In theoretical physics, thematrix theory is aquantum mechanical model proposed in 1997 byTom Banks,Willy Fischler,Stephen Shenker, andLeonard Susskind; it is also known asBFSS matrix model, after the authors' initials.[1]
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-dimensionalsupergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent toM-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-branes inType IIA string theory.[2]
In geometry, it is often useful to introducecoordinates. For example, in order to study the geometry of theEuclidean plane, one defines the coordinatesx andy as the distances between any point in the plane and a pair ofaxes. 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,xy =yx. This property of multiplication is known as thecommutative law, and this relationship between geometry and thecommutative algebra of coordinates is the starting point for much of modern geometry.[3]
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 whichxy is not necessarily equal toyx). 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.[4]
In a paper from 1998,Alain Connes,Michael R. Douglas, andAlbert Schwarz showed that some aspects of matrix models and M-theory are described by anoncommutative quantum field theory, a special kind of physical theory in which the coordinates on spacetime do not satisfy the commutativity property.[5] 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.[6][7]
Another notable matrix model capturing aspects ofType IIB string theory, theIKKT matrix model, was constructed in 1996–97 by N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya.[8][9]
Recently, the relationship to Nambu dynamics is discussed.(seeNambu dynamics#Quantization)
