String theory
In physics, string theory is a theoretical
framework in which the point-like particles
of particle physics are replaced by one-dimensional
objects called strings. In string theory,
the different types of observed elementary
particles arise from the different quantum
states of these strings. In addition to the
types of particles postulated by the standard
model of particle physics, string theory naturally
incorporates gravity, and is therefore a candidate
for a theory of everything, a self-contained
mathematical model that describes all fundamental
forces and forms of matter. Aside from this
hypothesized role in particle physics, string
theory is now widely used as a theoretical
tool in physics, and it has shed light on
many aspects of quantum field theory and quantum
gravity.
The earliest version of string theory, called
bosonic string theory, incorporated only the
class of particles known as bosons, although
this theory developed into superstring theory,
which posits that a connection (a "supersymmetry")
exists between bosons and the class of particles
called fermions. String theory requires the
existence of extra spatial dimensions for
its mathematical consistency. In realistic
physical models constructed from string theory,
these extra dimensions are typically compactified
to extremely small scales.
String theory was first studied in the late
1960s as a theory of the strong nuclear force
before being abandoned in favor of the theory
of quantum chromodynamics. Subsequently, it
was realized that the very properties that
made string theory unsuitable as a theory
of nuclear physics made it an outstanding
candidate for a quantum theory of gravity.
Five consistent versions of string theory
were developed before it was realized in the
mid-1990s that these theories could be obtained
as different limits of a conjectured eleven-dimensional
theory called M-theory.
Many theoretical physicists (among them Stephen
Hawking, Edward Witten, and Juan Maldacena)
believe that string theory is a step towards
the correct fundamental description of nature.
This is because string theory allows for the
consistent combination of quantum field theory
and general relativity, agrees with general
insights in quantum gravity such as the holographic
principle and black hole thermodynamics, and
because it has passed many non-trivial checks
of its internal consistency. According to
Hawking in particular, "M-theory is the only
candidate for a complete theory of the universe."
Other physicists, such as Richard Feynman,
Roger Penrose, and Sheldon Lee Glashow, have
criticized string theory for not providing
novel experimental predictions at accessible
energy scales and say that it is a failure
as a theory of everything.
Overview
The starting point for string theory is the
idea that the point-like particles of elementary
particle physics can also be modeled as one-dimensional
objects called strings. According to string
theory, strings can oscillate in many ways.
On distance scales larger than the string
radius, each oscillation mode gives rise to
a different species of particle, with its
mass, charge, and other properties determined
by the string's dynamics. Splitting and recombination
of strings correspond to particle emission
and absorption, giving rise to the interactions
between particles. An analogy for strings'
modes of vibration is a guitar string's production
of multiple distinct musical notes. In this
analogy, different notes correspond to different
particles.
In string theory, one of the modes of oscillation
of the string corresponds to a massless, spin-2
particle. Such a particle is called a graviton
since it mediates a force which has the properties
of gravity. Since string theory is believed
to be a mathematically consistent quantum
mechanical theory, the existence of this graviton
state implies that string theory is a theory
of quantum gravity.
String theory includes both open strings,
which have two distinct endpoints, and closed
strings, which form a complete loop. The two
types of string behave in slightly different
ways, yielding different particle types. For
example, all string theories have closed string
graviton modes, but only open strings can
correspond to the particles known as photons.
Because the two ends of an open string can
always meet and connect, forming a closed
string, all string theories contain closed
strings.
The earliest string model, the bosonic string,
incorporated only the class of particles known
as bosons. This model describes, at low enough
energies, a quantum gravity theory, which
also includes (if open strings are incorporated
as well) gauge bosons such as the photon.
However, this model has problems. What is
most significant is that the theory has a
fundamental instability, believed to result
in the decay (at least partially) of spacetime
itself. In addition, as the name implies,
the spectrum of particles contains only bosons,
particles which, like the photon, obey particular
rules of behavior. Roughly speaking, bosons
are the constituents of radiation, but not
of matter, which is made of fermions. Investigating
how a string theory may include fermions led
to the invention of supersymmetry, a mathematical
relation between bosons and fermions. String
theories that include fermionic vibrations
are now known as superstring theories; several
kinds have been described, but all are now
thought to be different limits of a theory
called M-theory.
Since string theory incorporates all of the
fundamental interactions, including gravity,
many physicists hope that it fully describes
our universe, making it a theory of everything.
One of the goals of current research in string
theory is to find a solution of the theory
that is quantitatively identical with the
standard model, with a small cosmological
constant, containing dark matter and a plausible
mechanism for cosmic inflation. It is not
yet known whether string theory has such a
solution, nor is it known how much freedom
the theory allows to choose the details.
One of the challenges of string theory is
that the full theory does not yet have a satisfactory
definition in all circumstances. The scattering
of strings is most straightforwardly defined
using the techniques of perturbation theory,
but it is not known in general how to define
string theory nonperturbatively. It is also
not clear as to whether there is any principle
by which string theory selects its vacuum
state, the spacetime configuration that determines
the properties of our universe (see string
theory landscape).
Strings
The motion of a point-like particle can be
described by drawing a graph of its position
with respect to time. The resulting picture
depicts the worldline of the particle in spacetime.
In an analogous way, one can draw a graph
depicting the progress of a string as time
passes. The string, which looks like a small
line by itself, will sweep out a two-dimensional
surface known as the worldsheet. The different
string modes (giving rise to different particles,
such as the photon or graviton) appear as
waves on this surface.
A closed string looks like a small loop, so
its worldsheet will look like a pipe. An open
string looks like a segment with two endpoints,
so its worldsheet will look like a strip.
In a more mathematical language, these are
both Riemann surfaces, the strip having a
boundary and the pipe none.
Strings can join and split. This is reflected
by the form of their worldsheet, or more precisely,
by its topology. For example, if a closed
string splits, its worldsheet will look like
a single pipe splitting into two pipes. This
topology is often referred to as a pair of
pants (see drawing at right). If a closed
string splits and its two parts later reconnect,
its worldsheet will look like a single pipe
splitting to two and then reconnecting, which
also looks like a torus connected to two pipes
(one representing the incoming string, and
the other representing the outgoing one).
An open string doing the same thing will have
a worldsheet that looks like an annulus connected
to two strips.
In quantum mechanics, one computes the probability
for a point particle to propagate from one
point to another by summing certain quantities
called probability amplitudes. Each amplitude
is associated with a different worldline of
the particle. This process of summing amplitudes
over all possible worldlines is called path
integration. In string theory, one computes
probabilities in a similar way, by summing
quantities associated with the worldsheets
joining an initial string configuration to
a final configuration. It is in this sense
that string theory extends quantum field theory,
replacing point particles by strings. As in
quantum field theory, the classical behavior
of fields is determined by an action functional,
which in string theory can be either the Nambu–Goto
action or the Polyakov action.
Branes
In string theory and related theories such
as supergravity theories, a brane is a physical
object that generalizes the notion of a point
particle to higher dimensions. For example,
a point particle can be viewed as a brane
of dimension zero, while a string can be viewed
as a brane of dimension one. It is also possible
to consider higher-dimensional branes. In
dimension p, these are called p-branes. The
word brane comes from the word "membrane"
which refers to a two-dimensional brane.
Branes are dynamical objects which can propagate
through spacetime according to the rules of
quantum mechanics. They have mass and can
have other attributes such as charge. A p-brane
sweeps out a (p+1)-dimensional volume in spacetime
called its worldvolume. Physicists often study
fields analogous to the electromagnetic field
which live on the worldvolume of a brane.
In string theory, D-branes are an important
class of branes that arise when one considers
open strings. As an open string propagates
through spacetime, its endpoints are required
to lie on a D-brane. The letter "D" in D-brane
refers to the fact that we impose a certain
mathematical condition on the system known
as the Dirichlet boundary condition. The study
of D-branes in string theory has led to important
results such as the AdS/CFT correspondence,
which has shed light on many problems in quantum
field theory.
Branes are also frequently studied from a
purely mathematical point of view since they
are related to subjects such as homological
mirror symmetry and noncommutative geometry.
Mathematically, branes may be represented
as objects of certain categories, such as
the derived category of coherent sheaves on
a Calabi–Yau manifold, or the Fukaya category.
Dualities
In physics, the term duality refers to a situation
where two seemingly different physical systems
turn out to be equivalent in a nontrivial
way. If two theories are related by a duality,
it means that one theory can be transformed
in some way so that it ends up looking just
like the other theory. The two theories are
then said to be dual to one another under
the transformation. Put differently, the two
theories are mathematically different descriptions
of the same phenomena.
In addition to providing a candidate for a
theory of everything, string theory provides
many examples of dualities between different
physical theories and can therefore be used
as a tool for understanding the relationships
between these theories.
S-, T-, and U-duality
These are dualities between string theories
which relate seemingly different quantities.
Large and small distance scales, as well as
strong and weak coupling strengths, are quantities
that have always marked very distinct limits
of behavior of a physical system in both classical
and quantum physics. But strings can obscure
the difference between large and small, strong
and weak, and this is how these five very
different theories end up being related. T-duality
relates the large and small distance scales
between string theories, whereas S-duality
relates strong and weak coupling strengths
between string theories. U-duality links T-duality
and S-duality.
M-theory
Before the 1990s, string theorists believed
there were five distinct superstring theories:
type I, type IIA, type IIB, and the two flavors
of heterotic string theory (SO(32) and E8×E8).
The thinking was that out of these five candidate
theories, only one was the actual correct
theory of everything, and that theory was
the one whose low energy limit, with ten spacetime
dimensions compactified down to four, matched
the physics observed in our world today. It
is now believed that this picture was incorrect
and that the five superstring theories are
related to one another by the dualities described
above. The existence of these dualities suggests
that the five string theories are in fact
special cases of some more fundamental theory
called M-theory.
Extra dimensions
Number of dimensions
An intriguing feature of string theory is
that it predicts extra dimensions. In classical
string theory the number of dimensions is
not fixed by any consistency criterion. However,
to make a consistent quantum theory, string
theory is required to live in a spacetime
of the so-called "critical dimension": we
must have 26 spacetime dimensions for the
bosonic string and 10 for the superstring.
This is necessary to ensure the vanishing
of the conformal anomaly of the worldsheet
conformal field theory. Modern understanding
indicates that there exist less trivial ways
of satisfying this criterion. Cosmological
solutions exist in a wider variety of dimensionalities,
and these different dimensions are related
by dynamical transitions. The dimensions are
more precisely different values of the "effective
central charge", a count of degrees of freedom
that reduces to dimensionality in weakly curved
regimes.
One such theory is the 11-dimensional M-theory,
which requires spacetime to have eleven dimensions,
as opposed to the usual three spatial dimensions
and the fourth dimension of time. The original
string theories from the 1980s describe special
cases of M-theory where the eleventh dimension
is a very small circle or a line, and if these
formulations are considered as fundamental,
then string theory requires ten dimensions.
But the theory also describes universes like
ours, with four observable spacetime dimensions,
as well as universes with up to 10 flat space
dimensions, and also cases where the position
in some of the dimensions is described by
a complex number rather than a real number.
The notion of spacetime dimension is not fixed
in string theory: it is best thought of as
different in different circumstances.
Nothing in Maxwell's theory of electromagnetism
or Einstein's theory of relativity makes this
kind of prediction; these theories require
physicists to insert the number of dimensions
manually and arbitrarily, and this number
is fixed and independent of potential energy.
String theory allows one to relate the number
of dimensions to scalar potential energy.
In technical terms, this happens because a
gauge anomaly exists for every separate number
of predicted dimensions, and the gauge anomaly
can be counteracted by including nontrivial
potential energy into equations to solve motion.
Furthermore, the absence of potential energy
in the "critical dimension" explains why flat
spacetime solutions are possible.
This can be better understood by noting that
a photon included in a consistent theory (technically,
a particle carrying a force related to an
unbroken gauge symmetry) must be massless.
The mass of the photon that is predicted by
string theory depends on the energy of the
string mode that represents the photon. This
energy includes a contribution from the Casimir
effect, namely from quantum fluctuations in
the string. The size of this contribution
depends on the number of dimensions, since
for a larger number of dimensions there are
more possible fluctuations in the string position.
Therefore, the photon in flat spacetime will
be massless—and the theory consistent—only
for a particular number of dimensions. When
the calculation is done, the critical dimensionality
is not four as one may expect (three axes
of space and one of time). The subset of X
is equal to the relation of photon fluctuations
in a linear dimension. Flat space string theories
are 26-dimensional in the bosonic case, while
superstring and M-theories turn out to involve
10 or 11 dimensions for flat solutions. In
bosonic string theories, the 26 dimensions
come from the Polyakov equation. Starting
from any dimension greater than four, it is
necessary to consider how these are reduced
to four-dimensional spacetime.
Compact dimensions
Two ways have been proposed to resolve this
apparent contradiction. The first is to compactify
the extra dimensions; i.e., the 6 or 7 extra
dimensions are so small as to be undetectable
by present-day experiments.
To retain a high degree of supersymmetry,
these compactification spaces must be very
special, as reflected in their holonomy. A
6-dimensional manifold must have SU(3) structure,
a particular case (torsionless) of this being
SU(3) holonomy, making it a Calabi–Yau space,
and a 7-dimensional manifold must have G2
structure, with G2 holonomy again being a
specific, simple, case. Such spaces have been
studied in attempts to relate string theory
to the 4-dimensional Standard Model, in part
due to the computational simplicity afforded
by the assumption of supersymmetry. More recently,
progress has been made constructing more realistic
compactifications without the degree of symmetry
of Calabi–Yau or G2 manifolds.
A standard analogy for this is to consider
multidimensional space as a garden hose. If
the hose is viewed from sufficient distance,
it appears to have only one dimension, its
length. Indeed, think of a ball just small
enough to enter the hose. Throwing such a
ball inside the hose, the ball would move
more or less in one dimension; in any experiment
we make by throwing such balls in the hose,
the only important movement will be one-dimensional,
that is, along the hose. However, as one approaches
the hose, one discovers that it contains a
second dimension, its circumference. Thus,
an ant crawling inside it would move in two
dimensions (and a fly flying in it would move
in three dimensions). This "extra dimension"
is only visible within a relatively close
range to the hose, or if one "throws in" small
enough objects. Similarly, the extra compact
dimensions are only "visible" at extremely
small distances, or by experimenting with
particles with extremely small wavelengths
(of the order of the compact dimension's radius),
which in quantum mechanics means very high
energies (see wave–particle duality).
Brane-world scenario
Another possibility is that we are "stuck"
in a 3+1 dimensional (three spatial dimensions
plus one time dimension) subspace of the full
universe. Properly localized matter and Yang–Mills
gauge fields will typically exist if the sub-spacetime
is an exceptional set of the larger universe.
These "exceptional sets" are ubiquitous in
Calabi–Yau n-folds and may be described
as subspaces without local deformations, akin
to a crease in a sheet of paper or a crack
in a crystal, the neighborhood of which is
markedly different from the exceptional subspace
itself. However, until the work of Randall
and Sundrum, it was not known that gravity
can be properly localized to a sub-spacetime.
In addition, spacetime may be stratified,
containing strata of various dimensions, allowing
us to inhabit the 3+1-dimensional stratum—such
geometries occur naturally in Calabi–Yau
compactifications. Such sub-spacetimes are
D-branes, hence such models are known as brane-world
scenarios.
Effect of the hidden dimensions
In either case, gravity acting in the hidden
dimensions affects other non-gravitational
forces such as electromagnetism. In fact,
Kaluza's early work demonstrated that general
relativity in five dimensions actually predicts
the existence of electromagnetism. However,
because of the nature of Calabi–Yau manifolds,
no new forces appear from the small dimensions,
but their shape has a profound effect on how
the forces between the strings appear in our
four-dimensional universe. In principle, therefore,
it is possible to deduce the nature of those
extra dimensions by requiring consistency
with the standard model, but this is not yet
a practical possibility. It is also possible
to extract information regarding the hidden
dimensions by precision tests of gravity,
but so far these have only put upper limitations
on the size of such hidden dimensions.
Testability and experimental predictions
Although a great deal of recent work has focused
on using string theory to construct realistic
models of particle physics, several major
difficulties complicate efforts to test models
based on string theory. The most significant
is the extremely small size of the Planck
length, which is expected to be close to the
string length (the characteristic size of
a string, where strings become easily distinguishable
from particles). Another issue is the huge
number of metastable vacua of string theory,
which might be sufficiently diverse to accommodate
almost any phenomena we might observe at lower
energies.
String harmonics
One unique prediction of string theory is
the existence of string harmonics. At sufficiently
high energies, the string-like nature of particles
would become obvious. There should be heavier
copies of all particles, corresponding to
higher vibrational harmonics of the string.
It is not clear how high these energies are.
In most conventional string models, they would
be close to the Planck energy, which is 1014
times higher than the energies accessible
in the newest particle accelerator, the LHC,
making this prediction impossible to test
with any particle accelerator in the near
future. However, in models with large extra
dimensions they could potentially be produced
at the LHC, or at energies not far above its
reach.
Cosmology
String theory as currently understood makes
a series of predictions for the structure
of the universe at the largest scales. Many
phases in string theory have very large, positive
vacuum energy. Regions of the universe that
are in such a phase will inflate exponentially
rapidly in a process known as eternal inflation.
As such, the theory predicts that most of
the universe is very rapidly expanding. However,
these expanding phases are not stable, and
can decay via the nucleation of bubbles of
lower vacuum energy. Since our local region
of the universe is not very rapidly expanding,
string theory predicts we are inside such
a bubble. The spatial curvature of the "universe"
inside the bubbles that form by this process
is negative, a testable prediction. Moreover,
other bubbles will eventually form in the
parent vacuum outside the bubble and collide
with it. These collisions lead to potentially
observable imprints on cosmology. However,
it is possible that neither of these will
be observed if the spatial curvature is too
small and the collisions are too rare.
Under certain circumstances, fundamental strings
produced at or near the end of inflation can
be "stretched" to astronomical proportions.
These cosmic strings could be observed in
various ways, for instance by their gravitational
lensing effects. However, certain field theories
also predict cosmic strings arising from topological
defects in the field configuration.
Supersymmetry
If confirmed experimentally, supersymmetry
could also be considered circumstantial evidence,
because all consistent string theories are
supersymmetric. However, the absence of supersymmetric
particles at energies accessible to the LHC
would not necessarily disprove string theory,
since the energy scale at which supersymmetry
is broken could be well above the accelerator's
range.
AdS/CFT correspondence
The anti-de Sitter/conformal field theory
(AdS/CFT) correspondence is a relationship
which says that string theory is in certain
cases equivalent to a quantum field theory.
More precisely, one considers string or M-theory
on an anti-de Sitter background. This means
that the geometry of spacetime is obtained
by perturbing a certain solution of Einstein's
equation in the vacuum. In this setting, it
is possible to define a notion of "boundary"
of spacetime. The AdS/CFT correspondence states
that this boundary can be regarded as the
"spacetime" for a quantum field theory, and
this field theory is equivalent to the bulk
gravitational theory in the sense that there
is a "dictionary" for translating calculations
in one theory into calculations in the other.
Examples of the correspondence
The most famous example of the AdS/CFT correspondence
states that Type IIB string theory on the
product AdS5 × S5 is equivalent to N = 4
super Yang–Mills theory on the four-dimensional
conformal boundary. Another realization of
the correspondence states that M-theory on
AdS4 × S7 is equivalent to the ABJM superconformal
field theory in three dimensions. Yet another
realization states that M-theory on AdS7 × S4is
equivalent to the so-called (2,0)-theory in
six dimensions.
Applications to quantum chromodynamics
Since it relates string theory to ordinary
quantum field theory, the AdS/CFT correspondence
can be used as a theoretical tool for doing
calculations in quantum field theory. For
example, the correspondence has been used
to study the quark–gluon plasma, an exotic
state of matter produced in particle accelerators.
The physics of the quark–gluon plasma is
governed by quantum chromodynamics, the fundamental
theory of the strong nuclear force, but this
theory is mathematically intractable in problems
involving the quark–gluon plasma. In order
to understand certain properties of the quark–gluon
plasma, theorists have therefore made use
of the AdS/CFT correspondence. One version
of this correspondence relates string theory
to a certain supersymmetric gauge theory called
N = 4 super Yang–Mills theory. The latter
theory provides a good approximation to quantum
chromodynamics. One can thus translate problems
involving the quark–gluon plasma into problems
in string theory which are more tractable.
Using these methods, theorists have computed
the shear viscosity of the quark–gluon plasma.
In 2008, these predictions were confirmed
at the Relativistic Heavy Ion Collider at
Brookhaven National Laboratory.
Applications to condensed matter physics
In addition, string theory methods have been
applied to problems in condensed matter physics.
Certain condensed matter systems are difficult
to understand using the usual methods of quantum
field theory, and the AdS/CFT correspondence
may allow physicists to better understand
these systems by describing them in the language
of string theory. Some success has been achieved
in using string theory methods to describe
the transition of a superfluid to an insulator.
Connections to mathematics
In addition to influencing research in theoretical
physics, string theory has stimulated a number
of major developments in pure mathematics.
Like many developing ideas in theoretical
physics, string theory does not at present
have a mathematically rigorous formulation
in which all of its concepts can be defined
precisely. As a result, physicists who study
string theory are often guided by physical
intuition to conjecture relationships between
the seemingly different mathematical structures
that are used to formalize different parts
of the theory. These conjectures are later
proved by mathematicians, and in this way,
string theory has served as a source of new
ideas in pure mathematics.
Mirror symmetry
One of the ways in which string theory influenced
mathematics was through the discovery of mirror
symmetry. In string theory, the shape of the
unobserved spatial dimensions is typically
encoded in mathematical objects called Calabi–Yau
manifolds. These are of interest in pure mathematics,
and they can be used to construct realistic
models of physics from string theory. In the
late 1980s, it was noticed that given such
a physical model, it is not possible to uniquely
reconstruct a corresponding Calabi–Yau manifold.
Instead, one finds that there are two Calabi–Yau
manifolds that give rise to the same physics.
These manifolds are said to be "mirror" to
one another. The existence of this mirror
symmetry relationship between different Calabi–Yau
manifolds has significant mathematical consequences
as it allows mathematicians to solve many
problems in enumerative algebraic geometry.
Today mathematicians are still working to
develop a mathematical understanding of mirror
symmetry based on physicists' intuition.
Vertex operator algebras
In addition to mirror symmetry, applications
of string theory to pure mathematics include
results in the theory of vertex operator algebras.
For example, ideas from string theory were
used by Richard Borcherds in 1992 to prove
the monstrous moonshine conjecture relating
the monster group (a construction arising
in group theory, a branch of algebra) and
modular functions (a class of functions which
are important in number theory).
History
Early results
Some of the structures reintroduced by string
theory arose for the first time much earlier
as part of the program of classical unification
started by Albert Einstein. The first person
to add a fifth dimension to general relativity
was German mathematician Theodor Kaluza in
1919, who noted that gravity in five dimensions
describes both gravity and electromagnetism
in four. In 1926, the Swedish physicist Oskar
Klein gave a physical interpretation of the
unobservable extra dimension—it is wrapped
into a small circle. Einstein introduced a
non-symmetric metric tensor, while much later
Brans and Dicke added a scalar component to
gravity. These ideas would be revived within
string theory, where they are demanded by
consistency conditions.
String theory was originally developed during
the late 1960s and early 1970s as a never
completely successful theory of hadrons, the
subatomic particles like the proton and neutron
that feel the strong interaction. In the 1960s,
Geoffrey Chew and Steven Frautschi discovered
that the mesons make families called Regge
trajectories with masses related to spins
in a way that was later understood by Yoichiro
Nambu, Holger Bech Nielsen and Leonard Susskind
to be the relationship expected from rotating
strings. Chew advocated making a theory for
the interactions of these trajectories that
did not presume that they were composed of
any fundamental particles, but would construct
their interactions from self-consistency conditions
on the S-matrix. The S-matrix approach was
started by Werner Heisenberg in the 1940s
as a way of constructing a theory that did
not rely on the local notions of space and
time, which Heisenberg believed break down
at the nuclear scale. While the scale was
off by many orders of magnitude, the approach
he advocated was ideally suited for a theory
of quantum gravity.
Working with experimental data, R. Dolen,
D. Horn and C. Schmid developed some sum rules
for hadron exchange. When a particle and antiparticle
scatter, virtual particles can be exchanged
in two qualitatively different ways. In the
s-channel, the two particles annihilate to
make temporary intermediate states that fall
apart into the final state particles. In the
t-channel, the particles exchange intermediate
states by emission and absorption. In field
theory, the two contributions add together,
one giving a continuous background contribution,
the other giving peaks at certain energies.
In the data, it was clear that the peaks were
stealing from the background—the authors
interpreted this as saying that the t-channel
contribution was dual to the s-channel one,
meaning both described the whole amplitude
and included the other.
The result was widely advertised by Murray
Gell-Mann, leading Gabriele Veneziano to construct
a scattering amplitude that had the property
of Dolen-Horn-Schmid duality, later renamed
world-sheet duality. The amplitude needed
poles where the particles appear, on straight
line trajectories, and there is a special
mathematical function whose poles are evenly
spaced on half the real line— the Gamma
function— which was widely used in Regge
theory. By manipulating combinations of Gamma
functions, Veneziano was able to find a consistent
scattering amplitude with poles on straight
lines, with mostly positive residues, which
obeyed duality and had the appropriate Regge
scaling at high energy. The amplitude could
fit near-beam scattering data as well as other
Regge type fits, and had a suggestive integral
representation that could be used for generalization.
Over the next years, hundreds of physicists
worked to complete the bootstrap program for
this model, with many surprises. Veneziano
himself discovered that for the scattering
amplitude to describe the scattering of a
particle that appears in the theory, an obvious
self-consistency condition, the lightest particle
must be a tachyon. Miguel Virasoro and Joel
Shapiro found a different amplitude now understood
to be that of closed strings, while Ziro Koba
and Holger Nielsen generalized Veneziano's
integral representation to multiparticle scattering.
Veneziano and Sergio Fubini introduced an
operator formalism for computing the scattering
amplitudes that was a forerunner of world-sheet
conformal theory, while Virasoro understood
how to remove the poles with wrong-sign residues
using a constraint on the states. Claud Lovelace
calculated a loop amplitude, and noted that
there is an inconsistency unless the dimension
of the theory is 26. Charles Thorn, Peter
Goddard and Richard Brower went on to prove
that there are no wrong-sign propagating states
in dimensions less than or equal to 26.
In 1969, Yoichiro Nambu, Holger Bech Nielsen,
and Leonard Susskind recognized that the theory
could be given a description in space and
time in terms of strings. The scattering amplitudes
were derived systematically from the action
principle by Peter Goddard, Jeffrey Goldstone,
Claudio Rebbi, and Charles Thorn, giving a
space-time picture to the vertex operators
introduced by Veneziano and Fubini and a geometrical
interpretation to the Virasoro conditions.
In 1970, Pierre Ramond added fermions to the
model, which led him to formulate a two-dimensional
supersymmetry to cancel the wrong-sign states.
John Schwarz and André Neveu added another
sector to the fermi theory a short time later.
In the fermion theories, the critical dimension
was 10. Stanley Mandelstam formulated a world
sheet conformal theory for both the bose and
fermi case, giving a two-dimensional field
theoretic path-integral to generate the operator
formalism. Michio Kaku and Keiji Kikkawa gave
a different formulation of the bosonic string,
as a string field theory, with infinitely
many particle types and with fields taking
values not on points, but on loops and curves.
In 1974, Tamiaki Yoneya discovered that all
the known string theories included a massless
spin-two particle that obeyed the correct
Ward identities to be a graviton. John Schwarz
and Joel Scherk came to the same conclusion
and made the bold leap to suggest that string
theory was a theory of gravity, not a theory
of hadrons. They reintroduced Kaluza–Klein
theory as a way of making sense of the extra
dimensions. At the same time, quantum chromodynamics
was recognized as the correct theory of hadrons,
shifting the attention of physicists and apparently
leaving the bootstrap program in the dustbin
of history.
String theory eventually made it out of the
dustbin, but for the following decade all
work on the theory was completely ignored.
Still, the theory continued to develop at
a steady pace thanks to the work of a handful
of devotees. Ferdinando Gliozzi, Joel Scherk,
and David Olive realized in 1976 that the
original Ramond and Neveu Schwarz-strings
were separately inconsistent and needed to
be combined. The resulting theory did not
have a tachyon, and was proven to have space-time
supersymmetry by John Schwarz and Michael
Green in 1981. The same year, Alexander Polyakov
gave the theory a modern path integral formulation,
and went on to develop conformal field theory
extensively. In 1979, Daniel Friedan showed
that the equations of motions of string theory,
which are generalizations of the Einstein
equations of General Relativity, emerge from
the Renormalization group equations for the
two-dimensional field theory. Schwarz and
Green discovered T-duality, and constructed
two superstring theories—IIA and IIB related
by T-duality, and type I theories with open
strings. The consistency conditions had been
so strong, that the entire theory was nearly
uniquely determined, with only a few discrete
choices.
First superstring revolution
In the early 1980s, Edward Witten discovered
that most theories of quantum gravity could
not accommodate chiral fermions like the neutrino.
This led him, in collaboration with Luis Alvarez-Gaumé
to study violations of the conservation laws
in gravity theories with anomalies, concluding
that type I string theories were inconsistent.
Green and Schwarz discovered a contribution
to the anomaly that Witten and Alvarez-Gaumé
had missed, which restricted the gauge group
of the type I string theory to be SO(32).
In coming to understand this calculation,
Edward Witten became convinced that string
theory was truly a consistent theory of gravity,
and he became a high-profile advocate. Following
Witten's lead, between 1984 and 1986, hundreds
of physicists started to work in this field,
and this is sometimes called the first superstring
revolution.
During this period, David Gross, Jeffrey Harvey,
Emil Martinec, and Ryan Rohm discovered heterotic
strings. The gauge group of these closed strings
was two copies of E8, and either copy could
easily and naturally include the standard
model. Philip Candelas, Gary Horowitz, Andrew
Strominger and Edward Witten found that the
Calabi–Yau manifolds are the compactifications
that preserve a realistic amount of supersymmetry,
while Lance Dixon and others worked out the
physical properties of orbifolds, distinctive
geometrical singularities allowed in string
theory. Cumrun Vafa generalized T-duality
from circles to arbitrary manifolds, creating
the mathematical field of mirror symmetry.
Daniel Friedan, Emil Martinec and Stephen
Shenker further developed the covariant quantization
of the superstring using conformal field theory
techniques. David Gross and Vipul Periwal
discovered that string perturbation theory
was divergent. Stephen Shenker showed it diverged
much faster than in field theory suggesting
that new non-perturbative objects were missing.
In the 1990s, Joseph Polchinski discovered
that the theory requires higher-dimensional
objects, called D-branes and identified these
with the black-hole solutions of supergravity.
These were understood to be the new objects
suggested by the perturbative divergences,
and they opened up a new field with rich mathematical
structure. It quickly became clear that D-branes
and other p-branes, not just strings, formed
the matter content of the string theories,
and the physical interpretation of the strings
and branes was revealed—they are a type
of black hole. Leonard Susskind had incorporated
the holographic principle of Gerardus 't Hooft
into string theory, identifying the long highly
excited string states with ordinary thermal
black hole states. As suggested by 't Hooft,
the fluctuations of the black hole horizon,
the world-sheet or world-volume theory, describes
not only the degrees of freedom of the black
hole, but all nearby objects too.
Second superstring revolution
In 1995, at the annual conference of string
theorists at the University of Southern California
(USC), Edward Witten gave a speech on string
theory that in essence united the five string
theories that existed at the time, and giving
birth to a new 11-dimensional theory called
M-theory. M-theory was also foreshadowed in
the work of Paul Townsend at approximately
the same time. The flurry of activity that
began at this time is sometimes called the
second superstring revolution.
During this period, Tom Banks, Willy Fischler,
Stephen Shenker and Leonard Susskind formulated
matrix theory, a full holographic description
of M-theory using IIA D0 branes. This was
the first definition of string theory that
was fully non-perturbative and a concrete
mathematical realization of the holographic
principle. It is an example of a gauge-gravity
duality and is now understood to be a special
case of the AdS/CFT correspondence. Andrew
Strominger and Cumrun Vafa calculated the
entropy of certain configurations of D-branes
and found agreement with the semi-classical
answer for extreme charged black holes. Petr
Hořava and Witten found the eleven-dimensional
formulation of the heterotic string theories,
showing that orbifolds solve the chirality
problem. Witten noted that the effective description
of the physics of D-branes at low energies
is by a supersymmetric gauge theory, and found
geometrical interpretations of mathematical
structures in gauge theory that he and Nathan
Seiberg had earlier discovered in terms of
the location of the branes.
In 1997, Juan Maldacena noted that the low
energy excitations of a theory near a black
hole consist of objects close to the horizon,
which for extreme charged black holes looks
like an anti-de Sitter space. He noted that
in this limit the gauge theory describes the
string excitations near the branes. So he
hypothesized that string theory on a near-horizon
extreme-charged black-hole geometry, an anti-deSitter
space times a sphere with flux, is equally
well described by the low-energy limiting
gauge theory, the N = 4 supersymmetric Yang–Mills
theory. This hypothesis, which is called the
AdS/CFT correspondence, was further developed
by Steven Gubser, Igor Klebanov and Alexander
Polyakov, and by Edward Witten, and it is
now well-accepted. It is a concrete realization
of the holographic principle, which has far-reaching
implications for black holes, locality and
information in physics, as well as the nature
of the gravitational interaction. Through
this relationship, string theory has been
shown to be related to gauge theories like
quantum chromodynamics and this has led to
more quantitative understanding of the behavior
of hadrons, bringing string theory back to
its roots.
Criticisms
Some critics of string theory say that it
is a failure as a theory of everything. Notable
critics include Peter Woit, Lee Smolin, Philip
Warren Anderson, Sheldon Glashow, Lawrence
Krauss, and Carlo Rovelli. Some common criticisms
include:
Very high energies needed to test quantum
gravity.
Lack of uniqueness of predictions due to the
large number of solutions.
Lack of background independence.
High energies
It is widely believed that any theory of quantum
gravity would require extremely high energies
to probe directly, higher by orders of magnitude
than those that current experiments such as
the Large Hadron Collider can attain. This
is because strings themselves are expected
to be only slightly larger than the Planck
length, which is twenty orders of magnitude
smaller than the radius of a proton, and high
energies are required to probe small length
scales. Generally speaking, quantum gravity
is difficult to test because gravity is much
weaker than the other forces, and because
quantum effects are controlled by Planck's
constant h, a very small quantity. As a result,
the effects of quantum gravity are extremely
weak.
Number of solutions
String theory as it is currently understood
has a huge number of solutions, called string
vacua, and these vacua might be sufficiently
diverse to accommodate almost any phenomena
we might observe at lower energies.
The vacuum structure of the theory, called
the string theory landscape (or the anthropic
portion of string theory vacua), is not well
understood. String theory contains an infinite
number of distinct meta-stable vacua, and
perhaps 10520 of these or more correspond
to a universe roughly similar to ours—with
four dimensions, a high planck scale, gauge
groups, and chiral fermions. Each of these
corresponds to a different possible universe,
with a different collection of particles and
forces. What principle, if any, can be used
to select among these vacua is an open issue.
While there are no continuous parameters in
the theory, there is a very large set of possible
universes, which may be radically different
from each other. It is also suggested that
the landscape is surrounded by an even more
vast swampland of consistent-looking semiclassical
effective field theories, which are actually
inconsistent.
Some physicists believe this is a good thing,
because it may allow a natural anthropic explanation
of the observed values of physical constants,
in particular the small value of the cosmological
constant. The argument is that most universes
contain values for physical constants that
do not lead to habitable universes (at least
for humans), and so we happen to live in the
"friendliest" universe. This principle is
already employed to explain the existence
of life on earth as the result of a life-friendly
orbit around the medium-sized sun among an
infinite number of possible orbits (as well
as a relatively stable location in the galaxy).
Background independence
A separate and older criticism of string theory
is that it is background-dependent—string
theory describes perturbative expansions about
fixed spacetime backgrounds which means that
mathematical calculations in the theory rely
on preselecting a background as a starting
point. This is because, like many quantum
field theories, much of string theory is still
only formulated perturbatively, as a divergent
series of approximations.
Although the theory, defined as a perturbative
expansion on a fixed background, is not background
independent, it has some features that suggest
non-perturbative approaches would be background-independent—topology
change is an established process in string
theory, and the exchange of gravitons is equivalent
to a change in the background. Since there
are dynamic corrections to the background
spacetime in the perturbative theory, one
would expect spacetime to be dynamic in the
nonperturbative theory as well since they
would have to predict the same spacetime.
This criticism has been addressed to some
extent by the AdS/CFT duality, which is believed
to provide a full, non-perturbative definition
of string theory in spacetimes with anti-de
Sitter space asymptotics. Nevertheless, a
non-perturbative definition of the theory
in arbitrary spacetime backgrounds is still
lacking. Some hope that M-theory, or a non-perturbative
treatment of string theory (such as "background
independent open string field theory") will
have a background-independent formulation.
