Editing M-theory (section)

===Noncommutative geometry===

{{main article|Noncommutative geometry|Noncommutative quantum field theory}}

In geometry, it is often useful to introduce [[coordinate system|coordinates]]. For example, in order to study the geometry of the [[Euclidean plane]], one defines the coordinates {{math|''x''}} and {{math|''y''}} as the distances between any point in the plane and a pair of [[Cartesian coordinate system|axes]]. In ordinary geometry, the coordinates of a point are numbers, so they can be multiplied, and the product of two coordinates does not depend on the order of multiplication. That is, {{math|''xy'' {{=}} ''yx''}}. This property of multiplication is known as the [[commutative law]], and this relationship between geometry and the [[commutative algebra]] of coordinates is the starting point for much of modern geometry.<ref>Connes 1994, p. 1</ref>

[[Noncommutative geometry]] is a branch of mathematics that attempts to generalize this situation. Rather than working with ordinary numbers, one considers some similar objects, such as matrices, whose multiplication does not satisfy the commutative law (that is, objects for which {{math|''xy''}} is not necessarily equal to {{math|''yx''}}). One imagines that these noncommuting objects are coordinates on some more general notion of "space" and proves theorems about these generalized spaces by exploiting the analogy with ordinary geometry.<ref>Connes 1994</ref>

In a paper from 1998, [[Alain Connes]], [[Michael R. Douglas]], and [[Albert Schwarz]] showed that some aspects of matrix models and M-theory are described by a [[noncommutative quantum field theory]], a special kind of physical theory in which the coordinates on spacetime do not satisfy the commutativity property.<ref name="Connes, Douglas, and Schwarz 1998"/> This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.<ref>Nekrasov and Schwarz 1998</ref><ref>Seiberg and Witten 1999</ref>