Hiding in the Mirror (17 page)

It turns out that it is precisely the infinite
sum I discussed earlier that implies this weird need for twenty-six
dimensions to preserve unitarity. Considering scattering processes
between strings, “virtual strings” could be exchanged, with the
possibility of having an infinite number of virtual strings
contributing to the scattering process. Now it turns out that the
result of performing this sum yields a term that screws up the
calculation of probabilities, of the following form: [1 + 1⁄2 (
D
− 2) (1 + 2 + 3 + 4 + 5 + . . . )], with
D
representing the dimension of space-time.
Now, if
D
= 26, and the infinite series in
the second term sums up to − 1⁄12, the total result for this
offending contribution to physical scattering is precisely zero.
Now, you may recall that when Kaluza postulated the existence of a
hypothetical mathematical fifth dimension, he did so sheepishly,
noting “all the physical and epistemological difficulties.” He
essentially suggested that this extra dimension was primarily a
mathematical trick, a way of unifying two disparate theories. But
Kaluza’s proposal was nothing compared to what appeared to be
required for the consistency of dual string models—namely, that the
universe must be not five-dimensional, but
twenty-six-dimensional.

You might wonder whether a mathematical trick
is sufficient reason to believe in twenty-two new dimensions of
space, and no doubt many physicists at the time did, too. However,
nature ultimately came to the rescue to resolve the debate, so that
no one had to worry about this issue. Or rather, a much simpler
theory than dual strings came along to completely explain the
strong force.

The first inklings that dual strings might not
provide the answer to the puzzling nature of the strong interaction
came from experiments performed within a year or so of the time
that Veneziano first proposed his mathematical solution for duality.
If duality held true, then at high energies the rates of scattering
of strongly interacting particles off of one another that would
produce particles that flew off at a fixed angle should decline
dramatically as the energy increased. But the observed falloff,
while it did exist, was much less severe than the prediction. It
turned out that this finding provided clear support for an idea first
proposed at the beginning of the decade by the brilliant
theoretical physicist Murray Gell-Mann, who between the mid-1950s
and the mid1960s seemed to have an unerring sense of what
directions might prove fruitful for unraveling the experimental
confusion in elementary particle physics. Gell-Mann suggested in
1961 that one could classify the existing strongly interacting
particle states into a very attractive mathematical pattern, which
he called the eightfold way. What made this classification system
more than mere taxonomy was that one of its first predictions is
that new particles would have to exist in order to fill out some
parts of the pattern that had not yet been seen. In one of the most
remarkably prescient combinations of experiment and theory in
recent times, in 1964 one of those new particles, called the
omega-minus, was discovered, more or less exactly as Gell-Mann and
his collaborators had predicted.

By 1964 Gell-Mann—and independently, George
Zweig—had recognized that this underlying mathematical framework
could have a physical basis if all of the dozens of strongly
interacting particles, now called “hadrons,” were composed of yet
more fundamental particles, which GellMann, the consummate scholar
and linguist, dubbed “quarks” in honor of a term from James Joyce’s
Ulysses. 
Quarks themselves remained a
purely theoretical construct that nevertheless proved remarkably
useful in classifying all the observed hadrons. However, in the
late 1960s the reality of quarks as physical entities was suggested
when the scattering experiments that killed the dual string picture
proved instead to be completely compatible with the notion that
hadrons were themselves composed of pointlike particles acting
almost independently. On its own, however, the quark model was not
sufficient to explain the data. If quarks existed, why had they not
been directly observed in highenergy scattering experiments? What
force or forces might bind them into hadrons, and how could one
explain hadron properties in terms of quark properties? And most
confusing of all, if hadrons were strongly interacting, which meant
that quarks had to be as well, why did the pointlike particles that
appeared to make up hadrons act independently, as if they were
almost noninteracting, in these high-energy scattering
experiments?

Well, I already gave the punch line away
several chapters earlier. In 1972–74 a series of remarkable
theoretical breakthroughs basically resolved almost all the
outstanding problems in elementary particle physics, as it was then
understood. In particular, in a last-ditch effort to potentially
put an end to what had become known as “quantum field theory,” which
is the theoretical framework that results when one
straightforwardly combines quantum mechanics and relativity using
familiar fundamental particles, David Gross at Princeton, who had
been a student of Geoffrey Chew’s at Berkeley during the heyday of
the bootstrap model, and his own student Frank Wilczek were
exploring the mathematical behavior of a type of quantum field
theory called a Yang-Mills theory, named after the two physicists
who had first proposed it way back in 1954. Yang-Mills theories have
another, more technical, name that is even more daunting:
nonabelian gauge theories. What this term means is that these
theories are similar to electromagnetism, which has a mathematical
property called gauge invariance, a form of which was first explored
by the mathematician Hermann Weyl in his efforts to unify
electromagnetism and gravity.

An equation is said to possess a certain
symmetry, or be invariant under some change, whenever that change
does not alter its meaning. For example, if
A
=
B
, then 
A
+2 =
B
+2. Adding 2 to each
side of an equation leaves the meaning of the equation invariant.
If
A
and
B
represent positions in space, for example, then adding 2 to both
sides of the equation would be equivalent to translating both
A
and
B
by two
units in some direction. Each point would still be at the same
position as the other point. This transformation is called a
“translation,” and the equation is said to be “translationally
invariant,” or possess a “translation symmetry.”

Similarly, the fundamental equations of both
gravity and electromagnetism remain invariant when one changes
certain quantities in the theory—in the case of gravity, these
include the coordinates used to measure the distance between
points. As pointed out earlier the specific coordinates one uses to
describe some space are chosen for convenience. The underlying
physical properties, like curvature, do not depend upon the choice
of coordinates. For electromagnetism however, the quantity one can
freely change is related to an intrinsic characteristic of charged
objects, associated, it turns out, with multiplying all charged
quantities by a complex number. Weyl thought one could also make
this latter quantity appear as if it were a kind of coordinate
transformation, achieved by changing the scale (or “gauge”) of
disance measurements. One could thus “unify” the “symmetries” of
electromagnetism and gravity as being associated with different
kinds of coordinate transformation, but he was wrong. Nevertheless,
it turns out that the separate symmetries of these two theories
imply that gravity and electromagnetism share one feature in
common: In both, the strength of the force between (massive or
charged, respectively) objects falls off with the square of the
distance between them. It turns out that when one attempts to turn
these theories into quantum theories, this particular force law,
which means the force is long ranged, requires, via the uncertainty
principle, the existence of a massless particle that can be
exchanged between objects and by which the force is transmitted. In
the case of electromagnetism this particle is called the photon,
and in gravity we call the (not yet directly measured) particle the
graviton.

However, in nonabelian or Yang-Mills theories,
because the transformations that can leave the equations the same
are more complex, instead of having only one massless force carrier
field, like the photon in electromagnetism, these theories can have
numerous such fields. Moreover, in electromagnetism the photon,
while it is emitted and absorbed by objects that carry electric
charge, does not itself carry an electric charge. But in Yang-Mills
theories the force carriers themselves are charged and thus
interact with one another as well as with matter. These theories
had begun to have newfound currency in the late 1960s after it was
proposed—it later turned out correctly—independently by Glashow,
Weinberg, and Salam, who later shared the Nobel Prize for their
insight, that one such nonabelian gauge theory could correctly
describe all aspects of the weak interaction that converted protons
into neutrons, and was responsible for the decay of neutrons into
protons, electrons, and neutrinos. Gross, Wilczek, and
independently David Politzer, a graduate student of Sidney
Coleman’s at Harvard, each turned his attention to another
nonabelian gauge theory whose form ultimately turned out to have
certain properties that suggested it might be appropriate to
describe the interactions between quarks that bound them together
into hadrons. Recall that Gross, who was trained in Chew’s
“bootstrap” group at Berkeley, was exploring this theory in hope of
ruling it out as the last possible quantum field theory—and hence
the last theory that was based on elementary particles as the
fundamental quantities of interest—that might explain the exotic
properties that seemed to be required to result in the high-energy
scattering behavior of hadrons.

Much to his surprise, however, when he,
Wilczek, and also Politzer completed their calculations, which
explored precisely how virtual particles and antiparticles in this
theory might affect how the force between quarks evolved as the
quarks got closer together, it turned out that a miracle occurred.
As Gross later put it: “For me, the discovery of asymptotic freedom
was totally unexpected. Like an atheist who has just received a
message from a burning bush, I became an immediate true
believer.”

The theory, which we now call quantum
chromodynamics, or QCD for short, had precisely the property needed
to explain the experimental data: Namely, the force between quarks
would grow weaker as the quarks got closer—which implies,
naturally, that as one pulled them farther apart the force would
get stronger. This could explain why in high-energy scattering
experiments the individual quarks close together inside the proton
might appear almost noninteracting, while at the same time no
scattering experiment had yet been successful in knocking a single
quark apart from its neighbors. Discovering the property that quark
interactions grew weaker with closer proximity—which they dubbed
asymptotic freedom—enabled them, and since then many other
researchers, to calculate and predict very precisely the behavior
of strongly interacting particles in high-energy collisions.
Needless to say, the predictions have all been correct. The
converse property, which suggests that the force between quarks
continues to grow without bound as you try to separate them, and
which has since been dubbed confinement, has not yet been fully
proven to arise from QCD. However, numerical calculations with
computers all suggest that it is indeed a property of the theory
that is now known to describe the strong force. Gross, Wilczek, and
Politzer were hence awarded the Nobel Prize in 2004 for their
discovery of asymptotic freedom thirty years earlier. Thus, out of
the incredible experimental confusion of the 1940s, ’50s, and ’60s
had ultimately arisen a beautiful set of theories, now called the
standard model, that described all the known, nongravitational
forces in nature in terms of rather elegant mathematical quantum
field theories called gauge theories. The simplest extension of the
basic laws of nature, involving quantum mechanics, relativity, and
electromagnetism, had ultimately triumphed over the competing
mathematical elegance of exotic ideas such as dual string models,
along with their exciting, if somewhat daunting, requirement of
extra dimensions.

But the game was far from over. The fatal warts
of dual strings, at least as far as explaining the strong
interaction, would later be turned into beauty marks in a much more
ambitious program to unify gravity with the other three forces in
nature. And the very properties of gauge fields and the matter that
couples to them, combined with the remarkable theoretical successes
that had been achieved by studying them, would lead theorists to
once again revisit the very first effort to unify the first known
gauge theories: gravity and electromagnetism. In so doing they
would once again be driven to reconsider whether extra dimensions
might be the key to understanding nature.

. . . the banality of
existence has been so amply demonstrated, there is no
need for us to discuss it any further here. The
brilliant Cerebron, attack-
ing the problem
analytically, discovered three distinct kinds of dragon: the
mythical, the chimeral, and the purely
hypothetical. They were all, one
might say,
nonexistent, but each nonexisted in an entirely different
way.

—Stanislaw Lem,
The Third
Sally

I
f physicists have been
fickle in their intermittent love affair with extra dimensions,
turning hot and cold as their whims and desires evolved, artists
and writers have been much more faithful with their affections.
Through good times and bad, a literary fascination with another
world beyond the reach of our senses has held steadfast. There is
an unbroken string of writing with this focus, stretching from
Lewis Carroll’s
Through the Looking Glass
(1872) to C. S. Lewis’s
The Lion, The Witch,
and the
Wardrobe
(1950) and beyond.
These books, written almost a century apart, were ostensibly
created for children by austere British academics, but both reach
out far more broadly to that primal yearning to answer with a
resounding
no,
Peggy Lee’s plaintive cry:
“Is that all there is?”

Despite the gap in time and intentions of the
writers (Carroll was, among other things, a satirist who poked fun
at both authority and thenmodern mores, while C. S. Lewis wrote his
tale as an allegory to promote his deep religious convictions),
there is a remarkable similarity in their choice of dramatic
method. Alice is transported through a looking glass to a new
three-dimensional world that exists inside of the glass, but
clearly not
behind
it. Lewis’s Lucy
similarly enters a wardrobe, which again has a well-defined back
when seen from the outside, but instead of encountering a wooden
frame, she stumbles into the snowy night of that other
threedimensional world, Narnia. From a mathematical perspective
(and Carroll, at least, was a mathematician), what both young girls
traverse is a mystical intersection between two completely separate
three-dimensional worlds. To enter one is to disappear from the
other . . . or at least in Alice’s case to disappear, once she
turns the corner out of view of those peering into the mirror. And
two separate and distinct three-dimensional worlds can intersect
only if the underlying space is at least four-dimensional. As I
have previously alluded, their experience is strangely reminiscent,
if less terrifying perhaps, than little Christie’s experience in
the
Twilight
Zone
episode “Little Lost Girl.” Actually, this 1962 screenplay derived
from an earlier short story by Richard Matheson that appeared in
the science fiction magazine
Amazing Stories
in November 1953. The contemporaneous appearance of Matheson’s
piece and Lewis’s allegory is perhaps not surprising, for just as
the world of elementary particle physics was turning topsy-turvy
during the 1940s and ’50s, so, too, did this period witness a
resurgence of interest among writers, artists, and now filmmakers in
a possible fourth spatial dimension. During this era and the
decades that followed, the extradimensional imagination of artists
and writers happily moved from the purely “mythical, chimerical,
and hypothetical,” as per Stanislaw Lem’s fanciful science fiction
story, to a sensibility that was more closely attuned to emerging
scientific themes. What began as a rather unrealistic fascination
with the mathematical properties of a purely hypothetical fourth
spatial dimension through the 1940s and ’50s eventually progressed
to topics like space and time travel, a host of possible new
dimensions, and issues such as how information might leak in and
out of our world.

Perhaps the first and best known among the
modern science fiction writers who helped rekindle popular
fascination with a fourth dimension was Robert Heinlein. His
classic short story “And He Built a Crooked House,” written in 1940
and published in the February 1941 issue of the science fiction
monthly
Astounding Science Fiction,
tells
the tale of an unfortunate California architect, Quintus Teal, who
designs a revolutionary house based on a tesseract, which you will
recall is a four-dimensional version of a cube. Teal has a
brilliant idea to save space. If you could build a tesseract house,
then its footprint in our three-dimensional world could be a simple
cube. But, since the full 4D tesseract has eight 3D cubical faces
(as a 3D ube has six 2D square faces, you will recall), one could
have an eightroom house on land with only enough space for a single
room. (I understand that in later editions of
Superman
comic books, his Fortress of Solitude had a
similar design, for a similar reason.)

Of course, not having access to four
dimensions, Teal does the next best thing: He builds an
unfolded
tesseract. Again, just as you could unfold
a cube by cutting along its edges to lay out on a piece of paper
the six squares that make it up, say as follows,

so too, you could imagine unfolding a tesseract
and projecting onto a three-dimensional space the eight cubes that
form its surface:

This projection, which is also called a “net,”
was Teal’s construction—that is, until an earthquake accidentally
causes the structure to fold back up into its four-dimensional
form, nearly trapping Teal and the new owners in another
three-dimensional space forever removed from our own. Heinlein’s
fascination with hypercubes was not novel. Charles Hinton’s fixation
with four dimensions caused him to imagine and present a host of
ways of visualizing four-dimensional objects such as tesseracts (or
hypercubes, as they are also known) in all of his many writings at
the turn of the last century. In the 1920s short stories continued
to focus on the fourth dimension as a way to move in and out of
interesting threedimensional spaces. Both Richard Hughes’s humorous
“The Vanishing Man” (1926) and Miles Breuer’s “The Appendix and the
Spectacles” (1928) focus on the opportunities and problems that
result from the fact that moving into a fourth spatial dimension
would allow one to visit and explore the insides of objects,
including human beings, without ever having to actually travel
through their outer surfaces. But I suspect it was Heinlein’s work
(in particular “Crooked House”) and later writing (such as Madeline
L’Engle’s children’s classic
A Wrinkle in
Time,
in which a tesseract is used as a portal to reach faraway
distances and times in a folded space) that brought the idea to
popular attention, and made the term
tesseract
a familiar one in popular culture.
(Heinlein continued his fascination with a fourth dimension up
through his 1963 story “Glory Road,” which involved a
hyperdimensional packing case that was bigger inside than
outside.)

Coincidentally, at almost the same time as
Heinlein’s and Matheson’s work was permeating popular culture, one
of the twentieth century’s most prolific and imaginative artists,
Salvador Dali, who had moved well beyond cubism to help spearhead
surrealism, produced his classic painting,
Crucifixion, Corpus Hypercubus,
which reproduces the
tesseract net I displayed earlier.

While modern art has itself moved well beyond
surrealism, so that concern with the three-dimensional notion of
form has been replaced by such interests as color—or in the most
extreme forms of conceptual art, no form at all—the inclusion in
1954 of a tesseract as a surrealist object of interest is part of a
pattern in popular culture that I find particularly intriguing.
Recall that, at the turn of the century, before, during, and
immediately after the introduction of Einstein’s work, fascination
with a fourth spatial dimension existed entirely independently of
special relativity. Yet almost a half-century after Einstein’s
revolutionary theories, the notion that the fourth dimension of
that theory was not a spatial dimension still had not fully filtered
down to the popular level. Or alternatively, even if it had, the
recognition of our existence within a four-dimensional space-time
continued to inspire at least a hope that other spatial dimensions
might actually exist. In particular, in both Matheson’s story and
Heinlein’s, and in much other contemporaneous writing—such as Mark
Clifton’s charming short story “Star Bright” (1952) about a
brilliant young girl (evolved in mental powers well beyond those of
her father) who starts studying about mobius strips, Klein bottles,
and tesseracts, and ultimately wills herself to step into a fourth
dimension—the protagonists assume that a fourth dimension actually
exists, and is common knowledge. In both “Little Lost Girl” and
“Star Bright” it was implied that such a fourth dimension was a
genuine concern of the physics of that period. The bewildered
father in “Star Bright” exclaims: “The Moebius Strip, the Klein
Bottle, the unnamed twisted cube—Einsteinian Physics. Yes, it was
possible.” And the physicist Bill in “Little Lost Girl” talks to
the terrified couple about a portal to the fourth dimension as if it
were something that everyone should be familiar with, although he
does add a cautionary note: “I’m not an expert in this. . . . Who
is?”

By the 1960s, however, one finds a growing and
more realistic use of the intimate connection between space and
time exposed by special and general relativity. Perhaps this was
driven in part by the new opportunities for creative expression as
special effects in movies began to blossom in the 1950s and ’60s,
and as television emerged as a key medium. With new graphic
opportunities came new stories that exploited them. I suspect that
one of the strongest driving forces, however, was the new popular
fascination that began, following the 1957 launch of
Sputnik,
with the opportunities it promised of space
travel. Once attention was focused on the apparently infinite
expanse of space, it was natural for scientists, and for writers
and filmmakers, to speculate both about ways that one might traverse
vast distances and about the large-scale nature of space itself.
Stories began to proliferate in which not only does time travel
become possible via transport into a fourth dimension, but also a
curved space, including a curved fourth dimension, can provide
spatial transport to otherwise inaccessibly distant locations. The
famous 1963 French science fiction novel by Pierre Boulle,
Monkey
Planet,
became an
even more famous American film,
Planet of the
Apes,
filmed in 1968. In the cinematic version, an astronaut who
has gone on a long voyage to a distant planet later discovers that
he has merely traveled in time but not in space.

Planet of the Apes
makes a vague inference that somehow the long time lapse on Earth
might be related to the remarkable fact, arising from special
relativity, that objects moving near the speed of light relative to
observers watching them have clocks that appear to be running
slowly relative to the observer’s clocks. This connection between
space and time in relativity implies that in principle, if one was
traveling at speeds close to light speed one could cross the galaxy
in a single human lifetime, even if observers on the ground would
measure the time elapsed for such a voyage to be many thousands of
years. This fact, which (again, in principle) allows human
interstellar travel without exceeding the speed of light, has
become a staple of science fiction writing about space travel over
the years. Indeed, as I have noted in
The
Physics of Star Trek,
even the
Star
Trek
writers took this fact into account, inventing a
Federation “impulse drive” Speed Limit of less than half the speed
of light for extended periods, so that Federation ships would not
get out of time synch with their home bases.