Why Beauty is Truth Page 20

  In the summer of 1886, Killing visited Lie and Engel at Leipzig, where both now worked. Unfortunately, there was some friction between Lie and Killing; Lie never really appreciated Killing’s work and generally tried to play down its significance.

  Killing quickly discovered that his original conjecture about simple Lie algebras was wrong, for he discovered a new one, whose corresponding Lie group is now known as G2. It had 14 dimensions, and unlike the special linear and orthogonal Lie algebras, it did not seem to belong to an infinite family. It was a lone exception.

  If this was strange, the final classification, which Killing completed in the winter of 1887, was stranger. To the two infinite families Killing added a third, the Lie algebras sp(2n) of what are now known as the symplectic groups Sp(2n). (Nowadays, we split the orthogonal groups into two different subfamilies, those acting on spaces of even dimension and those acting on spaces of odd dimension, yielding four families. There are reasons for doing this.) And now the exception G2 had acquired five companions: two of dimension 56, and a short family that petered out, with dimensions 78, 133, and 248.

  Killing’s classification proceeded by a lengthy algebraic argument, which reduced the entire question to a beautiful problem in geometry. From a hypothetical simple Lie algebra he conjured up a configuration of points in a multidimensional space, known today as a root system. For exactly three of the simple Lie algebras, the root system lives in a space of two dimensions. These root systems look like this:

  The root systems in two dimensions.

  These patterns have a great deal of symmetry. In fact, they are reminiscent of the patterns you see in a kaleidoscope, where two mirrors set at an angle create multiple reflections. The similarity is no coincidence, because root systems have wonderful, elegant symmetry groups. Now known as Weyl groups (unfair, since they were invented by Killing), they are multidimensional analogues of the patterns formed by reflecting objects in a kaleidoscope.

  The underlying structure of Killing’s proof is that the search for all possible simple Lie algebras can be carried out by breaking the algebras into nice pieces, analogous to structures found in su(n). The classification then reduces to the geometry of those pieces, using their wonderful symmetries. Having sorted out the geometry of those pieces, you can now bootstrap your results back to the problem you really wanted to solve: finding the possible simple Lie algebras.

  As Killing put it, “The roots of a simple system correspond to a simple group. Conversely, the roots of a simple group can be regarded as determined by a simple system. In this way one obtains the simple groups. For each l there are four structures, supplemented for l = 2, 4, 6, 7, 8 by exceptional simple groups.”

  Here “group” was a shortened form of “infinitesimal group,” which we now call a Lie algebra, and l is the dimension of the root system.

  The four structures that Killing refers to are the Lie algebras su(n), so(2n), so(2n+1), and sp(2n) corresponding to families of groups SU(n), SO(2n), SO(2n+1), and Sp(2n): the unitary groups, the orthogonal groups in spaces of even dimension, the orthogonal groups in spaces of odd dimension, and the symplectic groups in spaces of even dimension. The symplectic groups are the symmetries of the position–momentum variables introduced by Hamilton in his formulation of mechanics, and the number of dimensions there is always even because the variables come in position–momentum pairs. Aside from these four families, Killing claimed that exactly six other simple Lie algebras exist.

  He was nearly right. In 1894, the French geometer Élie Cartan noticed that Killing’s two 56-dimensional algebras are really the same algebra viewed in two different ways. That means that there are only five exceptional simple Lie algebras, corresponding to five exceptional simple Lie groups: Killing’s old friend G2, and four others now called F4, E6, E7, and E8.

  This is an exceedingly curious answer. The infinite families are reasonable enough; they are all related to various natural types of geometry in any number of dimensions. But the five exceptional Lie groups seem unrelated to anything geometric, and their dimensions are bizarre. Why are spaces of dimensions 14, 56, 78, 133, and 248 special? What is so unusual about those numbers?

  It’s a bit like wanting to list all possible shapes for a brick, and finding an answer something like this:

  • Oblong blocks of size 1, 2, 3, 4, . . .

  • Cubes of size 1, 2, 3, 4, . . .

  • Slabs of size 1, 2, 3, 4, . . .

  • Pyramids of size 1, 2, 3, 4, . . .

  Which would be very neat and tidy, except that the list continues:

  • A tetrahedron of size 14.

  • An octahedron of size 52.

  • A dodecahedron of size 78.

  • A dodecahedron of size 133.

  • A dodecahedron of size 248.

  And that’s it, there’s nothing else.

  Why do bricks with these strange shapes and sizes exist? What are they for?

  It seemed completely mad.

  It seemed so mad, in fact, that Killing was rather upset that the exceptional groups existed, and for a time he hoped they were a mistake that he could eradicate. They spoiled the elegance of his classification. But they were there, and we are finally beginning to understand why they are there. In many ways, the five exceptional Lie groups now look much more interesting than the four infinite families. They seem to be important in particle physics, as we will see; they are definitely important in mathematics. And they have a secret unity, not yet fully uncovered, relating them all to Hamilton’s quaternions and an even more curious generalization, the octonions. Of which more, in due course.

  It’s a wonderful series of ideas, and Killing had all of them. To be sure, his work included a few mistakes—some proofs that didn’t quite work. But the mistakes were all repaired long ago.

  That is how the greatest mathematical paper of all time went. What did Killing’s contemporaries think of it?

  Not a lot. It didn’t help that Lie poured derision on Killing’s magnum opus. He had fallen out with Killing for unknown reasons, and as far as he was concerned, Killing would never do anything important. Worse, of course, this was a theorem that Lie himself would have dearly loved to prove. Having been beaten to the punch, he resorted to the age-old technique of sour grapes. Anything in the area not done by Lie, said Lie, was rubbish. Though he wasn’t quite that blatant.

  It helped even less that Killing underestimated the value of his own theorem. To him it was a pale shadow of something far more important, which he had failed to achieve: classifying all Lie groups. Killing was a modest man, and Lie did his best to make him more so.

  In any case, Killing was ahead of his time. Very few mathematicians saw how important Lie theory was going to become. To most, it was a rather technical branch of geometry associated with differential equations.

  Finally, Killing was a staunch Catholic with a strong sense of duty and humility. He took St. Francis of Assisi as his model, and at the age of 39 he and his wife entered the Third Order of the Franciscans. He seems to have been a thoroughly decent man who worked tirelessly on behalf of his students. He was a conservative and a patriot, greatly saddened by the extreme social dissolution of Germany after World War I. His feelings were made worse by the deaths of his two sons in 1910 and 1918.

  The true worth of Killing’s researches became apparent in 1894, when Élie Cartan rederived the whole theory in his PhD thesis, and took it a big step further by classifying not just the simple Lie algebras but their representations in terms of matrices. Cartan was scrupulous in giving credit to Killing for nearly all of the ideas; he just tidied everything up, plugged a few gaps (some serious), and modernized the terminology. But a myth quickly grew up to the effect that Killing’s work was riddled with holes and the real credit should go to Cartan. Mathematicians are seldom good historians, and they tend to cite work that they know rather than the earlier work that led up to it. So Cartan’s name became attached to many of Killing’s ideas.

  Anyone who reads Kill
ing’s papers quickly discovers that the myth is just that. The ideas are clear and well formed, the proofs are perhaps old-fashioned but nearly all correct. Most importantly, the overall sweep of the ideas is beautifully chosen to produce the desired result. It is mathematics of the highest order, and it is not anyone else’s.

  Unfortunately, hardly anyone read Killing’s papers. They read Cartan and ignored the credit he gave to Killing. But eventually, Killing’s work began to achieve proper recognition. In 1900 he won the Lobachevsky Prize of the Kazan Physico-Mathematical Society. This was the second time the prize had been awarded: the first one went to Lie.

  Killing died in 1923. Even today, his name is not as well known as it deserves to be. He was one of the greatest mathematicians who ever lived. His legacy, at least, is immortal.

  11

  THE CLERK FROM THE PATENT OFFICE

  By the beginning of the twentieth century, groups were starting to show up in fundamental physics, a field they would transform just as radically as they had transformed mathematics.

  In the golden year of 1905, the man who would become the most iconic scientist of his time published three papers, each of which revolutionized a separate branch of physics. He was not at that time a professional scientist. He had studied at university but had not been able to obtain a teaching position and was working as a clerical official in the patent office in Bern, Switzerland. His name, of course, was Albert Einstein.

  If any one person can symbolize modern physics, it is Einstein. To many, he also symbolizes mathematical genius, but in fact he was merely a competent mathematician, not a creative one on the level of Galois or Killing. Einstein’s creativity lay not in producing new mathematics but in an extraordinarily rigorous intuition about the physical world, which he was able to express through remarkable uses of existing mathematics. Einstein also had a flair for the right philosophical standpoint. He drew radical theories from the simplest of principles and was guided by a sense of elegance rather than a wide knowledge of experimental facts. The important observations, he believed, could always be distilled into a few key principles. The gateway to truth was beauty.

  Acres of print and many lifetimes of scholarly study have been devoted to Einstein’s life and works. A single chapter cannot hope to compete in either completeness or erudition. But he is a key figure in the history of symmetry: it was Einstein, above all others, who set in motion the web of events that turned the mathematics of symmetry into fundamental physics. I don’t think Einstein saw it that way: to him, the mathematics was a servant of physics—often a rather disobedient one. Only later, following the trail that Einstein had blazed and tidying up the tangled, broken vegetation that his pioneering efforts had strewn across the path, did another generation uncover the elegant and deep mathematical concepts upon which his work was based.

  So we must retell the main outlines of the astonishing rise to fame of this minor patent clerk—technical expert third class, to be precise, and on a trial basis at that. Since he is but one part of our story, I will select only the relevant events. If you who want a more comprehensive, unbiased assessment of Einstein’s career, you should read Abraham Pais’s Subtle Is the Lord.

  Subtle, yes—but not, as Einstein once remarked, malicious.

  Einstein, who had little interest in religion, devoted his life to the principle that the universe is comprehensible and that it runs along mathematical lines. Many of his most famous sayings invoke the deity, but as a symbol of the orderliness of the universe, not as a supernatural being with a personal interest in human affairs. He worshipped no god and practiced no religious rituals.

  Einstein is generally seen as the natural successor to Newton. Earlier scientists had made additions to Newton’s “system of the world,” as his Mathematical Principles of Natural Philosophy was subtitled, but Einstein was the first to make significant changes to that vision. The most important of the earlier theorists was James Clerk Maxwell, whose equations for electromagnetism brought magnetic and electric phenomena, especially light, within the Newtonian purview. Einstein went much further, making major changes. Ironically, the changes that led to a revised theory of gravity came about as consequences of the Maxwellian theory of electromagnetic waves—light and its relatives. Even more ironically, a fundamental feature of that theory, the wave nature of light, played a key role, yet Newton denied that light could be a wave. To cap it all, one of the most elegant experiments now used to demonstrate that light is a wave was first carried out by Newton.

  Scientific interest in light goes back at least to Aristotle, who, though really a philosopher, asked the kind of question that scientists would find natural. How do we see? Aristotle suggested that when we look at some object, that object affects the medium between itself and the onlooking eye. (We now call this medium “air.”) The eye then detects this change in the medium, and the result is the sensation of sight.

  In medieval times this explanation was reversed. It was thought that our eyes emitted some kind of ray, which illuminated whatever we looked at. Instead of the object transmitting signals to the eye, the eye left eye-tracks all over the object.

  Eventually, it was understood that we see objects by means of reflected light, and that in daily life the main source of light is the Sun. Experiments showed that light travels in straight lines, forming “rays.” Reflection occurs when a ray bounces off a suitable surface. So the Sun sends light rays to everything that is not shadowed by something else, the rays bounce all over the place, some enter an observer’s eye, the eye receives a signal from that direction, the brain processes the incoming information from the eye, and we see whatever object the ray bounced off.

  The main question was, what is light? Light does a number of puzzling things. Not only does it reflect; it can also refract—change direction abruptly at the interface between two different media, such as air and water. This is why a stick poked into a pond looks bent, and also why lenses work.

  Even more puzzling is the phenomenon of diffraction. In 1664, the scientist and polymath Robert Hooke, whose career repeatedly clashed with Newton’s, discovered that if he placed a lens on top of a flat mirror and then looked through the lens, he saw tiny concentric colored rings. These rings are now known as “Newton’s rings” because Newton was the first person to analyze their formation. Today we consider this experiment a clear demonstration that light is a wave: the rings are interference fringes, where waves do or do not cancel each other out when they overlap. But Newton didn’t believe light was a wave. Because light traveled in straight lines, he believed it had to be a stream of particles. According to his Opticks, completed in 1705, “Light is composed of tiny particles, or corpuscles, emitted by luminous bodies.” The particle theory could explain reflection very simply: the particles bounced when they hit a (reflecting) surface. It encountered difficulties explaining refraction, and pretty much fell apart when it came to diffraction.

  Thinking about what could cause light rays to bend, Newton decided that the medium, not light, must be the root cause. This led him to suggest the existence of some “aethereal medium” which transmitted vibrations faster than light. He convinced himself that radiant heat was evidence in favor of these vibrations, because he had established that heat radiation could traverse a vacuum. Something in the vacuum must be carrying the heat and causing refraction and diffraction. In Newton’s words:

  Is not the Heat of the warm Room convey’d through the Vacuum by the Vibrations of a much subtiler Medium than Air, which after the Air was drawn out remained in the Vacuum? And is not this Medium the same with that Medium by which Light is refracted and reflected, and by whose Vibrations Light communicates Heat to Bodies, and is put into Fits of easy Reflexion and easy Transmission?

  When I read these words I cannot help thinking of my friend Terry Pratchett, whose series of fantasy novels set on “Discworld” satirize our own world, and whose assorted wizards, witches, trolls, dwarves, and people poke fun at human foibles. Light on Discworld trave
ls at roughly the speed of sound, which is why the light of dawn can be seen approaching across the fields. A necessary counterpart to light is dark—on Discworld almost everything is reified—and dark evidently travels faster than light because it has to get out of light’s way. It all makes excellent sense, even in our world, aside from the disappointing fact that none of it is true.

  Newton’s theory of light suffers from the same defect. Newton wasn’t being stupid: his theory seemed to answer a number of important questions. Unfortunately, these answers were based on a fundamental misunderstanding: he thought radiant heat and light were two different things. He believed that when light hits a surface, it excites heat vibrations. These were variants of the same vibrations that he thought caused light to refract and diffract.

  Thus was born the concept of the “luminiferous aether,” which proved remarkably persistent. Indeed, when it later turned out that light is a wave, the aether provided just the right medium for it to be a wave in. (We now think that light is neither wave nor particle exclusively but a bit of both—a wavicle. But I’m getting ahead of myself.)

  What, though, was the aether? Newton is perfectly frank: “I do not know what this Aether is.” He argued that if the aether is also composed of particles, then they must be much smaller and lighter than particles of air or even of light—essentially for the Discworldly reason that they have to be able to get out of light’s way. “The exceeding smallness of its Particles,” Newton says of the aether, “may contribute to the greatness of the force by which those Particles may recede from one another, and thereby make that Medium exceedingly more rare and elastick than Air, and by consequence exceedingly less able to resist the motions of Projectiles, and exceedingly more able to press upon gross Bodies, by endeavoring to expand itself.”