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

Washington Taylor

doi:10.1103/RevModPhys.73.419

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

Washington Taylor

Rev. Mod. Phys. 73, 419 – Published 8 June 2001

Abstract

This article reviews the matrix model of M theory. M theory is an 11-dimensional quantum theory of gravity that is believed to underlie all superstring theories. M theory is currently the most plausible candidate for a theory of fundamental physics which reconciles gravity and quantum field theory in a realistic fashion. Evidence for M theory is still only circumstantial—no complete background-independent formulation of the theory exists as yet. Matrix theory was first developed as a regularized theory of a supersymmetric quantum membrane. More recently, it has appeared in a different guise as the discrete light-cone quantization of M theory in flat space. These two approaches to matrix theory are described in detail and compared. It is shown that matrix theory is a well-defined quantum theory that reduces to a supersymmetric theory of gravity at low energies. Although its fundamental degrees of freedom are essentially pointlike, higher-dimensional fluctuating objects (branes) arise through the non-Abelian structure of the matrix degrees of freedom. The problem of formulating matrix theory in a general space-time background is discussed, and the connections between matrix theory and other related models are reviewed.

DOI:https://doi.org/10.1103/RevModPhys.73.419

Authors & Affiliations

Washington Taylor^a

Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

^aElectronic address: wati@mit.edu; Permanent address: Center for Theoretical Physics, MIT, Bldg. 6-306, Cambridge, MA 02139.
^bWe denote space-time indices in 11 dimensions by capital roman letters $I, J, K, \dots \in {0, 1, \dots, 8, 9, 11},$ and indices in 10 dimensions by Greek letters $μ, ν, \dots \in {0, 1, \dots, 9} .$
^cThe 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.
^dNote 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).
^eThe validity of this approximation is discussed byTafjord and Periwal (1998).
^fPrior 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 byAharony 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), andMassar and Troost (2000).
^gAspects of these fermionic contributions to the two-graviton interaction potential were studied byBarrio, 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), andNicolai and Plefka (2000).
^hPrevious partial results on the three-graviton problem had been found byDine and Rajaraman (1998),Echols and Gray (1998),Fabbrichesi, Ferretti, and Iengo (1998), andTaylor and Van Raamsdonk (1998b). Further work on this problem is described byMcCarthy, Susskind, and Wilkins (1998),Helling et al. (1999),Dine, Echols, and Gray (2000), andRefolli, Terzi, and Zanon (2000).
ⁱAn alternative suggestion was made bySerone (1998).
^jRelated work appeared inHoppe (1997b).
^kSimilar calculations were performed previously byClaudson and Halpern (1985) and byde Wit, Hoppe, and Nicolai (1988); in these earlier analyses, however, terms such as $Tr [X^{i} {, X}^{j}]$ and $Tr X^{[i} X^{j} X^{k} X^{l]}$ were dropped since they vanish for finite $N .$
^lIn 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.
^mThese includeBanks and Motl (1997),Danielsson and Ferretti (1997),Hořava (1997),Kachru and Silverstein (1997),Lowe (1997a,1997b),Motl (1996),Motl and Susskind (1997),Rey (1997a), andKrogh (1999a,1999b).
ⁿFurther details regarding string interactions in matrix string theory can be found in the articles ofWynter (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), andGrignani et al. (2000). Other aspects of matrix string theory were discussed byBonora and Chu (1997),Verlinde (1997),Kostov and Vanhove (1998),Baulieu, Laroche, and Nekrasov (1999),Billó et al. (1999),Brax and Wynter (1999), andSugino (1999).
^oSee, 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), andYoneya (2000).
^pSee, for example,Banks et al. (1998),Kabat and Lifschytz (2000) and references therein.
^qThese matrix models were first developed byWitten (1997),Aharony, Berkooz, and Seiberg (1998),Aharony et al. (1998), andGanor and Sethi (1998); for a review of this work and further developments in this direction seeBanks (1999).