheless, people didn't go and research deeper into it and find some way of making that right. They just went forward. They went forward instead of down. And eventually, they used Newton's matrix to calculate the orbits of the planets. Well, that turns out to be very hard. You can't do it in closed form. So, they expanded them in perturbation series. And Laplace, when they worked out these perturbation series, he sustainedly moved. And he went and gave a series of lectures about it. And Cauchy was there. And then Koshy went home, and the next day he came back and he invented a technique for figuring out whether series converge. And then, so 150 years after Newton, people finally had the basis for making a rigorous foundation for the mathematics. And they didn't do it by going back and by, you know, staring at the mathematics in itself. They went out into the world and somehow their contact with the world told them the next thing to do. So I think we have to do that again. That's the way it works. And that sounds a little bit like, well, I have to admit there is some connection.

12:30 Okay, so foundational issues. The two subjects have in common, but they don't really have a foundation. So, to begin with, the foundations of mathematics, more or less, I mean there are small branches of mathematics where you can avoid this, but the important ones depend on the definition of the real numbers. And as I think people are generally aware, we don't really know what we mean when we say the real numbers, or sometimes people say the real real numbers. We don't know what the real real numbers are. Now, the problem is not with the real numbers we know. Any real number that has a name isn't much of a problem. We can define it, in principle at least, and even though it would take an infinite number of steps to calculate it exactly, we know what we mean. But that's only a countable subset, which means it's a measure zero. Although it's not recursively enumerable in any system, but... So most real numbers are just given to us, but we have no way of naming them, and the only way we could describe them would be by an infinite process, and it wouldn't be recursive. You'd just have to write down an infinite number of decimals. So any attempt to actually write down any of those real numbers would require an infinite number of storage spaces in a computer. So they are very imponderable things. And there are many different models for them. One is as good as another, and theorems of the true and one are false in the other. So we don't know what we mean by the real, real numbers.

15:00 And if you ask most mathematicians what's the foundation of mathematics, they'll tell you what Herb Pilgrim told you, that mathematics is rules for how you combine symbols. And then if you press them and say, well, actually, the Hilbert program failed, we know that's not right. But pretend. Don't look down. You survive as a mathematician the way you cross a tightrope. You don't look down. That's math. But notice that the problem that we don't know how to understand is to find a scale structure. Passing from mathematics to physics, we now discover that modern physics, as it's currently developed, has two branches, which we don't know how to unify, and the other is general relativity. So let's discuss the foundations of both of those. And so the moral here is don't look down. The way we understand quantum field theory is by expanding a Lagrangian into terms called Feynman graphs. The Feynman graphs correspond to integrals. The integrals correspond to all configurations of the graph, all the way to map the graph into spacetime. And when we study this, we find that in all physically interesting cases, the integrals are divergent. So, and the divergence is what matters, is what's called the ultraviolet divergence, which means the graph might have loops in it, and as the loops shrink to zero, the integral has an infinite oscillatory part that we've done on the Maxenza. Now, looking down to the point, and this is an ultraviolet divergence. Now, the state of the art is that we have a bunch of tricks for extracting numbers from... And the one that we actually think is right, the Manabee-Lean-Yaniel theory, the one that actually lies at the foundation of particle physics, and therefore of our understanding of the world as it is now, for about 15 years people thought it didn't work because there weren't enough tricks, you couldn't cheat a finite answer out of it, and then Tuft came along and came up with dimensional regularization, and when you do it that way, you can. And so you can extract numbers from it and compare them to reality.

17:30 And sometimes it works and sometimes it really doesn't work all that well. But when it doesn't work all that well, there are I think that the theory, as it currently stands, should be considered a working model and not a real theory by pragmatists if they know how to get the answer they're having. But there's a deeper understanding of that. And the reason, the understanding is originally people thought that they could work back from these Feynman diagrams and reconstruct the theory. That was a program called Constructive Field Theory. When I was in graduate school, I said, I think I want to work in that. And people think they don't do it. And the reason is, it doesn't work. Constructive field theory turned into destructive field. No foundation. Those theories don't exist. Now, there's reasons why this doesn't work. And the reason is that people adopted what's called renormalization group picture. They said, do the quantum field theory only on a lattice. So you cut out the ultraviolet and then ask what happens as you expand the lattice, regrouping things. And you get a flow on the coefficients, which is called the renormalization group flow. And what they discovered is that if you put in a term which is non-renormalizable, The coefficient in front of it flows to zero. So, we are many, many, many doublings away from the fundamental scale of nature. So, the averaging process, it's sort of like a non-linear, very sophisticated version of the law of large numbers. In fact, if you do a free field theory, you find out that the propagator has to be a Gaussian, and that's nothing else but the law of large numbers. So, this is the law of large numbers with interactions. And the rule is, most interactions that you might write down just cancel out. They average out to zero. They disappear when you do this.

20:00 They're what's called irrelevant. And in fact, there is one interaction in particle physics. Two kaons collide and produce three pi. And this is not mediated by any renormalizable term in the Lagrangian. So what people believe is, oh, there is a non-renormalizable term. And the reason that the reaction is very rare is that it's almost completely killed by the renormalization. So people actually believe this picture is right. So we can't take the theory that we have, that we calculate with, and turn it into a fundamental theory. But that's because at a fine scale, there's something completely different happening. So the good news is most of that averages out anyhow. We can do our job. The bad news is many, many different fine theories will all converge to one here. Most of the information is gone anyhow. So it doesn't help us much trying to find the fundamental theory. The fundamental theory has to live in the continuum. You have to think there's some quantum field which contains the ones we see, which is renormalizable, blah blah blah blah blah. And there almost isn't any possibility. The only thing anybody could think of was supersymmetry. The trick there is that a fermion does the same thing as a boson, plus minus one, so if you glue them together they cancel out. And so people thought that was the only way you could try to finish quantum field theory without tackling the fine-scale problem of the continuum. Now, when string theory was just, which comes out of supersymmetry, the only direction in supersymmetry you can see. I was at the Institute for Fed Study, and I was talking with Ed Witten, and I said, well, I don't think that's probably the right answer. Look at the fine scale structure of spacetime. And he sort of grinned at me and he said, well, you're trying to take heroic measures. The argument is this is much easier, so we should look at that. Well, it's been 20 years and every possible thing has been done with string theory. It's really not important at all. So I think we really have, my take on the moral of the development of particle physics is we've got to go back and look at the fine scale structure of spacetime. You can figure out why renormalizable quantum field theory doesn't happen. In other words, when you get down to this very fine scale, quantum gravity is going to be important.

22:30 And we can't really solve the problem without taking it into account. Hundreds and hundreds of people disagree with string theory instead. Nevertheless, that's the state. So, problem. And then B, the other half. Now, general relativity, there's also no problem. is something called the singularity theorems. So there are two classes of singularity theorems. One has to do with the universe which is expanding and you go back into the past, and the other has to do with something like the black hole. If you have enough matter in a small enough region. And in either case you can prove that there are timeline curves of finite length which end. There has to be some place where the sidewalk ends. And these situations are absolutely generic and physical. So there has to be a place at the beginning of our universe, speakers say, where general relativity breaks down because you just go back and then you just can't go back any farther. And if you have a star that's bigger than the Chandrasekhar limit and collapses, same story. These general theorems, which are due to Penrose and Hawking and Girouche, show that there has to be a place, there's a finite length per... And then you can't extend it any further. So you might think, oh, it's a manifold with boundaries. People tried to put a boundary on it. And it didn't really work. There are several techniques. They don't really agree. And you don't end up getting a nice boundary. If you try to add on sort of virtual points at infinity by taking equivalence classes, the obvious thing a mathematician would do, you get a very, very funny thing. It's not houseboat. It's definitely not a manifold with boundaries. And the points don't seem physically meaningful. So, it seems that general relativity tells you that certain physical processes that really would exist, that in principle we could do if we just had a big enough experimental algorithm, would come to a point where there's no more points.

25:00 And in fact, if Hawking is right that black holes radiate, eventually at the end one of these would stick there and you could see it in your past. You know, what we call the naked singularity. And so the theory breaks down. So there's no foundation there either, and it's not something you can finesse. It's really something very generic that ought to happen and that we believe happened in our past. So, singularities.