Download by Margaret L. Silbar

by Margaret L. Silbar
24 MOSAIC Volume 17 Number 4 Winter 1986/7
A theory of point-sized
oscillating superstrings is
giwing quark theory and the
onion (or wheel-in-a-wheel)
model of the physical
universe a run for their
money. The race is not
necessarily heading toward
achieving a model that has
the largest number either of
dimensions or of supposedly
fundamental particles.
Physicists have recently looked
again at their favorite theories of
the universe's substance and
structure, and many of them now say
that the basic building blocks may not be
pointlike particles with no size at all. Instead, the blocks may be tiny, oscillating, extended, one-dimensional strings.
N a m e d superstrings because they
embody a supersymmetry that relates
otherwise disparate subatomic particles
to each other, these strings are so short
that they cannot ordinarily be distinguished from points. There is nonetheless a subtle distinction: Just as a harpsichord string vibrates in a series of harmonics, so a superstring can be excited
from one vibrating state into another.
The identity of the observed particle
that a string most resembles at any moment depends on the mode in which
the string is oscillating.
"The theory of superstrings can encompass all the known forces in na-
splitting of strings is effectively the
same as the exchange of gauge particles,
which, in conventional theory, are the
carriers of the fundamental forces. Says
another of string theory's founders,
Michael B. Green of Queen Mary College in London, England, "It is an unsolved mystery why gauge particles
emerge from superstring theories in
such a natural way/
Superstring theories do not involve
arbitrary parameters, as do the pointparticle field theories physicists are
accustomed to using. The standard
point-particle description of elementary
particles contains some 20 different parameters that must be arduously
adjusted to make the theory work. Superstrings also generally avoid such absurdities as the infinite answers that
plague the older theories. Indeed, the
superstring theories may well signal the
end of the picture of the universe as a
never-ending onion—the chain of rea-
He thought he saw a Banker's Clerk
Descending from a bus.
He looked again, and found it was
A Hippopotamus.
—Lewis Carroll, Sylvie and Bruno
ture" says one of the founders of string
theory, John H. Schwarz of the California Institute of Technology. Indeed, the
theory may finally bring gravity, the
weakest and longest-known of these
forces, or interactions, into the fold
along with the other fundamental interactions—electromagnetism and the
short-ranged weak and strong forces
that operate within the atomic nucleus.
"There is really only one kind of Interaction," says Schwarz, "and it occurs
when two strings touch and join to form
one string, or when a single string splits
into t w o /
At low energies, this joining and
soning that says that molecules are
made of atoms that are in turn made of
electrons and nuclei that are themselves
made of protons and neutrons consisting of quarks and gluons, and so on.
Even at the quark and gluon stage,
the most basic level that Is so far known,
it has been clear for some time that a
description of nature that needs some
three dozen or more supposedly elementary particles and associated arbitrary parameters must surely not be
very fundamental. The equations that
describe a proton in terms of quarks and
gluons are, moreover, even more difficult to solve in practice than those de-
MOSAIC Volume 17 Number 4 Winter 1986/7 25
scribing a nucleus in terms of protons
and neutrons. Thus, many physicists
welcome the possibilities being opened
up by the theory of superstrings. They
hope that understanding of root phenomena and concepts will grow as dramatically as it did at the birth of quantum mechanics in the 1920s. That was
when the paradigm of light as wave motion was replaced by the model of light
as photons having both wave and particle properties.
No one can yet make detailed predictions with superstring theory. Nor does
anyone yet grasp an underlying principle that explains why strings make
sense as nature's basic building blocks.
For example, superstring theory is not
yet based on a geometrical principle. As
Green says, 'This is not very satisfying
for a theory which includes gravity."
The accepted classical theory of gravity,
Einstein's 70-year-old general theory of
relativity, describes gravity as a distortion of the geometry of space-time due
to the presence of matter. In contrast,
the other three forces act upon matter in
space-time, so that for now at least,
there is a big disparity between these
two kinds of interactions.
"Providing string theories with a true
geometrical foundation may require a
new kind of geometry not yet invented," says Schwarz. That may turn out to
be "some fancy generalization of
Riemannian geometry, the geometry
which lies at the heart of general relativity." Just as general relativity theory incorporates Newton's force law for gravity as a special, limiting case valid for a
space-time with small curvature, the superstring theories (and their new presumed geometry) must reduce to the
concepts and framework of general relativity at low energies or long distances.
the forces binding protons and neutrons. Hadrons also include a large
number of short-lived, unstable particles, the resonances created in particle
accelerators.
The roots of the idea
String theories have had a bizarre history. They were originally proposed in
the late 1960s for the modest purpose of
bringing some order into the taxonomy
of those elementary particles that participate in the strong interactions. Known
as hadrons, this group of particles includes the familiar proton and neutron
and also the pion, a short-lived particle
once thought to be the main source of
Silbar, who writes frequently on physics, was
the author of "The Pursuit of Parallelism/'
Mosaic Volume 16 Number 3, a special issue on advanced scientific computing.
28 MOSAIC Volume 17 Number 4 Winter 1986/7
proponents of this theory hoped to finesse some of the lingering problems of
quantum field theory. The backers of
this theory were among the first to embrace an interesting generalization of
the concept of angular momentum.
Just as angular momentum resides in
the spinning sun and the planets that
revolve around it, so is it also a characteristic of an elementary particle. Its
value can be measured and is expressed
as a quantum number. It was the Italian
physicist Tullio Regge who showed in
1959 that each hadron, or strongly interacting particle, can be considered as a
member of a family characterized as
moving along a certain trajectory in a
complex, angular m o m e n t u m plane.
Various points on the diagonal represent the proton and its brethren in certain angular m o m e n t u m and energy
states. Particles on a particular trajectory
are alike in all respects other than energy and angular momentum; the larger
their spin value, the larger their massenergy will be.
Democracy among particles
Not long after their introduction,
string theories were pushed from center
stage by the notion that hadrons are
made of quarks. The quark model, in its
way, was a more clever way of grouping
similar particles into families. Together
with the development of a concomitant
field theory of quarks, quantum chromodynamics, it resulted in a loss of interest in superstrings. Schwarz a n d
Green remained loyal to the string idea,
however, and they worked for many
years outside of the mainstream of modern physics, independently at first, joining forces in 1979. Their faith seems to
have been vindicated by several recent
and surprising developments, including
discovery of the way the superstring
theories conspire to preserve some experimentally sacrosanct conservation
laws. It is this conformity to established
law that causes Edward Witten of
Princeton University in New Jersey, initially a skeptic, to say now that he is
convinced that "these are not just accidents" and that there really is "a profound theory lurking there."
The grandparent of today's superstring theory is S-Matrix theory. (The S
stands for scattering.) In the early 1960s,
The concept of a Regge trajectory suggested the possibility that any given elementary particle could be understood as
a dynamical result of the interactions of
all the others. Perhaps all particles He on
indefinitely rising trajectories, and the
number of particles within a given family is infinite. Each member of that family
would be equally fundamental. Among
the cognoscenti, this picture became
known as nuclear democracy. In such a
case, there is nothing special about the
proton except that it simply happens to
be the state with the lowest mass and
spin angular momentum on a particular
Regge trajectory.
With the discovery in 1968 of a clever
mathematical model by Gabriele
Venezlano, now at CERN, the European
Center for Nuclear Research, the concept of Regge trajectories evolved into
what are called dual resonance models.
These models were originally regarded
as simplified versions of S-Matrix theory; they gave simple, approximate formulas for describing the way one hadron might scatter off another. The models made heavy use of experimental data
to fix the various parameters. They
nonetheless contained a precise mathematical statement about the structure of
the scattering. The statement, known as
duality, is similar to the hypothesis that
the existence of any one particle can be
derived-—bootstrapped—from the interactions of all the others.
While these dual resonance models
were being developed, recalls John
Schwarz,»~"it was certainly known they
had something to do with relativistic
quantum strings. However, it was unclear to many of us whether this was
just a vague analogy or.a precise fact."
Subsequently, it was shown that the
quantum-mechanical states of the relativistic string are equivalent to the states
of the dual resonance model, and they
happen to fall on Regge.trajectories.
This work was pioneered by Leonard
Susskind of Stanford University and
Holger-B. Nielsen of the Niels Bohr Institute in Copenhagen then brought to
its. accepted form today by Yoichiro
Nambu of the University of Chicago
and the late T. Goto.
Some theoretical problems
By 1972, therefore, the physical picture resulting from the relativistic string
idea was that a pion, for example, was a
string with a quark attached to one end
and an antiquark to the other. Because
these strings were supposed to account
for the hadrons, they were about as long
as the proton or pion is wide, 10^13:cen-
timeters. Such a string is oscillating like
a violin string, transverse to its linear
dimension. If it is excited—by being
stretched, for example—the energized
string represents the next state on the
Regge trajectory of the pion, a highermass particle known as the rho meson.
This was an attractive picture. The dual
resonance models predicted that hadrons lie on nearly linear and parallel
Regge trajectories, and experiments verified this picture.
There were, .however, some problems. To mesh with quantum mechanics
and with special relativity, the spacetime in which the relativistic string
moves around must have an uncommon
number of dimensions-—for example,
10, or perhaps as many as 26. Otherwise, the theory predicts that some
processes would occur with negative
probabilities. This, says John Schwarz,
is "patent nonsense/' At first, physicists
were unwilling to take seriously the
idea of such a. large number of spatial
dimensions. Four dimensions—three
for space and one for time—are enough
to digest sometimes.
The simplest string models, moreover, failed to account for the basic particles, the quarks. To do this required ex-
panding the theories to incorporate the
important concept of supersymmetry,
which relates bosons (such as the gauge
particles that carry the. forces) to fermions (such as the quarks). This new feature is reflected i n the nomenclature by
the prefix super in many nouns, including the super in superstri-ngs.
Supersymmetry has a lunny status in
physics, says Michael Green. 'There's
not one shred of experimental evidence
for It, and yet It's been almost universally accepted as.something one would like
to have. Physicists who want to fit experimental data—in particular the masses of the particles one sees in nature—
have to explain why these are so Incredibly small [compared to the gravitational
mass scale]. Invoking supersymmetry
has the virtue that it makes the masses
essentially zero."
Yet another problem plaguing the
early string theorists was their prediction that one of the states in the models
is a massless particle with two units of
spin angular momentum. Hadrons are.
by no means massless! After some time,
people finally began to ask what John
Schwarz has called-"the obvious question": To what extent does this massless
particle behave like a graviton? The
MOSAIC Volume 17 Number 4 Winter 1986/7 27
graviton is the postulated spin-2 gauge
particle presumed to be exchanged between bodies bound to each other gravitationally, just as spin-1 photons hold
electrons in place around the nucleus.
String theory had inadvertently done
what all earlier grand unification,
schemes had not: It had produced the
quantum of gravity.
In 1974, Schwarz and the late Joel
Sherk proposed that strings be used for
unification of gravity and all the other
forces rather than as a theory of hadrons. As Michael Green points out, the
inconsistencies lurking in such a theory
seemed overwhelming.
In order to-suggest that a dual string
. model could also be a theory of gravity
a drastic change in viewpoint was needed. 'It required strings to.be 20 orders of
magnitude shorter than we had become
accustomed to," says Schwarz. The
length scale thus had to be changed to
units that could conceivably explain the
quantum effects of gravity It is believed
that only at very short distances (which
is, by the quantum-mechanical uncertainty principle, equivalent to very high
energies) would gravitons interact appreciably with matter. The energy under
consideration is that of the Planck mass,
some 1019 times the mass of the proton.
The argument that quantum effects
must become important In gravity at energies near, the Planck mass is as follows: Coulomb's law for the electric
force and Newton's law for the gravitational force both say the respective
forces are Inversely proportional to the
square of the distance between the interacting particles. The strength of the
electric interaction between a proton
and an electron is very precisely known
experimentally; It is alpha, a dimensionless constant whose value is 1/137. This
constant is small enough that a classical
theory of electromagnetism can and
does make good sense most of the time.
There is a corresponding dimensionless coupling constant in gravity that
can be formed from Newton's gravitational constant and the masses of the
two attracting bodies. For two protons,
for example, this coupling constant is a
very small number indeed, and a classical approach to gravity is even more valid. However, when the dimensionless
gravitational constant becomes a number as large as one,, classical theory no
longer applies. This happens w h e n the
energies of the two interacting particles
are equal to the Planck mass.
28 MOSAIC Volume 17 Number 4 Winter 1986/7
The very large Planck mass gives a
correspondingly small Planck length,
which is 10~33 centimeters. To probe
smaller and smaller distances, physicists must use more and more energetic
particles; thus, to probe distances as
small as the Planck length (with current
technology) would take an accelerator
the size of the galaxy Because of such
untestability, not everyone, appreciated
the shrinking of the relativis.tic string
from something the size of a proton to
something the size of the Planck length.
Some people have suggested that as
long as superstring theories can be used
to make predictions of new phenomena
only at the Planck scale, one might regard them and the mathematics they entail as simply recreational. Schwarz
feels, however, that many physicists
who work on superstring theory, expect
that predictions at measurable energies
will eventually become possible.
Worlds of many dimensions
With the possibility of dealing with
gravity in a quantum theory, however,
physicists began to take, the idea of
those extra spatial dimensions more seriously. The idea is actually quite an old
one. Georg Friedrich Bernhard Riemann,
in his famed 1854 paper on the founda-
Harvey (left) and Gross. With Martinec and
Rohm, they found a finite superstring theory.
tions of geometry, had suggested that
familiar three-dimensional space could
be embedded in another multidimensional space.
This idea was exploited in the 1920s
by Theodor Kaluza and Oskar Klein,
who separately proposed to geometrize
electromagnetism and gravity in a fivedimensional "cylinder world." The fifth
dimension was to be experimentally invisible; it was picturesquely conjectured
that light is a manifestation of this fifth
dimension. Even though several physicists worked on this approach in the
succeeding decades, they never came
up with a realistic unification of'gravity
and electromagnetism. Nor were they
able to address- the philosophical problem, first pointed .out by Einstein: Why
is the space-time continuum apparently
restricted to four dimensions?
Today, however, superstrings have
brought these -extra dimensions back
into vogue. The unseen dimensions
must "compactify" to a very small size,
only to "open up" if probed at energies
on the order of the Planck mass, energies that presumably existed only in the
first instants after the big bang. If each
additional direction is construed as being a tiny little rolled-up cylinder, a microbicyclist pedaling down a s u p e r string and trying to turn off in such a
new direction would return to the starting point so quickly he would fail to notice any deviation from the main course.
In such a picture, all space dimensions might have originally started out
in a symmetrical way, after which the
universe evolved into, the present asymmetrical situation. That is, three spatial
dimensions expanded to become those
apparent today and all the others contracted into those about which one can,
only conjecture. As Michael Green says,
however, "No one has proved that six of
the ten dimensions should curl up."
In the context of any point-particle
field theory, says Green, the extra dimensions only make the problems of
quantum infinities worse than they are.
in four dimensions. Infinities are ascribed to quantum fluctuations, and the
more dimensions there are, the more
complicated the fluctuations can be.
Remarkably, even though superstring
theories are 10-dimensional, infinities
do not seem to arise. This is one of the
reasons that superstring theories are
now capturing the imagination of theoretical physicists. These theories are
MOSAIC Volume 17 Number 4 Winter 1986/7 29
found to be, in the jargon of physics, finite "at the one-loop level" and may
continue to be finite at even more complex levels of analysis. (See "One-loop
finiteness,/ accompanying this article.)
A marriage of strings
Internal symmetry groups define
families of particles, the individuals of
which are differentiated by quantum
number labels. For example, electric
charge is a label that differentiates the
proton from the neutron. The proton
and neutron form a particular family of
a fairly small internal symmetry group
called su(2). In contrast, current superstring theories involve enormous
groups with members numbering in the
hundreds—496, to be precise.
The number of consistent superstring
theories was at first limited to just one
Internal symmetry group. Discovered
by Michael Green and John Schwarz in
1984, this, superstring theory was based
on the symmetry group so(32). In this
theory, the gauge particles mediating
the strong and electroweak force were
associated with open strings, the graviton with closed strings.
Later in 1984 came the birth of the heterotic string. This is the invention of
four physicists at Princeton University,
now often referred to as the Princeton
String Quartet. David J. Gross, Jeffrey
A. Harvey, Emil Martinec, and Ryan
Rohm. The heterotic string is always a
closed loop; it has no free ends. Previously quantum numbers were associated with the open string's ends; now
they could now be thought of as
"spread out" along It.
As Harvey explains, he and his colleagues "married the 26-dimensional
string with the 10-dImensional string,
Desperately- seeking superstrings?
by Paul Ginsparg and Sheldon Glashow
Why is the smart money all tied u p in strings? Why Is so
much theoretical capital expended upon, the properties of
supersymmetric systems of quantum strings propagating
in ten-dimensional space-time? The good news is that superstring theory may-have the right stuff to explain the
"low-energy phenomena" of high-energy physics and
gravity as well." In" the context, of possible 'quantum theories of gravity, each of the few currently known superstring theories may even be u n i q u e , finite, and selfconsistent. In principle, a superstring theory ordains what
particles exist and what properties they have, using no
arbitrary or adjustable parameters." The bad news is that
years of Intense effort by dozens of the best and .the
brightest have yielded not one verifiable prediction; nor
should any soon be expected. Called "the new physics" by
its promoters, it is not even known to encompass the old
and established standard model.
In lieu of the traditional confrontation between theory
and experiment, superstring theorists pursue an inner
harmony where elegance, uniqueness, and beauty define
truth. The theory depends for its existence upon magical
coincidences, miraculous cancellations, and relations
among seemingly unrelated (and possibly undiscovered)
fields of mathematics. Are these properties reasons to accept the reality of superstrings? Do mathematics and aesthetics supplant and transcend mere experiment? Will the
mundane phenomenological problems that we know as
physics simply come out in the wash in some distant tomorrow? Is further experimental endeavor not only difficult and expensive b u t u n n e c e s s a r y and irrelevant?
Contemplation of superstrings may evolve into an activity
as remote from conventional particle physics as particle
physics Is from chemistry, an activity to be conducted at
schools of divinity by future equivalents of medieval theo-
30 MOSAIC Volume 17 Number 4 Winter 1986/7
logians. For the first time since the Dark Ages, we can see
how our noble search may end, with faith replacing science once again. Superstring sentiments eerily recall "arguments from design" for the existence of a supreme
being. Was it only in jest that a leading string theorist
suggested that "superstrings may prove as successful as
God, Who has after-all lasted for'millennia and is still
invoked in some quarters as a Theory of Nature"?
The trouble began.with quantum chromodynamics, an
Integral part of the standard model that underlies the
quark structure of nucleons and the nuclear force itself.
QCD is not merely a theory but, within a certain context,
the theory of the strong force: It offers a complete' description of nuclear and particle physics at accessible energies.
While most questions are computationally too difficult for
QCD to answer fully, it has had many qualitative (and a few
quantitative) confirmations. That QCD is almost certainly
"correct" suggests and affirms the belief that elegance and
uniqueness—in this case, reinforced by experiment—are
criteria for truth.
No observed phenomenon disagrees with or demands
structure beyond the standard model. No internal contradictions and few loose ends remain, but there are some
vexing puzzles: Why Is the gauge group what it Is, and
what provides the mechanism for its breakdown? Why are
there three families of fundamental fermions w h e n one
would seem to suffice? Aren't 17 basic particles and 17
tunable parameters too many? What about a q u a n t u m
theory of gravity? Quantum field theory doesn't address
these questions, and one can understand its greatest past
triumphs without necessarily regarding it as fundamental. Field theory is clearly not the end of the story, so
something smaller and better is needed: Enter the superstring.
getting something new in the process/'
The "something new" is the internal
symmetry group E8 x E8, pronounced "Eeight cross E-eight." It also gives rise to a
finite superstring theory, the theory
now favored by almost everyone.
In E8 x E8 symmetry, there are 16 quantum number labels that can be used to
distinguish and label all known particles. Moreover, the internal symmetry
group allows one to make continuous
transformations from one particle to another among all these different particles
within a family without any of the essential physics changing. "Different colored quarks can be 'rotated' into each
other, quarks into electrons, electrons
into neutrinos, observed particles into
as-yet-unobserved particles," explains
Jeffrey Harvey
Why is it that the E8 x E8 heterotic
string theory is now preferred to the SO
The trouble is that most of superstring physics lies up at
the Planck mass—about 1019 Gev—and it is a long and
treacherous road down to where we can see the light of
day. A naive comparison of length scales suggests that to
calculate the electron mass from superstrings would be a
trillion times more difficult than to explain human behavior in terms of atomic physics. Superstring theory, unless it allows an approximation scheme for yielding useful
and testable physical information, might be the sort of
thing that Wolfgang Pauli would have said is "not even
wrong." It would continue to attract newcomers to the
field simply because it is the only obvious alternative to
explaining why certain detectors light up like video games
near the end of every funding cycle.
In the old days we moved up in energy step by step,
seeing smaller and smaller structures. Observations led to
theories or models that suggested further experiments.
The going is getting rougher: Colliders are inordinately
expensive, detectors have grown immense, and interesting collisions are rare. Not even a politically popular "Superstring Detection Initiative" with a catchy name like
"String Wars" could get us to energies where superstrings
are relevant. We are stuck with a gap of 16 orders of magnitude between theoretical strings and observable particles, unbridgeable by any currently envisioned experiment. Conventional grand unified theories, which also
depend on a remote fundamental energy scale (albeit one
extrapolated upward from known phenomena rather than
downward from abstract principle), retain the grand virtue that at least in their simplest form, they were predictive enough to be excluded—by our failure to observe
proton decay.
How tempting is the top-down approach! How satisfying and economical to explain everything in one bold
stroke of our aesthetic, mathematical, or intuitive sensibilities, thus displaying the power of positive thinking
without requiring tedious experimentation! But a priori
arguments have deluded us from ancient Greece on.
Without benefit of the experimental provocation that led to
Maxwell's equations and, inevitably, to the special theory
of relativity, great philosophers pondering for millennia
failed even to suspect the basic kinematical structure of
space-time. Pure thought could not anticipate the quantum. And even had Albert Einstein succeeded in the quest
that consumed the latter half of his life, somehow finding
a framework for unifying electromagnetism and gravity,
we would by now have discarded his theory in the light of
experimental data to which he had no access. He had to
fail, simply because he didn't know enough physics. Today we can't exclude the possibility that micro-unicorns
might be thriving at a length scale of 10 "18 centimeter.
Einstein's path, the search for unification now, is likely to
remain fruitless.
Having a potentially plausible candidate "theory of
everything" does dramatically alter the situation. But we
who are haunted by the lingering suspicion that superstrings, despite all the hoopla, may be correct are likely to
remain haunted for the foreseeable future. Only a continued influx of experimental ideas and data can allow the
paths from top and bottom to meet. The theory of everything may come in its time, but not until we are certain
that Nature has exhausted her bag of performable
tricks. •
Glashow is Higgins Professor of Physics and Ginsparg Associate
Professor and Particle Physicist at Harvard University.
Reprinted from the May 1986 Physics Today.
MOSAIC Volume 17 Number 4 Winter 1986/7 31
(32) theory? According to Harvey,
"None of the first string theories had a
very nice way of incorporating any fundamental gauge interactions other than
gravity/' With the advent of the heterotic
string, all interactions correspond geometrically to the splitting and joining of
closed strings in ten space-time dimensions. "Closed strings/' says Schwarz,
"are the most interesting case. It is precisely closed strings that involve gravity,
and that's where you get into geometry."
Worlds without anomalies
The two symmetry groups SO(32) and
E8 x E8 have an additional very important
property that led many people such as
Princeton's Edward Witten to jump on
the superstring bandwagon. This is the
so-called cancellation of anomalies. Although many scientific p a p e r s have
been written about anomalies, they are
generally ill-understood. "My only interest in anomalies was in how to get rid
of them," Schwarz says, "since theories
with anomalies are inconsistent."
"In Newton's day, there were no such
things as anomalies," says Witten, "but
we are past the point where we can rely
on ordinary intuition. As physics progresses and explains more and more
from simpler and simpler principles,
logical consistency is harder to obey and
also more important."
One-loop finiteness
In quantum field theory, an elementary particle is allowed a specific mass,
spin, and charge, as well as certain other more abstruse, though measurable,
characteristics that are described by quantum numbers. The ways in which
particles can interact with one another are also prescribed by the theory. From
this, one can in principle deduce what will happen if one particle is fired at
and bounces off another with some given velocity.
In the case of quantum electrodynamics, it was not clear initially that even
the simplest interaction, such as that between two electrons, could be calculated consistently. The simplest thing one can imagine happening in such a
case is that one electron emits a photon, the particle of light, and the other
electron absorbs that photon. The photon is virtual, a consequence of the
uncertainty principle. Its emission (and absorption) violate the law of energy
conservation, but the violation lasts such a short time that it does not count.
While this contribution to the scattering interaction is simple, an infinite
number of more complicated possibilities exist. Then how can those possibilities be calculated?
Richard Feynman of the California Institute of Technology developed a
special kind of diagrammatic method, now named after him, to do just this.
In this way, contributions are systematically tabulated. Not all Feynman diagrams are as simple as the one involving the exchange of a photon. Some
include loops, as in the case where a second virtual photon is emitted and
later reabsorbed by the same electron. Loop diagrams—or, rather, the physical processes behind them—generally lead to infinities in point-particle quantum field theories. In quantum electrodynamics, these infinities can be swept
under the rug by a set of tricks called renormalization, but other quantum
field theories do not enjoy this luxury.
Calculating what can happen for complicated Feynman diagrams is a tedious process. Theorists have done all the calculations for the superstring models for diagrams involving one—and only one—loop. They have found, for
the theories based on the groups so(32) and E8 x EH, no infinities. Hard work
lies ahead to calculate what happens in more complicated (higher-order loop)
situations, and much of the current research is now being directed at the
structure of multiloop diagrams.
"Nobody working in the field, however, imagines any problems will arise
beyond the one-loop level," says Michael Green of London's Queen Mary
College. In conventional quantum field theories, he explains, the fact that a
given theory is one-loop finite is absolutely irrelevant to what happens at two
or three loops. "But, the infinities that come from string theories are of a
completely different sort. If a theory is finite at one loop, it is probably finite at
two or three as well." •
32 MOSAIC Volume 17 Number 4 Winter 1986/7
For reasons of beauty and simplicity,
the mathematical equations of a theory
of the universe are expected to be highly
symmetrical. However, the very act of
quantizing the theory often destroys its
symmetry. Because the terms In the
equations that break the symmetry surprised their discoverers, they were
called anomalous. If these anomalies are
not somehow canceled away, various sacred conservation laws (e.g., the one ascribed to Ben Franklin that says "electric
charge is conserved") would have to be
sacrificed. No one, however, is willing
to make this sacrifice.
Anomalies (like infinities) attack the
very consistency of any theory, whether
of the superstring or point-particle type.
The requirement, then, that anomalies
be canceled away is an extremely powerful tool for limiting possible theories.
In addition to anomaly cancellation
and finiteness, physicists are pleased
with E8 x E8 for another reason. The
quantum number labels of E8 form a natural decomposition chain that ties in
very nicely with the groups that, In four
dimensions, are expected to lead to the
unification of the strong, electromagnetic, and weak forces. The chain involves the subgroups E6, SO(10), and su
(5). These in turn must ultimately reduce to the symmetry group SU(3) x SU
(2) x u(l) that applies to these forces.
Here, su(3) represents the strong force
and su(2) x u(l) the partially unified
electro weak force. "We can't yet elucidate how these bigger symmetry groups
might break down," says Witten, "but
that's probably related to our overall ignorance, still, about string theory."
Very recently, a third candidate internal symmetry group, so(16) x SO(16),
reached the marketplace. "In a sense,
this new group may be just a different
aspect of the other ones," says Green.
"It may be just a different solution of the
same underlying theory. I n d e e d , by
mathematically changing E8 x E8 In only a
slight way, it becomes so(16) x SO(16)."
This new group, Green notes, is "superficially disastrous." While it cancels
away anomalies, the underlying theory
does not, like its predecessors, possess
"ordinary supersymmetry." In addition,
SO(16) x so(16) is distinct in another way.
It.is not a finite theory, like SO(32) or E8 x
E8, but is "one-loop divergent." Green
points out, however, that finiteness may
not, after all, be as necessary as had
been thought previously: "The sorts of
Infinities that occur in SO(16) x so(16)
show that space-time is unstable, that it
wants to collapse. And that's exactly
what we want!"
All of today's calculations, Green explains, start with the approximation that
space and time are flat in ten dimensions. "We know that approximation is
silly/' Green says. Ultimately, if these
theories are correct, six dimensions
must curve and curl up. It may be that a
group that is not one-loop finite will
open the way to understanding just this
sticky point. •
Margaret L. Silbar, Volume 15 Number 2;
and "Grand Unification: An Elusive Grail"
by Arthur Fisher, Volume 10 Number 5.
Articles related to this one in previous issues of Mosaic include "Physicists' Symmetries" by Margaret L. Silbar, Volume 16
Number 2; "Gravity, the Fourth Force" by
The National Science Foundation contributes
to the support of research discussed in this
article through several programs in its Division of Physics.
MOSAIC Volume 17 Number 4 Winter 1986/7 33