Hiding in the Mirror (22 page)

In superstring language there is another way of
viewing this effect, and that is that the string has a fundamental
symmetry, called “conformal invariance.” This symmetry would imply
that the physical nature of string interactions is independent of
how one might stretch the string. Thus, for example, two strings
that might seem to be otherwise close together during an
interaction can in fact be stretched farther apart, and one would
still get the same answer for the contribution of this process to
physically measurable quantities. But, as we have seen, if the
interaction points are spread out in space, then the dangerous
infinities tend to be removed. This conformal, or stretching
symmetry turns out to have unexpected implications when strings
interact in certain exotic spaces. For example, in the particular
case where one has closed string loops moving on a space that looks
like a donut, called a torus, then it turns out that a string loop
having a very small size around one circle of the donut behaves
identically to a large loop stretched around the other circle (the
circle spanning the horizontal direction around the donut).

This was a remarkable result and its
implication was very important in the attempt to understand why
strings might universally tame quantum infinities. For if there is a
symmetry that says that string loops of radius smaller than some
quantity—say,
R
₀
—produce
identical physical effects to those of strings of a size much
bigger than
R
₀
, the
implication is that
R
₀
represents some fundamental physical scale below which distances
have no physical meaning. If you do try to probe smaller scales
using strings that appear to be smaller in size, you end up
producing phenomena that could instead be equally well pictured as
involving strings of a much larger size. This “duality” between
large and small strings, as it is called, can therefore be seen as
providing a clear physical cutoff on how small a region can be over
which virtual processes can occur. Once again, this small-scale
cutoff has the effect of rendering otherwise potentially infinite
virtual processes finite. While spreading out the interactions of
gravitons is one way to turn gravity from a quantum theory beset
with infinities to a quantum theory that is apparently finite, having
a
finite theory does not imply that one has
the
finite theory. A host of other issues,
both physical and mathematical, must be addressed before we might
gain confidence that this is the case. This brings us to the truly
unexpected string miracle. It was also discovered in 1984 that the
quantum theory of supersymmetric strings in ten dimensions can, in
certain circumstances, naturally avoid another type of more subtle
and dangerous mathematical inconsistency I mentioned earlier, which
physicists call an “anomaly.” An anomaly occurs when quantum
mechanical virtual processes destroy the mathematical symmetries
that one would otherwise expect a theory to possess. It is as if
one produced a theory that predicted the earth should be a perfect
sphere without any imperfections, so that any place on the planet
would be identical to any other place, but when one considered
quantum mechanical effects one would instead find that on small
scales the sphere would contain mountains and valleys, so that some
of its points would be very different than other points. Thus, the
beautiful spherical symmetry of the theory would be destroyed.

Such nasty quantum mechanical anomalies have
been found to generically occur in one particular type of quantum
theory: that which distinguishes left from right. Unfortunately, as
we have seen, the weak interaction is precisely such a theory, in
which “left-handed” electrons behave differently than
“right-handed” electrons. To step back a bit, it was somewhat of a
surprise that strings in higher dimensions even allow for such a
possibility of “handedness” in the first place. Careful studies of
Kaluza-Klein theories in higher dimensions by Ed Witten, in
particular, had earlier demonstrated “no-go” theorems implying that
there was no straightforward way to distinguish left-and
right-handed fermions in higher-dimensional theories.

It turned out, however, that one can avoid
these no-go theorems if one changes the rules a bit. Namely, if
instead of pure Kaluza-Klein gravity in the higher dimensions, one
supplements the theory by having extra YangMills fields living in
these higher dimensions—precisely the situation that, I remind you,
arises in supersymmetric string theories in ten dimensions—then
these fields can impact upon the fermions living on strings in
complicated new ways in order to produce right-handed and
left-handed objects that behave differently.

But with this realization came the concern
about anomalies. In general, once left-and right-handed fermions
behave differently, then the quantum mechanical contributions of
virtual left-and right-handed particles to various processes can
destroy the very symmetries that are required in order to keep the
theory mathematically consistent. These anomalies essentially undo
the very careful cancellations of various otherwise infinite
quantities that are ensured by the Yang-Mills symmetries, as well
as resulting in a host of other nonsensical predictions. Actually,
things are even worse in ten dimensions than in four, because not
only can the Yang-Mills symmetries get destroyed by anomalies, so
can the symmetries that underlie general relativity. Thus, there is
actually a greater chance that any given theory of gravity will
prove to be nonsensical as a quantum theory in ten dimensions than
it will in four dimensions. What Green and Schwarz showed in 1984
was that for two specific kinds of supersymmetric string theories in
ten dimensions, the theory was not only finite, but even with
left-and right-handed fermions acting differently, all anomalies
disappear. What might result therefore could be a completely finite
and consistent quantum theory of gravity. Within months of the
Green and Schwarz discovery, feverish activity by two different
groups produced two more dramatic developments that ultimately
generated enough excitement to induce much of the rest of the
particle physics community to drop what they were working on and
begin to explore this new possible Theory of Everything.

The first development involved a group led by
David Gross, who, you may recall, helped to kill the first
incarnation of dual string models when he discovered the phenomenon
of asymptotic freedom, which demonstrated that QCD, and not a dual
string model, was the proper theory of the strong interaction. As I
indicated earlier, David’s graduate career had begun at Berkeley,
and continued at Princeton, where he did important work on dual
string models with Neveu, Scherk, and Schwarz. His return to this
subject, after having abandoned it a decade earlier, was nothing
short of triumphant, and he has taken it up again with all the
fervor, as he himself suggested, of a converted atheist.

Gross, along with his colleague Jeff Harvey and
students Emil Martinec and Ryan Rohm, developed, in a tour de
force, something with the memorable name of “heterotic string.” The
name does not derive from the word
erotic,
but rather from the root
heterosis,
although there is also no doubt that the model is kinky, both
metaphorically and literally. Indeed, it is so imaginative as to be
considered sexy by many theorists, which says something either
about the model or about theorists.

When Green and Schwarz discovered that
superstring theories in ten dimensions could be consistent, finite,
and anomaly-free, they identified two possible symmetries of strings
that would allow this. They explicitly demonstrated three different
sorts of superstring solutions that exhibited one type of symmetry,
but none that exhibited the other type, which for a number of
technical reasons seemed like it might produce more interesting
grand unified scenarios. The heterotic string, on the other hand,
could work with either symmetry and thus was of special interest.
What made this particular string theory so exciting, however, was
not merely that it could produce potentially more interesting
Yang-Mills symmetries, but that the existence of this Yang-Mills
symmetry was forced upon it, not by the seemingly ad hoc need for
anomaly cancellation, but by the requirements of formulating the
string theory itself. This suggested some potentially deep
connection between the possible existence of strings in ten
dimensions and the observed Yang-Mills symmetries of nature in four
dimensions.

The heterotic string model involves closed
string loops, which on first glance is unusual, because closed
strings, while they incorporate gravity, do not generally
incorporate Yang-Mills symmetries. Gross and his collaborators,
however, realized that if one is bold enough then this limitation
can be circumvented. In particular, on a closed string, the
vibrations that travel in one direction around the string are
completely decoupled—that is, they do not interact with the
vibrations that travel in the other direction around the string.
There is a classical analogy for this: If you take a regular
string, and jiggle it from one end to send a wave down it, while at
the same time jiggling it from the other end to send a wave in the
opposite direction, you will be able to see the two waves pass
directly through each other at the center of the string. The two
wave modes do not interact. Now for an amazing feat of mathematical
sleight-of-hand: It is possible to imagine a sort of “hybrid”
string in which the left-moving and rightmoving vibrations on a
string are quite different. In fact, Gross and his coworkers argued
that these different modes could actually be pictured as living in
different sets of dimensions!

For the ten right-moving sets of vibrations on
strings, Gross and colleagues treated them precisely as Green and
Schwarz did for their tendimensional superstring: with ten normal
coordinates, and with sixteen of those strange Grassmann
anticommuting coordinates. Recall that the effect of this
construction is to produce equal numbers of fermion and boson
excitations on the string.

Gross and coworkers then imagined that the poor
left-moving vibrational excitations were bereft of supersymmetry.
You may recall, however, that the quantum mechanics of vibrating
strings without supersymmetry can only be formulated consistently
in twenty-six dimensions. In a leap of creative chutzpa that is
hard to beat, Gross and his colleagues then simply imagined that
the left-moving vibrations on strings act as if they live in a
twenty-six-dimensional space!

It may seem strange to you that some of the
vibrational modes on a closed string live in one number of
dimensions, while others live in another, much larger set of
dimensions. Actually, this little technicality was not lost on the
creators of the model, who pointed out an apparently
straightforward, if equally bold, solution. Simply curl up sixteen
of the dimensions on which the left-moving vibrations operate into
very small regions. In this case, then, just as happened in
Kaluza-Klein theory to make the fifth dimension invisible, on scales
too large to resolve the extra sixteen dimensions, one would appear
to be left with only ten remaining leftmoving modes to go along
with the ten right-moving modes. In a way this mathematical
wizardry is also reminiscent of what happened in the original the
Kaluza-Klein model. There, degrees of freedom in the extra
curled-up dimension end up looking, in the four-dimensional world,
like photons (i.e., particles associated with the gauge symmetry of
electromagnetism). In the new model, one could show that the extra
sixteen left-moving modes associated with the curled-up sixteen
dimensions end up appearing as welcome extra Yang-Mills symmetries
and fields on the remaining ten-dimensional closed string.

Incidentally, if this isn’t strange enough for
you, it turns out that there is a way to frame the heterotic string
in which the extra sixteen left-moving modes are not associated
with sixteen extra spatial dimensions at all, but rather with
thirty-two extra weird Grassmann anticommuting coordinates on a
ten-dimensional string! In string theory, it seems, as we shall see
again later, the existence of extra hidden dimensions may actually
depend upon the eye of the beholder.