Hiding in the Mirror (19 page)

Up until the 1980s, the many extra dimensions
proposed by science fiction and spiritual literature were
essentially completely divorced from anything being considered by
the scientific community. As late as 1981, for example, the idea
that somehow the nature of particles and fields at the smallest
scales might somehow be related to extra dimensions appeared in a
story written by Craig Harrison and, in 1986, turned into a movie
called
Quiet Earth.
In it, a scientist
produces a fundamental change in the basic structure of matter in
his laboratory, but as a result he transports almost all of the
human populace to another dimension.

The nature of the confluence of
extra-dimensional speculations in science and science fiction began
to change, however, as notions that started to arise in elementary
particle physics made their way into popular culture.

As we shall see, it was precisely the study of
elementary particle physics that caused physicists to reconsider,
at about the same time as Harrison’s story was published, the
existence of five, six, and even twenty-six dimensional spaces. And
by the 1990s, after various popular accounts of the emerging
research interest in the possibility of extra dimensions had
appeared, one finds numerous science fiction stories—for example,
“Eula Makes up Her Mind,” which was featured in a recent science
fiction anthology competition that I happened to judge—in which the
extra dimensions of string theory play a key role. In a recent New
York play the heroine somehow uses lessons from string theory in
twenty-six dimensions to help her sort out her confusing love
life!

As the latter example makes clear, in spite of
the cross-pollination of ideas, there nevertheless remains a
certain cognitive dissonance between explorations in physics and
the literary allusions. I imagine that this is inevitable, and that
one need not bemoan it. One of the purposes of science is to
inspire people to pose questions about the universe, and if the
inspiration that results is often off the mark, the effort should
still be welcomed—that is, as long as people don’t confuse art and
reality too strongly.

Consider, after all, that from the time of
Klein in the 1920s to the resurgence of interest in the topic in
the 1980s and ’90s, physicists were concentrating on
microscopically tiny extra dimensions, so small that nothing of
real interest on human scales could transpire within them or emerge
from them. As I wrote in
The Physics of Star
Trek,
while extra dimensions might exist, if they did, they
were thought to be far too small for aliens to abduct us into
them.

But, once again, life is appearing to imitate
art, and to some extent science is playing catch-up. As I shall
describe, possibly infinitely large extra dimensions and even
parallel universes that might house everything from stars and
planets to aliens have become topics that physicists now actually
discuss seriously. The story of how we got to this strange place
will occupy us for the rest of this book. Whether or not the
current speculations about large, or even small, extra dimensions
are any more firmly grounded in reality than the extra dimensions
imagined by More in 1671 to house spirits or those imagined by the
Star Trek
writers from which hostile aliens
might emerge, or whether instead they resemble the fictional
Cerebron’s analytical discovery of three different kinds of
dragon—the mythical, the chimeral, and the purely hypothetical—is,
of course, the million-dollar question.

My soul is an entangled
knot,

Upon a liquid vortex
wrought

By Intellect in the Unseen
residing.

And think doth like a
convict sit,

With marlinspike untwisting
it,

Only to find its knottiness
abiding;

Since all the tools for its
untying

In four-dimensional space
are lying.

—James Clerk Maxwell

W
hile the 1960s proved
to be a period of discovery and confusion in elementary particle
physics, as I have described, the 1970s were one of exultation,
vindication, and ultimately, hubris. We began the decade mired in
confusion about the quantum mechanical nature of every known force
except for electromagnetism, and we completed it with a beautiful
and perfectly accurate microscopic formulation of three of the four
known forces in nature, with the hope of one day joining them into
a single Grand Unified Theory (GUT).

It is within this historical framework that we
should view the developments that have taken place since the 1970s.
While the dual string theories of the late 1960s caused some
physicists to take what so far appears to have been a dead-end
detour to explore how microscopic extra dimensions might explain
the physics of strongly interacting particles, the subsequent
remarkable advances of the 1970s ultimately emboldened physicists
to attempt to address the “really big” questions. Just as
Einstein’s great success gave him the hubris, and stamina, to
devote the final thirty years of his life to an (ultimately futile)
effort to produce a unified theory of all interactions, so, too, in
the 1980s did physicists begin to reexamine ideas ranging from the
Kaluza–Klein higher-dimensional framework to the mathematical
miracles of the dual string model in an effort to once again
attempt to reach Einstein’s elusive goal of a unified theory.

Like all grand and ambitious campaigns,
perhaps, this one began via a series of independent and sometimes
serendipitous developments on seemingly unrelated fronts. These all
converged in the mid-1980s in an explosion of excitement and
activity that has transformed much of the focus of fundamental
physics ever since.

In 1971 a young Dutch physicist, Gerardus
’tHooft, working on his PhD with his professor, Martinus Veltman,
made one of those rare discoveries that changed the way physicists
thought about fundamental physics. When Veltman had first met young
’tHooft, he told him to read the classic 1954 paper by Yang and
Mills that proposed the now-famous Yang-Mills theories—the
generalizations of electromagnetism that I wrote about earlier.
While at the time the formalism proposed by Yang and Mills was
essentially purely mathematical—there were no systems in nature
that it could clearly describe—its elegance had raised the interest
of several key theoretical physicists. One was the Nobel laureate
Julian Schwinger, who around 1959 advised his graduate student
Sheldon Glashow to consider how one might use these ideas to study
the weak interactions, which ultimately led to Glashow’s 1961 paper
for which he would win the Nobel prize. Another was Veltman, who
was convinced that the symmetry associated with the Yang-Mills
theories was too beautiful to not be applicable to nature.

The problem with these theories was that if one
tried to use them to describe physical phenomena, such as those
associated with the weak interactions, then mathematical infinities
appeared to result, which were not too different than those that
caused physicists working on the strong interaction to first resort
to the study of dual string models. Indeed, the model proposed by
Glashow, and independently by Weinberg and Salam in 1967, appeared
to suffer from just such infinities, so it is interesting to note
that from the period 1961 to 1971 the papers that ultimately unified
the weak and electromagnetic interactions were cited in the
literature by physicists less than a dozen times.

However, ’tHooft, working under Veltman’s
guidance, discovered in 1971 that the infinities that appeared to
plague the electroweak model of Glashow, Weinberg, and Salam could
cleverly be removed so that the theories made mathematical sense
and their predictions could be compared with experiment to
arbitrarily high precision—if one had sufficient energy to do the
calculations. Within two years it was understood that both the
strong and weak interactions were described by Yang-Mills
“nonabelian gauge theories.” Three of the four forces in nature
were now understood as full quantum theories. All that remained to
conquer was gravity!

In the twenty years or so following Yang and
Mills’s work, a handful of physicists had explored the possibility
that one might extend the work of Kaluza and Klein in unifying
electromagnetism and gravity to the possibility of unifying gravity
and Yang-Mills theories. The rationale for this was not evident,
except that it was an interesting mathematical problem. It was
immediately clear that these theories, which you may recall involve
more than one “photonlike” force carrier, would require a
generalization to more than five dimensions. Remember that Kaluza
and Klein had been able to reproduce the force of electromagnetism
in four dimensions by making the photon field a part of a
five-dimensional gravitational field, with the one extra dimension
invisible to us.

By 1975 or so the problem had finally been
worked out by various physicists, with a complete derivation by
Peter Freund and collaborator Y. M. Cho. The result was what one
might expect: Namely, as one could incorporate one photon by having
a gravitational field in one extra (i.e., a fifth) dimension, so one
could accommodate more than one “photonlike”

field, as occurs in Yang-Mills theories, by
adding one extra dimension for each field, and having general
relativity operate in the full multidimensional space. This model,
however, did not attract much, if any, attention, for a variety of
reasons. Most important was the fact that unlike the Kaluza-Klein
theory, a complete solution of whose equations allows three “large”
and relatively flat spatial dimensions along with a compactified and
thus “small” fourth spatial dimension, it turned out that the
solutions of the higher-dimensional theories were not so
simple.

Since the world we happen to live in is
manifestly both large and threedimensional, one might expect that
the fact that these higher-dimensional unification models did not
predict such a universe would kill any interest in them whatsoever.
However, as I have pointed out in another context, putting aside
some mathematical ideas is like trying to put the toothpaste back
in the tube after you have squirted it out. Once they are out
there, they tend to take on lives of their own.

Indeed, within a year, it was recognized that
if one added additional particles and forces beyond those
associated with gravity in the higherdimensional framework, one
could produce the desired compactification to a large, flat,
three-dimensional space and smaller extra compact dimensions. Of
course, in so doing one was deviating from the spirit of Kaluza and
Klein, who hoped that all the forces in nature might arise from a
single gravitational field in higher dimensions. Once additional
particles and fields are introduced in these extra dimensions, much
of the beauty and economy of the proposal would at first seem to
fade. But beauty is in the eye of the beholder, and it would turn
out that there were other, equally mathematically elegant reasons
to consider such additions. For the moment though, let us return to
the spirit of Kaluza and Klein and ask, if the mathematical
Yang–Mills symmetries associated with the known forces in nature
were to result from the geometric properties of some underlying
extra-dimensional space, how many
extra
dimensions would we need to accommodate all the known forces? The
answer turns out to be seven, leading to a total of eleven
space-time dimensions. Thus, at the very least, the physics of the
past fifty years tells us that if extra dimensions are to be the key
to understanding all of the known forces in nature, there have to
be a lot of them!

Eleven dimensions may seem like a lot to
accommodate, but there are some good things associated with doing
so. First, it is fewer than twentysix dimensions, which is what the
dual string models naively seemed to require. At the same time, it
turned out that there was an independent reason to consider spaces
as large as eleven dimensions in physics, coming from a
consideration of the differences between the nature of matter and
the matter of nature. When we classify all the forces in nature,
one fact stands out clearly: All of these forces appear to result
via the exchange of virtual particles called “bosons.” Recall that
in quantum mechanics various properties of elementary objects, such
as energy and momentum, can take on only various discrete
“quantized” values. Bosons are elementary particles whose quantum
angular momentum, or “spin” as we call it, comes in integer
multiples of some basic fundamental value. However, when we look at
matter, there is no such restriction. The basic particles that make
up matter—electrons and quarks—all have spin values that are
half-integer multiples of that fundamental value, and are called
“fermions.” Composite objects, made up of combinations of quarks,
can have either half-integer or integer spin.

Now, one may wonder about this asymmetry in
nature (i.e., why forces are associated with bosons, and matter is
associated with both fermions and bosons). The investigation of
this asymmetry took a long and convoluted trail that ultimately
ended up in—you guessed it—extra dimensions. It began in 1970, when
it was realized, even before they were dashed by the development of
QCD, that the dual string models in twenty-six dimensions that
appeared to be consistent models actually had a serious flaw. These
theories predict particles called “tachyons.”

Tachyons may be familiar to people who like to
watch
Star Trek,
but in the real universe
of physics, tachyons are bad news. As the name suggests, they have
something to do with time. Strictly speaking, tachyons are
particles that can appear to move backward in time, which is
something that at the very least is embarrassing. Alternatively, it
turns out that one can think of this behavior as due to the fact
that they are particles that are restricted to always travel faster
than the speed of light. Because of the relation between relative
time and velocity for different observers in special relativity, it
turns out that particles that somehow are forever moving faster
than the speed of light (nothing can cross the threshold from
slower to faster in the theory) would behave to other observers as
if they are moving backward in time.

Now, it turns out that the laws of classical
physics do not forbid such unusual particles to exist, but all
sensible theories tend not to predict them (not to mention the fact
that no tachyons have ever been observed in nature). Generally, if
a theory predicts a tachyonic particle, it is usually a
mathematical indication of some instability in the ground state of
the theory—a reflection of the fact that one has somehow
misidentified what the true stable particles are. If the instability
is removed, so is the tachyon. So, on the surface, the 1970s would
seem to have been a very bad time for string theory. First, QCD
came along as the correct theory of the strong interaction, and
second, the dual string model appeared to be unstable, anyway. But,
as has happened numerous times since, string theory has
demonstrated an almost chameleon-like ability to morph into
something new, its flaws transforming into virtues. The roots of
such a novel version of string theory date back to 1971, when
physicists André Neveu and John Schwarz, and independently Pierre
Ramond, investigated ways of allowing the incorporation of
halfinteger spin particles (fermions) into dual string models.
Their motivation at the time was to enable these models to
incorporate quarks, which by then had been demonstrated to exist
inside of protons and neutrons and other strongly interacting
particles. If the dual models were supposed to describe strongly
interacting particles, then they would have to allow for the
existence of such objects.

The mechanism for doing this is somewhat
technical and may seem rather unusual on first, and probably second,
glance. Normally we describe distances along a string, or any other
object, in terms of regular numbers. We would say, for example:
“Move 5.5 units (i.e., feet, miles, whatever) along the string.”
However, the mechanism that Neveu, Schwarz, and Ramond investigated
did not involve using normal numbers to describe such distances
along the strings but instead quantities called Grassmann
variables, which obey rather strange relations. For normal numbers,
say, 5 and 4, 5
x
4 = 4
x
5. However, for two such Grassmann quantities,
A
and
B,
it turns
out that
AB
= −
BA
.
Moreover, since this same relation must be true for the individual
quantities
A
and
B,
this means that
A
²
= −
A
²
= 0 and
B
²
= −
B
²
= 0. 

I mention this not because it is particularly
illuminating, but because it gives a sense of the sometimes highly
nonintuitive mathematical manipulations associated with some string
miracles, many of which seem unphysical, at least until one gets
used to them. In any case, one of the first important developments
that occurred when fermions were added to strings using this
strange mechanism is that it was realized that the critical
dimension on which quantum dual string theories might make sense
could be reduced from twenty-six to ten dimensions. Now, ten is
close to eleven, which is the number of dimensions that pure
Kaluza-Klein-type arguments seemed to favor, as I discussed
earlier, but as the saying goes, close is only useful in horseshoes
and hand grenades. However, this development was not the end of the
story. Once fermions were added to strings, it was realized that
another remarkable bit of mathematical wizardry was possible: There
could exist a brand-new symmetry that related bosons (integer spin)
on the string to fermions (half-integer spin) on the string.
Interestingly, it had previously been thought to be impossible to
have such a symmetry interchanging bosons and fermions in one’s
description of nature, and in fact a theorem to this effect had
been proved in 1967 by the brilliant physicist and raconteur Sidney
Coleman at Harvard (who you may recall was David Politzer’s
supervisor) and his student Jeffrey Mandula.