Why Beauty is Truth Page 26
In 1926, he was contacted by a crystallographer at the Kaiser Wilhelm Institute who wanted a research assistant. The duties would combine both of Wigner’s interests, in a chemical context. The project had a huge influence on Wigner’s career, and thus on the course of nuclear physics, because it introduced him to group theory—the mathematics of symmetry. The first major application of group theory to physics had been the classification of all 230 possible crystal structures. Wigner wrote, “I received a letter from a crystallographer who wanted to find out why the atoms occupy positions in the crystal lattices which correspond to symmetry axes. He also told me that this had to do with group theory and that I should read a book on group theory and then work it out and tell him.”
Perhaps no less dismayed than his son by Jen’s foray into the tanning trade, Antal Wigner agreed to allow the research assistantship. Jen started by reading a few of Heisenberg’s papers on quantum theory, and developed a theoretical method to calculate the spectrum of an atom with three electrons. But he also realized that his methods would prove extraordinarily complicated for more electrons than three. At this point, he turned for advice to his old acquaintance von Neumann, who suggested he read about group representation theory. This area of mathematics was heavily laden with the algebraic concepts and techniques of the time, notably matrix algebra. But thanks to his studies in crystallography and his familiarity with a leading algebra textbook of the period—Heinrich Weber’s Lehrbuch der Algebra—matrices posed no problem for Wigner.
Von Neumann’s advice proved sound. If an atom possesses some number of electrons, then since all electrons are identical, the atom does not “know” which electron is which. In other words, the equations describing the radiation emitted by that atom must be symmetric under all permutations of those electrons. Using group theory, Wigner developed a theory of the spectrum of atoms with any number of electrons.
To that point, his work had taken place within the traditional realm of classical physics. But quantum theory was where the excitement was. Now he embarked on his life’s masterwork, the application of group representation theory to quantum mechanics.
Ironically, he did so despite, not because of, his new job. David Hilbert, the elder statesman of German mathematics, had developed a keen interest in the mathematical principles behind quantum theory and required the services of a research assistant. In 1927, Wigner went to Göttingen to join Hilbert’s research group. His ostensible role was to provide physical insight to inform Hilbert’s vast mathematical expertise.
It didn’t quite work out as planned. The two met only five times in the course of a year. Hilbert was old, tired, and increasingly reclusive. So Wigner went back to Berlin, gave lectures on quantum mechanics, and continued to put together his most famous book: Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra.
He had been partly anticipated by Hermann Weyl, who had also written a book about groups in quantum theory. But Weyl’s main focus was on foundational issues, whereas Wigner wanted to solve specific physical problems. Weyl was after beauty, Wigner was seeking truth.
We can understand Wigner’s approach to group theory in a simple, classical context, the vibrations of a drum. Musical drums are usually circular, but in principle they can be any shape. When you hit a drum with a stick, the skin vibrates and makes a noise. Different shapes of drum produce different sounds. The range of frequencies a drum can produce, called its spectrum, depends in a complex manner on the drum’s shape. If the drum is symmetrical, we might expect the symmetry to show up in the spectrum. It does, but in a subtle way.
Imagine a rectangular drum—you don’t see these often outside of mathematics departments. The typical patterns of vibration for such a drum divide it into a number of smaller rectangles, for example:
Two patterns of vibration of a rectangular drum.
Here we see two different vibrational patterns with two different frequencies. The pictures are snapshots of the patterns, taken at one instant. The dark regions are displaced downward, the white ones upward.
The symmetries of the drum have implications for the patterns, because any symmetry transformation of the drum can be applied to a possible pattern of vibration to produce another possible pattern of vibration. So the patterns come in symmetrically related sets. However, individual patterns need not have the same symmetries as the drum. For instance, a rectangle is symmetric under rotation through 180°. If we apply this symmetry transformation to the two patterns above, they become:
The same two patterns after rotating the drum through 180°.
The left-hand pattern is unchanged, so it shares the rotational symmetry of the drum. But the right-hand pattern has swapped dark regions for light. This effect is called spontaneous symmetry-breaking, and it is very common in physical systems: it occurs when a symmetric system has less-symmetric states. The left-hand pattern does not break symmetry, but the right-hand one does. Let’s focus on the right-hand pattern and see what effect its broken symmetry has.
Although the pattern and its rotation are different, they both vibrate at the same frequency, because rotation is a symmetry of the drum and hence of the equations that describe its vibrations. So the spectrum of the drum contains this particular frequency “twice.” It may seem difficult to detect that effect experimentally, but if you make small changes in the drum that destroy its rotational symmetry—say by making a small indentation along one edge—then the two frequencies drift slightly apart, and you can spot that there are two of them, very close together. This would not have happened if the frequency had occurred only once for the symmetric drum.
Wigner realized that the same effect arises with symmetric molecules, atoms, and atomic nuclei. The sounds made by the drum become vibrations of the molecules, and the spectrum of sounds is replaced by the spectrum of emitted or absorbed light. In the quantum world, the spectrum is created by transitions between different energy states, and the atom emits photons whose energy—hence frequency, thanks to Planck—corresponds to that difference. Now the spectrum can be detected using a spectroscope. Again, some of the frequencies—observed as spectral lines—may be double (or multiple) because of the symmetry of the molecule, atom, or nucleus.
How can we detect this multiplicity? We can’t make an indentation in the molecule, as we did for the drum. But we can place the molecule in a magnetic field. This also destroys the underlying symmetry and splits the spectral lines. Now you can use group theory—more strictly, group representation theory—to calculate the frequencies and how they split.
Representation theory is one of the most beautiful and powerful mathematical theories, but it is also technically demanding and full of hidden pitfalls. Wigner turned it into a high art. Others struggled to follow his lead.
By 1930, Wigner had secured a part-time post in America, at the Institute for Advanced Study, and he shuttled between Princeton and Berlin. In 1933, the Nazis passed laws forbidding Jews to hold university jobs, so Wigner moved permanently to the United States—mainly at Princeton, where he anglicized his name to Eugene Paul. His sister Margit joined him in Princeton. There she met Dirac, who was visiting, and in 1937 the two were married, to everyone’s amazement.
Margit’s marriage worked out fine, but Eugene’s job did not. In 1936, Wigner wrote, “Princeton dismissed me. They never explained why. I could not help feeling angry.” Actually, Wigner resigned, apparently because he was not advancing sufficiently quickly. Presumably he believed that Princeton’s refusal to promote him had effectively forced him to resign, so he felt as though he had been fired.
He quickly found a new job at the University of Wisconsin, took US citizenship, and met a physics student named Amelia Frank. They married, but Amelia had cancer and died within a year.
At Wisconsin, Wigner turned his attention to nuclear forces and discovered that they are governed by the symmetry group SU(4). He also made a basic discovery concerning the Lorentz group, published in 1939. But group theory was not t
hen a standard part of a physicist’s training, and its main application was still to the rather specialized area of crystallography. To most physicists, group theory looked complicated and unfamiliar, a fatal combination. The quantum physicists, appalled by what was invading their patch, described the development as the “Gruppenpest,” or “group disease.” Wigner had triggered an epidemic and his colleagues did not want to catch it. But Wigner’s views were prophetic. Group-theoretic methods came to dominate quantum mechanics, because the influence of symmetry is all-pervasive.
In 1941, Wigner began his second marriage, to a teacher named Mary Annette. They had two children, David and Martha. During the war, Wigner, like von Neumann and a great many top mathematical physicists, worked on the Manhattan Project to construct an atomic bomb. He was awarded the Nobel Prize in Physics in 1963.
Despite living for years in the USA, Wigner always longed for his homeland. “After 60 years in the United States,” he wrote in his declining years, “I am still more Hungarian than American. Much of American culture escapes me.” He died in 1995. The physicist Abraham Pais described him as “a very strange man . . . one of the giants of 20th century physics.” The viewpoint he developed is revolutionizing the twenty-first century as well.
13
THE FIVE-DIMENSIONAL MAN
By the late twentieth century, physics had made extraordinary advances. The large-scale structure of the universe seemed to be very well described by general relativity. Remarkable predictions such as the existence of black holes—regions of space-time from which light can never escape, created by the collapse of massive stars under their own gravity—were supported by observations. The small-scale structure of the universe, on the other hand, was described in extraordinary detail and with exquisite precision by quantum theory, in its modern form of quantum field theory, which incorporates special but not general relativity.
There were two serpents in the physicist’s paradise, however. One was a “philosophical” serpent: these two wildly successful theories disagreed with each other. Their assumptions about the physical world were mutually inconsistent. General relativity is “deterministic”—its equations leave no room for randomness. Quantum theory has inherent indeterminacy, captured by Heisenberg’s uncertainty principle, and many events, such as the decay of a radioactive atom, happen at random. The other serpent was “physical”: the quantum-based theories of elementary particles left a number of important issues unresolved—such as why particles have particular masses or indeed, why they have mass at all.
Many physicists believed that both serpents could be expelled from their Garden of Eden by the same bold action: unify relativity and quantum theory. That is, devise a new theory, a logically consistent one, that agrees with relativity on large scales and with quantum theory on small scales. This was what Einstein had tried to do for half his life—and failed. With typical modesty, physicists christened this unified view a “Theory of Everything.” The hope was that the whole of physics could be boiled down to a set of equations simple enough to be printed on a T-shirt.
It wasn’t such a wild idea. You can certainly get Maxwell’s equations on a T-shirt, and I currently own one with the equations of special relativity, with the slogan “Let there be light” in Hebrew. A friend bought it for me in the Tel Aviv airport. Less frivolously, major unifications of apparently disparate physical theories have been achieved before. Maxwell’s theory united magnetism and electricity, once thought to be entirely distinct natural phenomena powered by entirely different forces of nature, into a single phenomenon: electromagnetism. The name may be awkward, but it accurately reflects the process of unification. A more modern instance, less well known except to the physics community, is the electroweak theory, which unified electromagnetism with the weak nuclear force—see below. A further unification with the strong nuclear force has left just one thing missing from the mix: gravity.
Given this history, it is entirely reasonable to hope that this final force of nature can be brought into line with the rest of physics. Unfortunately, gravity has awkward features that make this process difficult.
It could be that no Theory of Everything is possible. Although mathematical equations—“laws of nature”—have so far been very successful as explanations of our world, there is no guarantee that this process must continue. Perhaps the universe is less mathematical than physicists imagine.
Mathematical theories can approximate nature very well, but it is not certain that any piece of mathematics can capture reality exactly. If not, then a patchwork of mutually inconsistent theories might provide workable approximations valid in different domains—and there might not be a single overriding principle that combines all of those approximations and works in all domains.
Except, of course, for the trivial list of if/then rules: “If speeds are small and scales are big, use Newtonian mechanics; if speeds are large and scales are big use special relativity,” and so on. Such a mix-and-match theory is horribly ugly; if beauty is truth, then mix-and-match can only be false. But perhaps at root the universe is ugly. Perhaps there is no root to be at. These are not appealing thoughts, but who are we to impose our parochial aesthetic on the cosmos?
The view that a Theory of Everything must exist brings to mind monotheist religion—in which, over the millennia, disparate collections of gods and goddesses with their own special domains have been replaced by one god whose domain is everything. This process is widely viewed as an advance, but it resembles a standard philosophical error known as “the equation of unknowns” in which the same cause is assigned to all mysterious phenomena. As the science fiction writer Isaac Asimov put it, if you are puzzled by flying saucers, telepathy, and ghosts, then the obvious explanation is that flying saucers are piloted by telepathic ghosts. “Explanations” like this give a false sense of progress—we used to have three mysteries to explain; now we have just one. But the one new mystery conflates three separate ones, which might well have entirely different explanations. By conflating them, we blind ourselves to this possibility.
When you explain the Sun by a sun-god and rain by a rain-god, you can endow each god with its own special features. But if you insist that both Sun and rain are controlled by the same god, then you may end up trying to force two different things into the same straitjacket. So in some ways fundamental physics is more like fundamentalist physics. Equations on a T-shirt replace an immanent deity, and the unfolding of the consequences of those equations replaces divine intervention in daily life.
Despite these reservations, my heart is with the physical fundamentalists. I would like to see a Theory of Everything, and I would be delighted if it were mathematical, beautiful, and true. I think religious people might also approve, because they could interpret it as proof of the exquisite taste and intelligence of their deity.
Today’s quest for a Theory of Everything has its roots in an early attempt to unify electromagnetism and general relativity—at the time, the whole of known physics. This attempt was made only fourteen years after Einstein’s first paper on special relativity, eight years after his prediction that gravity could bend light, and four years after the finished theory of general relativity was revealed to a waiting world. It was such a good attempt that it could easily have diverted physics onto a new course entirely, but unfortunately for its inventor, his work coincided with something that did set physics on a new course: quantum mechanics. In the ensuing gold rush, physicists lost interest in unified field theories; the world of the quantum offered far richer pickings, with far more chance of making a major discovery. It would be sixty years before the idea behind that first attempt was revived.
It began in the city of Königsberg, then the capital of the German province of East Prussia. Königsberg is now Kaliningrad, the administrative center of a Russian exclave lying between Poland and Lithuania. This city’s surprising influence on the development of mathematics began with a puzzle. Königsberg lay on the river Pregel (now Pregolya), and seven bridges linked t
he two banks of the river to each other and to two islands. Did there exist a route that would permit the citizens of Königsberg to walk across every bridge in turn, never crossing the same bridge twice? One of those citizens, Leonhard Euler, developed a general theory of such questions, implying that in this case the answer was no, and thereby took one of the first steps toward the area of mathematics now called topology. Topology is about geometrical properties that remain unchanged when a shape is bent, twisted, squashed, and generally deformed in a continuous manner—no tearing or cutting.
Topology has become one of the most powerful developments in today’s mathematics, with many applications to physics. It tells us the possible shapes of multidimensional spaces, a growing theme both in cosmology and particle physics. In cosmology we want to know the shape of space-time on the largest scale, that of the entire universe. In particle physics we want to know the shape of space and time on small scales. You might think the answer is obvious, but physicists no longer do. And their doubts also trace back to Königsberg.
In 1919, Theodor Kaluza, an obscure mathematician at the University of Königsberg, had a very strange idea. He wrote it up and sent it to Einstein, who apparently was struck speechless. Kaluza had found a way to combine gravity and electromagnetism in a single coherent “unified field theory,” something that Einstein had been trying to do for many years without success. Kaluza’s theory was very elegant and natural. There was just one disturbing feature: the unification required space-time to have five dimensions, not four. Time was the same as always, but space had somehow acquired a fourth dimension.