In theoretical physics, the matrix theory is a

quantum mechanical Quantum mechanics is the fundamental physical 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 the foundation of a ...

model proposed in 1997 by Tom Banks, Willy Fischler, Stephen Shenker, and

Leonard Susskind Leonard Susskind (; born June 16, 1940)his 60th birth anniversary was celebrated with a special symposium at Stanford University.in Geoffrey West's introduction, he gives Suskind's current age as 74 and says his birthday was recent. is an Americ ...

; it is also known as BFSS matrix model, after the authors' initials.

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 In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...

. These calculations led them to propose that the BFSS matrix model is exactly equivalent to

M-theory In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1 ...

. 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 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 ...

s in Type IIA string theory.

Noncommutative geometry

In geometry, it is often useful to introduce

coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...

. For example, in order to study the geometry of the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

, one defines the coordinates and as the distances between any point in the plane and a pair of

axes Axes, plural of ''axe'' and of ''axis'', may refer to * ''Axes'' (album), a 2005 rock album by the British band Electrelane * a possibly still empty plot (graphics) See also * Axis (disambiguation) An axis (: axes) may refer to: Mathematics ...

. 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, . This property of multiplication is known as the

commutative law In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...

, and this relationship between geometry and the

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

of coordinates is the starting point for much of modern geometry.

Noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some g ...

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 is not necessarily equal to ). 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. In a paper from 1998,

Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the , , Ohio State University and Vanderbilt University. He was awar ...

, 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. 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.

Related models

Another notable matrix model capturing aspects of Type IIB string theory, the IKKT matrix model, was constructed in 1996–97 by N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya.IKKT matrix model in nLab
/ref> Recently, the relationship to Nambu dynamics is discussed.(see Nambu dynamics#Quantization)

Notes

References

* * * * * {{String theory topics , state=collapsed String theory Leonard Susskind