Hiding in the Mirror (21 page)

In a similar vein, if supersymmetry were
somehow a “broken” symmetry, then perhaps the superpartners of
ordinary particles could behave differently than the particles we
know. If, for example, they were much more massive—too massive,
say, to have been created in current particle accelerators—then
that might explain why none of them has yet been discovered. Here
one might wonder what the point is of inventing a new symmetry in
nature and then coming up with a reason why it doesn’t seem to
apply to what we see. If this were all that were involved, the
whole process would resemble intellectual masturbation. (I am
motivated here perhaps by an infamous quip by Richard Feynman that
physics is to mathematics as sex is to masturbation.) It is more
than this, however—at least probably more than this—in part because
the existence of broken supersymmetry might resolve the hierarchy
problem.

One of the many interesting aspects of virtual
particles is that their indirect effects on physically measurable
quantities depend upon the spins of the virtual particles—that is,
whether they are bosons (integral spin) or fermions (half-integral
spin). Given otherwise identical fermions and bosons (i.e., masses,
charges, etc.), the fermions will produce contributions identical
in magnitude to the bosons, but opposite in sign. This means that
in a fully supersymmetric world, virtual particles can yield
zero
quantum mechanical corrections to
physical quantities because for every boson there is a fermion of
identical mass and charge, and the two sets of particles can
produce equal and opposite contributions to all the processes.
Thus, the effects of virtual particles at GUT scales, or at Planck
scales, can disappear, so the low-energy mass scale of the
particles we observe will be protected.

Of course, we do not live in a fully
supersymmetric world. If supersymmetry exists, it is broken, and we
would expect the fermionic partners of bosons, and vice versa, to
have large masses. However, if the masses of the superpartners of
ordinary matter are not too much larger than the masses of the
heaviest particles we have now measured, then it is still possible
for naturalness to be maintained even with a large hierarchy
between the GUT scale and the scale of ordinary particles.

This is because the same virtual particle
cancellation that in the fully supersymmetric world yields zero now
yields an inexact cancellation. The magnitude of its inexactness
will be precisely of the order of the mass difference between
particles and their superpartners. If this mass difference is much
smaller than the GUT scale, and is on the order of the weak scale
masses, say, then virtual particle corrections will not induce
masses for ordinary particles that are much larger than the weak
scale. The hierarchy between the GUT scale and the weak scale then,
while still uncomfortable, would at least be technically
natural.

In the same year that Witten presented his
argument regarding supersymmetry and the hierarchy problem, another
calculation was performed that further bolstered the argument for
both broken supersymmetry and grand unification. Recall that when
one calculated the strengths of the three nongravitational forces
as a function of scale assuming a desert between presently observed
scales and the GUT scale, the strengths of the three forces would
not converge together precisely at a single scale. However, if
instead one assumes that a whole new set of superpartners of
ordinary particles might appear with masses close to the weak
scale, this would change the calculation. One then finds, given the
current best-measured strengths of the three forces, a beautiful
convergence together at a single GUT scale. There are other
indirect arguments that suggest that broken supersymmetry may
actually be a property of nature. For example, it turns out that,
in supersymmetric models, various otherwise apparently puzzling
features of measured elementary particle properties can be
explained. These include most importantly the strange fact that the
so-called top quark (the heaviest known quark) is 175 times heavier
than the proton, and almost 40 times heavier than the next heaviest
quark, the bottom quark, and the fact that a predicted particle
called the Higgs particle, associated with the breaking of the
symmetry between the weak and electromagnetic interactions, has
both thus far escaped detection but yet still could yield the
quantum mechanical corrections necessary in the weak interaction to
preserve agreement between theoretical predictions and observation.
Finally, broken supersymmetry rather naturally predicts the
existence of heavy, stable, weakly interacting particles that might
make up the dark matter inferred to dominate the mass of our galaxy
and all other galaxies.

But even before all of this—indeed, within a
few years of the first GUT proposal and of Wess and Zumino’s
elucidation of the possibility of supersymmetry in our
four-dimensional universe—there was another reason proposed for
considering a supersymmetric universe, but this time not in four
dimensions, but rather in eleven dimensions.

As I keep stressing, the development of GUTs
set the stage for far more ambitious theoretical speculations about
nature. Once scientists were seriously willing to consider scales a
million billion times smaller than current experiments could
directly measure, why not consider scales a
billion
billion times smaller? This scale is the
Planck scale, where as I have mentioned one must come face to face
with the problems of trying to unite gravity and quantum mechanics.
Thus it was that from 1974 onward, a growing legion of physicists
began to turn their attentions to this otherwise esoteric legacy of
Einstein.

Recall that one of the issues that led to the
development of supersymmetry, in the context of dual strings, was
the realization that there appeared to be an unfortunate asymmetry
in nature, wherein forces seem to be transmitted by bosons, while
matter is made up of both bosons and fermions. In the context of
general relativity this asymmetry is exacerbated. Namely, general
relativity relates force (i.e., gravity) as a geometric quantity on
the one hand, to the energy of matter on the other. Thus, force and
matter are integrally related, and one might wonder if apparent
distinctions between them are actually artificial. One step in this
direction was taken in 1978 by Eugene Cremmer, Bernard Julia, and
Joel Scherk, who were following up on work a few years earlier
exploring the possibility of “local supersymmetry,” or, as it has
become known, “supergravity,” as a symmetry of nature. In the case
of supergravity, the duality between bosons and fermions is
extended to the case of the gravitational force. If one tries to
model gravity as a quantum theory like electromagnetism, then the
carrier of the gravitational force should be a massless particle
called the graviton. It is a boson, like the photon, but instead of
having spin one, it has spin two. Indeed, it would be the only
known fundamental particle of spin two in nature, which is why
gravity behaves so differently than the other forces. Now, if
supersymmetry is also a symmetry appropriate to gravity, then there
would be a fermionic partner of the graviton, which is
conventionally called the gravitino. This particle would couple to
all other matter just as the graviton does, except that, being a
fermion, it would be more comparable to the particles that make up
ordinary matter, such as electrons and quarks, rather than the
particles that carry forces, such as photons and gravitons.

Thus, in supergravity, the moment one
introduces a graviton to carry the gravitational force, one also
automatically must include a matter particle whose interactions are
also determined purely by the gravitational force. Cremmer, Julia,
and Scherk realized that this relation between matter particles and
the gravitational force in supergravity is in fact dimension
dependent. In some sense one can think of this as being due to the
fact that in many more dimensions, there are many more axes that a
spinning particle can spin about, so that there are many more
independent states in which a particle with fixed spin can be. In
four dimensions, a particle of spin zero can only be in one state,
a particle of spin one-half can exist in two spin states (which we
often label up and down), and so on. It turns out that in precisely
eleven dimensions only a single type of supergravity theory is
allowed. The mathematical relationships between particles of
different spins that are determined by supersymmetry in this case
is so restrictive that only one combination of particles that can
include the graviton is possible if one is to achieve mathematical
consistency. In eleven dimensions the graviton (which I remind you
is a boson with quantum mechanical spin value equal to 2) has 44
independent states, and the gravitino (a fermion with quantum
mechanical spin value of 3/2) has 128 independent states. Since
supersymmetry implies that if a graviton exists, so must its
fermionic partner. This presents a problem, because the total
number of fermionic states and bosonic states are not the same, as
is also required by supersymmetry. Therefore, there must be
eighty-four other bosonic states that can partner with the
fermions, which one can think of as making up all the allowed
particles of matter in this theory.

Eleven-dimensional supergravity can be thought
of, therefore, as an ultimate theory, in which gravity and
supersymmetry together determine all the allowed particles. Force
and matter are uniquely determined. Of course, once again the
astute reader will note that in our fourdimensional universe there
are many particles which have nongravitational interactions. Well,
it turns out that in ten dimensions—which, as you may recall,
happens to be the critical allowed number for dual strings with
fermions included—gravity and supersymmetry almost completely
constrain everything, but there turns out to be just enough wiggle
room to have additional particles and their superpartners, which in
fact can have Yang–Mills interactions.

By the early 1980s, therefore, there were
numerous independent reasons for serious physicists to actually
consider ten or maybe eleven dimensions as real possibilities in
theories that might unify gravity and the other interactions in
nature. (The independent argument I mentioned earlier—that eleven
dimensions might be necessary for a Kaluza-Klein theory
incorporating all known forces—was actually derived much later.)
The circle was at that point almost complete; just one more
ingredient was needed to close it.

Once again, we return to 1974. In that fateful
year, two pioneers of dual string models, Joel Scherk and John
Schwarz, realized that while these models proved a failure for
describing the strong interaction, they had even greater potential.
Recall that what dual string models did so well was get rid of
pesky apparent infinities in the calculation of processes where
particles of higher and higher spin were involved. Remember also
that one of the negative features of the dual string models,
besides producing incorrect predictions for scattering rates, was
that they predicted a number of massless particles that had not yet
been seen—in particular, a massless spin two particle.

Scherk and Schwarz argued that dual strings
still might be the correct solution, but that perhaps they had been
looking at the wrong problem: Maybe the apparently beautiful
feature of dual strings could be combined with one of their
negative features, not to describe a theory of the strong
interaction, but instead to unify gravity and quantum
mechanics!

After all, one of the reasons that gravity
confounded all attempts to quantize it was that it involved a
series of infinities in calculations because of the exchange of a
massless spin two particle, the graviton. Here, string theory not
only provided a possible way to remove such infinities, but also
automatically
predicted the existence of a
particle with precisely the properties of a graviton. Indeed, as
Richard Feynman had first demonstrated, any relativistic theory
involving the exchange of a massless spin two particle could be
shown to reproduce precisely Einstein’s equations of general
relativity.

Moreover, if dual strings were instead to be
viewed as models of quantum gravity, then one more of those
notorious warts in the theory could be turned into a beauty mark.
Remember that dual strings require higher dimensions to make
sense—either twenty-six or ten, depending upon the type of model.
As applied to a theory of the strong interaction, this strained the
bounds of credibility. However, as we have seen, ever since the
time of Kaluza and Klein, efforts to unify gravity and other forces
had focused on the possible existence of extra dimensions. In this
sense, Scherk and Schwarz could claim they were following a noble
tradition, rather than heading down a blind alley.

So it was that by 1981 all the independent
ingredients were now in the air: GUTs, strings, supersymmetry, and
a newfound desire to unify
all
the forces
in nature. It would take some years, and a few more miracles,
before many people other than Scherk (who sadly died in 1980),
Schwarz, and a few other diehards would join in the harvest, but
the seeds had been planted. A growing group of physicists began to
seriously believe that our four-dimensional universe really might
be just the tip of a cosmic iceberg, with six or seven hidden
dimensions lying, literally, just beneath the surface. The new love
affair with extra dimensions had begun.

If it is possible that
there could be regions with other dimensions, it is
very likely that God has somewhere brought them into
being.

—Immanuel Kant

T
he year in which many
in the particle physics community first experienced a “conversion”
was a full decade after the apparent 1974 demise of dual
strings—and, perhaps appropriately, a century after the publication
of Edwin Abbott’s
Flatland
. The twentieth
century had brought more change in our technology and our
fundamental understanding of the universe than anyone could have
imagined in 1884. Yet at the beginning of the twenty-first century
our fascination with hidden extra dimensions has, if anything,
become even stronger, in large part because of the remarkable
resurrection of an idea left for dead.

The road from Yang-Mills theories in 1954 to
the proposal that strings might be a theory of gravity in the 1970s
to the rise of supersymmetry in the early 1980s was, as I have
described, a long and winding one. Most of all, it was not a road
from which the destination was clearly visible on the horizon. Many
different aspects of the problem were being explored independently
by separate individuals and groups, and it was certainly not at all
obvious in advance, in spite of the natural way in which gravity
appeared to be embeddable in string theory, that much would amount
from this effort.

First, as the saying goes, “Once bitten, twice
shy.” Many physicists had already seen how dual strings, the
dominant fad of the late 1960s because of their great potential to
resolve apparent mathematical inconsistencies of the strong
interaction, had in fact been almost completely off the mark. Given
this, it was understandable that they would be hesitant to embrace
the theory again, even when applied in a different context. In the
second place, dual strings still suffered from embarrassing
problems in 1974. While the theory might predict a graviton, it
still also appeared to predict a tachyon, for example. And finally,
no one had yet shown that it would produce fully consistent quantum
mechanical predictions for either gravity or other forces in
nature. And of course, there was still the question of those pesky
extra dimensions.

What is remarkable is that, as we have seen,
piece by piece, different components seemed to fall into place to
make the theory less unattractive and, at the same time, less
removed from the rest of particle physics. Supersymmetry seemed to
be needed once one put fermions on strings. GUTs suggested that the
goal of unification itself was worth exploring, and then
supersymmetry again seemed to offer the most attractive, and
viable, scenarios for GUTs. Finally, applying supersymmetry to
gravity seemed to once again suggest that extra dimensions might be
called for. Still, even with this growing level of attraction,
string theories needed serious work to resolve their outstanding
issues, which required the dedicated efforts of a small cadre of
individuals, two of whom we have already encountered: Joel Scherk
and John Schwarz.

John Schwarz appeared twice in the previous
chapter: once associated with the effort to put fermion modes on
strings, and once with the initial proposal (along with Scherk),
suggesting that strings might yield a quantum theory of gravity.
But his role will be even more significant in what follows. For a
full decade during which much of the rest of the community was
focused elsewhere, Schwarz and various collaborators—notably Joel
Scherk, who tragically died before string theories truly achieved
wide recognition—continued to do work on the theory, convinced that
it must have something to do with nature. I have known John Schwarz
for over twenty years, and I am hard pressed to think of a time
when he wasn’t smiling, even when I was disagreeing with him. An
indefatigably cheerful individual, John always seems to be
optimistic. I believe, in fact, that his temperament has been an
essential part of his ultimate success. Were this not the case, it
is hard to imagine that he would have kept plugging away on what
was apparently such a long shot. From 1974 until 1984 he and other
string devotees labored in almost complete isolation on a model in
which, frankly, almost no one other than they was interested.
Without unflagging optimism they might have given up.

In any case, in 1977 string models received a
big boost when Ferdinando Gliozzi, Scherk, and David Olive
discovered a way to remove the tachyon from string theories in ten
dimensions. Their solution appeared to involve supersymmetry as a
symmetry not just on the string itself, as it had originally in
fact been discovered, but throughout the full tendimensional
space-time in which the strings moved. All of the particle states
on the string involved equal numbers of fermions and bosons, a
hallmark of space-time supersymmetry. Interestingly, this finding
appeared well before four-dimensional supersymmetric GUT models
were explored, four years later.

In 1981 John Schwarz and his collaborator
Michael Green, another well-established string theorist, actually
proved that Gliozzi and colleagues’ construction indeed involved
supersymmetry as a symmetry on the full ten-dimensional space, and
not just on the string itself. String theory had officially become
superstring theory.

The significance of this proof cannot be
overemphasized, because with the unphysical tachyon state done away
with, and with full supersymmetry in ten dimensions, a host of new
and elegant mathematical techniques could then be applied to the
problem of determining if the theory was fully consistent as a
possible quantum theory of gravity. Within two years Green and
Schwarz had their answer, and it rocked the physics world. In 1984
they submitted a paper to the European journal
Physics Letters
in which they demonstrated that
superstrings in ten dimensions could yield fermions, bosons,
Yang-Mills fields, and gravitons in a way in which all nasty
infinities appeared to be completely absent. It was a fully finite
quantum theory that in principle had the potential to be, as it
quickly became known, a Theory of Everything—the holy grail of
physics ever since Einstein had first set out to unify gravity with
the other forces in nature. Suddenly all the diverse pieces that
had occupied theorists over the past decade seemed to come together
in a most remarkable way. Perhaps the most unexpected result was
that this theory appeared to not only produce finite results instead
of infinite ones when dealing with what seemed otherwise intractable
physical processes, but if the ten-dimensional superstring had
attached to it a sufficiently large set of Yang-Mills fields, then it
turned out that it would be possible to break left–right symmetry.
As you will recall, this is required if the theory is ultimately to
incorporate in four dimensions the measured weak interaction—which
has no such symmetry—without producing a mathematical horror called
an
anomaly
. The response to these dramatic
results from the particle physics community was thunderous. The
first result—the lack of infinities—was perhaps not so surprising.
After all, strings had tamed infinities when they were proposed as
models of the strong interaction. Recall that the mechanism of
producing finite results was apparently based on a mathematical
trick: An infinite sum of terms can add up to a finite number even if
the individual terms appear to increase indefinitely. In the case of
strings, because an infinite set of states exist with every higher
energy, as vibrations of a string become more pronounced, the
possibility of infinite sums contributing to any physical process is
immediate. What was far less obvious was that the physical
conditions associated with the quantum mechanics of strings would
allow the infinite sums to, in fact, converge to a finite value. In
retrospect, there is a more concrete way of understanding this
particular string miracle. Remember that quantum mechanics and
relativity tell us that forces between particles occur via the
exchange of virtual particles—those objects that can appear
momentarily and then disappear so quickly that they cannot be
directly observed. In this case, a virtual particle can be emitted
by one object and absorbed by the other on an exceedingly small
timescale.

Now, the troublesome mathematical infinities
arise when virtual particles of arbitrarily high energy are
exchanged. Because the uncertainty principle tells us that if
virtual particles carry a great deal of energy, they can exist for
only a very short time, so the particles that can emit and absorb
them, respectively, must be very, very close together. High-energy
processes such as this are therefore really probing the nature of
very short distance scales. Strings solve this problem because on
very short distance scales what we would otherwise view as
elementary particles could instead be seen as excitations of
strings. Below some distance scale, then, elementary particles must
be treated as spread-out vibrations of a string. Thus, by changing
the rules at short distances, strings provide a new limit (or
“cutoff,” as it is referred to by physicists), thus taming the
otherwise potentially nasty short-distance, or high-energy,
behavior of virtual processes involving point particles. This kind
of smoothing mechanism actually has another precedent—in this case
arising not from earlier considerations of the strong interaction,
but rather from the weak interaction. Before the weak and
electromagnetic interactions were unified in a Yang-Mills–type
theory, Enrico Fermi developed an approximate theory that could be
used for calculating weak processes. While this theory was very
good at low energies, it was well known that it would eventually
produce nonsensical results if the energies involved got too high.
In the Fermi theory, weak interactions resulted from four different
particles interacting at a single point (for example, when a
neutron might decay into a proton, an electron, and an
antineutrino). In the refined electroweak theory, however, it was
seen that, in fact, what appeared at large distances to be four
particles interacting at a single point was really two particles
emitting a virtual particle that traveled a very short distance
before either being absorbed by or producing via its decay, the
other two particles. The short distance scale—at which this new
picture becomes manifest—provided a short distance cutoff in
calculations. Namely, the calculations of the old theory were only
valid if one considered processes on scales larger than this short
distance-limiting scale. On smaller scales new rules would apply,
which, in fact, turned the previously nonsensical results into
finite, sensible predictions that could be compared with
experiments. This new short-distance scale where the rules change,
called the “weak scale,” turns out to be precisely the scale below
which the particles that convey the weak force behave differently
than photons, which convey the electromagnetic force. On smaller
scales, the two forces would appear to behave quite similarly.

String theory had the potential to solve
similar nonsensical predictions of the naive quantum version of
general relativity. In this theory, recall, the gravitational force
occurs because of the exchange of virtual particles, called
gravitons. Because of the complicated structure of general
relativity, it turns out that there are an infinite tower of
possible interactions of gravitons with each other, so that one can
find interactions of three, four, five, or more gravitons at a single
point.

It turns out that, in a way similar to that in
which the interactions of four particles at a single point in the
weak interaction produced nonsensical results, these many-particle
interactions in general relativity ultimately produce a host of
infinities if one allows the energies involved to become arbitrarily
large.

But string theory offered a new opportunity to
once again change the rules at small distances. If the particle we
call the graviton is, at sufficiently small scales, resolved instead
to be a vibrating string, then what is allowed at small scales will
change. It turns out that, for technical reasons, a graviton is
required to be made up of a closed string loop rather than a string
segment whose two ends are not connected. In this case, one can
redraw what would otherwise appear at large distances to be an
interaction involving four gravitons at a single point. The picture
becomes more complicated than simply having two graviton particles
exchange some other particle with two other gravitons located some
distance away, as in the weak case. Rather, one imagines a more
complex process in which the vibrating string loops that masquerade
as gravitons at large distances bifurcate and exchange other
vibrating loops as shown in the second diagram below, which looks
like two pairs of trousers sewed together. But while this is more
complicated to draw, the effect is the same: The seemingly
pointlike interaction of gravitons is instead spread out over some
region of space, providing a new lower-scale cutoff that yields
results that are finite for such physical processes, even as the
energies of the particles involved become very large.