The Universe in Zero Words

by Mackenzie, Dana

  Nevertheless, Dirac was certainly appreciated by his colleagues. He inherited Newton’s professorship at Cambridge (the Lucasian Chair), and after he died in 1984, a memorial plaque in his name was placed in Westminster Abbey, not far from Newton’s grave. Appropriately for the man of few words, the plaque includes his equation. It is the only formula that has been preserved for posterity in the church.

  * * *

  * Experts may object to my identification of spinors with quaternions. In fact, the difference of a minus sign in their definitions has an important consequence: spinors exist in all dimensions, while quaternions are to some extent a low-dimensional fluke. However, in three and four dimensions the quibble is purely academic. Three-dimensional spinors are quaternions of length one, or “unit” quaternions. Four-dimensional spinors are pairs of unit quaternions.

  21

  the empire-builder the chern-gauss-bonnet equation

  Although it is impossible to summarize a whole century of mathematics in a few sentences, or even in a few pages, some trends can be discerned. The connection between physics and mathematics, which had always been close, in the twentieth century became deeper and more mysterious. Physicists, beginning with Einstein, were routinely startled to discover that mathematicians had already developed the tools they needed. Vice versa, mathematicians kept on discovering that the problems and equations of physics led to the most interesting and deepest mathematics.

  Another trend in the twentieth century, connected with the first, was the rise of geometry. Einstein’s theory of general relativity required space to be curved, and this demanded a non-Euclidean geometry whose curvature could vary from point to point. Gauss, Lobachevsky, and Bolyai had sowed the seeds of non-Euclidean geometry in the early 1800s, but their geometries had constant curvature. Bernhard Riemann made it possible to vary the curvature. Riemannian, or differential, geometry developed rather slowly for the first half of the twentieth century, but in the second half it exploded and became a central area of mathematics.

  A third important, and invigorating, trend in mathematics was its increasing globalization, especially after the Second World War. Many parts of ancient mathematics had been discovered in Asia, Egypt, or the Arab world before they reached western Europe. But from roughly 1500 to 1900, mathematics was mostly a game for European males. Now, with the inequality of opportunity decreasing, the next great discovery is now (almost) as likely to come from a Zhang or an Alice as from a Smith or a Bob.

  M represents an even-dimensional space or universe with no boundary. χ(M) is the Euler characteristic of the space, which in two dimensions tells you the number of holes it has. Ω is the curvature of the space. The formula allows you to deduce information about the overall shape of the universe if you know its curvature at every point.

  If I could choose one figure to exemplify all three of these trends, it would be Shiing-Shen Chern. Born in Jiaxing, China (near Shanghai) in 1911, Chern attended Nankai University near Beijing. He distinguished himself enough there to earn a scholarship to study in Europe. He studied for two years in Frankfurt with a geometer named Wilhelm Blaschke, then moved to Paris for a year to work with Elie Cartan.

  At that time, differential geometry was not a very fashionable subject. Looking at Einstein’s field equations on page 161 may give you a sense of why. In order to describe the geometry of a curved space (or “manifold”), you need to establish a set of coordinates on it. When equations are written in terms of these coordinates, they are festooned with symbols (like the indices μ and ν in the field equations) that act merely as bookkeeping devices. Michael Spivak, author of a classic textbook on differential geometry, calls it “the debauch of indices.”

  Ironically, the most important and interesting quantities in differential geometry are precisely those that do not depend on the choice of coordinates. In other words, we spend all this time keeping track of something that in the end we don’t care about! For example, in Einstein’s theory of general relativity, the independence of physical laws from the coordinate system was a fundamental tenet. Yet it took Einstein years to navigate the mathematics and find equations with the appropriate invariance.

  Cartan, Chern’s mentor, had pioneered an approach to differential geometry, called “moving frames,” which worked without coordinates. However, Cartan’s theory was extremely obscure and difficult to understand. Chern became his foremost interpreter for the rest of the world. In the process he transformed Cartan’s theory from a local one, suitable for describing small pieces of a curved space, into a global one that dealt with space as a whole. Chern’s first famous result, considered by many (including Chern himself) to be his greatest work, was a generalization of a nineteenth-century theorem about surfaces that had been named after Karl Freidrich Gauss and Pierre Ossian Bonnet. The Chern–Gauss–Bonnet theorem, as it is now known, reads as follows:

  What does this mean? From a top-level view, it means that if we live in a curved space or “manifold” (here denoted M), we can learn something about the global shape of our universe (here denoted χ(M), the Euler characteristic of M) by measuring the curvature (Ω) at every point. The Pfaffian (Pf) is a specific computation we must do with the curvature, and the integral sign (∫) means that we have to add up the curvatures of every point in the manifold. This is a global theorem par excellence.

  Let’s burrow a little bit deeper. In this formula the curved space M is assumed to be (2n)-dimensional. So in the simplest case, where n = 1, we are dealing with a two-dimensional space, or a surface. Surfaces have only one local geometric property that is independent of the coordinate frame, called the Gaussian curvature, K. If the surface is convex the Gaussian curvature is positive. If it is shaped like a potato chip, the curvature is negative. The total curvature within any region on the surface is a measure of how much the geometry differs from Euclidean. For instance, if the total curvature within a triangle is x, then the sum of the angles of the triangle will be:

  If the curvature is zero, the sum of the angles is 180 degrees, as Euclid had shown. For example, on a sphere, which is positively curved, you can find a triangle with three right angles. (See my discussion of Ant Geometry in Part Three, page 161.) The sum of the angles of this triangle is 90 + 90 + 90 = 270 degrees, and hence the total curvature inside it must be π/2.

  Let’s check this prediction against the Gauss–Bonnet formula. The curvature at every point on the sphere is

  The total curvature within the triangle is obtained by multiplying this curvature by the area of the triangle. The area of a sphere is 4πR2, and it takes eight right triangles to cover the sphere. Thus each triangle has an area of:

  Multiplying the area by the curvature gives a total curvature of:

  THE GAUSS-BONNET THEOREM demarcates a transition point between ancient geometry (What is the sum of the angles of a triangle?) and modern geometry (How do we describe the global properties of a curved surface?). From now on, we will leave ancient geometry behind. In order to make the Gauss–Bonnet theorem global, we need to add up the curvature not only over one triangle, but over our whole surface.

  When we do this, we make an extraordinary discovery. If the surface M is roughly ball-shaped—it can be a sphere, a football, or anything else without a hole—then its total curvature will always come out to 4π. That is:

  If we do the same computation on any surface M that is more or less torus-shaped—it can be a doughnut, a coffee cup, a vuvuzela, or anything else with one hole—then the total curvature is 0. That is,

  More generally, if the surface has g holes in it, then the total curvature allows us to detect the number of holes:

  The number χ(M) = 2 – 2g is the Euler characteristic of the surface. This formula matches the Chern–Gauss–Bonnet formula given above (with n = 1, of course).

  The classical Gauss–Bonnet theorem for surfaces is remarkable for two reasons. First, it means that a very smart ant, using only Ant Geometry, can determine what kind of surface it is crawling on (a ball,
a torus, or something more complicated). The curvature K is intrinsic to the surface, which means that an ant does not have to go outside the surface to measure it.

  Second, the total curvature of a surface is quantized. It is always a whole number times 2π. Thus this nineteenth-century formula foreshadows the preoccupation of twentieth-century mathematicians and physicists with quantization.

  IT IS NOW TIME to call Shiing-Shen Chern back from the wings. In 1943, Chern was extricated from Japanese-occupied China by the US Army and went to the Institute for Advanced Study in Princeton. While he was there, he heard that two other mathematicians, Andre Weil and Carl Allendoerfer, had proved a version of the Gauss–Bonnet theorem that worked for any even-dimensional curved space or manifold, not just two-dimensional surfaces. However, their proof was ugly and unenlightening. It made an extra assumption that was later proved to be unnecessary. It resembled the Disney story of Dumbo who learns to fly with the aid of a magic feather and then finds out that the feather was never magic, and he could fly all along.

  Above An artistic digital interpretation of a torus-shaped universe.

  In a short, six-page manuscript published in 1946, Chern gave a proof that had none of these flaws, and that set postwar geometry on a new course. He introduced a concept called a fiber bundle, which is like a castle that has the manifold M as its floor plan. Everything that happens in the manifold is merely a pale reflection of what happens in the fiber bundle above it. In particular, Chern discovered that the curvature integrand (Ω) lies at the base of a tower of similar integrands, called differential forms, in the fiber bundle. When the curvature is integrated “upstairs” in the fiber bundle, rather than downstairs in the manifold, the Chern–Gauss–Bonnet theorem becomes almost obvious.

  Almost, but not quite. Chern’s calculation was a tour de force, and his idea of doing the integral in the fiber bundle was a stroke of genius. The proof made it evident that fiber bundles contained an untapped wealth of information about a space. Not only the Euler characteristic, but also a variety of other invariants, now called Chern characteristics and Chern–Simons invariants, have now been constructed in this way.

  Chern’s work completed a cycle. Einstein and Dirac had shown that you could not do physics without geometry. Chern showed that you cannot do geometry without physics. The fiber bundle is the building in which a quantum field lives. To understand the shape of a space, you need to know what kinds of fiber bundles—or, what is essentially the same thing, what kinds of quantum fields—can be erected on that space.

  TWO DECADES LATER, in 1963, Michael Atiyah and Isadore Singer made the links between math and physics even more explicit. They gave a proof of the Chern–Gauss–Bonnet theorem (and quite a bit more) that proceeds directly from solutions to the Dirac equation!

  Why does the Chern–Gauss–Bonnet theorem matter? Because if we ever want to understand the kind of universe we live in, we do not have the option of going “outside” the universe. We will have to work from within, using the language that Chern pioneered.

  At the same time, I want to emphasize that mathematics is not only about the universe that we live in. To me, that is one of the main distinctions between mathematics and physics. Physics is supposed to be about our universe, and physical theories eventually have to be grounded at some point in experiment. On the other hand, mathematics is about all possible universes, the one we live in and those we do not. It is an amazing fact that to understand any possible universe (or at least any universe that is a smooth manifold with an even number of dimensions) you need the same language of fields and the same Dirac equation. For readers who are inclined to believe in a Creator, he (or she or it) must have been a very good mathematician!

  Chern’s later career straddled two continents. He returned to China after the war, but was forced to leave again before the Communist government took power in 1949. He enjoyed a long and successful career in the United States, first at the University of Chicago and then at the University of California at Berkeley. Together with Singer and Calvin Moore, he founded the Mathematical Sciences Research Institute in Berkeley, the first pure-math institute in the United States, and served as its first director. After he retired from MSRI in 1984, he devoted himself to reviving Chinese mathematics, which had suffered greatly during the Cultural Revolution. He traveled to China frequently and obtained opportunities for Chinese graduate students to study in America. He also founded the Nankai Institute of Mathematics in Tianjin (which was renamed the Chern Institute after his death in 2004). Remarkably, he achieved the same sort of rock-star celebrity in China that Einstein did in the United States.

  Robert Bryant, a later director of MSRI, tells the story of how Chern went to watch the world table tennis championships in Tianjin in 1994. “The TV cameras were all there, showing the prime minister as he was seated,” says Bryant. “Then Mr. and Mrs. Chern walked in. The cameras went straight to them and ignored the prime minister! He was this iconic figure, a great intellectual figure who had showed what Chinese could do in the outside world.”

  Like Dirac, Chern was very modest about his accomplishments, but unlike Dirac he was comfortable with people. He understood that mathematics advances not only by deriving formulas, but also by building institutions, such as MSRI and the Nankai Institute. According to Hung-Hsi Wu, a long-time colleague of his at Berkeley, “He was an empire-builder in the best sense of the word.”

  22

  a little bit infinite the continuum hypothesis

  Beginning in the 1870s, mathematicians began to realize that infinity actually comes in different sizes; a set can in fact be a little bit infinite or a whole lot infinite. The exploration of these different kinds of infinity has led to some of the most profound and paradoxical discoveries of twentieth-century mathematics.

  For most of the nineteenth century, mathematicians did their best to finesse the whole issue of infinity. There was good reason for that; as we have seen in Part One, the notion of infinity had been confounding mathematicians at least since the days of Zeno. In 1831, Gauss expressed this interdiction in a letter to Heinrich Schumacher: “I must protest most vehemently against your use of the infinite as something consummated, as this is never permitted in mathematics. The infinite is but a figure of speech …”

  However, toward the end of the century, a consensus began to form that sets, rather than numbers, are the fundamental building blocks of mathematics. And you just can’t get around the fact that some sets are infinite: for example, the set of positive integers, {1, 2, 3, …}, or the set of numbers that form successive approximations to pi {3.1, 3.14, 3.141, 3.1415, …}.

  Great scientists, like Einstein, are often the ones who are willing to face the inconvenient facts that other scientists would prefer to avoid. In the case of set theory, it was another German mathematician named Georg Cantor who blazed a trail into the strange world of infinity.

  ℵ0 is the “size” or cardinality of the smallest infinite set (the integers). ℵ1 is the cardinality of the next smallest. If true, this formula would mean that the real numbers are the next smallest set after the integers.

  To understand Cantor, we first need a way to describe what we mean by the “size,” or cardinality, of a set. First, Cantor suggested the following rule, a comparative or relative approach to defining the meaning of cardinality: If we can find a one-to-one matching between two sets A and B, so that each element of A corresponds to a unique element of B and vice versa, then the two sets have the same cardinality.

  A good example (suggested by David Hilbert) is to think of A as the set of rooms in a hotel and B as the set of guests wanting a room. Ideally, we would like to place every guest in a separate room. In addition, if we are the proprietors of the hotel, we would like to fill up all the rooms. If we can do this, then the “number of guests” is the same as the “number of rooms.”

  Let’s suppose that the hotel has infinitely many rooms, which are labeled 1, 2, 3, and so on. If a set of guests arrives that can fill up
the hotel without overfilling it, the set is countably infinite. With this preamble, we can list the surprising facts that Cantor discovered about countably infinite sets:

  1. If you add one element to a countably infinite set, you get a set of the same size! For instance, suppose you have filled your hotel, and then one more guest arrives. You do not have to turn him away! You simply bump the guest in room 1 to room 2, the guest in room 2 to room 3, and so on. Presto, room 1 is now available for your new guest.

  2. If you combine two countably infinite sets, you get another set of the same size. Suppose the hotel is filled with a countably infinite number of guests, but then a second countably infinite party arrives. No problem! You can accommodate them too. Just bump the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and so on. Now rooms 1, 3, 5, etc., are free for your infinite party of new guests.

  3. Proceeding in similar fashion, the union of a countably infinite number of countably infinite sets is still countably infinite. This is a little bit tricky to explain in words, but the idea is shown visually below.

  All of these facts are different from our everyday experience with finite sets, and frankly they seem a little bit like magic. But perhaps this should not surprise us too much. Physicists had to leave common sense behind when they encountered quanta, and infinite sets are also a strange new world.