Editing Matrix theory (physics)

{{Short description|Quantum mechanical model based on mathematical matrices}}
{{String theory}}
In theoretical physics, the '''matrix theory''' is a [[quantum mechanics|quantum mechanical]] model proposed in 1997 by [[Tom Banks (physicist)|Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], and [[Leonard Susskind]]; it is also known as '''BFSS matrix model''', after the authors' initials.<ref>Banks et al. 1997</ref>

==Overview==
This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional [[supergravity]]. These calculations led them to propose that the BFSS matrix model is exactly equivalent to [[M-theory]]. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting. The BFSS matrix model is also considered the worldvolume theory of a large number of D0-[[brane]]s in [[Type IIA]] string theory.<ref>[https://ncatlab.org/nlab/show/BFSS%20matrix%20model BFSS matrix model in nLab]</ref>

==Noncommutative geometry==

{{main|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>Connes, Douglas, and Schwarz 1998</ref> 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>

==Related models<!--'IKKT matrix model' redirects here-->==
Another notable matrix model capturing aspects of [[Type IIB]] string theory, the '''IKKT matrix model'''<!--boldface per WP:R#PLA-->, was constructed in 1996–97 by N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya.<ref>N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, "A Large-N Reduced Model as Superstring", ''Nucl.Phys.'' '''B498''' (1997), 467-491 (arXiv:hep-th/9612115).</ref><ref>[https://ncatlab.org/nlab/show/IKKT+matrix+model IKKT matrix model in nLab]</ref>

Recently, the relationship to Nambu dynamics is discussed.(see [[Nambu dynamics#Quantization]])

==See also==
* [[Matrix string theory]]

==Notes==
{{reflist|2}}

==References==
* {{cite journal |last1=Banks |first1=Tom |last2=Fischler |first2=Willy |last3=Schenker |first3=Stephen |last4=Susskind |first4=Leonard |date=1997 |title=M theory as a matrix model: A conjecture |journal=Physical Review D |volume=55 |issue=8 |pages=5112–5128 |doi=10.1103/physrevd.55.5112 |bibcode=1997PhRvD..55.5112B |arxiv = hep-th/9610043 |s2cid=13073785 }}
* {{cite book |last=Connes |first=Alain |date=1994 |title=Noncommutative Geometry |publisher=[[Academic Press]] |isbn=978-0-12-185860-5 |url-access=registration |url=https://archive.org/details/noncommutativege0000conn }}
* {{cite journal |last1=Connes |first1=Alain |last2=Douglas |first2=Michael |last3=Schwarz |first3=Albert |date=1998 |title=Noncommutative geometry and matrix theory |journal=Journal of High Energy Physics |volume=19981 |issue=2 |pages=003 | doi = 10.1088/1126-6708/1998/02/003 |bibcode=1998JHEP...02..003C |arxiv = hep-th/9711162 |s2cid=7562354 }}
* {{cite journal |last1=Nekrasov |first1=Nikita |last2=Schwarz |first2=Albert |date=1998 |title=Instantons on noncommutative {{math|'''R'''<sup>4</sup>}} and (2,0) superconformal six dimensional theory |journal=Communications in Mathematical Physics |volume=198 |issue=3 |pages=689–703 |doi=10.1007/s002200050490|bibcode=1998CMaPh.198..689N |arxiv = hep-th/9802068 |s2cid=14125789 }}
* {{cite journal |last1=Seiberg |first1=Nathan |last2=Witten |first2=Edward |date=1999 |title=String Theory and Noncommutative Geometry |journal=Journal of High Energy Physics |volume=1999 | doi = 10.1088/1126-6708/1999/09/032 | page=032 | issue = 9 |bibcode=1999JHEP...09..032S |arxiv = hep-th/9908142 |s2cid=668885 }}
{{String theory topics |state=collapsed}}

[[Category:String theory]]
[[Category:Leonard Susskind]]