## Friday, November 13, 2020

### The matrix model is too sexy for Millennials, New York, and Japan

'Cause I'm a Motl, you know what I mean

I came to the U.S. (primarily) after I wrote rather influential papers on the BFSS matrix model of M-theory. That original paper by the four physicists (three of whom were high) has over 2800 citations but I think that especially by 2020, it is a heavily understudied line of research. Maldacena's first AdS/CFT paper shows over 16,000 citations at Inspire now (and the video above has over half a billion views!). Yes, I do think that the papers should be comparable because both describe holography of some sort; and both include a new description of quantum gravitational physics that differs from the "effective quantum field theory in the bulk".

They are not comparable because the matrix model is too sexy for... especially for the younger generation. What has really happened was that lots of people have joined the field who mentally want to stay at the level of the undergraduate or early graduate school quantum field theory (throughout their lives). Even if they want to pretend to be at the cutting edge, they don't really want to go beyond the cutting edge as defined in 1930 or at most the 1970s. Most of the papers on AdS/CFT work in that mental framework.

But the matrix model is so new. It is pretty much demonstrably an extension of "quantum mechanics of point-like particles", i.e. a 0+1-dimensional quantum field theory, that nevertheless contains all of M-theory in 11 large dimensions, a theory with the graviton supermultiplet, supersymmetry, gravitational waves, general relativity at low energies, M2-branes, M5-branes, and black holes including their full and consistent thermodynamics.

New matrix models have to be derived for other superselection sectors such as type IIA string theory and heterotic $$E_8\times E_8$$ string theory which have $$D=10$$ large dimensions. Those contain strings with all the right boundary conditions and projections (if any), level-matching conditions, and the split-and-join string interactions. And the D-branes with all the allowed dimensions and types, along with their open strings. A BMN deformation of the BFSS matrix model describes the pp-wave limit of the $$AdS_{4/7}\times S^{7/4}$$ compactification that appears in the AdS/CFT.

What's new is that all these models may be considered a borderline violation of the standard proposition that "every relativistic quantum theory has to be a local quantum field theory". If we combine the principles of quantum mechanics and special relativity, it seems almost unavoidable that particles are quanta of quantum fields that have to exist, so there are natural operators like $$\phi(x,y,z,t)$$ that (super)commute with each other at space-like separation (and that can create or annihilate particle).

AdS/CFT violates this conclusion because it admits the "boundary CFT" description of the bulk where these operators are only defined at the boundary of the anti de Sitter space (it is the AdS holography). The BFSS matrix model violates the conclusion a bit differently. It extends the regular description of non-relativistic quantum mechanics. However, one of the momentum components (along a null direction)$P^+ = \frac{ P^{0} + P^{10} }{\sqrt{2}} = \frac{N}{R}$ is defined as the size $$N$$ of the matrices (the total Hilbert space is the direct sum of all the Hilbert spaces with non-negative integer $$N$$). The remaining 9 spatial coordinates of the particle in the 11-dimensional spacetime are encoded in the matrices of quantum mechanical positions $$X^i_{m,n}$$ and their conjugate momenta $$\Pi^i_{m,n}$$ where $$i=1,2,\dots ,9$$ and $$m,n=1,2, \dots , N$$. The dynamics (evolution in time) is fully encoded by the Hamiltonian (the component of energy-momentum along the "other" chosen null direction)\begin{align} P^- &= \frac{ P^{0} - P^{10} }{\sqrt{2}}=\\ &= R\,\,{\rm tr}\left\{ \frac{\Pi_i \Pi_i}{2} - \frac{1}{4} [Y^i,Y^j]^2 +\theta^T \gamma_i [\theta,Y^i] \right\} \end{align} It is the dimensional reduction of the 10-dimensional supersymmetric Yang-Mills theory to 0+1 dimensions; the fermion $$\theta$$ has a hidden fermionic index $$a=1\dots 16$$ transforming in a 16-dimensional real representation of $$Spin(9)$$. The trace goes over the $$m,n=1\dots N$$ indices. If you assumed that the trace is a sum of $$N$$ terms, you could say that this Hamiltonian describes the motion of $$N$$ particles. Well, it was derived from the dynamics of $$N$$ D0-branes in type IIA string theory. So $$N$$ could be identified with the "number of excitations". However, the degrees of freedom have not only the $$N$$ diagonal entries but also the off-diagonal ones. And they bring this matrix model to a new level.

The off-diagonal elements of the matrices $$X,\Pi,\theta$$ are actually responsible for all the interactions. You may describe a composite system by assuming that all the matrices are nearly block-diagonal; the blocks near the block diagonal describe the objects just like they would in the smaller systems (and the total Hamiltonian is approximately the sum of the Hamiltonians of the parts, just like you expect); the rectangular off-diagonal blocks are nearly zero but their fluctuations around zero guarantee that the diagonal blocks interact with each other. Nontrivially, these interactions are exactly right to allow the matrix model to become a Lorentz-covariant theory for large $$N$$.

These off-diagonal degrees of freedom also guarantee that the small particles may create arbitrarily large bound states that have the same properties but that carry $$P^- = k/R$$ instead of $$P^- = 1/R$$, for any $$k\in \ZZ$$. That's why the size of the matrices shouldn't be interpreted as the "number of particles" but as the total momentum $$P^- = N/R$$. Here, $$R$$ is a radius of the null direction $$x^-$$. It is "marginally consistent" to compactify null directions (it's in between the compactification of spatial dimensions which is encouraged; and the compactification of timelike dimensions which is forbidden by your murdered grandfather before he had his first sex).

The fully decompactified 11-dimensional spacetime, the normal M-theory in this case, is obtained as the limit $$R\to \infty$$. To keep the momenta $$P^\pm$$ finite in the Planck units, you need to send $$N\to\infty$$, too. So the truly "normal" physics in an infinite spacetime is only seen in the physics of infinitely large matrices. But the finite $$N$$ matrix models must be considered to be consistent sectors of quantum gravity by themselves, as I and then Leonard Susskind pointed out.

It's amazing that this simple model with the polynomial, at most quartic, Hamiltonian is precise but it encodes all the physics that the effective quantum field theory hides in lots of complications. For example, the model surely includes all the microstates of Schwarzschild black holes. In principle, you could identify the individual microstates and calculate the special correlations between the Hawking quanta that are emitted by such a microstate. In an effective field theory, it is an impossible task so far because you can't really describe the evaporating black hole by a well-defined bulk Lagrangian or Hamiltonian. Quantum gravity is non-renormalizable, even if you figured out what to do with all the higher-derivative terms, you shouldn't hope to get the precise description because the black hole interior must violate the locality of the spacetime, at least to some extent, so you know that the effective field theory in the bulk (as we know it) must ultimately be wrong for questions such as "the precise probability amplitudes for the Hawking decay of particular black hole microstates". But it seems like a demonstrated fact that the matrix model knows all these answers (although it's a terribly difficult computational task to solve these problems in quantum mechanics of the large matrices).

But again, the model has everything that you expect in a particular superselection sector. Models for other sectors exist (but not for all the Minkowski vacua of string/M-theory); I've done lots of work to show that they have the desired properties, indeed. What doesn't work too well in the BFSS matrix model is the "transition between superselection sectors". To have a model, you need to identify the size of matrices $$N$$ with the null momentum component $$P^+ \cdot R$$. If you don't have a flat spacetime, it doesn't quite work. If you don't fix the boundary conditions of the spacetime, you don't know what precise matrix model you should use. So you can't really draw a map the landscape of string/M-theory by that approach right now. A covariant extension of the matrix model could help but it is still not known whether it exists and how it works.

However, there are lots of questions that have known final answers but our understanding of the proof remains poor. For example, the very proof that the Poincaré invariance is recovered in the $$N\to \infty$$ limit doesn't exist. Even the proof that the spectrum becomes independent of $$N$$ in the limit, which is basically a proof of the invariance under $$J^{+-}$$, a generator of the boosts, is highly incomplete right now. The proof of the proper behavior of non-supersymmetric objects within the model remain rather unsatisfactory, and so on. Many of the proofs should be matrix counterparts of the world sheet conformal field theory (either covariant or light-cone gauge CFT) but the right way to do these "matrix counterparts" of all the elementary perturbative string theory remain largely obscure.

Matrix theory is clearly an important example for me but there have been tons of other programs that simply required the researchers to torture their brains and go beyond the mental framework of the undergraduate-or-early-graduate quantum field theory. All the minirevolutions that used to explode once in a year or so; but they have gone silent. Like the noncommutatative geometry and their derivation from D-branes with B-fields. Like the BMN pp-wave limit of AdS/CFT in which the strings arise from string bits was special because it demanded to find the properties of very complex or composite operators in a quantum field theory. That's very different from the cheesy average AdS/CFT paper (or even ABJM, SYK, JT... sorry) which is "just doing something ordinary like QCD and claiming that it says something about black holes or something" (there are surely thousands of redundant papers of this type). Or various games with the twistors and their combinations with string theory. You can't get too far if you pre-constrain yourself not to go too far.

I am convinced that a much more general, troubling conclusion is correct: the very desire of the people to write "seminal or at least much better than average or much better than mediocre" papers has been largely killed. Some people above 50 have still enjoyed pushing the cutting edge and some of them earned millions of dollars for that, too. But I believe that the young people below 30 are no longer rewarded for being very good which unavoidably leads to the situation that they don't seem to be very good. Mediocrity has won and it has killed stellar, world-class research. And some smart people may have even chosen another field such as "how to design a good cryptocurrency scam".

It's not hard to understand the pressure. Take another example: string phenomenology. Just construct some heterotic models (e.g. with Calabi-Yau manifolds, perhaps with some free fermionic description of the compactified dimensions, or braneworld models). Even if you have all the required talents, you really need some 5 years beyond the undergraduate degree to become a real expert. The number of experts who still know how to extract the low-energy spectrum of the particular heterotic string models from the Calabi-Yau topology and other things is "at most dozens". In the whole fudging world. It is very hard to write such a paper and that's also why the paper is likely to get "at most dozens" of citations in recent years if not decades. If someone accidentally found the right compactification, which is surely possible in principle, the number of people who could verify the big claim would be homeopathic.

Just try to imagine how isolated and brave these people are. Meanwhile, you have dozens if not hundreds of really lousy people who have no talent but who make living out of claiming that the string phenomenologists (less than 1 person per average U.S. state!) are living in some amazing group think. Idiotic laymen, and most laymen are gullible and idiotic when it comes (not only) to modern theoretical physics, tend to buy it. As a result, the amount of money that goes to the talent-free antiscience demagogues is probably much higher in 2020 than the amount of money that goes to the brilliant young physicists. And the potential young brilliant physicists know that which is why they don't want to try to become brilliant physicists anymore. It's easier to join the decay of the society. To earn dollars by emitting idiotic lies about the group think of people whose density is about "2-3 per continent"; earn money from fairy-tales about the suppression of some minorities, or similar stunning garbage.

Of course, these days, worries about the destruction of quality theoretical physics as an occupation seem secondary because the far left movement wants to destroy pretty much everything, including all of the culture, business, Christmas, families, nation states, professional sports, manliness, women's beauty, ... anything that has had any value. From some perspective, this general intentional decay of the Western society has some silver lining. The Millennials' inability to do things like Matrix theory looks like a tiny problem relatively to the overall worries about mankind – which may be interpreted as good news or bad news by a matrix model fan, depending on the precise psychological attitude (depending on the ratio of importance assigned to matrix models relatively to the Western civilization). ;-)