M(atrix) theory: matrix quantum mechanics as a fundamental theory

Washington Taylor^*
Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

^*Electronic address: wati@mit.edu; Permanent address: Center for Theoretical Physics, MIT, Bldg. 6-306, Cambridge, MA 02139.

We denote space-time indices in 11 dimensions by capital roman letters I,J,K,…∈{0,1,…,8,9,11}, and indices in 10 dimensions by Greek letters μ,ν,…∈{0,1,…,9}.

The term “M theory” is usually used to refer to an 11-dimensional quantum theory of gravity that reduces to N=1 supergravity at low energies. It is possible that a more fundamental description of this 11-dimensional theory and string theory can be given by a model in terms of which the dimensionality of space-time is either greater than 11 or is an emergent aspect of the dynamics of the system. Generally the term M theory does not refer to such models, but usage varies. In this article we mean by M theory a consistent quantum theory of gravity in 11 dimensions.

Note that de Wit, Lüscher, and Nicolai did not resolve the question of whether a state existed with identically vanishing energy H=0 (see Sec. V.A).

The validity of this approximation is discussed by Tafjord and Periwal (1998).

Prior to and following the proof of this general result, the agreement between one-loop matrix calculations and leading long-distance interactions due to linearized supergravity was verified in specific examples of two-body backgrounds by Aharony and Berkooz (1997), Balasubramanian and Larsen (1997), Berenstein and Corrado (1997), Chepelev and Tseytlin (1997, 1998a), Lifschytz (1997, 1998a), Lifschytz and Mathur (1997), Pierre (1997, 1998), Billó, Di Vecchia, Frau, Lerda, Pesando, et al. (1998) Brandhuber et al. (1998), Fatollahi, Kaviani, and Parvizi (1998), Gopakumar and Ramgoolam (1998), Hari Dass and Sathiapalan (1998), Kabat and Taylor (1998a), Keski-Vakkuri and Kraus (1998a, 1998b), Maldacena (1998a, 1998b), Hyun, Kiem, and Shin (1999b), and Massar and Troost (2000).

Aspects of these fermionic contributions to the two-graviton interaction potential were studied by Barrio, Helling, and Polhemus (1998), Harvey (1998), Kraus (1998), McArthur (1998), Morales, Scrucca, and Serone (1998a, 1998b), Plefka, Serone, and Waldron (1998a), Hyun, Kiem, and Shin (1999b, 1999c), and Nicolai and Plefka (2000).

Previous partial results on the three-graviton problem had been found by Dine and Rajaraman (1998), Echols and Gray (1998), Fabbrichesi, Ferretti, and Iengo (1998), and Taylor and Van Raamsdonk (1998b). Further work on this problem is described by McCarthy, Susskind, and Wilkins (1998), Helling et al. (1999), Dine, Echols, and Gray (2000), and Refolli, Terzi, and Zanon (2000).

An alternative suggestion was made by Serone (1998).

Related work appeared in Hoppe (1997b).

Similar calculations were performed previously by Claudson and Halpern (1985) and by de Wit, Hoppe, and Nicolai (1988); in these earlier analyses, however, terms such as Tr[Xⁱ,X^j] and TrX^[iX^jX^kX^l] were dropped since they vanish for finite N.

In the study of solitons (for example, magnetic monopoles), states that preserve some supersymmetry are particularly interesting. For example, they solve first-order rather than second-order differential equations, and one can often find explicit solutions. In addition, their masses are frequently related to their charges under various gauged symmetries. Such solutions are called “BPS” (for Bogolmony, Prasad, and Sommerfield). D-branes are examples of BPS objects, and more generally such supersymmetric solutions have played an important role in enhancing our understanding of dualities in field theories and in string theory.

These include Banks and Motl (1997), Danielsson and Ferretti (1997), Hořava (1997), Kachru and Silverstein (1997), Lowe (1997a, 1997b), Motl (1996), Motl and Susskind (1997), Rey (1997a), and Krogh (1999a, 1999b).

Further details regarding string interactions in matrix string theory can be found in the articles of Wynter (1997, 1998, 2000), Bonelli, Bonora, and Nesti (1998, 1999), Bonelli, Bonora, Nesti, and Tomasiello (1999), Giddings, Hacquebord, and Verlinde (1999), Grignani and Semenoff (1999), Hacquebord (1999), Brax (2000), and Grignani et al. (2000). Other aspects of matrix string theory were discussed by Bonora and Chu (1997), Verlinde (1997), Kostov and Vanhove (1998), Baulieu, Laroche, and Nekrasov (1999), Billó et al. (1999), Brax and Wynter (1999), and Sugino (1999).

See, for example, Balasubramanian, Gopakumar, and Larsen (1998), Hyun (1998), Itzhaki et al. (1998), Silva (1998), Chepelev (1999), de Alwis (1999), Hyun and Kiem (1999), Jevicki and Yoneya (1999), Martinec and Sahakian (1999), Sekino and Yoneya (1999), and Yoneya (2000).

See, for example, Banks et al. (1998), Kabat and Lifschytz (2000) and references therein.

These matrix models were first developed by Witten (1997), Aharony, Berkooz, and Seiberg (1998), Aharony et al. (1998), and Ganor and Sethi (1998); for a review of this work and further developments in this direction see Banks (1999).