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-branes in Type IIA string theory.[2]
In geometry, it is often useful to introduce coordinates. For example, in order to study the geometry of the Euclidean plane, one defines the coordinates x and y as the distances between any point in the plane and a pair of 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, 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.[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 which xy is not necessarily equal to 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.[4]
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.[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]
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.[8][9]
Banks, Tom; Fischler, Willy; Schenker, Stephen; Susskind, Leonard (1997). "M theory as a matrix model: A conjecture". Physical Review D. 55 (8): 5112–5128. arXiv:hep-th/9610043. Bibcode:1997PhRvD..55.5112B. doi:10.1103/physrevd.55.5112. S2CID 13073785.
Connes, Alain; Douglas, Michael; Schwarz, Albert (1998). "Noncommutative geometry and matrix theory". Journal of High Energy Physics. 19981 (2): 003. arXiv:hep-th/9711162. Bibcode:1998JHEP...02..003C. doi:10.1088/1126-6708/1998/02/003. S2CID 7562354.
Nekrasov, Nikita; Schwarz, Albert (1998). "Instantons on noncommutative R4 and (2,0) superconformal six dimensional theory". Communications in Mathematical Physics. 198 (3): 689–703. arXiv:hep-th/9802068. Bibcode:1998CMaPh.198..689N. doi:10.1007/s002200050490. S2CID 14125789.
Seiberg, Nathan; Witten, Edward (1999). "String Theory and Noncommutative Geometry". Journal of High Energy Physics. 1999 (9): 032. arXiv:hep-th/9908142. Bibcode:1999JHEP...09..032S. doi:10.1088/1126-6708/1999/09/032. S2CID 668885.
March 18, 2023
matrix, theory, physics, theoretical, physics, matrix, theory, quantum, mechanical, model, proposed, 1997, banks, willy, fischler, stephen, shenker, leonard, susskind, also, known, bfss, matrix, model, after, authors, initials, contents, overview, noncommutati. In theoretical physics the matrix theory is a quantum mechanical model proposed in 1997 by Tom Banks Willy Fischler Stephen Shenker and Leonard Susskind it is also known as BFSS matrix model after the authors initials 1 Contents 1 Overview 2 Noncommutative geometry 3 Related models 4 See also 5 Notes 6 ReferencesOverview EditThis 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 branes in Type IIA string theory 2 Noncommutative geometry EditMain articles Noncommutative geometry and Noncommutative quantum field theory In geometry it is often useful to introduce coordinates For example in order to study the geometry of the Euclidean plane one defines the coordinates x and y as the distances between any point in the plane and a pair of 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 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 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 which xy is not necessarily equal to 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 4 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 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 Related models EditAnother 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 8 9 See also EditMatrix string theoryNotes Edit Banks et al 1997 BFSS matrix model in nLab Connes 1994 p 1 Connes 1994 Connes Douglas and Schwarz 1998 Nekrasov and Schwarz 1998 Seiberg and Witten 1999 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 IKKT matrix model in nLabReferences EditBanks Tom Fischler Willy Schenker Stephen Susskind Leonard 1997 M theory as a matrix model A conjecture Physical Review D 55 8 5112 5128 arXiv hep th 9610043 Bibcode 1997PhRvD 55 5112B doi 10 1103 physrevd 55 5112 S2CID 13073785 Connes Alain 1994 Noncommutative Geometry Academic Press ISBN 978 0 12 185860 5 Connes Alain Douglas Michael Schwarz Albert 1998 Noncommutative geometry and matrix theory Journal of High Energy Physics 19981 2 003 arXiv hep th 9711162 Bibcode 1998JHEP 02 003C doi 10 1088 1126 6708 1998 02 003 S2CID 7562354 Nekrasov Nikita Schwarz Albert 1998 Instantons on noncommutative R4 and 2 0 superconformal six dimensional theory Communications in Mathematical Physics 198 3 689 703 arXiv hep th 9802068 Bibcode 1998CMaPh 198 689N doi 10 1007 s002200050490 S2CID 14125789 Seiberg Nathan Witten Edward 1999 String Theory and Noncommutative Geometry Journal of High Energy Physics 1999 9 032 arXiv hep th 9908142 Bibcode 1999JHEP 09 032S doi 10 1088 1126 6708 1999 09 032 S2CID 668885 Retrieved from https en wikipedia org w index php title Matrix theory physics amp oldid 1145265231, wikipedia, wiki, book, books, library,
