Matrix theory (physics)
| String theory |
|---|
 |
| Fundamental objects |
| Perturbative theory |
| Non-perturbative results |
| Phenomenology |
| Mathematics |
In theoretical physics, the BFSS matrix model or matrix theory is a quantum mechanical model proposed by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind in 1997.[1]
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-branes in type IIA string theory.Cite error: A <ref> tag is missing the closing </ref> (see the help page).
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.[2]
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.[3] 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.[4][5]
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.[6][7]
See also
Notes
- ^ Banks et al. 1997
- ^ 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 nLab
References
- Banks, Tom; Fischler, Willy; Schenker, Stephen; Susskind, Leonard (1997). "M theory as a matrix model: A conjecture". Physical Review D. 55 (8): 5112. arXiv:hep-th/9610043. Bibcode:1997PhRvD..55.5112B. doi:10.1103/physrevd.55.5112.
- 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.
- 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.
- 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.