String theory is a developing branch of
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. Its central core is mathematical physics,[Sometimes mathematical physics and theoretical physics are used synonymously to refer to the...]
that combines
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
and
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
into a
quantum theory of gravityQuantum gravity is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity and which reduces to...
. The
stringA string is one of the main objects of study in string theory, a branch of theoretical physics. There are different string theories, many of which are unified by M-theory...
s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of
points-M Theory:In physics, M theory is a fundamental formulation of M-theory as a Random matrix model. It is written in terms of interacting D0-branes in infinite momentum frame. It was proposed by Banks, Fischler, Shenker, and Susskind in 1996 [1]...
or surfaces too.
Since its inception as the
dual resonance modelA dual resonance model is a term used in theoretical physics which refers to the early investigation of string theory as an S-matrix theory of the strong interaction....
which described the strongly interacting
hadronIn particle physics, a hadron is a particle made of quarks held together by the strong force . Hadrons are either mesons or baryons...
s as strings, the term
string theory has changed to include any of a group of related
superstring theoriesSuperstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
which unite them. One shared property of all these theories is the
holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon...
. String theory itself comes in many different formulations, each one with a different mathematical structure, and each best describing different physical circumstances. But the principles shared by these approaches, their mutual logical consistency, and the fact that some of them easily include the standard model of particle physics, has led many physicists to believe that the theory is the correct fundamental description of nature. In particular, string theory is the first candidate for the
theory of everythingThe theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories...
(TOE), a way to describe the known
fundamental forceIn physics, fundamental interactions are the ways that the simplest particles in the universe interact with one other...
s (gravitational,
electromagneticElectromagnetism is the physics of the electromagnetic field, a field that exerts a force on particles with the property of electric charge and is reciprocally affected by the presence and motion of such particles....
,
weakThe weak interaction is one of the four fundamental interactions of nature. In the Standard Model of particle physics, it is due to the exchange of the heavy W and Z bosons...
and
strongIn particle physics, the strong interaction holds quarks and gluons together to form protons, neutrons and other particles. The strong interaction is one of the four fundamental interactions, along with gravitation, the electromagnetic force and the weak interaction...
interactions) and matter (
quarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never found in...
s and
leptonLeptons are a family of elementary particles, alongside quarks and gauge bosons . Like quarks, leptons are fermions and are subject to the electromagnetic force, the gravitational force, and weak interaction, but unlike quarks, leptons do not participate in the strong interaction.There are six...
s) in a mathematically complete system.
Many detractors criticize string theory as it has not provided quantitative experimental predictions. Like any other quantum theory of gravity, it is widely believed that testing the theory directly would require prohibitively expensive feats of engineering. Whether there are stringent indirect tests of the theory is unknown.
String theory is of interest to many
physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
s because it requires new mathematical and physical ideas to mesh together its very different mathematical formulations. One of the most inclusive of these is the 11-dimensional
M-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds the dimensionality of five superstring theories in 10 dimensions, it is believed that the 11-dimensional theory unifies all string theories...
, which requires
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
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 not described by a real number, but by a completely different type of mathematical quantity. So the notion of spacetime dimension is not fixed in string theory: it is best thought of as different in different circumstances.
String theories include objects more general than strings, called branes. The word
brane, derived from "membrane", refers to a variety of interrelated objects, such as D-branes, black p-branes and Neveu-Schwarz 5-branes. These are extended objects that are charged sources for
differential formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. A differential form of degree k, or k-form, on a smooth manifold M is a smooth section of the kth exterior power of the...
generalizations of the
vector potentialIn vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
electromagnetic field. These objects are related to one-another by a variety of dualities. Black hole-like black p-branes are identified with D-branes, which are endpoints for strings, and this identification is called Gauge-gravity duality. Research on this equivalence has led to new insights on
quantum chromodynamicsIn theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
, the fundamental theory of the
strong nuclear forceIn particle physics, the strong interaction holds quarks and gluons together to form protons, neutrons and other particles. The strong interaction is one of the four fundamental interactions, along with gravitation, the electromagnetic force and the weak interaction...
.
Overview
String theory posits that the
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s and
quarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never found in...
s within an
atomThe atom is a basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
are not 0-dimensional objects, but 1-dimensional strings. These strings can move and vibrate, giving the observed particles their flavor,
chargeIn physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...
,
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
and
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
. The strings make closed loops unless they encounter surfaces, called
D-braneIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989...
s, where they can open up into 1-dimensional lines. The endpoints of the string cannot break off the D-brane, but they can slide around on it.
String theory is a theory of gravity, an extension of
General RelativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, and the classical interpretation of strings and branes is that they are
quantum mechanicalQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
vibrating, extended charged
black holeIn general relativity, a black hole is a region of space in which the gravitational field is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come...
s. The overarching physical insight behind string theory is the
holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon...
, which states that the description of the oscillations of the surface of a
black holeIn general relativity, a black hole is a region of space in which the gravitational field is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come...
must also describe the spacetime around it. Holography demands that a low-dimensional theory describing the fluctuations of a horizon will end up describing everything that can fall through, which can be anything at all. So a theory of a black hole horizon is a theory of everything.
Finding even one consistent holographic description,
a priori, seems like a long shot, because it would be a disembodied nonlocal description of quantum gravity. In string theory, not only is there one such description, there are several different ones, each describing fluctuations of horizons with different charges and dimensions, and all of them logically fit together. So the same physical objects and interactions can be described by the fluctuations of
one-dimensional black hole horizonsA dual resonance model is a term used in theoretical physics which refers to the early investigation of string theory as an S-matrix theory of the strong interaction....
, or by
three-dimensional horizonsIn physics, the AdS/CFT correspondence , sometimes called the Maldacena duality, is the conjectured equivalence between a string theory defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more...
, or by
zero-dimensional horizons-M Theory:In physics, M theory is a fundamental formulation of M-theory as a Random matrix model. It is written in terms of interacting D0-branes in infinite momentum frame. It was proposed by Banks, Fischler, Shenker, and Susskind in 1996 [1]...
. The fact that these different descriptions describe the same physics is evidence that string theory is consistent.
An ordinary astronomical black hole does not have a convenient holographic description, because it has a
Hawking temperatureHawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects...
. String theories are formulated on cold black holes, which are those which have as much charge as possible. The first holographic theory described the scattering of one-dimensional strings, tiny loops of vibrating horizon charged with a
two-formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. A differential form of degree k, or k-form, on a smooth manifold M is a smooth section of the kth exterior power of the...
vector potential which makes a charged black hole a one-dimensional line. Fluctuations of this line horizon describe all matter, so every
elementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
can be described by a mode of oscillation of a very small segment or loop of
stringA string is one of the main objects of study in string theory, a branch of theoretical physics. There are different string theories, many of which are unified by M-theory...
. The string-length is approximately the
Planck lengthIn physics, the Planck length, denoted ℓP, is a unit of length, equal to . It is a base unit in the system of Planck units. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, Planck's constant, and the gravitational constant...
, but can be significantly bigger when the strings are weakly interacting.
All string theories predict the existence of
degrees of freedomDegrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters...
which are usually described as extra dimensions. Without fermions, bosonic strings can vibrate in a flat but unstable 26-dimensional space time. In a
superstring theorySuperstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
with fermions, the weak-coupling (no-interaction) limit describes a flat stable 10-dimensional space time. Interacting superstring theories are best thought of as configurations of an 11 dimensional supergravity theory called
M-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds the dimensionality of five superstring theories in 10 dimensions, it is believed that the 11-dimensional theory unifies all string theories...
where one or more of the dimensions are curled up so that the line-extended charged black holes become long and light.
Long, light strings can vibrate at different
resonantIn physics, resonance is the tendency of a system to oscillate at larger amplitude at some frequencies than at others. These are known as the system's resonant frequencies . At these frequencies, even small periodic driving forces can produce large amplitude vibrations, because the system...
frequencies, each such
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
describing a different elementary particle. So in string limits, any
elementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
should be thought of as a tiny vibrating line, rather than as a point. The string can vibrate in different modes just as a guitar string can produce different notes, and every mode appears as a different particle:
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
,
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
,
gluonGluons are elementary expressions of quark interaction, and are indirectly involved with the binding of protons and neutrons together in atomic nuclei...
, etc.
The only way in which strings can interact is by splitting and combining in a smooth way. It is impossible to introduce arbitrary extra matter, like point particles which interact with strings by collisions, because the particles can fall into the black hole, so holography demands that it must show up as a mode of oscillation. The only way to introduce new matter is to find gravitational backgrounds where strings can scatter consistently, or to add boundary conditions, endpoints for the strings. Some of the backgrounds are called NS-branes, which are extreme-charged black hole sheets of different dimensions. Other charged black-sheet backgrounds are the D-branes, which have an alternate description as planes where strings can end and slide. When the strings are long and light, the branes are classical and heavy. In other limits where the strings become heavy, some of the branes can become light.
Since the string theory is widely believed to be a consistent theory of
quantum gravityQuantum gravity is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity and which reduces to...
, many hope that it correctly describes our universe, making it a
theory of everythingThe theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories...
. There are known configurations which describe all the observed fundamental forces and matter but with a zero cosmological constant and some new fields. There are other configurations with different values of the cosmological constant, which are metastable but long-lived. This leads many to believe that there is at least one metastable solution which is quantitatively identical with the
standard modelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
, with a small cosmological constant, which contains dark matter and a plausible mechanism for
inflationIn physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized exponential expansion of the universe at the end of the grand unification epoch, 10
−36 seconds after the Big Bang, driven by a negative-pressure vacuum energy density...
. It is not yet known whether string theory has such a solution, nor how much freedom the theory allows to choose the details.
The full theory does not yet have a satisfactory definition in all circumstances, since the scattering of strings is most straightforwardly defined by a
perturbation theoryIn quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
. The complete
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
of high dimensional branes is not easily defined, and the behavior of string theory in cosmological settings (time-dependent backgrounds) is not fully worked out. It is also not clear if there is any principle by which string theory selects its
vacuum stateIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
, the spacetime configuration which determines the properties of our Universe (see
string theory landscapeThe string theory landscape or anthropic landscape refers to the large number of possible false vacua in string theory. The "landscape" includes so many possible configurations that it is thought by some physicists that the known laws of physics, the Standard Model and General relativity with a...
).
Basic properties
String theory can be formulated in terms of an
actionIn physics, action is an attribute of the development of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result....
principle, either the
Nambu-Goto actionThe Nambu–Goto action is the simplest invariant action in bosonic string theory. It is the starting point of the analysis of string behavior, using the principles of Lagrangian mechanics...
or the
Polyakov actionIn physics, the Polyakov action is the two-dimensional action of a conformal field theory describing the worldsheet of a string in string theory...
, which describes how strings move through space and time. In the absence of external interactions, string dynamics are governed by tension and kinetic energy, which combine to produce oscillations. The
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
of strings implies these oscillations take on discrete vibrational modes, the
spectrumAn energy spectrum is a distribution energy among a large assemblage of particles. It is a statistical representation of the wave energy as a function of the wave frequency, and an empirical estimator of the spectral function...
of the theory.
On distance scales larger than the string radius, each oscillation mode behaves as a different species of particle, with its mass, spin and charge 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 but distinct musical notes. In the analogy, different notes correspond to different particles. The only difference is the guitar is only 2-dimensional, you can strum it up, and down. In actuality the guitar strings would be every dimension, and the strings could vibrate in any direction. Meaning that the particles could move through not only our dimension, but other dimensions as well.
String theory includes both
open strings, which have two distinct endpoints, and
closed strings making a complete loop. The two types of string behave in slightly different ways, yielding two different spectra. For example, in most string theories, one of the closed string modes is the
gravitonIn physics, the graviton is a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2 In physics, the graviton is a hypothetical elementary particle that mediates the force...
, and one of the open string modes is the
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
. Because the two ends of an open string can always meet and connect, forming a closed string, there are no string theories without closed strings.
The earliest string model, the
bosonic stringBosonic string theory is the original version of string theory, developed in the late 1960s.Although it has many attractive features, it has a pair of features that render it unattractive as a physical model. Firstly it predicts only the existence of bosons whereas many physical particles are...
, incorporated only
bosonIn particle physics, bosons are particles which obey Bose–Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein. In contrast to fermions, which obey Fermi-Dirac statistics, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the...
s. This model describes, in low enough energies, a quantum gravity theory, which also includes (if open strings are incorporated as well) gauge fields such as the photon (or, more generally, any
gauge theoryGauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
). However, this model has problems. Most importantly, the theory has a fundamental instability, believed to result in the decay (at least partially) of spacetime itself. Additionally, 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
fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
s in its spectrum led to the invention of
supersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
, a mathematical relation between bosons and fermions. String theories which include fermionic vibrations are now known as
superstring theoriesSuperstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
; several different kinds have been described, but all are now thought to be different limits of
M-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds the dimensionality of five superstring theories in 10 dimensions, it is believed that the 11-dimensional theory unifies all string theories...
.
Some qualitative properties of quantum strings can be understood in a fairly simple fashion. For example, quantum strings have tension, much like regular strings made of
twineTwine is a strong thread or string composed of two or more smaller strands or yarns twisted together. More generally, the term can be applied to any thin cord.Natural fibers used for making twine include cotton, sisal, jute, hemp, henequen, and coir...
; this tension is considered a fundamental parameter of the theory. The tension of a quantum string is closely related to its size. Consider a closed loop of string, left to move through space without external forces. Its tension will tend to contract it into a smaller and smaller loop. Classical intuition suggests that it might shrink to a single point, but this would violate
HeisenbergWerner Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
's
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...
. The characteristic size of the string loop will be a balance between the tension force, acting to make it small, and the uncertainty effect, which keeps it "stretched". Consequently, the minimum size of a string is related to the string tension.
World-sheet
A point-like particle's motion may be described by drawing a graph of its position (in one or two dimensions of space) against time. The resulting picture depicts the worldline of the particle (its 'history') in
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
. By analogy, a similar graph depicting the progress of a
string as time passes by can be obtained; the string (a one-dimensional object — a small line — by itself) will trace out a surface (a two-dimensional
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
), known as the
worldsheetIn string theory, the worldsheet is a two-dimensional manifold which describes the embedding of the string in spacetime. It is a direct generalization of the familiar worldline of a particle in special and general relativity....
. The different string modes (representing different particles, such as
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
or
gravitonIn physics, the graviton is a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2 In physics, the graviton is a hypothetical elementary particle that mediates the force...
) are surface waves on this manifold.
A closed string looks like a small loop, so its worldsheet will look like a pipe or, more generally, a
Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
(a two-dimensional
oriented manifoldIn mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
) with no boundaries (i.e. no edge).
An open string looks like a short line, so its worldsheet will look like a strip or, more generally, a
Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
with a boundary.
Strings can split and connect. This is reflected by the form of their worldsheet (more accurately, by its
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
). For example, if a closed string splits, its worldsheet will look like a single pipe splitting (or connected) to two pipes (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
torusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with and not touching the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid...
connected to two pipes (one representing the ingoing string, and the other — the outgoing one). An open string doing the same thing will have its worldsheet looking like a ring connected to two strips.
Note that the process of a string splitting (or strings connecting) is a global process of the worldsheet, not a local one: locally, the worldsheet looks the same everywhere and it is not possible to determine a single point on the worldsheet where the splitting occurs. Therefore these processes are an integral part of the theory, and are described by the same dynamics that controls the string modes.
In some string theories (namely, closed strings in Type I and some versions of the
bosonic stringBosonic string theory is the original version of string theory, developed in the late 1960s.Although it has many attractive features, it has a pair of features that render it unattractive as a physical model. Firstly it predicts only the existence of bosons whereas many physical particles are...
), strings can split and reconnect in an opposite orientation (as in a
Möbius stripThe Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It is also a ruled surface...
or a
Klein bottleIn mathematics, the Klein bottle is a certain non-orientable surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a two-dimensional surface with boundary, a Klein...
). These theories are called
unoriented. Formally, the worldsheet in these theories is a
non-orientable surfaceIn mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
.
Dualities
Before the 1990s, string theorists believed there were five distinct superstring theories: open type I, closed type I, closed type IIA, closed type IIB, and the two flavors of
heterotic stringIn physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...
theory (SO(32) and
E8×E8In mathematics, E
8 is the name given to an exceptional simple Lie group of dimension 248 ; the same notation is sometimes used for its root lattice,which has rank 8....
). The thinking was that out of these five candidate theories, only one was the actual correct
theory of everythingThe theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories...
, and that theory was the one whose low energy limit, with ten spacetime dimensions
compactifiedIn physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic....
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 connected to one another as if they are each a special case of some more fundamental theory (thought to be
M-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds the dimensionality of five superstring theories in 10 dimensions, it is believed that the 11-dimensional theory unifies all string theories...
). These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.
These dualities link quantities that were also thought to be separate. 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
field theoryField theory may refer to:*Field , the theory of the algebraic concept of field*Field theory , a physical theory which employs fields in the physical sense, consisting of two types:...
and quantum
particle physicsParticle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called high energy physics, because many elementary particles do not occur under normal circumstances in nature, but can be created and detected...
. 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.
| String theories |
| Type |
Spacetime dimensions
|
Details |
| Bosonic |
26 |
Only bosonIn particle physics, bosons are particles which obey Bose–Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein. In contrast to fermions, which obey Fermi-Dirac statistics, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the...
s, no fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
s, meaning only forces, no matter, with both open and closed strings; major flaw: a particleParticle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called high energy physics, because many elementary particles do not occur under normal circumstances in nature, but can be created and detected...
with imaginary mass, called the tachyonA tachyon is a hypothetical subatomic particle that travels faster than the speed of light...
, representing an instability in the theory. |
| I |
10 |
SupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
between forces and matter, with both open and closed strings; no tachyonA tachyon is a hypothetical subatomic particle that travels faster than the speed of light...
; group symmetry is SO(32) |
| IIA |
10 |
SupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
between forces and matter, with closed strings and open strings bound to D-braneIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989...
s; no tachyonA tachyon is a hypothetical subatomic particle that travels faster than the speed of light...
; massless fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
s are non-chiralA phenomenon is said to be chiral if it is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
|
| IIB |
10 |
SupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
between forces and matter, with closed strings and open strings bound to D-braneIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989...
s; no tachyonA tachyon is a hypothetical subatomic particle that travels faster than the speed of light...
; massless fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
s are chiralA phenomenon is said to be chiral if it is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
|
| HO |
10 |
SupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
between forces and matter, with closed strings only; no tachyonA tachyon is a hypothetical subatomic particle that travels faster than the speed of light...
; heterotic, meaning right moving and left moving strings differ; group symmetry is SO(32) |
| HE |
10 |
SupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
between forces and matter, with closed strings only; no tachyonA tachyon is a hypothetical subatomic particle that travels faster than the speed of light...
; heterotic, meaning right moving and left moving strings differ; group symmetry is E8×E8In mathematics, E 8 is the name given to an exceptional simple Lie group of dimension 248 ; the same notation is sometimes used for its root lattice,which has rank 8....
|
Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional spacetime (called the
bulk), while open strings have their ends attached to
D-braneIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989...
s, which are membranes of lower dimensionality (their dimension is odd — 1, 3, 5, 7 or 9 — in type IIA and even — 0, 2, 4, 6 or 8 — in type IIB, including the time direction).
Number of dimensions
An intriguing feature of string theory is that it involves the prediction of extra dimensions. The number of dimensions is not fixed by any consistency criterion, but flat spacetime solutions do exist in the so-called "critical dimension". Cosmological solutions exist in a wider variety of dimensionalities, and these different dimensions—more precisely different values of the "effective central charge", a count of degrees of freedom which reduces to dimensionality in weakly curved regimes—are related by dynamical transitions.
Nothing in
MaxwellJames Clerk Maxwell was a Scottish theoretical physicist and mathematician. His most significant achievement was the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity, magnetism and even optics into a...
's theory of
electromagnetismElectromagnetism is the physics of the electromagnetic field, a field that exerts a force on particles with the property of electric charge and is reciprocally affected by the presence and motion of such particles....
or
EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
's
theory of relativityThe theory of relativity, or simply relativity, generally refers specifically to two theories of Albert Einstein: special relativity and general relativity...
makes this kind of prediction; these theories require physicists to insert the number of dimensions "by hand", and this number is fixed and independent of potential energy. String theory allows one to relate the number of dimensions to scalar potential energy. Technically, this happens because a
gauge anomalyIn theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...
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
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
included in a consistent theory (technically, a particle carrying a force related to an unbroken
gauge symmetryIn gauge symmetry, 'gauge' means 'measure', and symmetry means 'stays the same'. Geometry is the study of the properties of objects that do not change when they move around. An object is symmetric if some motion leaves it looking the same, for instance, rotating an equilateral triangle through 120...
) must be massless. The mass of the photon which is predicted by string theory depends on the energy of the string mode which represents the photon. This energy includes a contribution from the
Casimir effectIn quantum field theory, the Casimir effect and the Casimir-Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, placed a few micrometers apart, without any external electromagnetic field...
, 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 fluxuations 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
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
.
Compact dimensions
Two different ways have been proposed to resolve this apparent contradiction. The first is to
compactifyIn physics, a theory in D spacetime dimensions can be redefined in a lower number of dimensions d, by taking all the fields to be independent of the location in the extra D-d dimensions....
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
holonomyIn differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...
. A 6-dimensional manifold must have SU(3) structure, a particular case (
torsionlessIn differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...
) of this being SU(3) holonomy, making it a Calabi-Yau space, and a 7-dimensional manifold must have
G2In mathematics, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group G2. The group is one of the five exceptional simple Lie groups...
structure, with G
2 holonomy again being a specific, simple, case. Such spaces have been studied in attempts to relate string theory to the 4-dimensional
Standard ModelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
, 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 a 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
wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s (of the order of the compact dimension's radius), which in
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
means very high energies (see wave-particle duality).
Brane-world scenario
Another possibility is that we are "stuck" in a 3+1 dimensional (i.e. three spatial dimensions plus the time dimension) subspace of the full universe. This subspace is supposed to be a
D-braneIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989...
, hence this is known as a
braneworldBrane cosmology refers to several theories in particle physics and cosmology motivated by, but not exclusively derived from, superstring theory and M-theory.-Brane and bulk:...
theory. Some believe that some combination of the two ideas — compactification and branes — will ultimately yield the most realistic theory.
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 manifoldIn mathematics and theoretical physics, Calabi–Yau manifolds or Calabi–Yau spaces are a certain important class of compact Kähler manifolds...
s, 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 modelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
, 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.
D-branes
Another key feature of string theory is the existence of D-branes. These are membranes of different dimensionality (anywhere from a zero dimensional membrane — which is in fact a point — and up, including 2-dimensional membranes, 3-dimensional volumes and so on).
D-branes are defined by the fact that
worldsheetIn string theory, the worldsheet is a two-dimensional manifold which describes the embedding of the string in spacetime. It is a direct generalization of the familiar worldline of a particle in special and general relativity....
boundariesIn topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
are attached to them. Thus D-branes can emit and absorb closed strings; therefore they have mass (since they emit
gravitonIn physics, the graviton is a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2 In physics, the graviton is a hypothetical elementary particle that mediates the force...
s) and — in
superstring theoriesSuperstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
—
chargeIn physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...
as well (since they emit closed strings which are gauge bosons).
From the point of view of open strings, D-branes are objects to which the ends of open strings are attached. The open strings attached to a D-brane are said to "live" on it, and they give rise to
gauge theoriesGauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
"living" on it (since one of the open string modes is a
gauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...
such as the photon). In the case of one D-brane there will be one type of a gauge boson and we will have an
AbelianAn abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
gauge theory (with the gauge boson being the
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
). If there are multiple parallel D-branes there will be multiple types of gauge bosons, giving rise to a
non-AbelianIn theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group G, the elements of which do not obey the commutative law when they are multiplied. The original choice of G in the physics of electromagnetism was U, which is commutative.For a...
gauge theory.
D-branes are thus gravitational sources, on which a gauge theory "lives". This gauge theory is
coupledIn physics, two systems are coupled if they are interacting with each other. Of special interest is the coupling of two vibratory systems by means of springs or magnetic fields, etc...
to gravity (which is said to exist in the
bulk), so that normally each of these two different viewpoints is incomplete.
Gauge-gravity duality
Gauge-gravity duality is a conjectured duality between a quantum theory of gravity in certain cases and
gauge theoryGauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
in a lower number of dimensions. This means that each predicted phenomenon and quantity in one theory has an analogue in the other theory, with a "dictionary" translating from one theory to the other.
Description of the duality
In certain cases the
gauge theoryGauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
on the D-branes is
decoupledIn physics, two systems are coupled if they are interacting with each other. Of special interest is the coupling of two vibratory systems by means of springs or magnetic fields, etc...
from the gravity living in the bulk; thus open strings attached to the D-branes are not
interactingIn physics, fundamental interactions are the ways that the simplest particles in the universe interact with one other...
with closed strings. Such a situation is termed a
decoupling limit.
In those cases, the D-branes have two independent alternative descriptions. As discussed above, from the point of view of closed strings, the D-branes are gravitational sources, and thus we have a gravitational theory on spacetime with some background fields. From the point of view of open strings, the physics of the
D-braneIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989...
s is described by the appropriate gauge theory. Therefore in such cases it is often conjectured that the gravitational theory on spacetime with the appropriate background fields is dual (i.e. physically equivalent) to the gauge theory on the boundary of this spacetime (since the subspace filled by the D-branes is the boundary of this spacetime). So far, this duality has not been proven in any cases, so there is also disagreement among string theorists regarding how strong the duality applies to various models.
Examples and intuition
The most well-known example and the first one to be studied is the duality between Type IIB
supergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity...
on
AdS5 S5
(a product space of a five-dimensional
Anti de Sitter spaceIn mathematics and physics, n-dimensional anti de Sitter space, sometimes written , is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. It is the Lorentzian analogue of n-dimensional hyperbolic space, just as Minkowski space and de Sitter space are the analogues of...
and a five-sphere) on one hand, and
N = 4
supersymmetricIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
Yang-Mills theory on the four-dimensional boundary of the Anti de Sitter space (either a flat four-dimensional spacetime
R3,1 or a three-
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
with time
S3 R). This is known as the
AdS/CFT correspondenceIn physics, the AdS/CFT correspondence , sometimes called the Maldacena duality, is the conjectured equivalence between a string theory defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more...
, a name often used for Gauge / gravity duality in general.
This duality can be thought of as follows: suppose there is a spacetime with a gravitational source, for example an
extremal black holeIn theoretical physics, an extremal black hole is a black hole with the minimal possible mass that can be compatible with the given charges and angular momentum....
. When particles are far away from this source, they are described by closed strings (i.e. a gravitational theory, or usually
supergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity...
). As the particles approach the gravitational source, they can still be described by closed strings; alternatively, they can be described by objects similar to
QCD stringIn quantum chromodynamics, or more generally, quantum gauge theories with a connection which are confining, stringlike degrees of freedom called QCD strings or QCD flux tubes form. These stringlike excitations are responsible for the confinement of color charges since they are always attached to at...
s, which are made of
gauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...
s (
gluonGluons are elementary expressions of quark interaction, and are indirectly involved with the binding of protons and neutrons together in atomic nuclei...
s) and other
gauge theoryGauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
degrees of freedom. So if one is able (in a
decoupling limit) to describe the gravitational system as two separate regions — one (the
bulk) far away from the source, and the other close to the source — then the latter region can also be described by a gauge theory on D-branes. This latter region (close to the source) is termed the
near-horizon limit, since usually there is an
event horizonIn general relativity, an event horizon is a boundary in spacetime, most often an area surrounding a black hole, beyond which events cannot affect an outside observer...
around (or at) the gravitational source.
In the gravitational theory, one of the directions in spacetime is the radial direction, going from the gravitational source and away (towards the bulk). The gauge theory lives only on the D-brane itself, so it does not include the radial direction: it lives in a spacetime with one less dimension compared to the gravitational theory (in fact, it lives on a spacetime identical to the boundary of the near-horizon gravitational theory). Let us understand how the two theories are still equivalent:
The physics of the near-horizon gravitational theory involves only on-shell states (as usual in string theory), while the
field theoryField theory may refer to:*Field , the theory of the algebraic concept of field*Field theory , a physical theory which employs fields in the physical sense, consisting of two types:...
includes also off-shell
correlation functionA correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
. The on-shell states in the near-horizon gravitational theory can be thought of as describing only particles arriving from the bulk to the near-horizon region and interacting there between themselves. In the gauge theory these are "projected" onto the boundary, so that particles which arrive at the source from different directions will be seen in the gauge theory as (off-shell) quantum fluctuations far apart from each other, while particles arriving at the source from almost the same direction in space will be seen in the gauge theory as (off-shell) quantum fluctuations close to each other. Thus the angle between the arriving particles in the gravitational theory translates to the distance scale between quantum fluctuations in the gauge theory. The angle between arriving particles in the gravitational theory is related to the radial distance from the gravitational source at which the particles interact: the larger the angle, the closer the particles have to get to the source in order to interact with each other. On the other hand, the scale of the distance between quantum fluctuations in a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
is related (inversely) to the energy scale in this theory. So small radius in the gravitational theory translates to low energy scale in the gauge theory (i.e. the IR regime of the
field theoryField theory may refer to:*Field , the theory of the algebraic concept of field*Field theory , a physical theory which employs fields in the physical sense, consisting of two types:...
) while large radius in the gravitational theory translates to high energy scale in the
gauge theoryGauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
(i.e. the UV regime of the field theory).
A simple example to this principle is that if in the gravitational theory there is a setup in which the
dilatonDilaton is a hypothetical particle appearing in string theory.It is a particle of a scalar field ; a scalar field that always comes with gravity...
field (which determines the strength of the
couplingIn physics, two systems are coupled if they are interacting with each other. Of special interest is the coupling of two vibratory systems by means of springs or magnetic fields, etc...
) is decreasing with the radius, then its dual field theory will be
asymptotically freeIn physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles, such as quarks, to become arbitrarily weak at shorter distances, i.e...
, i.e. its coupling will grow weaker in high energies.
Contact with experiment
This branch of string theory may lead to new insights on
quantum chromodynamicsIn theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
, a gauge theory which is the fundamental theory of the
strong nuclear forceIn particle physics, the strong interaction holds quarks and gluons together to form protons, neutrons and other particles. The strong interaction is one of the four fundamental interactions, along with gravitation, the electromagnetic force and the weak interaction...
. To this end, it is hoped that a gravitational theory dual to quantum chromodynamics will be found.
In fact, a vague contact with experiment has already been claimed to have been achieved, though currently the alternative explanation for
quark-gluon plasmaA quark–gluon plasma is a phase of quantum chromodynamics which exists at extremely high temperature and/or density. This phase consists of free quarks and gluons, which are the basic building blocks of matter...
behavior,
Lattice QCDLattice QCD is a well established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is formulated on a grid or lattice of points in space and time....
, is doing a much better job and has already made contact with experiments in various fields with good results, though the computations are
numericalNumerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square.Being able to compute the sides...
rather than analytic.
Other possible experiments for string theory have been proposed. One is the discovery of large cosmic strings in space, formed when the high energies in the Big Bang "stretched" some strings to astronomical proportions. Other possible avenues of experiment which could help provide evidence for string theory may take place at the newly built
Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator intended to collide opposing particle beams of either protons at an energy of 7 TeV per particle or lead nuclei at an energy of 574 TeV per nucleus...
. One is the measurement of the strength of gravity on a microscopic scale, which could provide evidence for extra dimensions; if gravitons (which are closed strings) leak off the membrane, at small scales the force of gravity should be much greater than at large scales where the gravitons would have ample chance to leak away into the bulk. The discovery of
supersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
could also be considered evidence since string theory was the first theory to require it, though other theories have managed to incorporate supersymmetry as well. Also, the absence of supersymmetric particles at energies accessible to the LHC would not necessarily disprove string theory, since supersymmetry could exist but still be outside the accelerator's range.
Problems and controversy
Although string theory comes from physics, some say that string theory's current untestable status means that it should be classified as more of a mathematical framework for building models as opposed to a physical theory.
Some go further, and say that string theory as a
theory of everythingThe theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories...
is a failure. This led to a public debate in 2007, with one commentator expressing this opinion:
"For more than a generation, physicists have been chasing a will-o’-the-wisp called string theory. The beginning of this chase marked the end of what had been three-quarters of a century of progress. Dozens of string-theory conferences have been held, hundreds of new Ph.D.s have been minted, and thousands of papers have been written. Yet, for all this activity, not a single new testable prediction has been made, not a single theoretical puzzle has been solved. In fact, there is no theory so far—just a set of hunches and calculations suggesting that a theory might exist. And, even if it does, this theory will come in such a bewildering number of versions that it will be of no practical use: a Theory of Nothing." -- Jim Holt.
Is string theory predictive?
String theory as a
theory of everythingThe theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories...
has been criticized as unscientific because it is so difficult to test by experiments. The controversy concerns two properties:
- It is widely believed that any theory of quantum gravity
Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity and which reduces to...
would require extremely high energies to probe directly, higher by orders of magnitude than those that current experiments such as the Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator intended to collide opposing particle beams of either protons at an energy of 7 TeV per particle or lead nuclei at an energy of 574 TeV per nucleus...
can reach.
- String theory as it is currently understood has a huge number of equally possible solutions, called string vacua and these vacua might be sufficiently diverse to explain almost any phenomena we might observe at lower energies.
If these properties are true, string theory as a theory of everything would have little or no
predictive powerThe predictive power of a scientific theory refers to its ability to generate testable predictions. Theories with strong predictive power are highly valued, because the predictions can often encourage the falsification of the theory...
for low energy particle physics experiments. Because the theory is so difficult to test, some theoretical physicists have asked if it can even be called a
scientific theoryThe term theory has two broad sets of meanings, one used in the empirical sciences and the other used in philosophy, mathematics, logic, and across other fields in the humanities. There is considerable difference and even dispute across academic disciplines as to the proper usages of the term...
. Notable critics include
Peter WoitPeter Woit is a mathematical physicist at Columbia University.-Career:Woit graduated in 1979 from Harvard University with bachelor's and master's degrees in physics...
,
Lee SmolinLee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo....
,
Philip Warren AndersonPhilip Warren Anderson is an American physicist and Nobel laureate. Anderson has made contributions to the theories of localization, antiferromagnetism and high-temperature superconductivity.- Biography :...
, Sheldon Glashow, Lawrence Krauss, and
Carlo RovelliCarlo Rovelli is an Italian physicist and cosmologist who has worked in Italy, the USA, and France. He was born in Verona, Italy in 1956. In 1981 he graduated with a BS/MS in Physics at the University of Bologna and in 1986 he obtained his PhD at the University of Padova, Italy...
.
All string theory models are quantum mechanical, Lorentz invariant, unitary, and contain Einstein's
General RelativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
as a low energy limit. So to falsify string theory, it suffices to falsify quantum mechanics, Lorentz invariance, or general relativity. Therefore string theory is falsifiable and meets the definition of
scientific theoryIn the sciences generally, a scientific theory comprises a collection of concepts, including abstractions of observable phenomena expressed as quantifiable properties, together with rules that express relationships between observations of such concepts...
according to the
PopperianSir Karl Raimund Popper, CH, FRS, FBA was an Austrian and British philosopher and a professor at the London School of Economics. He is considered one of the most influential philosophers of science of the 20th century, and also wrote extensively on social and political philosophy...
criterion. However to constitute a convincing potential verification of string theory, a prediction should be specific to it, not shared by any quantum field theory model or by General Relativity.
One such unique prediction is
string harmonics: at sufficiently high energies—probably near the quantum gravity scale—the string-like nature of particles would become obvious. There should be heavier copies of all particles corresponding to higher vibrational states of the string. But it is not clear how high these energies are. In the most likely case, they would be 10
15 times higher than those accessible in the newest
particle acceleratorA particle accelerator is a device that uses electric fields to propel ions or charged subatomic particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator...
, the
LHCThe Large Hadron Collider is the world's largest and highest-energy particle accelerator intended to collide opposing particle beams of either protons at an energy of 7 TeV per particle or lead nuclei at an energy of 574 TeV per nucleus...
, making this prediction impossible to test with a particle accelerator in the foreseeable future.
Swampland
In response to these concerns,
Cumrun VafaCumrun Vafa کامران وفا is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
and others have challenged the idea that string theory is compatible with anything. They propose that most possible theories of low energy physics lie in the swampland. The swampland is the collection of theories which could be true if gravity wasn't an issue, but which are not compatible with string theory. An example of a theory in the swampland is quantum electrodynamics in the limit of very small electron charge. This limit is perfectly fine in quantum field theory — in fact, in this limit, the perturbation theory becomes better and better. But in string theory, at the moment the charge of the lightest charged particle becomes less than the mass in natural units, the theory becomes inconsistent.
The reason is that two such charged massive particles will attract each other gravitationally more than they repel each other electrostatically, and could be used to form black holes. If there are no light charged particles, these black holes could not decay efficiently, barring improbable conspiracies or remnants. From the study of examples, and from the analysis of black-hole evaporation, it is now accepted that theories with a small charge quantum must come with light charged particles. This is only true within string theory—there is no such restriction in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
. This means that the discovery of a new gauge group with a small quantum of charge and only heavy charged particles would falsify string theory. Since this argument is very general—relying only on black-hole evaporation and the
holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon...
, it has been suggested that this prediction would be true of any consistent holographic theory of quantum gravity, although the phrase "consistent holographic theory of quantum gravity" might very well be synonymous with "string theory".
It is notable that all the gross features of the
Standard modelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
can be embedded within String theory, so that the standard model is not in the swampland. This includes features such as non-abelian gauge groups and chiral fermions which are hard to incorporate in non-string theories of quantum gravity.
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. Although the theory has some background-independence — topology change is an established process in string theory, and the exchange of gravitons is equivalent to a change in the background — mathematical calculations in the theory rely on preselecting a background as a starting point. This is because, like many
quantum field theoriesQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, much of string theory is still only formulated
perturbativeIn quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
ly, as a
divergent seriesIn mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....
of approximations. Although nonperturbative techniques have progressed considerably — including conjectured complete definitions in
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
s satisfying certain asymptotics — a full
non-perturbativeIn Mathematics and Physics, a non-perturbative function or process is one that cannot be accurately described by Perturbation theory. An example is the function....
definition of the theory is still lacking. Some see background independence as a fundamental requirement of a theory of quantum gravity, particularly since
General RelativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
is already background independent. Some hope that
M-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds the dimensionality of five superstring theories in 10 dimensions, it is believed that the 11-dimensional theory unifies all string theories...
, or a
non-perturbativeIn Mathematics and Physics, a non-perturbative function or process is one that cannot be accurately described by Perturbation theory. An example is the function....
treatment of string theory (
string field theoryString field theory is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory...
was thought to be non-perturbative in the 1980s) have a background-independent formulation.
Supersymmetry breaking
A central problem for applications is that the best understood backgrounds of string theory preserve much of the supersymmetry of the underlying theory, which results in time-invariant spacetimes: currently string theory cannot deal well with time-dependent, cosmological backgrounds. However, several models have been proposed to explain supersymmetry breaking, most notably the KKLT model, which incorporates branes and fluxes to make a metastable compactification.
The vacuum structure of the theory, called the
string theory landscapeThe string theory landscape or anthropic landscape refers to the large number of possible false vacua in string theory. The "landscape" includes so many possible configurations that it is thought by some physicists that the known laws of physics, the Standard Model and General relativity with a...
, is not well understood. String theory contains an infinite number of distinct meta-stable vacua, and perhaps 10
500 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.
Some physicists believe this is a good thing, because it may allow a natural
anthropic explanationIn physics and cosmology, the anthropic principle is the collective name for several ways of asserting that physical and chemical theories, especially astrophysics and cosmology, need to take into account that there is life on Earth, and that one form of that life, Homo sapiens, has attained...
of the observed values of
physical constantA physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...
s, in particular the small value of the
cosmological constantIn physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
. The argument is that most universes contain values for physical constants which do not lead to habitable universes (at least for humans), and so we happen to live in the most "friendly" 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). However, the cosmological version of the anthropic principle remains highly controversial because it would be difficult if not impossible to
PopperSir Karl Raimund Popper, CH, FRS, FBA was an Austrian and British philosopher and a professor at the London School of Economics. He is considered one of the most influential philosophers of science of the 20th century, and also wrote extensively on social and political philosophy...
falsifyFalsification may mean:*The act of disproving a proposition, hypothesis, or theory. *Forgery, the act of producing something that lacks authenticity with the intent to commit fraud or deception...
; so many do not accept it as scientific.
Other testability criteria
Many physicists strongly oppose the idea that string theory is not falsifiable, among them
Sylvester James GatesSylvester James Gates, Jr. is a noted American theoretical physicist. He received BS and PhD degrees from Massachusetts Institute of Technology, the latter in 1977. His doctoral thesis was the first thesis at MIT to deal with supersymmetry. Gates is currently the John S...
: "So, the next time someone tells you that string theory is not testable, remind them of the AdS/CFT connection ..." AdS/CFT relates string theory to gauge theory, and allows contact with low energy experiments in
quantum chromodynamicsIn theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
. This type of string theory, which only describes the strong interactions, is much less controversial today than string theories of everything (although two decades ago, it was the other way around).
In addition, Gates points out that the grand unification natural in string theories of everything requires that the coupling constants of the four forces meet at one point under renormalization group rescaling. This is also a falsifiable statement, but it is not restricted to string theory, but is shared by grand unified theories. The
LHCThe Large Hadron Collider is the world's largest and highest-energy particle accelerator intended to collide opposing particle beams of either protons at an energy of 7 TeV per particle or lead nuclei at an energy of 574 TeV per nucleus...
will be used both for testing AdS/CFT, and to check if the electroweakstrong unification does happen as predicted.
History
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 EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
. The first person to add a
fifth dimensionIn physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space. When N=5, the space consisting of all locations with a nonzero fifth number is called the fifth dimension....
to
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
was German mathematician
Theodor KaluzaTheodor Franz Eduard Kaluza was a German mathematician and physicist known for the Kaluza-Klein theory involving field equations in five-dimensional space...
in 1919, who noted that gravity in five dimensions describes both gravity and electromagnetism in four. In 1926, the Swedish physicist
Oskar KleinOskar Benjamin Klein was a Swedish theoretical physicist.Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Dr. Gottlieb Klein and Antonie Levy...
gave a physical interpretation of the unobservable extra dimension--- it is wrapped into a small circle. Einstein introduced a
non-symmetric geometric tensorIn mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric...
, 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
hadronIn particle physics, a hadron is a particle made of quarks held together by the strong force . Hadrons are either mesons or baryons...
s, the
subatomic particleIn physics, subatomic particles are the particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...
s like the
protonThe proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H
+...
and
neutronThe neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutron are usually found in atomic nuclei. The nuclei of most atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of protons in a...
which feel the
strong interactionIn particle physics, the strong interaction holds quarks and gluons together to form protons, neutrons and other particles. The strong interaction is one of the four fundamental interactions, along with gravitation, the electromagnetic force and the weak interaction...
. In the 1960s,
Geoffrey ChewGeoffrey F. Chew is an American theoretical physicist.He has worked as a professor of physics at the UC Berkeley since 1957 and has been an emeritus since 1991. Chew holds a PhD in theoretical particle physics from the University of Chicago. Between 1950 and 1956, he was a physics faculty member...
and
Steven FrautschiSteven Frautschi is an American theoretical physicist, Professor of Physics at the California Institute of Technology. He is known for his contributions to the bootstrap theory of the strong interactions....
discovered that the
mesonIn particle physics, mesons are subatomic particles composed of one quark and one antiquark. They are part of the hadron particle family—particles made of quarks. The other members of the hadron family are the baryons—subatomic particles composed of three quarks...
s make families called Regge trajectories with masses related to spins in a way that was later understood by
Yoichiro Nambuis a Japanese-born American physicist, currently a professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded the Nobel Prize in Physics in 2008 for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics.-Early...
,
Holger Bech NielsenHolger Bech Nielsen is a Danish theoretical physicist, professor at the Niels Bohr Institute, at the University of Copenhagen, where he started studying physics in 1961....
and
Leonard SusskindLeonard Susskind is the Felix Bloch professor of theoretical physics at Stanford University, whose research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...
to be the relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories which did not presume that they were composed of any fundamental particles, but would construct their interactions from
self-consistency conditionsIn physics, the term bootstrap model is used for a class of theories that use very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles...
on the S-matrix. The
S-matrix approachS-Matrix theory was a proposal for replacing local quantum field theory as the basic principle of elemetary particle physics.It avoided the notion of space and time by replacing it with abstract mathematical properties of the S-matrix...
was started by
Werner HeisenbergWerner Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
in the 1940s as a way of constructing a theory which 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. Schmidt developed some
sum ruleSum rule may refer to:*Sum rule in differentiation*Sum rule in integration*Rule of sum, a counting principle in combinatorics*Sum rule in quantum mechanics...
s 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 which 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-MannMurray Gell-Mann is an American physicist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles....
, leading
Gabriele VenezianoGabriele Veneziano is an Italian theoretical physicist and a founder of string theory. Between 1968 and 1972 he worked at MIT and CERN. In 1972 he became Amos de Shalit Professor of Physics at the Weizmann Institute of Science and in 1976 he was offered a position in the Theory Division at CERN in...
to construct a scattering amplitude which had the property of Dolen-Horn-Schmidt 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 functionIn mathematics, the Gamma function is an extension of the factorial function to real and complex numbers...
--- 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 which 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 which appears in the theory, an obvious self-consistency condition, the lightest particle must be a
tachyonA tachyon is a hypothetical subatomic particle that travels faster than the speed of light...
.
Miguel VirasoroMiguel Angel Virasoro is an Argentine physicist who did most of his work in Italy. The Virasoro algebra is named after him. Together with Giorgio Parisi and Marc Mezard he discovered theUltrametric organization of low temperature spin glass states....
and Joel Shapiro found a different amplitude now understood to be that of closed strings, while Ziro Koba and
Holger NielsenHolger Bech Nielsen is a Danish theoretical physicist, professor at the Niels Bohr Institute, at the University of Copenhagen, where he started studying physics in 1961....
generalized Veneziano's integral representation to multiparticle scattering. Veneziano and Sergio Fubini introduced an operator formalism for computing the scattering amplitudes which 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 ThornCharles Thorn is a Professor of Physics at University of Florida in Gainesville, Florida. He played an important role in the development of Dual Models and String Theory. Among his contributions is the proof of the non-existence of ghosts in certain aspects of the theory. The Goddard–Thorn theorem...
,
Peter GoddardPeter Goddard is a mathematical physicist who works in string theory and conformal field theory. Among his manycontributions to these fields is the no-ghost theorem ....
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 Nambuis a Japanese-born American physicist, currently a professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded the Nobel Prize in Physics in 2008 for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics.-Early...
,
Holger Bech NielsenHolger Bech Nielsen is a Danish theoretical physicist, professor at the Niels Bohr Institute, at the University of Copenhagen, where he started studying physics in 1961....
and
Leonard SusskindLeonard Susskind is the Felix Bloch professor of theoretical physics at Stanford University, whose research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...
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 GoddardPeter Goddard is a mathematical physicist who works in string theory and conformal field theory. Among his manycontributions to these fields is the no-ghost theorem ....
,
Jeffrey GoldstoneJeffrey Goldstone is a British-born theoretical physicist and an emeritus physics faculty at MIT Center for Theoretical Physics.He worked at the University of Cambridge until 1977....
, Claudio Rebbi and
Charles ThornCharles Thorn is a Professor of Physics at University of Florida in Gainesville, Florida. He played an important role in the development of Dual Models and String Theory. Among his contributions is the proof of the non-existence of ghosts in certain aspects of the theory. The Goddard–Thorn theorem...
, giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the
Virasoro conditionsIn mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in string theory.-Definition:...
.
In 1970,
Pierre RamondPierre Ramond is a Distinguished Professor of Physics at University of Florida in Gainesville, Florida.He played an important role in the development of superstring theory.- Academic career :...
added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. John Schwarz and
André NeveuAndré Neveu is a French physicist working on string theory and quantum field theory who coinvented the Neveu-Schwarz algebra and the Gross-Neveu model.-References:*...
added another sector to the fermi theory a short time later. In the fermion theories, the critical dimension was 10.
Stanley MandelstamStanley Mandelstam is a South African-born theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations...
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 Kakuis an American theoretical physicist specializing in string field theory, and a futurist. He is a popularizer of science, host of two radio programs and a best-selling author.-Early life and education:...
and Keiji Kikkawa gave a different formulation of the bosonic string, as a
string field theoryString field theory is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory...
, with infinitely many particle types and with fields taking values not on points, but on loops and curves.
In 1974,
Tamiaki YoneyaTamiaki Yoneya is a physicist. Independently of Joel Scherk and John H. Schwarz, he realized that string theory describes, among other things, the force of gravity. Yoneya has worked on the stringy extension of the uncertainty principle for many years....
discovered that all the known string theories included a massless spin-two particle which obeyed the correct Ward identities to be a graviton. John Schwarz and
Joel ScherkJoël Scherk was a physicist who studied string theory and supergravity. Together with John H. Schwarz, he figured out that string theory was a theory of quantum gravity in 1974...
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 chromodynamicsIn theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
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 the work of a handful of devotees. Ferdinando Gliozzi, Joel Scherk, and
David OliveDavid Olive FRS, is a British theoretical physicist. Olive made fundamental contributions to the string theory and duality theory. He was Professor of physics at Imperial College, London. He later moved to Swansea University to help set up the new theoretical physics group.He was awarded the Dirac...
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 GreenMichael Boris Green FRS is a British physicist and one of the pioneers of string theory. He is a professor of Theoretical Physics at Cambridge University in England. On 19 October 2009 he was confirmed as the next Lucasian Professor of Mathematics, succeeding Stephen Hawking on 1 November...
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 FriedanDaniel Friedan is an American theoretical physicist and is one of two sons of the feminist author and activist Betty Friedan. He earned his Ph.D. from the University of California, Berkeley in 1980 and was named a MacArthur Fellow in 1987....
showed that the equations of motions of string theory, which are generalizations of the Einstein equations of
General RelativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, emerge from the
Renormalization groupIn theoretical physics, renormalization group refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views it at different distance scales. In particle physics it reflects the changes in the underlying force laws as one varies the energy scale at...
equations for the two-dimensional field theory. Schwarz and Green discovered T-duality, and constructed two different 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.
In the early 1980s,
Edward WittenEdward Witten is an American theoretical physicist and professor at the Institute for Advanced Study, who is widely known as “the most brilliant physicist of his generation”, and "one of the world's greatest living physicists, perhaps even Einstein's successor". He is a leading researcher in...
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
anomaliesIn theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics–usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with some other fields. The adjective "gravitational" is derived from...
, concluding that type I string theories were inconsistent. Green and Schwarz, while working on the
Green-Schwarz mechanismThe Green-Schwarz mechanism is the main discovery that started the first superstring revolution in superstring theory.-Discovery:In 1984, Michael Green and John H...
, 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 revolutionIn physics, the first superstring revolution is a period of important discoveries in string theory roughly between 1984 and 1986. It was realised that string theory was capable of describing all elementary particles as well as the interactions between them. Hundreds of physicists started to work on...
.
During this period,
David GrossDavid Jonathan Gross is an American particle physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for his discovery of asymptotic freedom....
, Jeffrey Harvey,
Emil MartinecEmil Martinec is an American theoretical physicist born in 1958. He graduated from Northwestern University in 1979 and obtained his Ph.D. from Cornell University in 1984. He spent the last two years of his graduate education at SLAC after his dissertation advisor, Michael Peskin, left Cornell for...
, 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 HorowitzGary T. Horowitz is an American theoretical physicist, author, and professor at the University of California Santa Barbara who has been cited as a figure in the historical development of quantum gravity.-Professional overview:...
,
Andrew StromingerAndrew Strominger is an American theoretical physicist who works on string theory and son of Jack L. Strominger. He is currently a professor at Harvard University and a senior fellow at the Society of Fellows...
and Edward Witten found that the Calabi-Yau manifolds are the compactifications which 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 VafaCumrun Vafa کامران وفا is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
generalized T-duality from circles to arbitrary manifolds, creating the mathematical field of
mirror symmetryIn physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden...
.
David GrossDavid Jonathan Gross is an American particle physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for his discovery of asymptotic freedom....
and Vipul Periwal discovered that string perturbation theory was divergent in a way that suggested that new non-perturbative objects were missing.
In the 1990s,
Joseph PolchinskiJoseph Polchinski is a physicist working on string theory. He graduated from Canyon del Oro High School in Tucson, Arizona in 1971, obtained his B.S. degree from Caltech in 1975, and his Ph.D. from the University of California, Berkeley in 1980 under the supervision of Stanley Mandelstam...
discovered that the theory requires higher-dimensional objects, called
D-braneIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989...
s 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 SusskindLeonard Susskind is the Felix Bloch professor of theoretical physics at Stanford University, whose research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...
had incorporated the
holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon...
of
Gerardus 't HooftGerardus 't Hooft is a professor in theoretical physics at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with Martinus J. G. Veltman "for elucidating the quantum structure of electroweak interactions". Asteroid 9491 Thooft is named in his honor; he has written a...
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.
In 1995, at the annual conference of string theorists at the University of Southern California (USC),
Edward WittenEdward Witten is an American theoretical physicist and professor at the Institute for Advanced Study, who is widely known as “the most brilliant physicist of his generation”, and "one of the world's greatest living physicists, perhaps even Einstein's successor". He is a leading researcher in...
gave a speech on string theory that essentially united the five string theories that existed at the time, and giving birth to a new 11-dimensional theory called
M-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds the dimensionality of five superstring theories in 10 dimensions, it is believed that the 11-dimensional theory unifies all string theories...
. M-theory was also foreshadowed in the work of
Paul TownsendPaul Townsend is a British physicist, currently a Professor of Theoretical Physics in Cambridge University's Department of Applied Mathematics and Theoretical Physics. He is notable for his work on string theory....
at approximately the same time. The flurry of activity which began at this time is sometimes called the
second superstring revolutionThe second superstring revolution was the intense wave of breakthroughs in string theory that took place approximately between 1994 and 1997.The different versions of superstring theory were unified, as long hoped, by new equivalences. These are known as S-duality, T-duality, U-duality, mirror...
.
During this period,
Tom BanksTom Banks is a theoretical physicist at University of California, Santa Cruz and a professor at Rutgers University. His work centers around string theory and its applications to high energy particle physics and cosmology...
,
Willy FischlerWilly Fischler born in 1949 in Antwerpen, Belgium is a theoretical physicist and string theorist. He is currently the Jane and Roland Blumberg Centennial Professor of Physics at the University of Texas at Austin, where he is affiliated with the Weinberg theory group.Fischler is, among other things,...
Stephen ShenkerStephen Shenker is an American theoretical physicist who works on string theory. He is currently a professor at Stanford University and director of the Stanford Institute for Theoretical Physics...
and
Leonard SusskindLeonard Susskind is the Felix Bloch professor of theoretical physics at Stanford University, whose research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...
formulated a full holographic description of M-theory on IIA D0 branes, the first definition of string theory that was fully non-perturbative and a concrete mathematical realization of the
holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon...
.
Andrew StromingerAndrew Strominger is an American theoretical physicist who works on string theory and son of Jack L. Strominger. He is currently a professor at Harvard University and a senior fellow at the Society of Fellows...
and
Cumrun VafaCumrun Vafa کامران وفا is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
calculated the entropy of certain configurations of D-branes and found agreement with the semi-classical answer for extreme charged black holes.
Petr HoravaPetr Hořava is a Czech string theorist. He is well-known for his articles written with Edward Witten about the Hořava-Witten domain walls in M-theory. These articles demonstrated that the ten-dimensional heterotic string theory could be produced from 11-dimensional M-theory by making one of the...
and Edward 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 SeibergNathan "Nati" Seiberg, born in 1956, is an Israel-born American theoretical physicist who works on string theory. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, USA...
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 spaceIn mathematics and physics, n-dimensional anti de Sitter space, sometimes written , is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. It is the Lorentzian analogue of n-dimensional hyperbolic space, just as Minkowski space and de Sitter space are the analogues of...
. 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 theoryGauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
, the
N=4 supersymmetric Yang-Mills theory. This hypothesis, complemented by converging work due to
Steven GubserSteven S. Gubser is a professor of physics at Princeton University. His research focuses on theoretical particle physics, especially string theory, and the AdS/CFT correspondence. He is a widely cited scholar in these and other related areas....
,
Igor KlebanovIgor R. Klebanov is a theoretical physicist whose research is centered on relations between string theory and quantum gauge field theory. Born in Russia, he emigrated to the U.S. as a teenager. He received his undergraduate education at MIT, and his Ph.D. degree at Princeton University as a student...
and Alexander Polyakov, is called the
AdS/CFT correspondenceIn physics, the AdS/CFT correspondence , sometimes called the Maldacena duality, is the conjectured equivalence between a string theory defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more...
and it is now well-accepted. It is a concrete realization of the
holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon...
, which has far-reaching implications for
black holeIn general relativity, a black hole is a region of space in which the gravitational field is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come...
s,
localityIn physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...
and
informationInformation as a concept has many meanings, from everyday usage to technical settings. The concept of information is closely related to notions of constraint, communication, control, data, form, instruction, knowledge, meaning, mental stimulus, pattern, perception, and representation.The English...
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 chromodynamicsIn theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
and this has led to more quantitative understanding of the behavior of
hadronIn particle physics, a hadron is a particle made of quarks held together by the strong force . Hadrons are either mesons or baryons...
s, bringing string theory back to its roots.
In popular culture
- The book The Elegant Universe
The Elegant Universe is a book by Brian Greene published in 1999 which introduces string theory and provides a comprehensive though non-technical assessment of the theory and some of its shortcomings.-Themes:...
by Brian GreeneBrian Greene is an American theoretical physicist and one of the best-known string theorists. He has been a professor at Columbia University since 1996. Greene has worked on mirror symmetry, relating two different Calabi-Yau manifolds...
, Professor of Physics at Columbia UniversityColumbia University in the City of New York is a private university in the United States and a member of the Ivy League. Columbia's main campus lies in the Morningside Heights neighborhood in the borough of Manhattan, in New York City...
, on the subject of string theory, was adapted into a three-hour documentary for NovaNova is a popular science television series from the U.S. produced by WGBH Boston. It can be seen on the Public Broadcasting Service in the United States, and in more than 100 other countries...
.
- In The Big Bang Theory
The Big Bang Theory is an American sitcom created and executive produced by Chuck Lorre and Bill Prady. It premiered on CBS September 24, 2007....
, the character Sheldon CooperDr. Sheldon Cooper is a fictional character on the CBS television series The Big Bang Theory, portrayed by actor Jim Parsons. A Caltech theoretical physicist, Sheldon is Leonard Hofstadter's best friend, roommate, and colleague. He is cynical, believing that Leonard is only setting himself up for...
, being a theoretical physicist, has made references to string theory several times.
Popular books and articles
– An interview with
Leonard SusskindLeonard Susskind is the Felix Bloch professor of theoretical physics at Stanford University, whose research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...
, the theoretical physicist who discovered that string theory is based on one-dimensional objects and now is promoting the idea of
multiple universesThe multiverse is the hypothetical set of multiple possible universes that together comprise everything that physically exists: the entirety of space and time, all forms of matter, energy and momentum, and the physical laws and constants that govern them...
.
- Klebanov, Igor
Igor R. Klebanov is a theoretical physicist whose research is centered on relations between string theory and quantum gauge field theory. Born in Russia, he emigrated to the U.S. as a teenager. He received his undergraduate education at MIT, and his Ph.D. degree at Princeton University as a student...
and Maldacena, Juan (January 2009). Solving Quantum Field Theories via Curved Spacetimes. Physics TodayPhysics Today magazine, created in 1948, is the membership journal of the American Institute of Physics. It is provided to 130,000 members of twelve physics societies, including the American Physical Society...
.
- Taubes, Gary
Gary Taubes is an American science writer. He is the author of Nobel Dreams , Bad Science: The Short Life and Weird Times of Cold Fusion , and Good Calories, Bad Calories , which is titled The Diet Delusion in the UK...
(November 1986). "Everything's Now Tied to Strings" Discover Magazine vol 7, #11. (Popular article, probably the first ever written, on the first superstring revolutionIn physics, the first superstring revolution is a period of important discoveries in string theory roughly between 1984 and 1986. It was realised that string theory was capable of describing all elementary particles as well as the interactions between them. Hundreds of physicists started to work on...
.) – An easy nontechnical article on the very basics of the theory.
Two nontechnical books that are critical of string theory:
Textbooks
- Becker, Katrin, Becker, Melanie, and John H. Schwarz (2007) String Theory and M-Theory: A Modern Introduction . Cambridge University Press. ISBN 0-521-86069-5
- Binétruy, Pierre (2007) Supersymmetry: Theory, Experiment, and Cosmology. Oxford University Press. ISBN 978-0-19-850954-7.
- Dine, Michael (2007) Supersymmetry and String Theory: Beyond the Standard Model. Cambridge University Press. ISBN 0-521-85841-0.
- Gasperini, Maurizio (2007) Elements of String Cosmology. Cambridge University Press. ISBN 978-0-521-86875-4.
- Michael Green
Michael Boris Green FRS is a British physicist and one of the pioneers of string theory. He is a professor of Theoretical Physics at Cambridge University in England. On 19 October 2009 he was confirmed as the next Lucasian Professor of Mathematics, succeeding Stephen Hawking on 1 November...
, John H. Schwarz and Edward WittenEdward Witten is an American theoretical physicist and professor at the Institute for Advanced Study, who is widely known as “the most brilliant physicist of his generation”, and "one of the world's greatest living physicists, perhaps even Einstein's successor". He is a leading researcher in...
(1987) Superstring theory. Cambridge University Press. The original textbook.
- Vol. 1: Introduction. ISBN 0-521-35752-7.
- Vol. 2: Loop amplitudes, anomalies and phenomenology. ISBN 0-521-35753-5.
- Kiritsis, Elias (2007) String Theory in a Nutshell. Princeton University Press. ISBN 978-0-691-12230-4.
- Polchinski, Joseph
Joseph Polchinski is a physicist working on string theory. He graduated from Canyon del Oro High School in Tucson, Arizona in 1971, obtained his B.S. degree from Caltech in 1975, and his Ph.D. from the University of California, Berkeley in 1980 under the supervision of Stanley Mandelstam...
(1998) String Theory. Cambridge University Press.
- Vol. 1: An introduction to the bosonic string. ISBN 0-521-63303-6.
- Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4.
- Szabo, Richard J. (Reprinted 2007) An Introduction to String Theory and D-brane Dynamics. Imperial College Press. ISBN 978-1-86094-427-7.
- Zwiebach, Barton
Barton Zwiebach is a string theorist and professor at the Massachusetts Institute of Technology, born in Lima, Perú. His undergraduate work was in Electrical Engineering at the Universidad Nacional de Ingenieria in Peru, from which he graduated in 1977.His graduate work was in Physics, at the...
(2004) A First Course in String Theory. Cambridge University Press. ISBN 0-521-83143-1. Contact author for errata.
Technical and critical:
External links
- Superstring Theory Perimeter Institute for Theoretical Physics – A Three-Hour Miniseries with Brian Greene
Brian Greene is an American theoretical physicist and one of the best-known string theorists. He has been a professor at Columbia University since 1996. Greene has worked on mirror symmetry, relating two different Calabi-Yau manifolds...
by NOVA (original PBS Broadcast Dates: October 28, 8-10 p.m. and November 4, 8-9 p.m., 2003). Various images, texts, videos and animations explaining string theory. – A project by a string physicist explaining aspects of string theory to a broad audience.
- Dialogue on the Foundations of String Theory at MathPages – Online tutorial. – A moderated newsgroup for discussion of string theory (a theory of quantum gravity and unification of forces) and related fields of high-energy physics. – Four lectures, presented at the NATO
The North Atlantic Treaty Organization ); ), also called "the Atlantic Alliance", is an intergovernmental military alliance based on the North Atlantic Treaty which was signed on April 4, 1949...
Advanced Study Institute on Techniques and Concepts of High Energy Physics, St. Croix, Virgin IslandsThe Virgin Islands are an archipelago, part of the Leeward Islands in the Caribbean Sea. The Leeward Islands are the northern islands of the Lesser Antilles, where the Caribbean Sea meets the western Atlantic Ocean....
, in June 2000, and addressed to an audience of graduate students in experimental high energy physics, survey basic concepts in string theory. – Slides and audio from an Ed Witten lecture where he introduces string theory and discusses its challenges. – Invited Lecture at COSLAB 2004, held at Ambleside, Cumbria, United Kingdom, from 10 to 17 September 2004. – A guide to the string theory literature. – A comprehensive compilation of materials concerning string theory. Created by an international team of students. – A criticism of string theory. – A blog critical of string theory. – Scholar search}}
- A website dedicated to creative writing inspired by string theory. — An up-to-date and thorough review of string theory in a popular way.
- Woit, Peter. Not Even Wrong: The Failure of String Theory & the Continuing Challenge to Unify the Laws of Physics, 2006. ISBN 0-224-07605-1 (Jonathan Cape), ISBN 0-465-09275-6 (Basic Books)