Why Beauty is Truth (50 page)

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 then 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.

B
y 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 the 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.

Kaluza had not set out to unify gravity and electromagnetism. For some reason best known to him, he had been messing around with five-dimensional gravity, a kind of mathematician's warmup exercise, working
out how Einstein's field equations would look if space had that absurd extra dimension.

In four dimensions the Einstein equations have ten “components”—they boil down to ten separate equations describing ten separate numbers. These numbers jointly constitute the metric tensor, which describes the curvature of space-time. In five dimensions there are fifteen components, hence fifteen equations. Ten of them reproduce Einstein's standard four-dimensional theory, which is no surprise; four-dimensional space-time is embedded in five-dimensional space-time, so you would naturally expect the four-dimensional version of gravity to be embedded in the five-dimensional one. What about the remaining five equations? They could have been just some peculiar structure with no significance for our own world. But they weren't. Instead, they were very familiar, and that's what amazed Einstein. Four of Kaluza's remaining equations were precisely Maxwell's equations for the electromagnetic field, the ones that hold in
our
four-dimensional space-time.

The one remaining equation described a very simple kind of particle, which played an insignificant role. But no one, least of all Kaluza, had expected both Einstein's theory of gravity and Maxwell's theory of electromagnetism to emerge spontaneously from the five-dimensional analogue of gravity alone. Kaluza's calculation seemed to be saying that light is a vibration in an extra, hidden dimension of space. You could put gravity and electromagnetism together into a seamless whole, but only by supposing that space is really four-dimensional and space-time is five-dimensional.