Love and Math

by Frenkel, Edward

  My response: “Every formula we create is a formula of love.” Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries. My dream is that all of us will be able to see, appreciate, and marvel at the magic beauty and exquisite harmony of these ideas, formulas, and equations, for this will give so much more meaning to our love for this world and for each other.

  A Guide for the Reader

  I have made every effort to present mathematical concepts in this book in the most elementary and intuitive way. However, I realize that some parts of the book are somewhat heavier on math (particularly, some parts of Chapters 8, 14, 15, and 17). It is perfectly fine to skip those parts that look confusing or tedious at the first reading (this is what I often do myself). Coming back to those parts later, equipped with newly gained knowledge, you might find the material easier to follow. But that is usually not necessary in order to be able to follow what comes next.

  Some mathematical concepts in the book (especially, in the later chapters) are not described in every detail. My focus is on the big picture and the logical connections between different concepts and different branches of math, not technical details. A more in-depth discussion is often relegated to the endnotes, which also contain references and suggestions for further reading. However, although endnotes may enhance your understanding, they may be safely skipped (at least, at the first reading).

  I have tried to minimize the use of formulas – opting, whenever possible, for verbal explanations. But a few formulas do appear. I think that most of them are not that scary; in any case, feel free to skip them if so desired.

  A word of warning on mathematical terminology: while writing this book, I discovered, to my surprise, that certain terms that mathematicians use in a specific way actually mean something entirely different to non-mathematicians. For example, to a mathematician the word “correspondence” means a relation between two kinds of objects (as in “one-to-one correspondence”), which is not the most common connotation. There are other terms like this, such as “representation,” “composition,” “loop,” “manifold,” and “theory.” Whenever I detected this issue, I included an explanation. Also, whenever possible, I changed obscure mathematical terms to terms with more transparent meaning (for example, I would write “Langlands relation” instead of “Langlands correspondence”). You might find it useful to consult the Glossary and the Index whenever there is a word that seems unclear.

  Please check out my website http://edwardfrenkel.com for updates and supporting materials, and send me an e-mail to share your thoughts about the book (my e-mail address can be found on the website). Your feedback will be much appreciated.

  Chapter 1

  A Mysterious Beast

  How does one become a mathematician? There are many ways that this can happen. Let me tell you how it happened to me.

  It might surprise you, but I hated math when I was at school. Well, “hated” is perhaps too strong a word. Let’s just say I didn’t like it. I thought it was boring. I could do my work, sure, but I didn’t understand why I was doing it. The material we discussed in class seemed pointless, irrelevant. What really excited me was physics – especially quantum physics. I devoured every popular book on the subject that I could get my hands on. I grew up in Russia, where such books were easy to find.

  I was fascinated with the quantum world. Ever since ancient times, scientists and philosophers had dreamed about describing the fundamental nature of the universe – some even hypothesized that all matter consists of tiny pieces called atoms. Atoms were proved to exist at the beginning of the twentieth century, but at around the same time, scientists discovered that each atom could be divided further. Each atom, it turned out, consists of a nucleus in the middle and electrons orbiting it. The nucleus, in turn, consists of protons and neutrons, as shown on the diagram below.1

  And what about protons and neutrons? The popular books that I was reading told me that they are built of the elementary particles called “quarks.”

  I liked the name quarks, and I especially liked how this name came about. The physicist who invented these particles, Murray Gell-Mann, borrowed this name from James Joyce’s book Finnegans Wake, where there is a mock poem that goes like this:

  Three quarks for Muster Mark!

  Sure he hasn’t got much of a bark

  And sure any he has it’s all beside the mark.

  I thought it was really cool that a physicist would name a particle after a novel. Especially such a complex and non-trivial one as Finnegans Wake. I must have been around thirteen, but I already knew by then that scientists were supposed to be these reclusive and unworldly creatures who were so deeply involved in their work that they had no interest whatsoever in other aspects of life such as Art and Humanities. I wasn’t like this. I had many friends, liked to read, and was interested in many things besides science. I liked to play soccer and spent endless hours chasing the ball with my friends. I discovered Impressionist paintings around the same time (it started with a big volume about Impressionism, which I found in my parents’ library). Van Gogh was my favorite. Enchanted by his works, I even tried to paint myself. All of these interests had actually made me doubt whether I was really cut out to be a scientist. So when I read that Gell-Mann, a great physicist, Nobel Prize–winner, had such diverse interests (not only literature, but also linguistics, archaeology, and more), I was very happy.

  According to Gell-Mann, there are two different types of quarks, “up” and “down,” and different mixtures of them give neutrons and protons their characteristics. A neutron is made of two down and one up quarks, and a proton is made of two up and one down quarks, as shown on the pictures.2

  That was clear enough. But how physicists guessed that protons and neutrons were not indivisible particles but rather were built from smaller blocks was murky.

  The story goes that by the late 1950s, a large number of apparently elementary particles, called hadrons, was discovered. Neutrons and protons are both hadrons, and of course they play major roles in everyday life as the building blocks of matter. As for the rest of hadrons – well, no one had any idea what they existed for (or “who ordered them,” as one researcher put it). There were so many of them that the influential physicist Wolfgang Pauli joked that physics was turning into botany. Physicists desperately needed to rein in the hadrons, to find the underlying principles that govern their behavior and would explain their maddening proliferation.

  Gell-Mann, and independently Yuval Ne’eman, proposed a novel classification scheme. They both showed that hadrons can be naturally split into small families, each consisting of eight or ten particles. They called them octets and decuplets. Particles within each of the families had similar properties.

  In the popular books I was reading at the time, I would find octet diagrams like this:

  Here the proton is marked as p, the neutron is marked as n, and there are six other particles with strange names expressed by Greek letters.

  But why 8 and 10, and not 7 and 11, say? I couldn’t find a coherent explanation in the books I was reading. They would mention a mysterious idea of Gell-Mann called the “eightfold way” (referencing the “Noble Eightfold Path” of Buddha). But they never attempted to explain what this was all about.

  This lack of explanation left me deeply unsatisfied. The key parts of the story remained hidden. I wanted to unravel this mystery but did not know how.

  As luck would have it, I got help from a family friend. I grew up in a small industrial town called Kolomna, population 150,000, which was about seventy miles away from Moscow, or just over two hours by train. My parents worked as engineers at a large company making heavy machinery. Kolomna is an old town on the intersection of two rivers that was founded in 1177 (only thirty years after the founding of Moscow). There are still a few pretty churches and the city wall to attest to Kolomna’s storied past. But it’s not exactly an educational or intellectual cen
ter. There was only one small college there, which prepared schoolteachers. One of the professors there, a mathematician named Evgeny Evgenievich Petrov, however, was an old friend of my parents. And one day my mother met him on the street after a long time, and they started talking. My mom liked to tell her friends about me, so I came up in conversation. Hearing that I was interested in science, Evgeny Evgenievich said, “I must meet him. I will try to convert him to math.”

  “Oh no,” my mom said, “he doesn’t like math. He thinks it’s boring. He wants to do quantum physics.”

  “No worries,” replied Evgeny Evgenievich, “I think I know how to change his mind.”

  A meeting was arranged. I wasn’t particularly enthusiastic about it, but I went to see Evgeny Evgenievich at his office anyway.

  I was just about to turn fifteen, and I was finishing the ninth grade, the penultimate year of high school. (I was a year younger than my classmates because I had skipped the sixth grade.) Then in his early forties, Evgeny Evgenievich was friendly and unassuming. Bespectacled, with a beard stubble, he was just what I imagined a mathematician would look like, and yet there was something captivating in the probing gaze of his big eyes. They exuded unbounded curiosity about everything.

  It turned out that Evgeny Evgenievich indeed had a clever plan how to convert me to math. As soon as I came to his office, he asked me, “So, I hear you like quantum physics. Have you heard about Gell-Mann’s eightfold way and the quark model?”

  “Yes, I’ve read about this in several popular books.”

  “But do you know what was the basis for this model? How did he come up with these ideas?”

  “Well...”

  “Have you heard about the group SU(3)?”

  “SU what?”

  “How can you possibly understand the quark model if you don’t know what the group SU(3) is?”

  He pulled out a couple of books from his bookshelf, opened them, and showed me pages of formulas. I could see the familiar octet diagrams, such as the one shown above, but these diagrams weren’t just pretty pictures; they were part of what looked like a coherent and detailed explanation.

  Though I could make neither head nor tail of these formulas, it became clear to me right away that they contained the answers I had been searching for. This was a moment of epiphany. I was mesmerized by what I was seeing and hearing; touched by something I had never experienced before; unable to express it in words but feeling the energy, the excitement one feels from hearing a piece of music or seeing a painting that makes an unforgettable impression. All I could think was “Wow!”

  “You probably thought that mathematics is what they teach you in school,” Evgeny Evgenievich said. He shook his head, “No, this” – he pointed at the formulas in the book – “is what mathematics is about. And if you really want to understand quantum physics, this is where you need to start. Gell-Mann predicted quarks using a beautiful mathematical theory. It was in fact a mathematical discovery.”

  “But how do I even begin to understand this stuff?”

  It looked kind of scary.

  “No worries. The first thing you need to learn is the concept of a symmetry group. That’s the main idea. A large part of mathematics, as well as theoretical physics, is based on it. Here are some books I want to give you. Start reading them and mark the sentences that you don’t understand. We can meet here every week and talk about this.”

  He gave me a book about symmetry groups and also a couple of others on different topics: about the so-called p-adic numbers (a number system radically different from the numbers we are used to) and about topology (the study of the most fundamental properties of geometric shapes). Evgeny Evgenievich had impeccable taste: he found a perfect combination of topics that would allow me to see this mysterious beast – Mathematics – from different sides and get excited about it.

  At school we studied things like quadratic equations, a bit of calculus, some basic Euclidean geometry, and trigonometry. I had assumed that all mathematics somehow revolved around these subjects, that perhaps problems became more complicated but stayed within the same general framework I was familiar with. But the books Evgeny Evgenievich gave me contained glimpses of an entirely different world, whose existence I couldn’t even imagine.

  I was instantly converted.

  Chapter 2

  The Essence of Symmetry

  In the minds of most people, mathematics is all about numbers. They imagine mathematicians as people who spend their days crunching numbers: big numbers, and even bigger numbers, all having exotic names. I had thought so too – at least, until Evgeny Evgenievich introduced me to the concepts and ideas of modern math. One of them turned out to be the key to the discovery of quarks: the concept of symmetry.

  What is symmetry? All of us have an intuitive understanding of it – we know it when we see it. When I ask people to give me an example of a symmetric object, they point to butterflies, snowflakes, or the human body.

  Photo by K.G. Libbrecht

  But if I ask them what we mean when we say that an object is symmetrical, they hesitate.

  Here is how Evgeny Evgenievich explained it to me. “Let’s look at this round table and this square table,” he pointed at the two tables in his office. “Which one is more symmetrical?”

  “Of course, the round table, isn’t it obvious?”

  “But why? Being a mathematician means that you don’t take ‘obvious’ things for granted but try to reason. Very often you’ll be surprised that the most obvious answer is actually wrong.”

  Noticing confusion on my face, Evgeny Evgenievich gave me a hint: “What is the property of the round table that makes it more symmetrical?”

  I thought about this for a while, and then it hit me: “I guess the symmetry of an object has to do with it keeping its shape and position unchanged even when we apply changes to it.”

  Evgeny Evgenievich nodded.

  “Indeed. Let’s look at all possible transformations of the two tables which preserve their shape and position,” he said. “In the case of the round table...”

  I interrupted him: “Any rotation around the center point will do. We will get back the same table positioned in the same way. But if we apply an arbitrary rotation to a square table, we will typically get a table positioned differently. Only rotations by 90 degrees and its multiples will preserve it.”

  “Exactly! If you leave my office for a minute, and I turn the round table by any angle, you won’t notice the difference. But if I do the same to the square table, you will, unless I turn it by 90, 180, or 270 degrees.”

  Rotation of a round table by any angle does not change its position, but rotation of a square table by an angle that is not a multiple of 90 degrees does change its position (both are viewed here from above)

  He continued: “Such transformations are called symmetries. So you see that the square table has only four symmetries, whereas the round table has many more of them – it actually has infinitely many symmetries. That’s why we say that the round table is more symmetrical.”

  This made a lot of sense.

  “This is a fairly straightforward observation,” continued Evgeny Evgenievich. “You don’t have to be a mathematician to see this. But if you are a mathematician, you ask the next question: what are all possible symmetries of a given object?”

  Let’s look at the square table. Its symmetries1 are these four rotations around the center of the table: by 90 degrees, 180 degrees, 270 degrees, and 360 degrees, counterclockwise.2 A mathematician would say that the set of symmetries of the square table consists of four elements, corresponding to the angles 90, 180, 270, and 360 degrees. Each rotation takes a fixed corner (marked with a balloon on the picture below) to one of the four corners.

  One of these rotations is special; namely, rotation by 360 degrees is the same as rotation by 0 degrees, that is, no rotation at all. This is a special symmetry because it actually does nothing to our object: each point of the table ends up in exactly the same position as
it was before. We call it the identical symmetry, or just the identity.3

  Note that rotation by any angle greater than 360 degrees is equivalent to rotation by an angle between 0 and 360 degrees. For example, rotation by 450 degrees is the same as rotation by 90 degrees, because 450 = 360 + 90. That’s why we will only consider rotations by angles between 0 and 360 degrees.

  Here comes the crucial observation: if we apply two rotations from the list {90°, 180°, 270°, 360°} one after another, we obtain another rotation from the same list. We call this new symmetry the composition of the two.

  Of course, this is obvious: each of the two symmetries preserves the table. Hence the composition of the two symmetries also preserves it. Therefore this composition has to be a symmetry as well. For example, if we rotate the table by 90 degrees and then again by 180 degrees, the net result is the rotation by 270 degrees.

  Let’s see what happens with the table under these symmetries. Under the counterclockwise rotation by 90 degrees, the right corner of the table (the one marked with a balloon on the previous picture) will go to the upper corner. Next, we apply the rotation by 180 degrees, so the upper corner will go to the down corner. The net result will be that the right corner will go to the down corner. This is the result of the counterclockwise rotation by 270 degrees.

  Here is one more example:

  By rotating by 90 degrees and then by 270 degrees, we get the rotation by 360 degrees. But the effect of the rotation by 360 degrees is the same as that of the rotation by 0 degrees, as we have discussed above – this is the “identity symmetry.”

  In other words, the second rotation by 270 degrees undoes the initial rotation by 90 degrees. This is in fact an important property: any symmetry can be undone; that is, for any symmetry S there exists another symmetry S′ such that their composition is the identity symmetry. This S′ is called the inverse of symmetry S. So we see that rotation by 270 degrees is the inverse of the rotation by 90 degrees. Likewise, the inverse of the rotation by 180 degrees is the same rotation by 180 degrees.