Download Strings as hadrons

Superstrings by Leonard Susskind
Feature: November 2003
String theory is either a Theory of Everything (which automatically unites gravity with the other 3
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 space - superstring theory lives
in 10 dimensions, which means that six
of the dimensions have to be
"compactified" in order to explain why
we can only perceive four. The best
way to do this is to use a complicated 6D geometry called a Calabi-Yau
manifold in which all the intrinsic
properties of elementary particles are
hidden. Credit: A Hanson.
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
judgment. So 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.
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The Planck energy is equal to ħ5/G where ħ is Planck's constant divided by 2n, c is the speed-oflight, 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 10-43 seconds 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 4 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 3 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 4 familiar dimensions of space-time. ... …
Why quantum gravity?
Elementary particles have far too many properties (such as spin, charge, color, 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
meters. 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
neighborhood 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
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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 (i.e., 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 2 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, ħ, and 2ħ, whereas fermions have half-integer spins of ħ/2,
3ħ/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, 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. 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 4). 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 4
dimensions -- not 10.
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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 addition 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 mass-less particles. But most dangerous of all were
mass-less 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 mass-less 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 behavior of these particles without the bad side effects. For most highenergy theorists, string theory had lost its reason for existence. But a few bold souls saw opportunity in
the debacle. A mass-less spin-two field might not be good for hadronic physics. But it is just what was
needed for quantum gravity (albeit in 10-D). 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. 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 mass-less 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 E"verything 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 5th 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.
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One may ask whether particles move in the extra dimensions. For example, can a particle that
appears to be standing still in our usual 3-D space have velocity or momentum components in the
compact dimensions? The answer is 'yes'. And the corresponding components of momentum define
new conserved quantities. 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. 6 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.
>
>Figure 1. There are an infinite number of ways to wind a string around
an extra dimension. Each way is topologically described by an
integer called the winding number, which can be positive or
negative depending on the orientation of the string. The
wound string is stretched along the compact direction, but
from the point of view of ordinary 3D space it is located at a
point and therefore looks like a particle. The winding number
is simply a new kind of quantum number. String theory
therefore has lots of possibilities for describing the
complicated conservation laws and internal symmetries of
elementary particles.
Life in 6 dimensions
Much of the development of string theory is therefore concerned with 6-D 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 Tduality -- 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 also have completely different topologies -- was
completely unsuspected before physicists began studying quantum strings moving on them.
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I wish it was possible to draw a Calabi-Yau space. But they are tremendously complicated. They
are 6-dimensional -- which is 3 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 mass-less. No such massless
fields are known in Nature and, furthermore, such fields are very dangerous. Indeed, mass-less 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.
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Supersymmetry also stabilizes the vacuum against various hypothetical instabilities. It allows us to
make exact mathematical conclusions. Indeed, T-duality and mirror symmetry are examples of those
exact consequences.
Black Holes
Figure 2.
Originally it was
thought that there were five
distinct string theories in flat 10D space, each of which could
provide a starting point for
compactifications
to
four
dimensions. SO(32) Type I
theories are distinguished by
having open as well as closed
strings, where SO(32) represents
the symmetry group of the theory.
Type IIA and IIB theories have
only closed strings.
Closed
strings have no ends and are like
closed rubber bands with the
topology of a circle, while open
strings have 2 free ends that can
move. Waves on open stings
bounce back-and-forth between the ends. But waves on closed strings circulate endlessly around the
string in one of two possible directions (hence the two versions of Type II string theory). Finally,
there are 2 "heterotic" theories -- SO(32) heterotic and E8 x E8 heterotic -- which allow different
kinds of waves to move in the 2 possible directions. It is now thought that the 5 different types of
string theory are related to each other by deep symmetries such as T-duality, as if they were each the
"classical" limit of a more fundamental theory called M-theory.
Throughout the 1980s and early 1990s, progress in string theory largely consisted of working out the
detailed rules of perturbation theory for the 5 known versions of the theory which would allow theorists
to arrive at actual solutions (Figure 2). These perturbative rules were generalizations of the Feynman
diagrams of QED and QCD in which the "world-lines" of point particles are replaced by "world-sheets"
that are traced out by moving strings. The study of world-sheet physics created a huge body of
knowledge about 2D Quantum Field Theory. But it did not offer much insight into the inner workings
of quantum gravity. At best, string theory provided an especially consistent way to introduce a small
distance scale and thereby regulate the divergences that had plagued the older attempts at quantizing
gravity.
Personally, I found the whole enterprise dry, overly technical, and -- above all -- disappointing. I
felt that a quantum theory of gravity should profoundly affect our views of space-time, Quantum
Mechanics, the origin of the Universe, and the mysteries of black holes. But string theory was largely
silent about all these matters. Then in 1993, all this began to change. And the catalyst was the
awakening interest in Stephen Hawking's earlier speculations about black holes.
The starting point for Hawking's speculations was the thermal behavior of black holes, which built
on earlier work by Jacob Bekenstein of the Hebrew University in Israel. Rather than the cold, dead
objects that they were originally thought to be, black holes turned out to have a heat content and to glow
like blackbodies. Because they glow they lose energy and evaporate, and because they have a
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temperature and an energy, they also have an entropy. This entropy S is defined by the BekensteinHawking equation: S = AkBc3/4 ħG, where A is the surface area of the horizon and kB is Boltzmann's
constant.
After realizing that black holes must evaporate by the emission of blackbody radiation, Hawking
raised an extremely profound question. What happens to all the detailed information that falls into a
black hole? Once it falls through the horizon, it cannot subsequently reappear on the outside without
violating causality. That is the meaning of a 'horizon'. But the black hole will eventually evaporate,
leaving only photons, gravitons, and other elementary particles as products of the decay. Hawking
concluded that the information must ultimately be lost to our world. But one of the fundamental
principles of Quantum Mechanics is that information is never lost, because the information in the
initial state of a quantum system is permanently imprinted in the quantum state.
Hawking's view was that conventional Quantum Mechanics must be violated during the formation
and evaporation of the black hole. He rightly understood that if this is true, the rules of Quantum
Mechanics must be drastically modified as the Planck scale is approached. The importance of this for
particle physics -- particularly for unified theories -- should have been obvious. But initially Hawking's
idea generated little interest among high-energy theorists apart from myself and Gerard 't Hooft at the
University of Utrecht. We were convinced that by modifying the rules of Quantum Mechanics in the
way advocated by Hawking, "all hell would break loose" such as causing empty space to quickly heat up
to stupendous temperatures and energy densities. We were sure that Hawking was wrong. By the early
1990s, however, the issue was becoming critical -- especially to string theorists. String theory by its
very definition is based on the conventional rules of Quantum Mechanics. And if Hawking was right,
the entire foundation of the theory would be destroyed.
Over the last decade, the apparent clash between standard quantum principles and black-hole
evaporation has been resolved, favoring -- I should add -- the views of 't Hooft and myself. The
formation and evaporation of a black hole is similar to many other process in Nature in which a collision
between particles gives rise to a very rich and chaotic spectrum of intermediate states. In the case of a
black hole, the collisions are between the original protons, neutrons, and electrons in a collapsing star.
Roughly speaking, a black hole is nothing but a very excited string with a total length that is
proportional to the area of its horizon.
During the collision or collapse process, all the energy of the initial state goes into forming a single
long, tangled string. And the entropy of the configuration is the logarithm of the number of
configurations of a random-walking quantum string.
The correspondence between string configurations and black-hole entropy was checked for all of the
various kinds of charged and neutral black holes that occur in compactifications of string theory. In
most of the cases, the entropy of the string configuration could be estimated. It agreed with the
Bekenstein-Hawking entropy to within a factor of order unity.
But string theorists wanted to do better. The Bekenstein-Hawking formula for the entropy of a black
hole is very precise: the entropy is 1/4 of the horizon area (measured in Planck units) for every kind of
black hole be it static, rotating, charged, or even higher-dimensional. Surely the universal factor of a
quarter should be computable in string theory? The key to a precise calculation was obvious. Certain
black holes called extremal black holes -- which are the ground states of charged black holes that carry
electric and magnetic charges -- are especially tractable in a supersymmetric theory. The only problem
was that in 1993 no one knew how to build an extremal black hole out of the right type out of strings.
This had to wait a couple of years for the discovery of entities called D-branes.
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Brane World
In 1995, Joe Polchinski of the University of California in Santa Barbara electrified the string-theory
community with a major discovery that has subsequently impacted every field of physics. As we have
seen, T-duality is the strange symmetry that interchanges the Kaluza-Klein momenta and winding
numbers of a closed string (see Figure 1). But what happens to an open string? Obviously the idea of a
winding number does not make sense for such a string. What actually happens to open strings under Tduality is that the free ends become fixed on surfaces called D-branes.
>Figure 3. D-branes are
surfaces that "live" in
string theory, and they
come
in
various
dimensions. D2-branes,
for example, are 2dimensional and can also
be called membranes.
D0-branes are particlelike, and D1-branes are
string-like.
Higherdimensional objects can
exist as well. D-branes
are essential for making
string theory mathematically consistent and have far-reaching implications for a theory of quantum
gravity
D-branes come in various dimensions. 2D branes, for example, can also be called membranes.
They have an energy or mass per unit surface area and are localized physical objects in their own right.
In a sense, they seem to be no less fundamental than the strings themselves. To an outsider, D-branes
may seem to be arbitrary additions to the theory. But they are not. Their existence is absolutely
essential to the mathematical consistency of the theory. In addition to allowing T-duality to act on an
open string in Type I string theory, they are necessary for implementing the deep dualities that link the 5
different kinds of string theory together.
But from the point-of-view of black holes, the importance of D-branes is that you can build extremal
black holes from them. In fact, just by placing a large number of D-branes at the same location, you can
build an extremal supersymmetric black hole. And because of the special properties of supersymmetric
systems, the statistical entropy of that black hole can be precisely computed. The result -- which was
first derived by Andrew Strominger and Cumrun Vafa at Harvard in 1996 -- is that the entropy is equal
to exactly 1/4 of the horizon area in Planck units! This suggested that the microscopic degrees of
freedom implied by the Bekenstein-Hawking entropy are the degrees of freedom describing strings. It
was a major boost for the superstring community.
At about the same time as D-branes were discovered, another very important development took
place. As I mentioned, the coupling constant of string theory is not really a constant at all and in many
respects it is very similar to the compactification moduli. String theorists took a surprisingly long time
to make the connection. But it turns out that 10-D string theory is itself a Kaluza-Klein compactification
of an 11-D theory that became known as "M-theory".
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M-theory appears to underlie all string theories (Figure 2). The 5 different versions of string theory
are just different ways of compactifying its 11 dimensions. But M-theory is not itself a string theory. It
has membranes but no strings. And the strings only appear when the 11th dimension is compactified.
Furthermore, the momentum in the compact 11th direction (the Kaluza-Klein momentum) is identified as
the number of D0-branes (i.e., zero-dimensional branes or "points") in a particular type of string theory.
This connection between Kaluza-Klein momentum and D0-branes led to another breakthrough. In
1996, myself, Tom Banks, Steve Shenker (at Rutgers University), and Willy Fischler (at the University
of Texas) realized that M-theory could be cast in a form no more complicated than the quantum
mechanics of a system of non-relativistic particles (i.e., D0-branes). The resulting theory -- which is
called Matrix theory -- is an exact and complete quantum theory that describes the microscopic degrees
of freedom of M-theory. As such, it is the first precise formulation of a quantum theory of gravity.
Duality
Matrix theory was just one example of how D-branes can be used to formulate a theory of quantum
gravity. Soon after its discovery, Juan Maldacena (who is now at the Institute for Advanced Study
(IAS) in Princeton) came up with a new direction to explore. Ed Witten of the IAS and others had
previously shown that D-branes have their own dynamics. But it turned out that the fluctuations and
motions of a D-brane can be quantized in the form of a gauge theory that is restricted to the D-brane
itself. The theory that lives on a coincident collection of D3-branes, for example, is a supersymmetric
non-Abelian gauge theory. In other words, it is a supersymmetric version of QCD -- the theory
describing quarks and gluons. In a sense, string theory is returning to its roots as a possible description
of hadrons (See Physics World, May 2003, pp35-38).
Figure 4. Anti-de Sitter space is a solution to
Einstein's field equations that is negatively curved
everywhere. It is analogous to the geometry of
Eschers's "Angels and Devils" in which you have to
imagine that all the angels and devils are the same
size, but that they are distorted due to the negative
curvature of space (in the same way that Mercator's
projection of the globe misrepresents the area of
continents away from the Equator). This figure
depicts anti-de Sitter space at an instant; the vertical
dimension is time. The geometry is bounded by the
surface of the cylinder and the holographic principle
states that quantum gravity in the interior of the
space is described by a quantum field theory (such as
QCD) on the boundary.
Maldacena realized that in an appropriate limit, the theory of D3-branes should be a complete
description of string theory not just on the branes, but in the entire geometry in which the branes are
embedded. A gauge theory would therefore also be a description of quantum gravity in a particular
background space-time. This space-time is called anti-de Sitter space which, roughly speaking, is a
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"universe inside a cavity". The walls of the cavity behave like reflecting surfaces so that nothing
escapes it (Figure 4).
This "duality" between Quantum Field Theory and Gravity is an exact realization of what is called
the holographic principle. This strange principle -- formulated by 't Hooft and myself -- grew from our
debate with Hawking regarding the validity of Quantum Mechanics in the formation and evaporation of
black holes.
According to the holographic principle, everything that ever falls into a black hole can be described
by degrees of freedom that reside in a thin layer just above the horizon. In other words, the full 3-D
world inside the horizon can be described by the 2-D degrees of freedom on its surface. Even more
generally, it should be possible to describe the physics of any region of space in terms of holographic
degrees of freedom that reside on the boundary of that region. This leads to a drastic reduction of the
number of degrees of freedom in a field theory. Most theorists found it very hard to swallow until
Maldacena's work came along. Maldacena's duality replaces a gravitational theory in anti-de Sitter
space by a field theory that lives on its boundary in a very precise way. In other words, the 3 + 1
dimensional boundary field theory is a holographic description of the interior of 4 + 1 dimensional antide Sitter space.
The D-brane revolution has been very far reaching. Matrix theory and the Maldacena duality are
both formulations of quantum gravity that conform to the standard rules of Quantum Mechanics and
should therefore lay to rest any further questions about black holes violating these rules.
Googles of possibilities
I would like to end by discussing the future of string theory not as a mathematical subject, but as a
framework for particle physics and cosmology. The final evaluation of string theory will rest on its
ability to explain the facts of Nature and not on its own internal beauty and consistency. String theory is
well into its 4th decade. But so far, it has not produced a detailed model of elementary particles or a
convincing explanation of any cosmological observation. Many of the models that are based on specific
methods of compactifying either 10-D string theory or 11-D M-theory have a good deal in common with
the real World. They have bosons and fermions, for example, and gauge theories that are similar to
those in the Standard Model. Furthermore, unlike any other theory, they inevitably include gravity. But
"the devil is in the details". And so far, the details have eluded string theorists.
It is possible, of course, that string theory is the wrong theory. But I believe that would be a very
premature judgment. And probably incorrect. The problem does not seem to be a lack of richness but
rather the opposite. String theory contains too many possibilities. For most physicists, the ideal
physical theory is one that is unique and perfect in that it determines all that can be determined and that
it could not logically be any other way. In other words, it is not only a 'Theory of Everything', but it is
also the only 'Theory of Everything'. To the orthodox string theorist, the goal is to discover the one true
consistent version of the theory, and then to demonstrate that the solution manifests the known laws of
Nature such as the Standard Model of particle physics with its empirical set of parameters.
But the more we learn about string theory, the more non-unique it seems to be. There are probably
millions of Calabi-Yau spaces on which to compactify string theory. Each space has hundreds of
moduli and hundreds of subspaces on which branes can be wrapped, fluxes imposed upon, and so on. A
conservative estimate of the number of distinct vacua of the theory is in the googles -- that is, more than
10100. The space of possibilities is called the "Landscape". And it is huge. To mix metaphors, it is a
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stupendous haystack that contains googles of straws and only one needle. Worse still, the theory itself
gives us no hint about how to pick among the possibilities (see "The String-Theory Landscape").
This enormous variety may, however, be exactly what cosmology is looking for. A common theme
among cosmologists is that the observed universe may merely be a minuscule part of a vastly bigger
Universe that contains many local environments (or what Alan Guth at MIT calls "pocket universes").
According to this view, so many pocket universes formed during the early inflationary epoch (each of
which with its own vacuum structure) that the entire landscape of possibilities is represented. The
reasons for this view are not just idle speculation, but are rooted in the many accidental fine-tunings that
are necessary for a Universe that supports Life. Thus, it may be that the enormous number of possible
vacuum solutions -- which is the bane of particle physics -- may be just what the doctor ordered for
Cosmology.
Further information
T-duality
In a single compact dimension, there are 2 kinds of quantum numbers: momentum in the compact
direction and the winding number. Both of these are quantized in integer multiples of a basic unit, and
each has a certain energy associated with it. In the case of momentum, for example, the energy is just
the kinetic energy of motion in the compact direction. The energy of a particle with n units of compact
momentum is equal to n/R, where R is the circumference of the compact direction. Note that the energy
grows as the size of the compact space gets smaller. On the other hand, the winding modes also have
energy which is the potential energy needed to stretch the string around the compact coordinate. If we
call the winding number m, then the winding energy is equal to mR. In this case, the energy decreases
as the size of the compact direction decreases.
The surprising thing is that the spectrum of energies is unchanged if we change the compactification
radius from R to 1/R and at the same time interchange the Kaluza-Klein momentum and winding
modes. In other words, just by looking at the spectrum of energies, you could never tell the difference
between a theory that is compactified on a space of size R or on one of size 1/R. As you try to make the
compactification scale smaller than the natural string scale (i.e., the size of a vibrating string), the theory
begins to behave as if the compactification radius was getting bigger. Physically, the smallest
compactification value of R is the string scale. But from a mathematical viewpoint, 2 different spaces -one large, the other small -- are completely equivalent. This equivalence is called T-duality.
Author
Leonard Susskind is in the Department of Physics, Stanford University, 382 Via Pueblo Mall, CA
94305-4060, U.S., e-mail [email protected]
Further reading
J Maldacena, 1999. "The large N limit of superconformal field theories and supergravity", Int. J. Theor.
Phys. 38 1113-1133
J Polchinski, 1995. "Dirichlet-branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 4724-4727
J Polchinski, 1998. String Theory (volume 2): Superstring Theory and Beyond (Cambridge University
Press)
J H Schwarz et al. 1981. Superstring Theory (volume 1): Introduction (Cambridge University Press)
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A Strominger and C Vafa ,1996. "Microscopic origin of the Bekenstein-Hawking entropy", Phys. Lett.
B 379 99-104
The official string theory website: http://www.superstringtheory.com
What we've discovered in the last several years is that string theory has an incredible diversity -- a
tremendous number of solutions -- and allows different kinds of environments. A lot of the practitioners
of this kind of mathematical theory have been in a state of denial about it. They didn't want to recognize
it. They want to believe the Universe is an elegant universe. But it's not so elegant. It's different over
here; it's that over here. It's a Rube Goldberg machine over here. And this has created a sort of sense of
denial about the facts about the theory. The theory is going to win. And physicists who are trying to
deny what's going on are going to lose.
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