Superstring theory is an
attempt to explain all of the
particles and
fundamental forces of nature in one theory by modeling them as vibrations of tiny
supersymmetric strings.
'Superstring theory' is a shorthand for supersymmetric string theory because unlike
bosonic string theory, it is the version of
string theory that accounts for both
fermions and
bosons and incorporates
supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
to model gravity.
Since the
second superstring revolution, the five superstring theories (
Type I,
Type IIA,
Type IIB,
HO and HE) are regarded as different limits of a single theory tentatively called
M-theory.
Background
One of the deepest open problems in
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
is formulating a theory of
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
. Such a theory incorporates both the theory of
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, which describes gravitation and applies to large-scale structures, and
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
or more specifically
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, which describes the other three
fundamental forces that act on the atomic scale.
Quantum field theory, in particular the
Standard model
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
, is currently the most successful theory to describe fundamental forces, but while computing physical quantities of interest, naïvely one obtains infinite values. Physicists developed the technique of
renormalization to 'eliminate these infinities' to obtain finite values which can be experimentally tested. This technique works for three of the four fundamental forces:
Electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, the
strong force and the
weak force, but does not work for
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, which is non-renormalizable. Development of a
quantum theory of gravity therefore requires different means than those used for the other forces.
According to superstring theory, or more generally string theory, the fundamental constituents of reality are strings with radius on the order of the
Planck length (about 10
−33 cm). An appealing feature of string theory is that fundamental particles can be viewed as excitations of the string. The tension in a string is on the order of the
Planck force (10
44 newtons). The
graviton (the proposed
messenger particle of the gravitational force) is predicted by the theory to be a string with wave amplitude zero.
History
Investigating how a string theory may include fermions in its spectrum led to the invention of
supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
(in
the West) in 1971, a mathematical transformation between bosons and fermions. String theories that include fermionic vibrations are now known as "superstring theories".
Since its beginnings in the seventies and through the combined efforts of many different researchers, superstring theory has developed into a broad and varied subject with connections to
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
,
particle and
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
,
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
, and
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...
.
Absence of physical evidence
Superstring theory is based on supersymmetry. No supersymmetric particles have been discovered and initial investigation, carried out in 2011 at the
Large Hadron Collider (LHC)
and in 2006 at the
Tevatron has excluded some of the ranges.
For instance, the mass constraint of the
Minimal Supersymmetric Standard Model squarks has been up to 1.1 TeV, and
gluinos up to 500 GeV. No report on suggesting
large extra dimensions has been delivered from the LHC. There have been no principles so far to limit the number of vacua in the concept of a landscape of vacua.
Some particle physicists became disappointed by the lack of experimental verification of supersymmetry, and some have already discarded it.
Jon Butterworth at
University College London
University College London (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...
said that we had no sign of supersymmetry, even in higher energy regions, excluding the
superpartners of the top quark up to a few TeV. Ben Allanach at the University of Cambridge states that if we do not discover any new particles in the next trial at the LHC, then we can say it is unlikely to discover supersymmetry at
CERN
The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in Meyrin, western suburb of Gene ...
in the foreseeable future.
Extra dimensions
Our
physical space is observed to have
three large spatial dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
s and, along with
time
Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
, is a boundless 4-dimensional
continuum known as
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. However, nothing prevents a theory from including more than 4 dimensions. In the case of
string theory,
consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
requires spacetime to have 10 dimensions (3D regular space + 1 time(1 time dimension is not necessary, it may be multi-dimensional, according to F-theory) + 6D
hyperspace). The fact that we see only 3 dimensions of space can be explained by one of two mechanisms: either the extra dimensions are
compactified on a very small scale, or else our world may live on a 3-dimensional
submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
corresponding to a
brane
In string theory and related theories (such as supergravity), a brane is a physical object that generalizes the notion of a zero-dimensional point particle, a one-dimensional string, or a two-dimensional membrane to higher-dimensional objec ...
, on which all known particles besides gravity would be restricted.
If the extra dimensions are compactified, then the extra six dimensions must be in the form of a
Calabi–Yau manifold. Within the more complete framework of M-theory, they would have to take form of a
G2 manifold. A particular exact symmetry of string/M-theory called
T-duality (which exchanges momentum modes for
winding number
In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise aroun ...
and sends compact dimensions of radius R to radius 1/R), has led to the discovery of equivalences between different Calabi–Yau manifolds called
mirror symmetry.
Superstring theory is not the first theory to propose extra spatial dimensions. It can be seen as building upon the
Kaluza–Klein theory, which proposed a 4+1 dimensional (5D) theory of gravity. When compactified on a circle, the gravity in the extra dimension precisely describes
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
from the perspective of the 3 remaining large space dimensions. Thus the original Kaluza–Klein theory is a prototype for the unification of gauge and gravity interactions, at least at the classical level, however it is known to be insufficient to describe nature for a variety of reasons (missing weak and strong forces, lack of
parity violation, etc.) A more complex compact geometry is needed to reproduce the known gauge forces. Also, to obtain a consistent, fundamental, quantum theory requires the upgrade to string theory, not just the extra dimensions.
Number of superstring theories
Theoretical physicists were troubled by the existence of five separate superstring theories. A possible solution for this dilemma was suggested at the beginning of what is called the
second superstring revolution in the 1990s, which suggests that the five string theories might be different limits of a single underlying theory, called M-theory. This remains a
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
.
The five consistent superstring theories are:
* The
type I string has one supersymmetry in the ten-dimensional sense (16
supercharges). This theory is special in the sense that it is based on unoriented
open and
closed strings, while the rest are based on oriented closed strings.
* The
type II string theories have two supersymmetries in the ten-dimensional sense (32 supercharges). There are actually two kinds of type II strings called type IIA and type IIB. They differ mainly in the fact that the IIA theory is non-
chiral (parity conserving) while the IIB theory is chiral (parity violating).
* The
heterotic string theories are based on a peculiar hybrid of a type I superstring and a bosonic string. There are two kinds of heterotic strings differing in their ten-dimensional
gauge groups: the heterotic
''E''8×''E''8 string and the heterotic
SO(32) string. (The name heterotic SO(32) is slightly inaccurate since among the SO(32)
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, string theory singles out a quotient Spin(32)/Z
2 that is not equivalent to SO(32).)
Chiral
gauge theories can be inconsistent due to
anomalies. This happens when certain one-loop
Feynman diagrams cause a quantum mechanical breakdown of the gauge symmetry. The anomalies were canceled out via the
Green–Schwarz mechanism.
Even though there are only five superstring theories, making detailed predictions for real experiments requires information about exactly what physical configuration the theory is in. This considerably complicates efforts to test string theory because there is an astronomically high number—10
500 or more—of configurations that meet some of the basic requirements to be consistent with our world. Along with the extreme remoteness of the Planck scale, this is the other major reason it is hard to test superstring theory.
Another approach to the number of superstring theories refers to the
mathematical structure
In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...
called
composition algebra. In the findings of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
there are just seven composition algebras over the
field of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. In 1990 physicists R. Foot and G.C. Joshi in Australia stated that "the seven classical superstring theories are in one-to-one correspondence to the seven composition algebras".
Integrating general relativity and quantum mechanics
General relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
typically deals with situations involving large mass objects in fairly large regions of
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
(when it is applied to small distances it often conflicts with quantum mechanics) whereas
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
is generally reserved for scenarios at the atomic scale (small spacetime regions). The two are very rarely used together, and the most common case that combines them is in the study of
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s. Having ''peak density'', or the maximum amount of matter possible in a space, and very small area, the two must be used in synchrony to predict conditions in such places. Yet, when used together, the equations fall apart, spitting out impossible answers, such as imaginary distances and less than one dimension.
The major problem with their incongruence is that, at
Planck scale (a fundamental small unit of length) lengths, general relativity predicts a smooth, flowing surface, while quantum mechanics predicts a random, warped surface, which are nowhere near compatible. Superstring theory resolves this issue, replacing the classical idea of point particles with strings. These strings have an average diameter of the
Planck length, with extremely small variances, which completely ignores the quantum mechanical predictions of Planck-scale length dimensional warping. Also, these surfaces can be mapped as branes. These branes can be viewed as objects with a morphism between them. In this case, the morphism will be the state of a string that stretches between brane A and brane B.
Singularities are avoided because the observed consequences of "
Big Crunches" never reach zero size. In fact, should the universe begin a "big crunch" sort of process, string theory dictates that the universe could never be smaller than the size of one string, at which point it would actually begin expanding.
Mathematics
D-branes
D-branes are membrane-like objects in 10D string theory. They can be thought of as occurring as a result of a
Kaluza–Klein compactification of 11D M-theory that contains membranes. Because compactification of a geometric theory produces extra
vector fields the D-branes can be included in the action by adding an extra U(1) vector field to the string action.
:
In type I open string theory, the ends of open strings are always attached to D-brane surfaces. A string theory with more gauge fields such as SU(2) gauge fields would then correspond to the compactification of some higher-dimensional theory above 11 dimensions, which is not thought to be possible to date. Furthermore, the tachyons attached to the D-branes show the instability of those D-branes with respect to the annihilation. The tachyon total energy is (or reflects) the total energy of the D-branes.
Why five superstring theories?
For a 10 dimensional supersymmetric theory we are allowed a 32-component Majorana spinor. This can be decomposed into a pair of 16-component Majorana-Weyl (chiral)
spinors. There are then various ways to construct an invariant depending on whether these two spinors have the same or opposite chiralities:
The heterotic superstrings come in two types SO(32) and E
8×E
8 as indicated above and the type I superstrings include open strings.
Beyond superstring theory
It is conceivable that the five superstring theories are approximated to a theory in higher dimensions possibly involving membranes. Because the action for this involves quartic terms and higher so is not
Gaussian, the functional integrals are very difficult to solve and so this has confounded the top theoretical physicists.
Edward Witten has popularised the concept of a theory in 11 dimensions, called M-theory, involving membranes interpolating from the known symmetries of superstring theory. It may turn out that there exist membrane models or other non-membrane models in higher dimensions—which may become acceptable when we find new unknown symmetries of nature, such as noncommutative geometry. It is thought, however, that 16 is probably the maximum since SO(16) is a maximal subgroup of E8, the largest exceptional Lie group, and also is more than large enough to contain the
Standard Model
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
. Quartic integrals of the non-functional kind are easier to solve so there is hope for the future. This is the series solution, which is always convergent when a is non-zero and negative:
:
In the case of membranes the series would correspond to sums of various membrane interactions that are not seen in string theory.
Compactification
Investigating theories of higher dimensions often involves looking at the 10 dimensional superstring theory and interpreting some of the more obscure results in terms of compactified dimensions. For example,
D-branes
In string theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open string (physics), strings can end with Dirichlet boundary conditions, after which they are named.
D-branes are typically classified by their ...
are seen as compactified membranes from 11D M-theory. Theories of higher dimensions such as 12D F-theory and beyond produce other effects, such as gauge terms higher than U(1). The components of the extra vector fields (A) in the D-brane actions can be thought of as extra coordinates (X) in disguise. However, the ''known'' symmetries including
supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
currently restrict the
spinors to 32-components—which limits the number of dimensions to 11 (or 12 if you include two time dimensions.) Some physicists (e.g.,
John Baez et al.) have speculated that the exceptional
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Euclidean space, whereas ...
E
6, E
7 and E
8 having maximum orthogonal subgroups SO(10), SO(12) and SO(16) may be related to theories in 10, 12 and 16 dimensions; 10 dimensions corresponding to string theory and the 12 and 16 dimensional theories being yet undiscovered but would be theories based on 3-branes and 7-branes, respectively. However, this is a minority view within the string community. Since E
7 is in some sense F
4 quaternified and E
8 is F
4 octonified, the 12 and 16 dimensional theories, if they did exist, may involve the
noncommutative geometry based on the
quaternions and
octonions, respectively. From the above discussion, it can be seen that physicists have many ideas for extending superstring theory beyond the current 10 dimensional theory, but so far all have been unsuccessful.
Kac–Moody algebras
Since strings can have an infinite number of modes, the symmetry used to describe string theory is based on infinite dimensional Lie algebras. Some
Kac–Moody algebras that have been considered as symmetries for M-theory have been E
10 and E
11 and their supersymmetric extensions.
See also
*
AdS/CFT correspondence
*
dS/CFT correspondence
*
Grand unification theory
*
List of string theory topics
*
String field theory
References
Cited sources
*
*
{{DEFAULTSORT:Superstring Theory
String theory
Supersymmetry
Supersymmetric quantum field theory
Physics beyond the Standard Model