# M Theory As A Matrix Model: A Conjecture

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Referee this paper: arXiv:hep-th/9610043 by T. Banks, W. Fischler, S.H. Shenker, (show more)

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In this classic, which colloquially goes by the name "BFSS", Banks, Fischler, Susskind, and Shenker define the first complete nonperturbative formalism for string-theoretic quantum gravity. This paper for the first time allowed arbitrary processes in a particular string vacuum to be calculated to arbitrary accuracy without any limitations inherent to the calculation method. In other words, for 11 dimensional uncompactified M-theory, this paper provides a theory of everything. Not to imply that this paper is just a measly TOE. In addition, it also gave the first example of the physics of the AdS/CFT correspondence.

The theory is formulated as a cartoonishly simple nonrelativistic 0+1 highly supersymmetric quantum mechanics model, which superficially seems to only describe the near-light-cone oscillations of a finite number N of D0 brane positions, which move in a noncommutative geometry due to their mutual electromagnetic interactions. The holographic completeness of these black hole states forces the oscillations in the large N limit to reproduce the entire dynamics of the complete 11-dimensional theory, including gravitons, M2 and M5 branes, and all their mutual dynamics. The construction is using string bit methods pioneered by Susskind, and a precient M2 membrane construction considered without appreciation of the generality in 1980s work by deWit, Hoppe and Nicolai.

summarized
paper authored Oct 7, 1996 to bfss
edited Apr 25, 2015

This paper can hardly be reviewed: it is the best physics paper of all time, heck, it's possibly the greatest text produced by humans in the entire history of civilization. Reading it is like looking directly at the sun, first you go blind, then you go mad, then you feel that your life has lost all meaning, because nothing can ever be so beautiful again.

@RonMaimon: Every paper can be reviewed to make reading it interesting and possibly easier to read for those not knowing it. Please describe the content in a way that someone like me - who doesn't know much about string theory - can see the main features and gets a guide for browsing the paper with profit (and ultimately perhaps read it in detail).

@ArnoldNeumaier: You are right, of course, at least other people who are not in awe of this can review it. I am just too intimidated to review the paper, as it is like reviewing the Bible or Newton or Shakespeare, or something like that for me, it feels scary and inappropriate. I put up the summary, The best I can do to add something is forward links to future work, and I am not competent to say which is the best future work, there is a ton. The paper is so well written and clear, it is difficult for me to write about it. It's also extremely famous, I think you can read it with no prior knowledge, or very little prior knowledge of strings, as it builds up the M-theory essentially from scratch using physical principles. The only thing you need coming in is light-cone gauge and basic (1-d!) SUSY.

@RonMaimon: The paper cannot be read with very little prior knowledge of strings - it is full of string terminology beyond my command (I understand essentially nothing from the first 6 pages), and scarce in the development from first principles. At least it requires a guide to see the pieces that are independent of string notions and how they connect.

What is a BPS state? It is used in the paper without reference or explanation, and seems to play a crucial role, e.g., after (5.4)

How can the Hamiltonian system given by (4.6) which according to a few lines before (4.7) is Galilean invariant be Lorentz invariant? What are the Lorentz boost generators supposed to be?

In addition, it also gave the first example of the physics of the AdS/CFT correspondence.

Where?

@ArnoldNeumaier BFSS itself. See also Lubos's answer to one of my old questions.

@RonMaimon It's not really accurate to say that BFSS demonstrates the first example of AdS/CFT, BFSS demonstrates an example of holography, but as far as I know, the paper discusses M-theory on a Minkowski spacetime, not AdS.

@dimension10: It's a Minkowski spacetime, but the physical principles of the noncommutative matrices giving rise to a spacetime external to a black hole collection is similar to AdS/CFT. That's why I said "first example of the physics of AdS/CFT" rather than "first example of AdS/CFT", because it isn't precisely AdS/CFT. But it's even weirder. You can embed some sort of actual extended theory AdS/CFT in it using the deWitt Hoppe Nicolai M2 membrane construction which is in the paper and reviewed extremely well in Polchinski.

@ArnoldNeumaier: I was thinking of reviewing it now that I read it afresh, I am less intimidated and more inspired with every passing year. This paper completely blew my mind when I was in my 20s, as my first reaction is that it must be garbage, because you couldn't possibly have an 11 dimensional theory reconstructed from a trivial nonrelativistic matrix model. Later, I came to accept (like everyone else) that it had solved the major problem of constructing a fully consistent nonperturbative quantum gravity in at least one regime (this is the most difficult problem, once you have the first construction, you can generalize, the generalization was AdS/CFT).

The nonrelativistic Hamiltonian giving rise to relativistic physics is nothing special, it's just the light-cone frame (infinite momentum frame), which Susskind explains very pedagogically and insightfully in his lectures on string theory. The "BPS state" means a state which has mass=charge, it is usually an extremal black hole in string theory applications, but field theory contexts it can mean other things (the typical examples are usually dual to extremal black holes). In this paper, the "charge" is a Kaluza-Klein charge, so it is momentum in the 11th dimension (10th spatial dimension), and the BPS states in question are the extremal D0 branes, that is, point black holes boosted to infinite momentum in the 10th dimension and broken up into their elementary quanta, whose spectrum you know from IIA string theory. The reason they are called "BPS" is because they have special supersymmetry properties that protect their mass/charge relationship from corrections, but you don't need to know all the details of this. In this case, all you need to see is that in the infinite momentum frame, all states except for D0 branes don't appear as asymptotic states, because of the boosting to infinite momentum. Since asymptotic states are complete for a quantum description of gravity (the string theoretic S-matrix principle, if you like), this means that the D0 branes in their interactions reconstruct the full theory. The D0 interactions, however, are derived in weak-coupling IIA string theory limit from unexcited strings with endpoints on the branes, the result is a famous result of Witten from the mid 90s, the brane effective theory is just SU(N) gauge theory, in this case dimensionally reduced to 0 dimensions (this is an extension of the Chan Paton method of the 70s). So the resulting interactions of D0 branes don't reference any other degrees of freedom at all, they are consistently interacting only among themselves, is a gauge theory which you can reasonably extrapolate to strong coupling (large 11th dimension) because these states are BPS. The result is that you get an extremely plausible conjectural correspondence between this extremely simple matrix model, describing the physics of congealing and separating D0 branes and the full nonperturbatively complete M-theory, including full 11 dimensional SUGRA.

The description is in an infinite momentum frame, where there is infinite time-dilation, so every dynamical process is constructed from following the slow back-and-forth motion of D0 branes congealing and falling apart in a frame where they are highly boosted. And yet, you can explicitly construct the objects you expect in the M-theory from configurations of the D0 branes. For an explicit example, the reconstruction of the M2-brane from the D0 branes is reviewed in Polchinski, and predated the BFSS model (it was known in the late 1980s).

You can see the 10 dimensional gravity emerging from the effective interactions of the D0 branes, you get the entire space reconstructed from the motion of the branes and their coming together and falling apart. This description is extremely shocking to a person brought up with field theory, there are no loops, no divergences, nothing but matrix dynamics. The description is so crazy and the spacetime is noncommutative, due to the charges on the branes. The resulting theory is a complete description of gravity in this limit with no constraints on applicability from any calculational approximation--- the large N and large "R" limit is the only thing you need to take to recover the M-theory, and they are perfectly well defined. It is like the Wilsonian revolution in field theory, except in string theory. The next work was AdS/CFT proper.

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