Hiding in the Mirror (23 page)

This brings us to the final development of the
trilogy in 1984–85 that truly put string theory at the center of
the particle physics universe. With the excitement generated by
finite, consistent superstrings and the new heterotic possibility of
generating large and phenomenologically interesting Yang-Mills
symmetries in ten dimensions, there was only one tiny thing left to
do: Make some contact with the four-dimensional universe of our
experience!

Enter Ed Witten. While some time earlier,
Claude Lovelace at Rutgers had begun to examine what might happen
if one put strings on spaces that curve up into small balls, a
comprehensive analysis of how one might turn these hypothetical
hyperdimensional theories of everything into realistic models of
our world was carried out first by Witten, and then in a seminal
paper by Witten and his collaborators Philip Candelas, Gary
Horowitz, and Andy Strominger.

Witten first showed that one could in principle
“compactify” six of the ten dimensions associated with superstrings
into small, finite volumes in a way that would leave four large
dimensions left over while still preserving, in those four
dimensions, essential features such as the absence of anomalies.
Then, Witten and his colleagues, a second “string quartet,” (the
first being Gross and colleagues) explicitly demonstrated how this
might be done. The key was to rely on a new type of mathematics,
not then well known among physicists, called “Calabi-Yau
manifolds,” after the mathematicians who had first described them. A
“manifold,” in mathematics, is something like a rubber sheet.
Generalizing the properties of such smooth, pliable objects to
higher dimensions has allowed mathematicians to invent a host of
strange new objects. Calabi-Yau manifolds are one interesting
mathematical class of manifolds with exotic curvatures in many
dimensions that can be mathematically classified.

Remember that Kaluza and Klein had considered
the simple case where their single extra dimension was curled up
into a small circle (a very simple one-dimensional manifold). One
might likewise imagine that this concept could be applied to the
six extra dimensions in string theory, have them curl up into a
small six-dimensional sphere. This was the approach first explored
by Lovelace, but it turns out not to work. As Witten and
collaborators demonstrated, very specific conditions needed to be
imposed on this “compactified space” in order for the resulting
fourdimensional theory to remain sensible. Such spaces turned out
to have been investigated by the mathematicians Eugenio Calabi and
Shing-Tung Yau, and Witten (who would later win the most
prestigious award in mathematics, the Fields Medal, for his work
using string theory to illuminate the detailed mathematics of knot
theory) and his collaborators were able to use their results to
explore what kind of theories one might expect to produce in four
dimensions. The results were encouraging. It appeared to be
possible to produce theories with plausible grand unified Yang-Mills
symmetries, and with a spectrum of elementary particles, quarks,
electrons, muons, and so forth that could bear an eerie resemblance
to what we actually observe in our universe.

The reaction to the Candelas and coworkers’
paper by the physics community was astounding. Suddenly the
esoteric and mathematically complex field of string theory held the
promise of actually making contact with reality—and not just slight
contact. It opened up the possibility of providing a fundamental
explanation of why the world at its most basic scale looks like it
does, and the answer seemed to lie hidden in this extradimensional
Calabi-Yau universe. Within two to three years most major physics
departments had a group of brilliant young theorists working on
string theory, and in turn this group, usually tenured within a few
years of getting their PhDs, started training a new generation,
many of whom began their training with string theory, and had never
heard of such elementary particles as pions, which had started the
whole effort off in the first place. It was a common belief at the
time that even though the theory was so complex that the
approximations that had thus far been performed barely scratched
its surface, it was just a matter of time—and not much time,
perhaps—before all the details would be worked out and all the big
questions answered. For example, in order to approximate the
complex Calabi-Yau manifolds, physicists instead explored
approximations called “orbifolds,” which on the whole behave like
higher-dimensional generalizations of the nice, smooth rubber
sheets one can picture in one’s head, but which have, at a discrete
number of points, locations where the sheet gets warped into a
conelike shape, with a single point of very high (in a strict
mathematical sense, infinite) curvature. Thus, all of the
complexities of the Calabi-Yau manifolds could be relegated to what
might occur at a finite number of weird points in an otherwise
smooth and simple space. One hoped that big questions would be
insensitive to this dramatic approximation.

In the meantime, the world of elementary
particle physics underwent a sea change after 1984. In particular,
an interesting sociological phenomenon began to take place that
still has repercussions for the field today. The largely
mathematical questions underlying the new theories became for a
number of young physicists new to the field much more interesting
than trying to figure out such “trivial” low-energy details as how
grand unification might account for the actual physics that resulted
in a universe full of matter instead of antimatter, or why the
proton is two thousand times heavier than the electron. In short,
the as-of-yet hypothetical world of hidden extra dimensions had,
for many who called themselves physicists, ultimately become more
compelling than the world of our experience.

I never think of the
future. It comes soon enough.

—Albert Einstein

T
he theoretical
discoveries of 1984–85 energized theoretical particle physicists as
nothing had done in a long while. At the same time they produced a
remarkable optimism in those who had already begun to work on
string theory that the long-sought goal of a consistent unified
theory of all the fundamental interactions in nature was at hand,
if only the theory could be fully understood. What began as an
investigation of an idea that might incorporate gravity and quantum
mechanics had, precisely because of its enforced necessity of extra
dimensions, begun to appear as if it might explain why everything
else existed as well. The concluding sentence of the original
heterotic superstring paper stated, “Although much work remains to
be done, there seem to be no insuperable obstacles
to deriving all known physics
from the . . .
heterotic superstring.” (Italics mine.) With the realization that
the heterotic superstring literally required, for its internal
consistency, precisely those Yang-Mills symmetries that appeared
most promising to describe the real world, it seemed as if nature
was saying, “Build a string, and they will come.” If the
requirements for a consistent string theory in turn required a
specific Yang-Mills symmetry that might explain all of the observed
distribution of particles and forces in our four-dimensional
universe, then maybe we could finally resolve Einstein’s long-ago
query, “Did God have any choice in the creation of the Universe?”
The answer would be “No, not if she chose to create it via
strings!”

Along with the optimism came a sense of
astonishment: Within the course of less than a year it seemed as if
an almost insurmountable problem had largely been resolved. So it
was that one often heard the remark that, by means of a fortunate
accident (the development of dual string models to attempt to
explain the strong interactions) we had discovered what rightfully
should have been considered twenty-first-or twentysecond-century
physics in the twentieth century. We were truly living in the
future!

And if life were an impressionist painting, we
would have been. Seen with broad brush strokes, everything appeared
to be in order. However, there were still a number of nagging
details, not to mention the growing recognition that the theory was
nowhere near to being fully explored, let alone understood. Indeed,
it was not quite clear precisely what string theory actually
was
. In a prescient paper written in 1983,
shortly before the great string revolution, in which he guessed
that string theories might be candidates for a consistent theory of
quantum gravity, Ed Witten admitted, “What is really unsatisfactory
about string theory at the moment is that it isn’t yet a
theory.”

Unfortunately, the closer one looked, the
greater the problems became. The very richness of the string models
and compactification schemes, for example, appeared to undermine
claims for uniqueness and with it the hope that string theory would
prescribe a universe that simply
had
to
look precisely like the one we live in. Shin-Tung Yau had, for
example, elucidated over a hundred thousand different Calabi-Yau
manifolds, and compactifying six dimensions on each of them would
produce a different four-dimensional theory. Moreover, detailed
analysis of the approximations used to compactify the theory from
ten dimensions to four suggested that these operations might not be
well controlled, invalidating the attractive phenomenological
pictures that had first been presented. In an effort to check
whether the four-dimensional theories that appeared to result from
compactification really were consistent, theorists began to analyze
string theories in four dimensions from a new perspective. It turns
out that because a string is a one-dimensional object moving in
time, its “world sheet”—that is, the region of space-time it maps
out as it moves—is a two-dimensional surface. This is the case
whether the string is moving in four dimensions, ten dimensions, or
twenty-six dimensions. Adding new fields onto the world sheet, which
is what happens when fermions and Yang-Mills fields are added to
strings, therefore involves studying how fields behave on
two-dimensional surfaces.

Interestingly, this is an area of intense
interest in condensed matter physics, which studies the bulk
properties of real material, whether boiling water,
superconductors, or magnets. When such materials undergo a change
of phase—for example, water begins to boil, magnets become
magnetized—then near the point of this change the properties of the
material become particularly interesting and simple. The physics
turns out in some cases to depend almost entirely on phenomena
associated with twodimensional surfaces, such as bubble walls form
the boundary between different phases of boiling water. As a
result, condensed matter physicists have become experts on studying
such surfaces. Moreover, it turns out that as materials approach
the conditions where such phase transitions can occur, their nature
begins to look self-similar (i.e., the same phenomena like bubbles
seem to appear on all scales). This “scale invariance” is similar
to the conformal symmetry of the string theories, which implied
that the physics looked the same regardless of over what scales one
might stretch the strings.

In any case, studies of such condensed matter
systems had classified essentially all two-dimensional field
theories, and demonstrated that many of them had the properties
that one guessed they might have if they instead described string
world sheets obtained by compactifying from higher dimensional
theories. That was the good news. But at the same time it suggested
that perhaps one could consider string theories in four dimensions
without ever worrying about their ten-dimensional roots. Indeed,
are the ten dimensions necessary at all, or are the extra
dimensions just mathematical artifacts? This is the central
question that continues to haunt us.

It was clear that to go beyond the
impressionistic connection to the real world, one was going to have
to understand string theory a lot better than it was understood
thus far. And this was going to be hard work, involving the
development of new mathematics that could handle systems far more
complex than anything that had been heretofore studied. An army of
bright new physicists immediately launched a campaign to scour
every cave where interesting possibilities might lurk. Over the
next four years the line that had previously tended to separate
articles that appeared in physics journals from those that were
published in mathematics journals began to blur. Ed Witten, in
particular, worked furiously on a host of remarkable ideas.

But, in spite of this plethora of talent and
output, progress in actually answering questions about our
four-dimensional world was distinctly lacking. New insights about
the possible nature of string theory, field theory, and Yang-Mills
theories might have been accumulating, but solid physical
predictions were not.

Most embarrassing (from my point of view, at
least) was the apparent inability of string theory to address the
key physical paradoxes that seemed to be associated with a quantum
theory of gravity. Sure, the theory appeared to get rid of
infinities that might otherwise render predictions nonsensical, but
when it came to predicting such things as what the energy of empty
space might be (i.e., why the cosmological constant must be zero or
extremely small), the theory appeared to make no useful
predictions. Another area where strings had thus far shed no light
was the very question that Stephen Hawking raised that appeared to
result in a direct challenge to quantum theory itself in a world of
gravity. What happens to the information about what falls into
black holes if the black holes can ultimately evaporate away and
disappear? While not much had happened on these fronts, theoretical
progress in trying to understand the different varieties of
consistent string theories had begun to suggest that the five
different types of consistent string theories explored in ten
dimensions, might be related.

Might these apparently different theories
merely be different manifestations of some single “
über
” theory? As early as 1985, in fact, several
researchers had suggested this possibility. After all, this is
precisely the trend that had worked so well to simplify the physics
of the known world: Electricity and magnetism had been shown to be
different reflections of the same force, the weak and
electromagnetic interactions had been shown to be different
reflections of the same underlying physics, and so on.
Interestingly, however, when physicists began to explore such a
possible new connection between the different string theories,
hints began to appear that these different theories might well be
unified—but not in ten dimensions. Rather, they seemed as if they
might be different tendimensional reflections of an underlying
eleven-dimensional theory!

Alert readers may remember that eleven
dimensions had previously appeared in the grab bag of theoretical
physics, associated with a special theory of supergravity. In
eleven dimensions, all interactions and particles are specified by
gravity and supersymmetry alone, while in ten dimensions there is
much more freedom to choose extra Yang-Mills symmetries, fields, and
so on. Perhaps an eleven-dimensional theory might be unique, even
if a ten-dimensional theory wasn’t.

The first step on this road came from work by
Witten and collaborators in 1995, which suggested that all five
known consistent string theories were merely different versions of
a single underlying, more expansive theory. The next major
development in understanding this possible unification came from a
remarkable and unexpected observation in 1995 by Joe Polchinski, at
the Kavli Institute for Theoretical Physics at Santa Barbara.
Polchinski changed the whole nature of our understanding of what
was possible in string theory because he demonstrated that what
people had been exploring up to that point—indeed, the theory that
had been claimed to be a theory of
everything
—had in fact overlooked an infinite number
of things, including new objects in higher dimensions. For reasons
that will become clear, he called them
D-branes
.

His observation derived from considerations of
how open strings might behave in toroidal (i.e., donut-shaped)
spaces. As you will recall, insuch spaces it appeared that
shrinking one radius of the donut produced a theory that, for
closed strings that might wrap in different directions around the
donut, looked identical to one in which the same closed strings
were wrapping around a radius that became very large.

Open strings—that is, strings that do not close
back upon themselves, forming loops, but have two end points like a
regular piece of string—however, end up in this case leading to
another interesting phenomenon. Their ends are free to move about,
and it turns out that the surfaces comprising the set of points
along which their ends can move can themselves form a whole new
type of mathematical object, behaving like a sort of (mem)brane. In
three spatial dimensions, for example, a two-dimensional brane
could be a plane or a membrane surface like a rubber sheet. Open
strings would be attached at either end to this plane (as the
diagram shows). They could wiggle and move in the extra dimension,
but their ends would by definition, move about on the plane
(brane).

One might imagine that these structures are
called D-branes because they need not be two-dimensional, but can
be any number of dimensions, less than or equal to the total number
of dimensions of space-time itself. That would be too simple,
however. It turns out that they are called D-branes because of the
special mathematical conditions (called “Dirichlet boundary
conditions”) that one imposes, which Polchinski realized could
exist when a string ends on a surface. The different dimensional
D-branes are actually called “p-branes” (since the letter
D
was taken already), where
p
refers to the dimensionality and
D
to Dirichlet. A one-brane looks like a string, a
two-brane looks like a familiar membrane (like a rubber sheet), a
three-brane like our own three-dimensional space, and so on. What
is more notable about these new objects than their names is the
fact that they have their own type of dynamics. Recall that years
earlier, when dual strings were first being explored, physicists had
wondered whether one might generalize the underlying concepts to
yet higherdimensional objects. In a sense, Polchinski’s D-branes
are just these generalizations, but more interestingly, he showed
that they are
required
to arise when one
attempts to consider the full dynamics of string theories. They had
been previously missed for two reasons. First, almost all of the
previous analyses of strings had dealt with the simplest
approximation to the theory, the so-called weak coupling
limit—namely, when strings are almost noninteracting and their
wiggles are minimal. Second, fixing the ends of strings to lie on
some surface spoils some of the space-time symmetries of the theory
in ten dimensions. Physicists had tacitly assumed that keeping such
symmetries was essential. But they seemed to forget that the world
we experience is only four-dimensional, and what is important is
that the resulting theory have the observed space-time symmetries
in four dimensions that Einstein ultimately incorporated into
general relativity. D-branes, through the mathematical conditions
that occur when strings are connected to them, preserve these
latter symmetries, if not the full tendimensional symmetries. Once
D-branes are included in the theory, it becomes much richer and
more complex than it was before, with a host of possible new
phenomena. One might imagine that it was somewhat of an
embarrassment that string theorists had previously proclaimed that
they were on the verge of victory in creating a “theory of
everything,” when they had in fact virtually missed “almost
everything” in the theory. But in the everoptimistic string
worldview, there are no embarrassments. On a slightly less
facetious note, it is important to realize that devoting literally
decades of one’s career to a theoretical struggle, with unknown
odds for success, requires those who engage in it to have a deep
underlying faith in the validity of what they are attempting. For
these “true believers,” every new development provides an
opportunity to confirm one’s expectations that these ideas
ultimately reflect reality. What separates this from religion, or
what
should
separate this from religion,
however, is the willingness to give up these expectations if it
turns out that the theory makes predictions that disagree with
observations, or if it turns out that the theory is impotent and
makes no predictions.