Editing M-theory (section)

==Background==

===Quantum gravity and strings===

{{main article|Quantum gravity|String theory}}

[[Image:Open and closed strings.svg|right|thumb|alt=A wavy open segment and closed loop of string.|The fundamental objects of string theory are open and closed [[string (physics)|strings]]. ]]

One of the deepest problems in modern physics is the problem of [[quantum gravity]]. The current understanding of [[gravity]] is based on [[Albert Einstein]]'s [[general theory of relativity]], which is formulated within the framework of [[classical physics]]. However, [[fundamental interaction|nongravitational forces]] are described within the framework of [[quantum mechanics]], a radically different formalism for describing physical phenomena based on [[probability]].{{efn|For a standard introduction to quantum mechanics, see Griffiths 2004.}} A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics,{{efn|The necessity of a quantum mechanical description of gravity follows from the fact that one cannot consistently [[coupling (physics)|couple]] a classical system to a quantum one. See Wald 1984, p. 382.}} but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.{{efn|From a technical point of view, the problem is that the theory one gets in this way is not [[renormalizable]] and therefore cannot be used to make meaningful physical predictions. See Zee 2010, p. 72 for a discussion of this issue.}}

[[String theory]] is a [[mathematical theory|theoretical framework]] that attempts to reconcile gravity and quantum mechanics. In string theory, the [[point particle|point-like particles]] of [[particle physics]] are replaced by [[one-dimensional]] objects called [[string (physics)|strings]]. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can vibrate in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its [[mass]], [[charge (physics)|charge]], and other properties determined by the vibrational state of the string. In this way, all of the different elementary particles may be viewed as vibrating strings. One of the vibrational states of a string gives rise to the [[graviton]], a quantum mechanical particle that carries gravitational force.{{efn|For an accessible introduction to string theory, see Greene 2000.}}

There are several versions of string theory: [[type I string|type I]], [[type IIA string|type IIA]], [[type IIB string|type IIB]], and two flavors of [[heterotic string]] theory ({{math|[[special orthogonal group|''SO''(32)]]}} and {{math|[[E8 (mathematics)|''E''<sub>8</sub>×''E''<sub>8</sub>]]}}). The different theories allow different types of strings, and the particles that arise at low energies exhibit different [[symmetry (physics)|symmetries]]. For example, the type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while types IIA and IIB include only closed strings.<ref>Zwiebach 2009, p. 324</ref> Each of these five string theories arises as a special limiting case of M-theory. This theory, like its string theory predecessors, is an example of a quantum theory of gravity. It describes a [[force (physics)|force]] just like the familiar gravitational force subject to the rules of quantum mechanics.<ref name="Becker, Becker, and Schwarz 2007, p. 12">Becker, Becker, and Schwarz 2007, p. 12</ref>

===Number of dimensions===

{{main article|Extra dimensions|Compactification (physics)}}

[[File:Compactification example.svg|left|thumb|alt=A tubular surface and corresponding one-dimensional curve.|An example of [[compactification (physics)|compactification]]: At large distances, a two-dimensional surface with one circular dimension looks one-dimensional.]]

In everyday life, there are three familiar dimensions of space: height, width and depth. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to a four-dimensional [[spacetime]], three spatial dimensions and one time dimension. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime.<ref>Wald 1984, p. 4</ref>

In spite of the fact that the universe is well described by four-dimensional spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily.{{efn|For example, in the context of the [[AdS/CFT correspondence]], theorists often formulate and study theories of gravity in unphysical numbers of spacetime dimensions.}} There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in [[condensed matter physics]].<ref>Zee 2010, Parts V and VI</ref> Finally, there exist scenarios in which there could actually be more than four dimensions of spacetime which have nonetheless managed to escape detection.<ref>Zwiebach 2009, p. 9</ref>

One notable feature of string theory and M-theory is that these theories require [[extra dimensions]] of spacetime for their mathematical consistency. In string theory, spacetime is ''ten-dimensional'' (nine spatial dimensions, and one time dimension), {{anchor|11-dimensional spacetime}}while in M-theory it is ''eleven-dimensional'' (ten spatial dimensions, and one time dimension). In order to describe real physical phenomena using these theories, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.<ref>Zwiebach 2009, p. 8</ref>

[[Compactification (physics)|Compactification]] is one way of modifying the number of dimensions in a physical theory.{{efn|[[Dimensional reduction]] is another way of modifying the number of dimensions.}} In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles.<ref name=autogenerated1>Yau and Nadis 2010, Ch. 6</ref> In the limit where these curled-up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions.{{efn|This analogy is used for example in Greene 2000, p. 186.}}

===Dualities===

{{main article|String duality|S-duality|T-duality}}

[[File:StringTheoryDualities.svg|right|thumb|alt=A diagram indicating the relationships between M-theory and the five string theories.|upright=1.2|A diagram of string theory dualities. Yellow arrows indicate [[S-duality]]. Blue arrows indicate [[T-duality]]. These dualities may be combined to obtain equivalences of any of the five theories with M-theory.<ref>Becker, Becker, and Schwarz 2007, pp. 339–347</ref>]]

Theories that arise as different limits of M-theory turn out to be related in highly nontrivial ways. One of the relationships that can exist between these different physical theories is called [[S-duality]]. This is a relationship which says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently. [[Type I string theory]] turns out to be equivalent by S-duality to the {{math|''SO''(32)}} heterotic string theory. Similarly, [[type IIB string theory]] is related to itself in a nontrivial way by S-duality.<ref name="Becker, Becker, and Schwarz 2007">Becker, Becker, and Schwarz 2007</ref>

Another relationship between different string theories is [[T-duality]]. Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius {{math|''R''}} is equivalent to a string propagating around a circle of radius {{math|1/''R''}} in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, a string has [[momentum]] as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the [[winding number]]. If a string has momentum {{math|''p''}} and winding number {{math|''n''}} in one description, it will have momentum {{math|''n''}} and winding number {{math|''p''}} in the dual description. For example, [[type IIA string theory]] is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.<ref name="Becker, Becker, and Schwarz 2007"/>

In general, the term ''[[string duality|duality]]'' refers to a situation where two seemingly different [[physical system]]s turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be ''dual'' to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.<ref>Zwiebach 2009, p. 376</ref>

===Supersymmetry===

{{main article|Supersymmetry}}

Another important theoretical idea that plays a role in M-theory is [[supersymmetry]]. This is a mathematical relation that exists in certain physical theories between a class of particles called [[bosons]] and a class of particles called [[fermions]]. Roughly speaking, fermions are the constituents of matter, while bosons mediate interactions between particles. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa. When supersymmetry is imposed as a local symmetry, one automatically obtains a quantum mechanical theory that includes gravity. Such a theory is called a [[supergravity theory]].<ref name="Duff 1998, p. 64">Duff 1998, p. 64</ref>

A theory of strings that incorporates the idea of supersymmetry is called a [[superstring theory]]. There are several different versions of superstring theory which are all subsumed within the M-theory framework. At low [[energy|energies]], superstring theories are approximated by one of the three supergravities in ten dimensions, known as [[type I supergravity|type I]], [[type IIA supergravity|type IIA]], and [[type IIB supergravity|type IIB]] supergravity. Similarly, M-theory is approximated at low energies by supergravity in eleven dimensions.<ref name="Becker, Becker, and Schwarz 2007, p. 12"/>

===Branes===

{{main article|Brane}}

In string theory and related theories such as supergravity theories, a [[brane]] is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension {{math|''p''}}, these are called {{math|''p''}}-branes. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They can have mass and other attributes such as charge. A {{math|''p''}}-brane sweeps out a {{math|(''p''&nbsp;+&nbsp;1)}}-dimensional volume in spacetime called its ''worldvolume''. Physicists often study [[field (physics)|fields]] analogous to the [[electromagnetic field]] which live on the worldvolume of a brane. The word brane comes from the word "membrane" which refers to a two-dimensional brane.<ref>Moore 2005</ref>

In string theory, the fundamental objects that give rise to elementary particles are the one-dimensional strings. Although the physical phenomena described by M-theory are still poorly understood, physicists know that the theory describes two- and five-dimensional branes. Much of the current research in M-theory attempts to better understand the properties of these branes.{{efn|For example, see the subsections on the [[6D (2,0) superconformal field theory]] and [[ABJM superconformal field theory]].}}