Hiding in the Mirror (24 page)

In any case, what made D-branes a cause for
celebration rather than sullenness, was that they allowed a full
demonstration that the various consistent string theories in ten
dimensions were in fact different aspects of the same theory. In
order to establish this, the previously discovered “duality” of
open strings on donut-shaped toroidal spaces—in which large and
small radii of the different compactified dimensions are
exchanged—was essential. Once D-branes are included in the picture,
going to the small-radii limit in one type of string theory could
be seen as producing the same physics as the large-radii limit of
another theory.

D-branes are also of great interest because
charges can exist on them, like electric charges, that are the
source of fields like the electromagnetic field. Since D-branes are
the surfaces on which the two ends of open strings are fixed, and it
turns out that Yang-Mills charges can exist on the ends of open
strings, these charges are then fixed to the D-dimensional surface
of the brane. However, remember that closed strings, which have no
end points and thus are not fixed to branes, also incorporate all
the physics associated with gravitons, the particles that convey
gravitational forces. Thus gravity can operate throughout the
“bulk” ten-dimensional space both on and between the branes, while
the charges that are the source of Yang-Mills fields live on the
branes themselves. As we shall see, this can have dramatic
implications.

In any case, the presence of D-branes in string
theory also allowed theorists to explore the all-important domain
where strings might interact strongly with one another, an area
that could not be addressed using conventional techniques developed
to try to understand the theory. This was especially critical
because it was known that considering only the possibilities where
strings might interact more feebly with one another would yield a
picture of the theory that was not fully accurate, quantitatively
or qualitatively. In particular, it was discovered that there is a
new kind of “duality” in string theories with D-branes. Recall once
more that for strings living on toroidal (i.e., donut-shaped
spaces) the large-radius physics is equivalent, and thus “dual” to
the small-radius physics. Now, when D-branes are introduced into
the picture, a new and different sort of duality results that
connects what otherwise may seem to be disparate physical extremes,
obtained by interchanging strings and branes in the theory. This
interchange maps a part of the theories where strings may be
interacting strongly with each other, and where one cannot perform
calculations, with a part of the theories where the strings are
more weakly interacting, and their behavior can be more simply
followed. In this way, not only might one hope to explore new
features of the various different string theories, but it becomes
possible to demonstrate how different theories might be
related.

The good news is that a new relation between
formerly disparate theories was uncovered. The bad news is that
while previously there had been five distinct consistent string
theories—suggesting that string theory in ten dimensions, with six
dimensions ultimately being compactified to leave four large
dimensions, was not unique—there now appeared to be a continuum of
theories. Specifically, these different theories were related to one
another, but each theory represented a distinctly different
physical limit. These different theories could be continuously
transformed into each other, implying a continuously infinite number
of intermediate physical possibilities. There was a ray of hope,
however. When examining one of the string theories with branes when
the string interaction strength became large, the number of states
grew in such a way that it appeared as if some new, hidden
dimension was appearing. Recall that in the original Kaluza-Klein
theory, as long as one was considering distances much larger than
the radius of the circular fifth dimension, all the extra
five-dimensional degrees of freedom remained hidden. However, as the
radius of the fifth dimension becomes larger in this model, the
energy required to resolve these new states decreases. Ultimately,
as the radius goes to infinity, the infinite tower of new states
makes its presence known. Such behavior was precisely what was
being observed for the number of D-branes in this string theory as
one tuned up the string interaction strength. Suddenly an eleventh
dimension began to suggest itself. This apparent extra dimension
was not observed in the first decade following the superstring
revolution in 1984 precisely because the analysis of weakly
interacting strings could only reveal a small part of the theory.
It was now understood that this “weak-coupling” approximation was
really very similar to what our four-dimensional world is in the
original Kaluza-Klein model—namely, an approximation to reality
obtained when the size of the extra dimension is very small
compared to anything one might measure. It would have been missed,
just as a fifth dimension would be forever missed in the original
Kaluza-Klein model, if one always did experiments on scales much
larger than the extra compact dimension. This is as close as
anything can come to “physics irony.” Here we had an apparently
remarkable new paradigm for physical theory that in some sense had
ultimately been motivated by the suggestion of Kaluza and Klein
that the physics of our world might derive from the hidden physics
of extra dimensions. Yet hidden within the theory itself apparently
lies hidden physics of yet another hidden dimension!

The key questions then become: What is this new
hidden physics, and does the propagation of dimensions continue?
The answer to the first question was, and to some extent still is,
“Anyone’s guess.” Clearly the theory will in some limit in eleven
dimensions resemble supergravity, which forms the basis of much of
string theory. But at higher energies it is unlikely to resemble
either supergravity or string theory, but perhaps something even
more miraculous.

One thing is clear, however. If this picture is
correct, what string theorists had previously claimed were
fundamental
tiny strings wiggling in tiny
extra dimensions deep inside what we otherwise thought were
fundamental elementary particles, would in fact perhaps be tiny
membranes wrapped around yet other tiny extra dimensions, with yet
even more fundamental objects. They would be masquerading as
strings because, in the approximations that had been used to define
the string theories in question, the extra dimension was curled up
on a scale smaller than the string scale, so that a two-dimensional
surface would look like a one-dimensional string. Strings, in this
respect, need not therefore be the truly elementary objects in the
theory.

Even when a new theory might not be understood
fully, at least it can be labeled. This new eleven-dimensional
theory has become known as M-theory. What does the
M
stand for? Well, first we must recognize that the
term
M-theory
has evolved to encompass not
just the theory that the ten-dimensional theories each approach as
some parameters are varied, but the theory that encompasses all the
theories in all their limits! Thus, it is only partially facetious
to claim that the name stands for “mother of all theories.” I am
told that Ed Witten introduced the term and said it stands for
magical, or mysterious, but that may be apocryphal. Other proposals
exist: Membrane theory? Marvelous theory?

A somewhat more informed guess, however,
suggests that perhaps the
M
stands for
matrix
. The argument for this is based on
the fact that if one takes one of the string theories that appears
to suggest this hidden extra dimension, then as the string
interaction strength is varied, the quantities that would normally
be the coordinates describing the motion of the strings and branes
are not simple numbers but are instead described by mathematical
objects called matrices.

A matrix is like a table of numbers, arrayed in
rows and columns. Here are two examples:

Matrices can be treated like ordinary numbers
in that one can define for them operations such as multiplication
and addition. However, unlike normal numbers, matrix multiplication
is not commutative. That is, while 3 × 4 equals 12 whether or not
one multiplies 3 times 4 or 4 times 3, the product of two matrices
A
and
B
is
not
in general equal to the product of
B
times
A
. This is
because the rules for multiplying matrices are complicated. One
multiplies each term in the first row of one matrix times the term
in the corresponding column and then adds the sum to get the
corresponding term (upper-left-hand corner) in the new matrix.
Thus, for example, for the two matrices given above, the first term
in the corresponding matrix if I multiply the first matrix times the
second is [(1 × 2) + (5 × 1) + (7 × 4)] = 35. However, if I
multiply the second matrix times the first, the first term in the new
matrix is [(2 × 1)+ (4 × 3) + (5 × 2)] = 24.

What is interesting and at the same time odd
about this is that if matrices are the fundamental objects
describing the eleven-dimensional universe of M-theory, then each
point in the space is described by a matrix and not a mere number.
This means the eleven-dimensional universe of M-theory bears little
or no resemblance to the universe we experience. The coordinates
that describe where you are in this space don’t commute with each
other! As if eleven-dimensional ordinary space was not complicated
enough to think about. Equally important is the fact that in this
new eleven-dimensional space, neither strings nor D-branes may be
the truly fundamental objects. If this picture is correct, strings
in ten dimensions are just as much an approximate illusion of
reality as elementary particles in four dimensions were supposed to
be in the original string picture.

One might, of course, wonder if all of this
rampant breeding of new dimensions is any different from the
earlier rampant breeding of new elementary particles at ever higher
energies, which seemed so confusing and complex in the 1960s, and
which led, in a sense, to the original proposal for dual string
theories.

Nevertheless, there are reasons to suspect that
eleven dimensions are as far as one need go. After all, one cannot
have sensible supergravity symmetries in higher dimensions, and
supergravity is one of the hallmarks that is supposed to
characterize feasible and consistent string theories as candidates
for quantum gravitational theories. Readers with a fantastic memory
and remarkable attention for detail may remember that another
feature of eleven-dimensional supergravity theories was that
gravity determined all of the matter fields in the theory, and that
there was no room left over for Yang-Mills fields and all the other
paraphernalia that makes our world so interesting. So, what is the
difference in M-theory? It is that M-theory contains many more
objects than merely elementary particles and fields. It contains
things that look, in some limits, like strings and D-branes, and in
other limits, like matrices. And who knows what else? Finally,
after this seemingly miraculous convergence on an unknown M-theory
(I remind you that for some people everything in string theory is
miraculous), you might think that this fiddling with extra
dimensions would be over with. However, the next, and up to the
present time, last string miracle was yet to occur.

In 1997, a young Princeton graduate student
turned Harvard professor Juan Maldacena made a daring conjecture,
which once again completely changed the face of string theory.
Remember that strings in ten dimensions can host Yang-Mills gauge
fields, while in eleven dimensions at low enough energies,
gravitational degrees of freedom associated with supergravity are
all that can be detected. Maldacena suggested another kind of
dramatic correspondence appropriate for our understanding of
Yang-Mills theories in four dimensions (i.e., the world of our
experience). Using ideas based in ten-dimensional string theory,
Maldacena proposed that perhaps our fourdimensional world, full of
Yang-Mills gauge symmetries, might have a hidden five-dimensional
meaning. Specifically, he conjectured that a fourdimensional flat
space with quantum Yang-Mills fields and supersymmetry, which our
world might contain, could be completely equivalent to a somewhat
strange five-dimensional universe with just classical (super)gravity
and nothing else. If this sounds suspiciously like déjà-vu all over
again—namely, like a modern reframing of the original Kaluza
proposal of 1919, in which electromagnetism in four dimensions
arose from an underlying theory involving just gravity in five
dimensions—you are not that far off. But there is a fundamental and
critical difference. In Kaluza-Klein theory, and all subsequent
theories with extra dimensions, our four-dimensional universe is
merely the tip of the iceberg. We only see four dimensions because
our microscopes cannot resolve those tiny extra dimensions.
However, in Maldecena’s conjecture, four-dimensional space is not
just some large-distance approximation of the underlying
five-dimensional space. Rather, the two are precisely the same! All
the physical laws of one universe are equivalent to those of the
other universe!