Parallel Worlds (31 page)

Kaluza, a
previously obscure mathematician, wrote a letter to Einstein proposing to
formulate Einstein's equations in five dimensions (one dimension of time and
four dimensions of space). Mathematically, this was no problem, since
Einstein's equations can be trivially written in any dimension. But the letter
contained a startling observation: if one manually separated out the fourth-
dimensional pieces contained within the five-dimensional equations, you would
automatically find, almost by magic, Maxwell's theory of light! In other words,
Maxwell's theory of the electromagnetic force tumbles right out of Einstein's
equations for gravity if we simply add a fifth dimension. Although we cannot
see the fifth dimension, ripples can form on the fifth dimension, which
correspond to light waves! This is a gratifying result, since generations of
physicists and engineers have had to memorize Maxwell's difficult equations
for the past 150 years. Now, these complex equations emerge effortlessly as the
simplest vibrations one can find in the fifth dimension.

Imagine fish
swimming in a shallow pond, just below the lily pads, thinking that their
"universe" is only two-dimensional. Our three-dimensional world may
be beyond their ken. But there is a way in which they can detect the presence
of the third dimension. If it rains, they can clearly see the shadows of
ripples traveling along the surface of the pond. Similarly, we cannot see the
fifth dimension, but ripples in the fifth dimension appear to us as light.

(Kaluza's theory
was a beautiful and profound revelation concerning the power of symmetry. It
was later shown that if we add even more dimensions to Einstein's old theory
and make them vibrate, then these higher-dimensional vibrations reproduce the
W- and Z-bosons and gluons found in the weak and strong nuclear forces! If the
program advocated by Kaluza was correct, then the universe was apparently much
simpler than previously thought. Simply vibrating higher and higher dimensions
reproduced many of the forces that ruled the world.)

Although
Einstein was shocked by this result, it was too good to be true. Over the
years, problems were discovered that rendered Kaluza's idea useless. First, the
theory was riddled with divergences and anomalies, which is typical of quantum
gravity theories. Second, there was the much more disturbing physical question:
why don't we see the fifth dimension? When we shoot arrows into the sky, we
don't see them disappear into another dimension. Think of smoke, which slowly
permeates every region of space. Since smoke is never observed to disappear
into a higher dimension, physicists realized that higher dimensions, if they
exist at all, must be smaller than an atom. For the past century, mystics and
mathematicians have entertained the idea of higher dimensions, but physicists
scoffed at the idea, since no one had ever seen objects enter a higher
dimension.

To salvage the theory,
physicists had to propose that these higher dimensions were so small that they
could not be observed in nature. Since our world is a four-dimensional world,
it meant that the fifth dimension has to be rolled up into a tiny circle
smaller than an atom, too small to be observed by experiment.

String theory
has to confront this same problem. We have to curl up these unwanted higher
dimensions into a tiny ball (a process called compactification). According to
string theory, the universe was originally ten-dimensional, with all the forces
unified by the string. However, ten-dimensional hyperspace was unstable, and
six of the ten dimensions began to curl up into a tiny ball, leaving the other
four dimensions to expand outward in a big bang. The reason we can't see these
other dimensions is that they are much smaller than an atom, and hence nothing
can get inside them. (For example, a garden hose and a straw, from a distance,
appear to be one- dimensional objects defined by their length. But if one
examines them closely, one finds that they are actually two-dimensional surfaces
or cylinders, but the second dimension has been curled up so that one does not
see it.)

WHY STRINGS?

Although
previous attempts at a unified field theory have failed, string theory has
survived all challenges. In fact, it has no rival. There are two reasons why
string theory has succeeded where scores of other theories have failed.

First, being a
theory based on an extended object (the string), it avoids many of divergences
associated with point particles. As Newton observed, the gravitational force
surrounding a point particle becomes infinite as we approach it. (In Newton's
famous inverse square law, the force of gravity grows as
1/r
²
,
so that it soars to infinity as we approach the point
particle—that is, as
r
goes to zero,
the gravitational force grows as 1/0, which is infinite.)

Even in a
quantum theory, the force remains infinite as we approach a quantum point
particle. Over the decades, a series of arcane rules have been invented by
Feynman and many others to brush these and many other types of divergences
under the rug. But for a quantum theory of gravity, even the bag of tricks
devised by Feynman is not sufficient to remove all the infinites in the theory.
The problem is that point particles are infinitely small, meaning that their
forces and energies are potentially infinite.

But when we
analyze string theory carefully, we find two mechanisms that can eliminate
these divergences. The first mechanism is due to the topology of strings; the
second, due to its symmetry, is called supersymmetry.

The topology of
string theory is entirely different from the topology of point particles, and
hence the divergences are much different. (Roughly speaking, because the
string has a finite length, it means that the forces do not soar to infinity as
we approach the string. Near the string, forces only grow as
1/I
²
,
where
L
is the length
of the string, which is on the order of the Planck length of 10
^-33
cm. This length
I
acts to cut off
the divergences.) Because a string is not a point particle but has a definite
size, one can show that the divergences are "smeared out" along the
string, and hence all physical quantities become finite.

Although it
seems intuitively obvious that the divergences of string theory are smeared out
and hence finite, the precise mathematical expression of this fact is quite
difficult and is given by the "elliptic modular function," one of the
strangest functions in mathematics, with a history so fascinating it played a
key role in a Hollywood movie.
Good Will Hunting
is the story of a rough working- class kid from the
backstreets of Cambridge, played by Matt Damon, who exhibits astounding
mathematical abilities. When he is not getting into fistfights with
neighborhood toughs, he works as a janitor at MIT. The professors at MIT are
shocked to find that this street tough is actually a mathematical genius who
can simply write down the answers to seemingly intractable mathematical
problems. Realizing that this street tough has learned advanced mathematics on
his own, one of them blurts out that he is the "next Ramanujan."

In fact,
Good Will Hunting
is loosely based on the life of
Srinivasa Ramanujan, the greatest mathematical genius of the twentieth century,
a man who grew up in poverty and isolation near Madras, India, at the turn of
the last century. Living in isolation, he had to derive much of
nineteenth-century European mathematics on his own. His career was like a
supernova, briefly lighting up the heavens with his mathematical brilliance.
Tragically, he died of tuberculosis in 1920 at the age of thirty-seven. Like
Matt Damon in
Good Will Hunting,
he dreamed of
mathematical equations, in this case the elliptic modular function, which possesses
strange but beautiful mathematical properties, but only in twenty-four
dimensions. Mathematicians are still trying to decipher the "lost
notebooks of Ramanujan" found after his death. Looking back at Ramanujan's
work, we see that it can be generalized to eight dimensions, which is directly
applicable to string theory. Physicists add two extra dimensions in order to
construct a physical theory. (For example, polarized sunglasses use the fact
that light has two physical polarizations; it can vibrate left-right or
up-down. But the mathematical formulation of light in Maxwell's equation is
given with four components. Two of these four vibrations are actually
redundant.) When we add two more dimensions to Ramanujan's functions, the
"magic numbers" of mathematics become 10 and 26, precisely the
"magic numbers" of string theory. So in some sense, Ramanujan was
doing string theory before World War I!

The fabulous
properties of these elliptic modular functions explain why the theory must
exist in ten dimensions. Only in that precise number of dimensions do most of
the divergences that plague other theories disappear, as if by magic. But the
topology of strings, by itself, is not powerful enough to eliminate all the
divergences. The remaining divergences of the theory are removed by a second
feature of string theory, its symmetry.

SUPERSYMMETRY

The string
possesses some of the largest symmetries known to science. In chapter 4, in
discussing inflation and the Standard Model, we see that symmetry gives us a
beautiful way in which to arrange the subatomic particles into pleasing and
elegant patterns. The three types of quarks can be arranged according to the
symmetry SU(3), which interchanges three quarks among themselves. It is
believed that in GUT theory, the five types of quarks and leptons might be
arranged according to the symmetry SU(5).

In string
theory, these symmetries cancel the remaining divergences and anomalies of the
theory. Since symmetries are among the most beautiful and powerful tools at our
disposal, one might expect that the theory of the universe must possess the
most elegant and powerful symmetry known to science. The logical choice is a
symmetry that interchanges not just the quarks but all the particles found in
nature—that is, the equations remain the same if we reshuffle all the subatomic
particles among themselves. This precisely describes the symmetry of the
superstring, called supersym- metry.
It
is the only symmetry that
interchanges all the subatomic particles known to physics.
This makes it
the ideal candidate for the symmetry that arranges all the particles of the
universe into a single, elegant, unified whole.

If we look at
the forces and particles of the universe, all of them fall into two categories:
"fermions" and "bosons," depending on their spin. They act
like tiny spinning tops that can spin at various rates. For example, the
photon, a particle of light that mediates the electromagnetic force, has spin
1. The weak and strong nuclear forces are mediated by W-bosons and gluons,
which also have spin 1. The graviton, a particle of gravity, has spin 2. All
these with integral spin are called bosons. Similarly, the particles of matter
are described by subatomic particles with half-integral spin—1/2, 3/2, 5/2, and
so on. (Particles of half-integral spins are called fermions and include the
electron, the neutrino, and the quarks.) Thus, supersymmetry elegantly
represents the duality between bosons and fermions, between forces and matter.

In a supersymmetric
theory, all the subatomic particles have a partner: each fermion is paired with
a boson. Although we have never seen these supersymmetric partners in nature,
physicists have dubbed the partner of the electron the "selectron,"
with spin 0. (Physicists add an "s" to describe the superpartner of a
particle.) The weak interactions include particles called leptons; their
superpart- ners are called sleptons. Likewise, the quark may have a spin-0
partner called the squark. In general, the partners of the known particles
(the quarks, leptons, gravitons, photons, and so on) are called sparticles, or
superparticles. These sparticles have yet to be found in our atom smashers
(probably because our machines are not powerful enough to create them).

But since all
subatomic particles are either fermions or bosons, a supersymmetric theory has
the potential of unifying all known subatomic particles into one simple
symmetry.
We now have a symmetry large
enough
to
include the entire universe.

Think of a
snowflake. Let each of the six prongs of the snowflake represent a subatomic
particle, with every other prong being a boson, and the one that follows being
a fermion. The beauty of this "super snowflake" is that when we
rotate it, it remains the same. In this way, the super snowflake unifies all
the particles and their sparti- cles. So if we were to try to construct a
hypothetical unified field theory with just six particles, a natural candidate
would be the super snowflake.

Supersymmetry
helps to eliminate the remaining infinities that are fatal to other theories.
We mentioned earlier that most divergences are eliminated because of the
topology of the string—that is, because the string has a finite length, the
forces do not soar to infinity as we approach it. When we examine the
remaining divergences, we find that they are of two types, from the
interactions of bosons and fermions. However, these two contributions always
occur with the opposite signs, hence the boson contribution precisely cancels
the fermion contribution! In other words, since fermionic and bosonic
contributions always have opposite signs, the remaining infinities of the
theory cancel against each other. So supersymmetry is more than window
dressing; not only is it an aesthetically pleasing symmetry because it unifies
all the particles of nature, it is also essential in canceling the divergences
of string theory.