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Superstrings...

Superstrings... - Physics Forum

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Old 11-11-2003, 08:01 PM
Do Wah Ditty
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Superstrings
Feature: November 2003

String theory is either a theory of everything - which automatically unites
gravity with the other three forces in nature - or a theory of nothing, but
finding the correct form of the theory is like searching for a needle in a
stupendous haystack

As I sit down to write this article I feel that I have taken on a task
rather like trying to summarize the history of the world in 10 pages. It is
just too large a subject, with too many lines of thought and too many
threads to weave together. In the 34 years since it began, string theory has
developed into an enormous body of knowledge that touches on every aspect of
theoretical physics.


String theory is a theory of composite hadrons, an aspiring theory of
elementary particles, a quantum theory of gravity, and a framework for
understanding black holes. It is also a powerful technical tool for taming
strongly interacting quantum field theories and, perhaps, a basis for
formulating a fundamental theory of the universe. It even touches on
problems in condensed-matter physics, and has also provided a whole new
world of mathematical problems and tools.

All I can do with this gargantuan collection of material is to make my own
guess about which aspects of string theory are most likely to form the core
of a future physical theory, perhaps 100 years from now. It will come as no
surprise to my friends that my choice revolves around those things that have
most interested me in the last several years. No doubt many of them will
disagree with my judgement. Let them write their own articles.

String theory is considered to be a branch of high-energy or elementary
particle physics. However, a high-energy theorist from the 1950s, 1960s or
1970s would be surprised to read a recent string-theory paper and find not a
single Feynman diagram, cross-section or particle decay rate. Nor would
there be any mention of protons, neutrinos or Higgs bosons in the majority
of current literature. What the reader would find are black-hole metrics,
Einstein equations, Kaluza-Klein theories and plenty of fancy geometry and
topology. The energy scales of interest are not MeV, GeV or even TeV, but
energies at the Planck scale - the scale at which the classical concepts of
space and time break down.

The Planck energy is equal to h-bar5/G, where h-bar is Planck's constant
divided by 2?, c is the speed of light and G is the gravitational constant,
and it corresponds to masses that are some 19 orders of magnitude larger
than the proton mass. This is the energy of the universe when it was just
10-43s old, and it will probably be forever out of range of any particle
accelerator. To understand physics at the Planck scale we need a quantum
theory of gravity.

In the days when my career was beginning, a typical colloquium on
high-energy physics would often begin by stating that there are four forces
in nature - electromagnetic, weak, strong and gravitational - followed by a
statement that the gravitational force is much too weak to be of any
importance in particle physics so we will ignore it from now on. That has
all changed.

Today the other three forces are described by the gauge theories of quantum
chromodynamics (QCD) and quantum electrodynamics (QED), which together make
up the Standard Model of particle physics. These quantum field theories
describe the fundamental forces between particles as being due to the
exchange of field quanta: the photon for the electromagnetic force, the W
and Z bosons for the weak force, and the gluon for the strong force. In the
string-theory community, however, the electromagnetic, strong and weak
forces are generally considered to be manifestations of certain
"compactifications" of space from 10 or 11 dimensions to the four familiar
dimensions of space-time. But before I report on the status of string
theory, I want to tell you how it came about that so many otherwise sensible
high-energy theorists became interested in quantum gravity.

Why quantum gravity?
Elementary particles have far too many properties - such as spin, charge,
colour, parity and hypercharge - to be truly elementary. Particles obviously
have some kind of internal machinery at some scale. Protons and mesons
reveal their "parts" at the modestly small distance of about 10-15 m, but
quarks, leptons and photons hide their structure much more effectively.
Indeed, no experiment has ever seen direct evidence of size or structure for
any of these particles.

The first indication that the true scale of elementary particles might be
somewhere in the neighbourhood of the Planck scale came in the 1970s. Howard
Georgi and Sheldon Glashow, then at Harvard University, showed that the very
successful, but somewhat contrived, Standard Model could be elegantly
unified into a single theory by enlarging its symmetry group. The new
construction was astonishingly compact and most particle theorists assumed
that there must be some truth to it. But its predictions for the coupling
constants - the constants that describe the strengths of the strong, weak
and electromagnetic interactions - were wrong.

Georgi, along with Helen Quinn and Steven Weinberg, also at Harvard, soon
solved this problem when they realized that the coupling constants are not
really constants at all - they vary with energy. If the known couplings are
extrapolated they all intersect the predictions of the unified theory at
roughly the same scale. Moreover, this scale is close to the Planck scale.
The implication of this was clear: the scale of the internal machinery of
elementary particles is the Planck scale. And since the gravitational
constant, G, appears in the definition of the Planck energy, to many of us
this inevitably meant that gravitation must play an essential role in
determining the properties of particles.

The earliest attempts to reconcile gravity and quantum mechanics - notably
by Richard Feynman, Paul Dirac and Bryce DeWitt, who is now at the
University of Texas at Austin - were based on trying to fit Einstein's
general theory of relativity into a quantum field theory like the hugely
successful QED. The goal was to find a set of rules for calculating
scattering amplitudes in which the photons of QED are replaced by the quanta
of the gravitational field: gravitons. But gravitational forces become
increasingly strong as the energy of the participating quanta increases, and
the theory proved to be wildly out of control. Attempting to treat the
graviton as a point particle simply gave rise to far too many degrees of
freedom at short distances.

In a sense the failure of this "quantum gravity" theory was a good sign. The
theory itself gave no insight into the internal machinery of elementary
particles, and it offered no explanation for the other forces of nature. At
best it was more of the same: an effective (but not very) description of
gravitation with no deeper insight into the origin of particle properties.
At worst, it was mathematical nonsense.

Strings as hadrons
We all know that science is full of surprising twists, but the discovery of
string theory was particularly serendipitous. The theory grew out of
attempts in the 1960s to describe the interactions of hadrons - particles
that contain quarks, such as the proton and neutron. This was a problem that
had nothing to do with gravity. Gabriele Veneziano, now at CERN, and others
had written down a simple mathematical expression for scattering amplitudes
that had certain properties that were fashionable at that time. It was soon
discovered by Yoichiro Nambu of the University of Chicago and myself, and in
a slightly different form by Holger Bech Nielsen at the Niels Bohr
Institute, that these amplitudes were the solution of a definite physical
system that consists of extended 1D elastic strings.

For the two years that followed, string theory was the theory of hadrons.
One of the spectacular discoveries made in this early period was that the
mathematical infinities that occur in quantum field theory are completely
absent in string theory. However, from the very beginning there were big
problems in interpreting hadrons as strings. For example, the earliest
version of the theory could only accommodate bosons, whereas many hadrons -
including the proton and neutron - are fermions.

The distinction between bosons and fermions is one of the most important in
physics. Bosons are particles that have integer spins, such as 0, h-bar and
2h-bar, whereas fermions have half-integer spins of h-bar/2, 3h-bar/2 and so
on. All fundamental matter particles, such as quarks and leptons, are
fermions, while the particles that carry fundamental forces - the photon, W
and Z, and so on - are all bosons.

Fermionic versions of string theory were soon discovered and, moreover, they
turned out to have a surprising symmetry called supersymmetry that is now
totally pervasive in high-energy physics. In supersymmetric theories all
bosons have a fermionic superpartner and vice versa. The early development
of "superstring" theory was due to pioneering work by John Schwarz of
Caltech, Andrei Neveu of the University of Montpellier II, Michael Green of
Cambridge and Pierre Ramond of the University of Florida, and much of the
subsequent technical development was carried out in a famous series of
papers by Green and Schwarz in the 1980s.

Another apparently serious problem with the string theory of hadrons
concerned dimensions. Although the original assumptions in string theory
were simple enough, the mathematics proved internally inconsistent, at least
if the number of dimensions of space-time was four. The source of this
problem was quite deep, but, strangely, if space-time has 10 dimensions it
contrives to cancel out. The reasons were not at all easy to understand, but
the extraordinary mathematical consistency of superstring theory in 10
dimensions was compelling. However, so was the obvious fact that space-time
has four dimensions, not 10.

Thus by about 1972 theorists were beginning to question the relevance of
string theory for hadrons. In fact, there were other serious physical
shortcomings in ddition to the bizarre need for 10 dimensions. A
mathematical string can vibrate in many patterns, which represent a
different type of particle, and among these are certain patterns that
represent massless particles. But most dangerous of all were massless
particles with two units of spin angular momentum ("spin-two"). There are
certainly spin-two hadrons, but none that have anything like zero mass.
Despite all efforts, the massless spin-two particle could not be removed or
made massive.

Eventually, mathematical string theory gave way to QCD as a theory of
hadrons, which had its own explanation of the string-like behaviour of these
particles without the bad side effects. For most high-energy theorists,
string theory had lost its reason for existence. But a few bold souls saw
opportunity in the debacle. A massless spin-two field might not be good for
hadronic physics, but it is just what was needed for quantum gravity, albeit
in 10D. This is because just as the photon is the quantum of the
electromagnetic field, the graviton is the quantum of the gravitational
field. But the gravitational field is a symmetric tensor rather than a
vector, and this means the graviton is spin-two, rather than spin-one like
the photon. This difference in spin is the principal reason why early
attempts to quantize gravity based on QED did not work.

A theory of everything
The massless spin-two graviton led to a radical shift in perspective among
theorists. The focus of mainstream high-energy physics at the time was on
energy scales anywhere from the hadronic scale of a few GeV to the weak
interaction scale of a few hundred GeV. But to explore the idea that string
theory governs gravity, the energy scale of string excitations has to jump
from the hadronic scale to the Planck scale. In other words, with barely a
blink of the eye, string theorists would leapfrog 19 orders of magnitude,
and therefore completely abandon the idea that progress in physics proceeds
incrementally. Heady stuff, but also the source of much irritation in the
rest of the physics community.

Another reason for annoyance was somebody's idea to start referring to
string theory as a "theory of everything". Even string theorists found this
irritating, but there is actually a technical sense in which string theory
can either be a theory of everything or a theory of nothing. One of the
problems in describing hadrons with strings was that it proved impossible to
allow for the hadrons to interact with other fields, such as electromagnetic
fields, as they clearly do experimentally. This was a deadly flaw for a
theory of hadrons, but not for a theory in which all matter, including
photons, are strings. In other words, either all matter is strings, or
string theory is wrong. This is one of the most exciting features of the
theory.

But what about the problem of dimensions? Here again, a sow's ear was turned
into a silk purse. The basic idea goes back to Theodor Kaluza in 1919, who
tried to unify Einstein's gravitational theory with electrodynamics by
introducing a compact space-like fifth dimension. Kaluza discovered the
beautiful fact that the extra components of the gravitational field tensor
in 5 dimensions behaved exactly like the electromagnetic field plus one
additional scalar field. Somewhat later, in 1938, Oskar Klein and then
Wolfgang Pauli generalized Kaluza's work so that the single compact
dimension was replaced by a 2D space. If the 2D space is the surface of a
sphere then a remarkable thing happens when Kaluza's procedure is followed.
Instead of electrodynamics, Klein and Pauli discovered the first
"non-Abelian" gauge theory, which was later rediscovered by Chen Ning Yang
and Robert Mills. This is exactly the same class of theories that is so
successful in describing the strong and electromagnetic interactions in the
Standard Model.





One may ask whether particles move in the extra dimensions. For example, can
a particle that appears to be standing still in our usual 3D space have
velocity or momentum components in the compact dimensions? The answer is
yes, and the corresponding components of momentum define new conserved
quantities (figure 1). What is more, these quantities are quantized in
discrete units. In short, they are "charges" similar to electric charge,
isospin and all the other internal quantum numbers of elementary particles.
The answer to the problem of dimensions in string theory is obvious: six of
the 10 dimensions should be wrapped up into some very small compact space,
and the corresponding quantized components of momenta become part of the
internal machinery of elementary particles that determines their quantum
numbers.

Life in six dimensions
Much of the development of string theory is therefore concerned with 6D
spaces. These spaces, which can be thought of as generalized Kaluza-Klein
compactification spaces, were originally studied by mathematicians and are
known as Calabi-Yau spaces. They are tremendously complicated and are not
completely understood. But in the process of studying how strings move on
them, physicists have created an unexpected revolution in the study of
Calabi-Yau spaces.

In particular, it was discovered that a compactification radius of size R is
completely equivalent to a space with size 1/R from the point of view of
string theory. This connection, which is known as T-duality, has a
mathematically profound generalization called mirror symmetry, which states
that there is an equivalence between small and large spaces (see box above).
Mirror symmetry of Calabi-Yau spaces - which are not only of different sizes
but have completely different topologies - was completely unsuspected before
physicists began studying quantum strings moving on them.

I wish it was possible to draw a Calabi-Yau space but they are tremendously
complicated. They are six-dimensional, which is three more than I can
visualize, and they have very complicated topologies, including holes,
tunnels and handles. Furthermore, there are thousands of them, each with a
different topology. And even when their topology is fixed there are hundreds
of parameters called moduli that determine the shape and size of the various
dimensions. Indeed, it is the complexity of Calabi-Yau geometry that makes
string theory so intimidating to an outsider. However, we can abstract a few
useful things from the mathematics, one of them being the idea of moduli.

The simplest example of a modulus is just the compactification radius, R,
when there is only a single compact dimension. In more complicated cases,
the moduli determine the sizes and shapes of the various features of the
geometry. The moduli are not constants but depend on the geometry of the
space itself, in the same way that the radius of the universe changes with
time in a manner that is controlled by dynamical equations of motion. Since
the compact dimensions are too small to see, the moduli can simply be
thought of as fields in space that determine the local conditions. Electric
and magnetic fields are examples of such fields but the moduli are even
simpler: they are scalar fields (i.e. they have only one component), rather
than vector fields. String theory always has lots of scalar-field moduli and
these can potentially play important roles in particle physics and
cosmology.

All of this raises an interesting question: what determines the
compactification moduli in the real world of experience? Is there some
principle that selects a special value of the moduli of a particular
Calabi-Yau space and therefore determines the parameters of the theory, such
as the masses of particles, the coupling constants of the forces, and so on?
The answer seems to be no: all values of the moduli apparently give rise to
mathematically consistent theories. Whether or not this is a good thing, it
is certainly surprising.

Ordinarily we might expect the vacuum or ground state of the world to be the
state of lowest energy. Furthermore, in the absence of very special
symmetries, the energy of a region of space will depend non-trivially on the
values of the fields in that region. Finding the true vacuum is then merely
an exercise in computing the energy for a given field configuration and
minimizing it. This is, to be sure, a difficult task, but it is possible in
principle. In string theory, however, we know from the beginning that the
potential energy stored in a given configuration has no dependence on the
moduli fields.

The reason that the field potential is exactly zero for every value of the
moduli is that string theory is supersymmetric. Supersymmetry has both
desirable and undesirable consequences. Its most obvious drawback is the
requirement that for every fermion there is a boson with exactly the same
mass, which is clearly not a property of our world.

A more subtle difficulty involves the aforementioned fact that the vacuum
energy is independent of the moduli. As well as telling us that we cannot
determine the moduli by minimizing the energy, supersymmetry also tells us
that the quanta of the moduli fields are exactly massless. No such massless
fields are known in nature and, furthermore, such fields are very dangerous.
Indeed, massless moduli would probably lead to long-range forces that would
compete with gravity and violate the equivalence principle - the cornerstone
of general relativity - at an observable level.

On the plus side, the vanishing vacuum energy that is implied by
supersymmetry ensures that the cosmological constant vanishes. If it were
not for supersymmetry, the vacuum would have a huge zero-point energy
density that would make the radius of curvature of space-time not much
bigger than the Planck scale - a most undesirable situation. Supersymmetry
also stabilizes the vacuum against various hypothetical instabilities, and
it allows us to make exact mathematical conclusions. Indeed, T-duality and
mirror symmetry are examples of those exact consequences. --
Do Wah Ditty


"You're free. And freedom is beautiful. And, you know, it'll take time to
restore chaos and order - order out of chaos. But we will."

- Washington, D.C., April 13, 2003
- President Bushisms


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