Matrix theory: a novel alternative to second quantization etc.

I still consider Matrix theory (BFSS, 1996) one of the most conceptually original developments in theoretical physics of the last 20 years.

It is a relatively unusual way to describe physics in the 11-dimensional asymptotically flat vacuum of M-theory – and in other sectors of string theory. The physical phenomena are completely equivalent as in other descriptions; but the way how they're encoded in the mathematics looks very different.



There are several natural pedagogical ways to get to Matrix theory and I will try to sketch the following three of them:

1. try to study some obviously beautiful quantum mechanical models in extreme limits (the infinite number of colors) and try to find a simplified description of this limit;
2. start with M-theory whose explicit equations weren't known before BFSS 1996 and transform it via dualities and tricks into something that you may describe;
3. try to invent a completely new framework (different from quantum field theory and second quantization) to describe multiparticle states and interactions between the particles, among other things, that may lead to the same kind of physics.

It's the last approach that was used in the title of this blog entry.




Some hindsight is needed to describe the situation in this way and some of this hindsight is often missing in the pioneering papers; BFSS 1996 is no exception. But we have it now and we may look at the situation from various perspectives and avoid some speculative comments that were discussed at the beginning but that were found to be invalid.

The third approach seems to be the most conceptually valuable one but let me start from the first approach.

Maximally supersymmetric gauge quantum mechanics

This year, the \(\NNN=4\) gauge theory in \(d=4\) celebrated its 35th birthday. It's obviously a beautiful theory, the most supersymmetric non-gravitational theory in \(d=4\) you may find. It may be derived as a dimensional reduction of the 10-dimensional supersymmetric theory; as a low-energy limit of the dynamics of D3-branes; and in many other ways.

It has many cool symmetries including the \(SL(2,\ZZ)\) S-duality group (which is able to exchange electricity and magnetism, the weak coupling and the strong coupling, electrically charged particles with magnetic monopoles), the superconformal symmetry, the dual superconformal symmetry, their infinite-dimensional extended and completed union, the Yangian, and others. For those reasons, various scattering amplitudes, while nonzero, dramatically simplify relatively to the form you would expect in a field theory with a similar Lagrangian but fewer supercharges. Many of those features may be exposed in the twistor-based approaches.

As I have mentioned above, this 4-dimensional theory may be viewed as the dimensional reduction of the 10-dimensional gauge theory whose action looks like this:\[

\eq{
S &= \int \dd^{10} x\, \LL,\\
\LL &={\rm Tr}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i\bar\Psi D^\mu \gamma_\mu \Psi
\right]
}

\] It's an ordinary Yang-Mills Lagrangian for a gauge field associated with a gauge group we will take to be \(U(N)\), i.e. \(SU(N)\times U(1)\) including the \(U(1)\). The unitary groups are the least complicated infinite family of simple compact Lie groups, and for a Majorana-Weyl (real-and-chiral) fermion field (covariant derivatives have to be used). This innocent combination is enough to produce an exactly supersymmetric classical (or effective) field theory. It's non-renormalizable in \(d=10\) but its dimensional reductions to \(d\leq 4\) are renormalizable.

By dimensional reductions, we mean that the "excessive" dimensions are compactified on a torus – they are made periodic – and the radii of the torus are sent to zero. For this reason, finite-energy excitations are forced to be constant in these dimensions. Moreover, the rotational symmetry between the compactified and uncompactified dimensions becomes badly broken. It is no longer sensible to consider the components of the gauge field \(A_\mu\) along the compactified dimensions \(\mu\) to be components of a gauge field. It's better to consider them scalars.

Off-topic but fun: A reader has pointed out that in the new version of the Hartle-Hawking-Hertog paper, your humble correspondent and HB are thanked for the innocent sign error whose impact so far looks isolated (but I still believe that there must be some other errors unless the paper shows some really important loophole in the Ehrenfest "theorem" way of thinking about the evolution in quantum gravity).

For example, if we dimensionally reduce from \(d=10\) to \(d=4\), six of the components of the gauge field (whole matrices in the adjoint of the gauge group) will become six scalars: they're the source of the \(SO(6)\sim SU(4)\) "R-symmetry" of the \(d=4\) gauge theory. Once we get some scalars, some gauge couplings for fermions also produce the Yukawa couplings as a result. It's convenient to decompose the fields under the unbroken \(SO(d-1,1)\) i.e. \(SO(3,1)\) Lorentz symmetry acting on the large dimensions which is a subgroup of the original \(SO(9,1)\) symmetry.



This 2007 model was called Toyota Matrix M-theory. No kidding. Those East German engineers who refused the German reunification plan to produce a competing car, The Trouble With Trabant: Not Even Wrong.

The 35th anniversary article discussed the reduction to \(d=4\) which gives the prettiest "descendant" of the 10-dimensional gauge theory, one that has the most amazing properties at the quantum level. But we may dimensionally reduce the theory to lower dimensions, too. For example, we may reduce it to \(d=1\) spacetime dimension. That's the minimum you may naively think of even though the reduction to \(d=0\) may actually be important, too.

What does it mean to have a \(d=1\) quantum field theory? It has one spacetime dimension. We want at least one dimension of time. So a simple counting, \(1-1=0\), implies that time is the only dimension on which our fields depend. What does it mean? Well, it means that it's not a full-fledged "quantum field theory" anymore. It's a quantum mechanical model. Instead of fields such as \(\Phi^6(x,y,z,t)\), we have "fields" that may be renamed as \(X^6(t)\). They're quantum observables. You see that the field content is completely analogous to the textbook models of non-relativistic quantum mechanics.

In similar contexts, "quantum mechanics" is often used as a synonym for a 1-dimensional quantum field theory. (Of course, "quantum mechanics" also means – and mainly means – the general framework of modern physics given by the Copenhagen school's postulates that all quantum theories, including higher-dimensional quantum field theories and string theory, obey.)

You might argue that quantum mechanics has advantages over higher-dimensional quantum field theories. It doesn't need any renormalization and similar stuff. The wave function is just \(\psi(x_1,x_2,\dots)\) and obeys some partial differential equations in a finite number of variables. There's no room for infinities. Everything looks simple. Everything that annoys you about regularization, renormalization, renormalization group etc. is fundamentally absent here. (Well, it's not quite true because some of these things have nontrivial analogues in \(d=1\) but it's still true that you may choose a fundamentally unequivocal, finite description of quantum mechanical models.)

Fine, so what is the quantum mechanical model we are considering here? Its Lagrangian looks like this:\[

\eq{
S &= \int \dd t\,\LL\\
L &= \frac{1}{2g}{\rm Tr} \zav{
\dot X^i \dot X^i + 2\theta^T\dot\theta+\frac{1}{2}[X^i,X^j]^2-2\theta^T \gamma_i[\theta,X^i]
}
}

\] You see that the Lagrangian contains some Klein-Gordon kinetic terms for the nine scalars \(X^i\), Dirac-like kinetic terms for the 16 components of the fermions \(\theta\), a quartic commutator-squared-based potential for \(X^i\) that arises from the quartic terms in \(F_{\mu\nu}F^{\mu\nu}\) in the 10-dimensional gauge theory, and a Yukawa term. All the terms are traces over the \(N\times N\) matrices, just like in any other version of the \(U(N)\) gauge theory with fields in the adjoint.

I should be more explicit in explaining how many degrees of freedom the theory actually has. Well, all the fields transform as Hermitian matrices – i.e. adjoint representation – under the \(U(N)\) gauge group. Moreover, the bosonic fields \(X^i\) arising from \(A_\mu\) carry an extra index \(i=1,2,\dots,9\) which corresponds to the 9 spatial dimensions we dimensionally reduced. The remaining component \(A_0\) may be (but doesn't have to be) set to \(A_0=0\) by the gauge redundancy.

So the fields \(X^i\) are nine Hermitian \(N\times N\) matrices. Because we ultimately want to study the quantum theory, all the components of these matrices are operators. Much like in normal non-relativistic quantum mechanics, we find out that there are canonical momenta \(\Pi^i\), also nine Hermitian matrices, and they have the commutators\[

[X^i_{kl},\Pi^j_{mn} ] = i\hbar \delta^{ij} \delta_{nk}\delta_{lm}

\] where the pairing of the indices is determined by the \(U(N)\) symmetry: just appreciate that in the index pairs \(kl,mn\), one index is "lower" and one is "upper" (we suppressed the difference to streamline the notation only) and each Kronecker delta-symbol has to have one upper index and one lower index, too. I've restored \(\hbar\) to emphasize that this is a "modest" extension of the undergraduate quantum mechanics models.

Now, we must also discuss the fermionic degrees of freedom \(\theta\). They arose from the Majorana-Weyl (real chiral) spinor of the 10-dimensional gauge theory so they include 16 real components that now transform as a 16-component real spinor of \(SO(9)\), the manifest rotational symmetry acting on the scalars \(X^i\) as well. When promoted to the adjoint of \(U(N)\), each component becomes a Hermitian matric again. Much like in the usual quantization of Dirac fields, these degrees of freedom are Grassmann i.e. fermionic variables that are canonical momenta to themselves (because the kinetic term in the Lagrangian only contains one time derivative):\[

\{ \theta^a_{kl},\theta^b_{mn} \} = \hbar \delta^{ab}\delta_{nk}\delta_{lm}

\] where \(a,b=1,2,\dots, 16\) and the remaining commutators (all the anticommutators have already been written because they're only appropriate for pairs of fermionic objects) vanish.

So except for the high number of dimensions indicated by the indices such as \(i,j\) that take 9 values and except for the extra degeneracy of the coordinates given by the gauge indices \(k,l,m,n=1,2,\dots,N\), and except for the extra fermionic variables, this is a pretty normal quantum mechanical model similar to the non-relativistic toy models you know from your undergraduate course of quantum mechanics (or its equivalent if you are self-taught).

We really want to say that the state of the physical system described by this \(U(N)\) quantum mechanical model may be encoded in a wave function \(\psi(x,\theta)\). How many variables does it depend on?

Well, there are \(9N^2\) real components in the matrices \(X^i_{mn}\): note that the Hermiticity reduces the number of real (i.e. Hermitian, when treated as operators on the Hilbert space) components \(2N^2\) exactly to one-half of that. The canonical momenta \(\Pi^i_{mn}\) are just \(-i\hbar\) times the derivatives with respect to the variables \(X^i_{mn}\). And then we have the fermionic coordinates; out of the \(16N^2\) components of \(\theta^a_{mn}\), one-half of them are taken to be the coordinates and the remaining one-half may be taken to be the canonical momenta. So we have \(8N^2\) Grassmann variables.

Because the Taylor expansion in the Grassmann variables terminates, we may rewrite the wave function in terms of components and their number is simply \(2^{8N^2}\) because each of the \(8N^2\) variables is either present or absent in the given monomial in the power law expansion. For example, for \(N=5\), we have \(200\) fermionic coordinates and the number of component functions is therefore \(2^{200}\sim 10^{60}\). Each of these component functions depends on \(9N^2=225\) real bosonic variables.

To solve the quantum mechanical model for \(N=5\) by the brute force, you simply calculate some coupled linear partial differential equations for \(10^{60}\) functions that depend on 225 variables. It's a trivial task you've been doing in your kindergarten, at least in principle, and you may totally avoid any complications with regularization and renormalization.

I am partly kidding because the number of components and coordinates looks horrifyingly high and for realistic simulations of the large \(N\) limit, you really don't want \(N=5\) but something like \(N=100\). But it's just a problem for the naive brute-force approaches. If you understand the model well and you know some maths, you may find methods to calculate properties of the model which are doable on one line or two. Or several pages. It shouldn't shock you that the naive brute-force simulation is neither the only way nor the recommended way to attack a seemingly mathematically difficult problem although this claim might be controversial among many laymen including physics fans.

If it makes you happier, the number of independent variables on which the wave function for a physical state depends is effectively \(8N^2\) and not \(9N^2\) because all physical states have to be invariant under the \(N^2\) generators of \(U(N)\); well, only \(N^2-1\) of these conditions are nontrivial. After all, it's a gauge theory and therefore this gauge-invariance condition for the physical states comes from varying the action with respect to \(A_0\): the corresponding charge \(Q=J_0\) – which is an \(N\times N\) matrix – has to vanish.

Fine, we know a certain supersymmetric quantum mechanical model – a quantum field theory in 0+1 dimensions – that may be in principle studied. Remarkably enough, the union of the Hilbert spaces from all these \(U(N)\) models for all integer values of \(N\) gives you the Hilbert space of M-theory in 11 dimensions! It contains everything you expect in a consistent theory of quantum gravity, including gravitons and their superpartners, their interactions, gravitational force obeying the equivalence principle, the 10+1-dimensional Lorentz symmetry, evaporating black holes that preserve the information, M2-branes and M5-branes (and strings and D-branes in compactified, stringy versions of the matrix model, including the right stringy interactions), and other things.

The quantum mechanical model was known before it was conjectured by BFSS that the model was relevant for M-theory in 11 dimensions. It was known as the description of D0-branes, point-like particles moving in the 10-dimensional type IIA string theory, at low energies. Just like D3-branes in type IIB string theory give rise to the \(d=4\) 35-years-old gauge theory at long distances, a similar statement holds for the D0-branes.

However, you should only say that you qualitatively understand what the model describes for a finite and small enough value of \(N\). When some parameters such as \(N\), the size of the matrices, are sent to infinity, the most important (and lowest-energy) objects and processes in the theory may describe something that admits (if not requires) a qualitatively different language. In this case, the particles originally viewed as D0-branes in a 10-dimensional string theory (where they're just a part of physics and interact with other objects) actually become the complete description of an 11-dimensional theory of quantum gravity, namely M-theory.

This conclusion may look shocking – how an ordinary undergraduate quantum mechanical model (which isn't even a quantum field theory, so we don't see how it could contain multiparticle states with indistinguishable particles and/or Lorentz invariance) could contain all the marvels of quantum gravity including Lorentz invariance, graviton scattering, equivalence principle, black holes? But remarkably enough, it does.

I will postpone the explanation why the matrix model includes the right building blocks of M-theory and why there's no detectable contradiction to the final section of this blog entry – about the conceptual leap. But before we get there, let us look at a proof that the matrix model is correct.

Seiberg's derivation of Matrix theory in DLCQ

The October 1997 proof was originally presented by Nathan Seiberg and, in a slightly less complete version (but more extended in some other directions we're not interested in here), by Ashoke Sen. So why is the matrix model correct?

Consider M-theory in the asymptotically flat 11-dimensional Minkowski spacetime parameterized by \(X^0\dots X^{10}\). We want to find an explicit Lagrangian that describes all the objects in this 11-dimensional theory of quantum gravity – which may be obtained as the strong coupling limit of type IIA string theory or heterotic-E string theory.

It's useful to single out two light-like directions or coordinates,\[

X^\pm = \frac{X^{0}\pm X^{10}}{\sqrt{2}},

\] and use \(X^\pm\) instead of \(X^0\) and \(X^{10}\). The light-like coordinates are very useful in relativity. For example, the expression \(t^2-z^2\) appearing in the invariants may be rewritten as \((t+z)(t-z)\): the difference of two terms may be written as one term that factorizes. Correspondingly, it's useful to consider the light-like components of the momentum etc. such as \(P^+\).

The states in M-theory have continuous, non-negative values of \(P^+\). However, it's very convenient to put the theory in a certain "box" for the momentum to become discrete. However, we don't want the box to matter so its size has to be sent to infinity. More explicitly, we want to make the following identification of the light-like coordinate \(X^-\):\[

X^- \approx X^- + 2\pi R.

\] As long as \(R\) is huge – imagine that it is hundreds of billions of light years (although the citizens of the 11-dimensional empire don't necessarily use these units for the distances in their completely different world) – the identification doesn't affect the local physics. If a copy of your local experiment is happening hundreds of billions of light years away from you, it shouldn't matter in your lab. We implicitly use some "modest version" of locality in M-theory here.

However, the change of the \(X^-\) coordinate isn't a pure translation in space; it contains a translation in time by the same amount. In fact, you should be worried about the consistency of this light-like compactification. If we made this periodic identification for \(X^0\), a time-like coordinate, we would create closed time-like curves which would make the theory inconsistent. On the other hand, a spatial compactification of \(X^{10}\) would be harmless. What about the marginal case, the light-like compactification?

Well, we may take it as the limit of spatial compactifications that are OK which should be enough for you to believe that it's OK, too. So we're taking an M-theoretical spacetime and identify points that differ by something like the vector\[

(M,0,0,0,0,0,0,0,0,0,M+\epsilon)

\] in the usual coordinates \(X^{0\dots 10}\). It's a nearly null interval but it's a little bit spacelike because of the extra \(\epsilon\) in the last coordinate. We may scale the huge \(M\sim R\) and \(\epsilon\) in such a way that the the proper length of the vector above is tiny, much shorter than the 11-dimensional Planck length.

However, if that's so, we may use the Lorentz symmetry of the 11-dimensional M-theory – an assumption supported by lots of evidence, including the Lorentz symmetry of the 11-dimensional supergravity, the low-energy limit of M-theory – and boost the vector above to a simple vector of the form\[

(0,0,0,0,0,0,0,0,0,0,\epsilon')

\] whose last spatial coordinate is much shorter than the 11-dimensional Planck length. Note that the momenta of all the particles in the original M-theory get boosted by a correspondingly dramatic boost. But we know the spacetime in which these boosted objects propagate: it's M-theory compactified on a very short spatial circle given by \(\epsilon'\). But it's nothing else than type IIA string theory at a weak coupling!

Moreover, the null component of momentum \(P^+\) which is the complementary variable to \(X^-\) is quantized because \(X^-\) is compact, \[

P^+ = \frac{N}{R},\quad N\in\ZZ

\] We wanted to describe objects with a fixed and finite momentum in the 11-dimensional Planck units, i.e. \(P^+\) is finite, and because \(R\) needs to be sent to infinity, the integer \(N\) has to be sent to infinity for these interesting states, too.

However, as we boosted the light-like interval defining the light-like compactification to a spatial one, we have a new interpretation for \(N\): it's the number of units of momentum in the direction of the short compactified coordinate \(X^{10}\sim X^{10}+\epsilon'\) we mentioned at some point above. But the momentum in the extra 11th dimension is nothing else than the number of D0-branes. So the original state of M-theory is equivalent, via this boost and in the limit, to a state involving \(N\) D0-branes in type IIA string theory.

When you analyze the right role of the \(R\to\infty\), \(\epsilon'\to 0\) limit, the relevant estimated magnitude of the energy of the D0-branes (as a function of the finite momenta and energies in the original M-theory spacetimes), the precision we need for this energy in the limit, and the states and objects that may be neglected in this limit because they're infinitely times heavier, you will find out that what you need to describe the original states with \(P^+=N/R\) in the M-theory spacetime is nothing else than the low-energy limit of the dynamics of \(N\) D0-branes in type IIA string theory at a vanishingly low coupling (because of the small \(\epsilon'\)).

It's nothing else than the supersymmetric quantum mechanics model we have already written above; for some time, the model was also referred to as the DKPS model because of a pre-matrix-theory paper that studied it.

So the Hilbert space of the modestly light-like-compactified M-theory is the direct sum of the Hilbert spaces of the \(U(N)\) matrix quantum mechanical models for \(N\) D0-branes. The Hamiltonian of the matrix model gets directly translated as the light-like Hamiltonian (generator of evolution in a light-like direction) in M-theory. The relevant value of \(N\) for the generic states in M-theory with finite energies is the \(N\to\infty\) limit.

As we have seen, there exists a proof that the BFSS matrix model is an equivalent description of M-theory. The proof only relies on the Lorentz invariance of M-theory; its well-known relationship with type IIA string theory; some elementary derivable properties of D0-branes in type IIA string theory; and the irrelevance of the near-light-like compactification if the radius is sufficiently long.

But we may still be shocked by the claim that this seemingly naive non-relativistic quantum mechanical model with 9 bosonic matrices \(X^i\) describes an 11-dimensional, and not just 10-dimensional, theory that is moreover Lorentz-invariant in the large \(N\) limit and includes gravity, gravitons, gravitinos, multiparticle states with indistinguishable particles, membranes, fivebranes, black holes, and their interactions including all the right loop corrections that could be derived from effective quantum field theories, among many other things.

Can we see that those things are present in the theory? This brings me to the final "megapoint" I want to make, namely that Matrix theory is a really cool and original way to mathematically describe many processes that used to be described – and are still being described, in most cases – by totally different mathematics.

All roads lead to string theory: describing quantum gravity in totally new, revolutionary yet consistent languages

In order to understand what kind of physical phenomena Matrix theory allows and implies, it's useful to start with small values of \(N\). For such small values, the M-theoretical physics does depend on the details of the light-like compactification; we're away from the large \(N\) limit. However, many things acquire their expected properties already for small \(N\).

Start with \(N=0\). The degrees of freedom are \(0\times 0\) matrices. In other words, there are no degrees of freedom. There is a unique wave function (up to normalization) that depends on them, namely \(\psi() = 1\). This state vector may be identified with the vacuum state of M-theory as it carries no energy or momentum.

That was easy: too easy. Let's continue with \(N=1\). That's the last matrix model that will be easy and fully solvable, of course. This model should describe all states in M-theory that have \(P^+=1/R\), the minimal positive value of the light-like momentum. We are dealing with a \(U(1)\) gauge theory in 0+1 dimensions. The bosons \(X^i\) and \(\theta^a\) are \(1\times 1\) matrices so they're not really matrices at all.

Because \(U(1)\) is an Abelian group (you know it from electromagnetism), all the commutator terms vanish. So the Hamiltonian is just quadratic, namely\[

P^0_{\rm BFSS} = \frac{(\Pi^i)^2}{2}.

\] That's it. It's just like the non-relativistic kinetic energy – for a particle in 9 spatial dimensions. No potential because the potential terms vanish. No terms from fermions, either. You might be puzzled what this non-relativistic formula has to do with the relativistic physics in 10+1 dimensions. The answer is that the right interpretation of the energy in the quantum mechanical model is \(P^-\), a light-like component of the energy-momentum vector that is treated as energy in the light cone quantization.

Massless particles' energy and momentum obeys \[

(P^0)^2-(P^i)^2 - (P^{10})^2 = 0, \quad i=1,2,\dots ,9

\] which may be rewritten, using the light-like components as\[

2P^+ P^- - (P^i)^2 = 0, \quad P^- = \frac{(P^i)^2}{2P^+}

\] which has the simple quadratic, non-relativistic form! So the light-like description which is the ultimate limit of the "infinite momentum frame" which described ultrarelativistic particles is mathematically analogous to non-relativistic physics. Note that \(2P^+\) is just a constant in the sector with \(P^+=1/R\) i.e. \(N=1\).

You could even say that the light-like momentum component \(P^+\) plays the role of the non-relativistic mass \(m\). Also, if you use the light-like coordinates, you may find a copy of the Galilean group that is embedded, without any deformation, right into the Lorentz group. At low speeds, relativity reduces to non-relativistic physics approximately; but at the speed of light, a copy of non-relativistic physics is embedded in the relativistic one exactly!

We see that the \(N=1\) model describes particles with arbitrary values of the nine "totally transverse" components of the momentum \(P^i={\rm Tr}(\Pi^i)\). The tenth component treated as a spacelike one, the light-like component \(P^+\), is equal to \(1/R\) which is a fixed constant. That's OK because we're considering a sector of the Hilbert space only; the states with higher values of \(P^+\) will be found in the models with higher values of \(N\).

The remaining component, i.e. the other light-like component \(P^-\), is treated as energy and the BFSS Hamiltonian gives us the right dispersion relations already for \(N=1\). We shouldn't forget about the fermionic degrees of freedom. There are 16 Hermitian objects \(\theta^a\) transforming as the real spinor of \(SO(9)\). They're serving as momenta to themselves so you may combine them into 8 Grassmann variables and their 8 Grassmann derivatives.

By Taylor expanding the wave function in the 8 Grassmann variables, you get \(2^8=256\) components. That's exactly the number of polarizations in the graviton supermultiplet of 11-dimensional supergravity. Recall that these get decomposed to 128 fermionic components (of the gravitino) and 128=44+84 bosonic components including the physical polarizations of the graviton as well as those of the C-field three-form potential.

One may actually define the 32 real supercharges we expect in M-theory. 16 of them are nothing else than the 16 supercharges of the maximally supersymmetric gauge theory; the remaining 16 of them are trivial or kinematical and these generators are given simply by \({\rm Tr}(\theta^a)\). This decomposition of supercharges to the "complicated dynamical" ones (one-half) and the "trivial kinematic" ones (the other half) is a consequence of the light cone treatment. To get from the maximally supersymmetric gauge theory to the maximally supersymmetric supergravity, we needed to double the number of supercharges (from 16 to 32) and the existence of the kinematic or trivial supercharges did the job for us.

We have seen that already the non-interacting \(N=1\) model knows about the number (and representations under rotations) of the components of the supergraviton multiplet and the right dispersion relation (relationship between energy and momentum). That's quite a success for such a simple model. Note that we only have one particle; our derivations tell us that there can't exist any multiparticle or otherwise complicated states that would have the minimal value of the longitudinal momentum.



Screws and Nuts by Mandrage, a Pilsner band, has been among the Czech radios' top 3 songs for more than half a year. The observation that men and women are like screws and nuts obviously has a sexual connotation but most physicists don't appreciate that the Czech word for nuts in this sense, "matice", is the same word as one for "matrices", and moreover, matrix strings are screwing around the matrices as the monodromy introduces a permutation. This breathtaking ignorance of rudimentary Czech among most physicists and their lacking sense of humor has led to the widespread but flawed terminology "matrix strings" for what should actually be called "screwing strings". ;-)

But the real fun only starts with \(N=2\) when we obtain the first real matrices. Let us talk about all the values \(N\gt 1\) simultaneously. The first observation we want to make is that the \(U(N)\) matrix model does contain states composed of several objects.

In quantum field theory, one may create multiparticle states as\[

\ket\psi = a^\dagger_\text{here} a^\dagger_\text{there}\ket 0.

\] One simply acts with several creation operators at different places to create several objects. As long as the places are sufficiently distant, the Hilbert spaces associated with the places are independent and the whole Hilbert space isn't far from a tensor product of the Hilbert spaces associated with the individual regions.

However, the matrix model isn't a quantum field theory. It doesn't have any creation and annihilation operators for particles. How can it possibly contain multiparticle states? The right indistinguishability conditions and statistics? And other things? It looks like some fully well-defined but seemingly non-relativistic undergraduate quantum mechanical model.

The matrices are the answer. A funny thing about large (or any) matrices is that they may take special matrix values that are block-diagonal. Imagine that the matrices for all \(X^i\) and similarly \(\theta^a\) have the following form:\[

X^i = \pmatrix{ \text{WOLF}^i & \heartsuit^i \\ \heartsuit^{i,\dagger} & \text{BUNNY}^i }

\] Here, WOLF is a square matrix that contains the information about positions of particles that make up an object called WOLF. BUNNY is a similar square matrix whose size may be the same or different. And the hearts are some generally rectangular off-block-diagonal blocks.

We see that large enough matrices are capable of including several objects, such as WOLF and BUNNY. The longitudinal momentum \(P^+\) is linked to the linear size of the matrices so \(P^+\) of the WOLF-BUNNY composite states is simply the sum of the values of \(P^+\) that these two animals would carry separately. And indeed, if we set \(\heartsuit=0\) for a while, the Hamiltonian for the large matrices simply reduces to the sum of the Hamiltonians for both animals: just recall how a trace of a block-diagonal matrix behaves. The two animals evolve independently.

What about the heart? Because of the commutator terms in the Lagrangian, the off-diagonal elements of \(X^i\) in the off-block-diagonal rectangles \(\heartsuit\) (and their Hermitian conjugate ones) actually acquire a mass. By a mass, we mean the term in the Hamiltonian that you would associate with a massive field such as the W-boson field, \(m^2 \cdot \heartsuit^\dagger \heartsuit/2\). The value of the mass is actually proportional to the distance between WOLF and BUNNY, assuming that their internal sizes are much smaller than their relative distance.

In other words, these off-block-diagonal matrix elements become harmonic oscillators with huge frequencies. The further WOLF and BUNNY are, the higher frequencies we face, the further the equally spaced energy levels of the harmonic oscillators are from each other, and the more we may neglect the possibility that the harmonic oscillator could actually be found in an excited state. We may approximate the \(\heartsuit\) harmonic oscillators by assuming that they're in the ground state most of the time (or, classically, at the \(\heartsuit=0\) point). Only when WOLF and BUNNY get close enough to each other, a significant chance that the harmonic oscillator gets excited emerges. The virtual effects of this harmonic oscillator – its propagators – transmit influences between WOLF and BUNNY. Of course, the interaction between them is love, a reason I picked \(\heartsuit\). ;-)

Once again, we see that the matrices may have a block-diagonal form. The blocks on the diagonal remember the coordinates of particles in the subsystems while the off-block-diagonal blocks have \(X\) mostly equal to zero but their ability to get excited or nonzero is actually the one and only source of the interactions between the subsystems!

I won't be proving that the interactions induced in this way are exactly those you expect from M-theory, e.g. that they reduce to supergravity at very low energies. But it's true. Instead, let us discuss some simpler but still very interesting issues.

We said that there were many harmonic oscillators with frequencies scaling with the distance between WOLF and BUNNY. Don't they give us huge zero-point energies \(\hbar\omega/2\) that would also depend on the WOLF-BUNNY distance which would drive WOLF and BUNNY towards each other with a constant force? The answer is that such terms really do arise but they get exactly canceled!

The other compensating sources we haven't considered are the fermions. The off-block-diagonal elements of those 16 \(\theta^a\) also behave as harmonic oscillators, but the fermionic ones. The zero-point energies will scale as \(-\hbar\omega/2\) where \(\omega\) is again proportional to the BUNNY-WOLF distance. And if you're careful about all the factors, you will find out that those 16 Hermitian fermions exactly cancel the bosonic oscillators from 8 matrices \(X^i\) and that's exactly the right number because the ninth one is the longitudinal one, parallel to the BUNNY-WOLF separation, and it doesn't get massive at all. Supersymmetry actually plays a key role in making the individual separated objects independent and keeping their force low at very high separation. It's rather unlikely to find a non-supersymmetric matrix model that would have the same desirable properties.

That's great. We have multi-particle states. But do we also have gravitons and gravitinos with \(P^+\gt 1/R\)? The answer is Yes. They have \(P^+=N/R\) and are represented by the \(N\times N\) blocks. In other words, we may find such a graviton as the only object in a state of the \(U(N)\) matrix model. Now, \(U(N)\) and its adjoint representation roughly decomposes to \(SU(N)\times U(1)\). The \(U(1)\) part behaves much like the \(N=1\) model, except that the value of \(P^-\), our Hamiltonian, gets correctly rescaled: recall that \(P^-\) is inversely proportional to \(P^+\) and that's what a calculation yields (after the \(P^i\) momentum is fairly divided among the \(N\) entries to keep the trace constant).

The relative, interacting, \(SU(N)\) degrees of freedom describe the relative coordinates of those \(N\) D0-branes. And one may prove via index theorems that this model has exactly one state of a vanishing energy: there must exist a mathematically fascinating ground state wave function with a vanishing energy that solves a rather complicated differential equation in \(9(N^2-1)\) bosonic variables even though no one can write how this wave function looks like too explicitly (the most understandable argument in favor of the state's existence that I know is based on the interpolation between the BFSS model and screwing string theory which is "more solvable" in the weakly coupled limit). When this state is tensor-multiplied with the degrees of freedom of the \(U(1)\) model, we get the right single-graviton or single-gravitino states with all the required polarizations and all the required values of the momenta.

As you have noticed, the matrix model differs from non-relativistic multiparticle quantum mechanics by having the off-diagonal entries of all the matrices \(X^i_{mn}\). Otherwise the diagonal entries \(X^i_{nn}\) (no summing) behave in a very similar way to \(X^i_n\) in non-relativistic quantum mechanical models where \(i\) labels a direction in space and \(n\) labels a particle. But the BFSS matrix model also yields the off-diagonal entries, whole matrices! There is a sense in which this extension of position operators to matrices is analogous to the conceptual transformation that was imposed by the quantum revolution. But we're doing it at another level: each matrix entry \(X^i_{mn}\) in the BFSS model is an operator acting on the Hilbert space; we're just saying that there are many such operators so that one may organize them into new matrices whose matrix indices remotely resemble the indices labeling one of many particles in multiparticle quantum mechanics. But there are two such indices for each matrix of position operators!

Another thing you could worry about is the following issue: are the gravitons indistinguishable particles? And do they have the right statistics that depends on the spin? Once again, the answer is Yes. But the reasons are technically very different from those in quantum field theory. In quantum field theory, the multiparticle states are created by the creation operators constructed from the quantum fields. There's only one field for each particle species (and it commutes or anticommutes with itself at spatial separations) so the resulting states inevitably end up being symmetric or antisymmetric wave functions for commuting and anticommuting field operators, respectively.

How can the same symmetry and antisymmetry of the wave functions be realized in the matrix model? The answer is the \(U(N)\) gauge symmetry. Some time ago, I have mentioned that the physical states also have to be gauge-invariant: the BFSS matrix model is a gauge theory, after all. We said that this effectively reduced the number of bosonic coordinates from \(9N^2\) to \(8N^2\) – well, it is really \(8N^2+1\).

But we're dealing with a pretty large system and many interesting things follow from the \(U(N)\) symmetry, too. For example, consider a system of \(N\) gravitons such that each of them has the minimum longitudinal momentum \(P^+=1/R\). A funny thing of the \(U(N)\) group is that it has an \(S_N\) subgroup, the permutation group. The physical states must be invariant under it, too. If you consider states that are only supported by simultaneously diagonalizable matrices, the invariance under \(S_N\) condition simply means that the wave functions have to be symmetric! And if the graviton-like particles are actually gravitinos, you will get the antisymmetry because of some extra permutations of fermions that the permutation operator induces. You will get the right indistinguishability of the particles with otherwise identical quantum numbers. And the right spin-statistics relationship for the symmetry or antisymmetry emerges, too!

Needless to say, this (anti)symmetrization also applies to particles with larger values of \(P^+=N/R\), represented by larger blocks: the permutation group exchanging equally large blocks is a subgroup of \(U(N)\), too.

It's kind of incredible: the \(S_N\) permutation of particles was a purely discrete operation, a bookkeeping device, in quantum field theory. But in the BFSS matrix model, it is actually enhanced to a much more nontrivial structure, a whole continuous \(U(N)\) group. When particles of the same kind are far from each other, the \(U(N)\) symmetry is broken to \(S_N\) and the (anti)symmetry of the wave functions is the only residual condition. But Matrix theory shows that if the particles are coincident (or at least very close), the permutation group gets enhanced to a whole unitary group!

You may have believed that Nature is hiding some cool symmetries but they are broken to smaller symmetries. But would you be able to invent – by pure thought – that an important example of this phenomenon is a secret \(U(N)\) symmetry that is broken to the permutation group exchanging particles? Nature and mathematics are clearly more creative than we are. We were literally forced to discover the matrix description of indistinguishable particles (much like if we observed it experimentally); attempts by arrogant but limited self-described "seers" to social engineer it in a man-made way would have almost certainly failed. Humans are pretty smart but there are just many clever mysteries and mechanisms that are much more likely to be discovered only once our skulls hit them.

This is the kind of a revolutionary description that was promoted from the very title of this blog entry. You could look for theories extending general relativity that are as similar to non-relativistic quantum mechanics as possible. If you were creative and lucky, you would ultimately introduce matrices of degrees of freedom, with the right commutators, and a simple enough supersymmetric Hamiltonians would be surprisingly found to describe not only a Lorentz-invariant theory in the large \(N\) limit but actually a theory that has all the wonders of a consistent theory of quantum gravity.

For a while, you would think that you have found a completely new framework to understand quantum gravity, a competitor of string/M-theory. However, at some time, you would realize that it's actually exactly equivalent to string/M-theory or at least one of its Hilbert space's superselection sectors! That's not a coincidence; despite the great diversity and amazing phenomena how various things are related and represented, all these structures are just projections of a single theory as long as they are consistent. As Joe Polchinski once stated, all roads lead to string theory.



A moving human constructed out of screwing string theory (sometimes incorrectly called "matrix string theory").

Because this blog entry has gotten pretty long, I will wrap it up at this point. Sometimes in the future, I plan to explain why other objects with the right features – including (spherical, toroidal, and other) membranes, \(E_8\) gauge bosons (and whole gauge supermultiplets) on Hořava-Witten domain walls, and strings with the right perturbative string interactions in the weakly coupled limit (in compactified versions of the BFSS matrix model, something I was fortunate to discover) emerge out of the matrix model description of string/M-theory.

So far, I thank you for your patience if you managed to penetrate up to this point...

FAQ on black holes and information

Of course, the black hole information puzzle has been discussed in dozens of older TRF blog entries

Backreaction has posted a flow diagram that can't hide its similarity with the scheme of a female brain. Unfortunately, among lots of wrong answers to questions about black holes, it doesn't include the right answer to the question what happens with the information stored in an evaporating black hole.




So here are the questions and answers.

• When a massive object of mass \(M\gg m_\text{Planck}\) collapses (let's assume \(Q=0\) and \(\vec J=0\) although we don't have to), does a horizon form?

Yes, it does. When Karl Schwarzschild originally presented his solution in 1916, Albert Einstein had doubts about its validity. He believed that some forces would prevent a collapsed star from developing the horizon and the black hole interior, although these features were clearly suggested by Schwarzschild's solution. Einstein used to think that only the exterior, mild enough geometry of the solution could be trusted.

However, no such forces can exist for a large enough black hole. If you have a really huge conglomerate of matter, the density may be as low as the density of water and the gravitational force becomes large enough so that the collapse is inevitable. To derive this collapse, we only need to rely on the validity of Einstein's equations in each region – and they may be, locally speaking, totally unspectacular reasons whose density only matches that of water and the curvature is correspondingly low.



The inevitability of the emergence of the event horizon was proved by the Hawking-Penrose singularity theorems of the 1970s. Why did they care about the singularity if it is the event horizon that defines the black hole? Well, the event horizon is the boundary of the black hole interior. And the black hole interior is composed of all the spacetime points whose future time-like geodesics can't get to the spatial infinity. They must get elsewhere; and the "elsewhere" means the singularity because there are not too many other options. (Let's assume that one can't form or connect to new infinite spacetimes by a collapse of a star.)

So they needed to prove that one actually forms a singularity, the alternative future fate of an observer. And indeed, such singularities arise rather generically when a massive enough object collapses. So black holes have to exist. Their existence has also been independently derived from string theory although this theory is based on an independent starting point from that of general relativity. And the evidence for astrophysical black holes – those in the real world – has become overwhelming, too.

• Do black holes radiate?

Yes, they do. However, in this case, we don't have any experimental detection of the radiation to boast: the radiation from large black holes is negligibly weak and there aren't too many smaller black holes around us. That's a pity; if there were some black holes of this kind on the market, Stephen Hawking would surely get his hugely deserved Nobel prize.

The thermal radiation – whose black body temperature is equal to the gravitational acceleration at the event horizon (surface gravity) in certain natural units – was originally derived by Stephen Hawking around the mid 1970s. A simplified calculation for the Rindler space – a wedge of the flat Minkowski spacetime as seen by a uniformly accelerating observer – was later derived by William Unruh.

Hawking's and Unruh's calculations only rely on the validity of the semiclassical approximation; for large black holes, one may consistently ignore all effects and corrections that are of higher order in Planck's constant than those that are considered. The existence of the Hawking radiation may also be partly independently derived from string theory.

In the operator formalism, one needs to discuss the Bogoliubov transformation mixing creation and annihilation operators which is needed if we redefine the Hamiltonian from an inertial observer's one to an accelerated observer's one (and if we have to redefine the ground state in a corresponding way). In the path integral approach, the black hole temperature may be derived using the Gibbons-Hawking method.

• Does the radiation at \(M\gg T\) i.e. for black hole masses much heavier than the typical mass/energy of the Hawking particles (dictated by the temperature) carry information?

Yes, it does. It's the key insight that got settled in the mid 1990s. According to Hawking's original approximate calculation, the information couldn't be getting out because that would violate causality. However, the exact analysis that goes beyond the semiclassical approximation changes the answer to this qualitative question. Quantum gravity allows the causal restrictions of the black hole background to be surpassed.

(Alternatively, the external observer always has the right to imagine that the infalling matter got stuck at [or right above] the event horizon and there is no interior at all.)

This is in no contradiction with the validity of the semiclassical approximation for operationally meaningful questions: the code in which the Hawking radiation stores the information is incredibly subtle and scrambled and using the fat and awkward probes that are compatible with the semiclassical approximation, we can't decode the information. The semiclassical approximation gives the right approximate answers to arbitrary quantitative questions which have the form of continuous numbers (with small errors); however, it may fail and it does fail to give the right Yes/No answers to some qualitative questions.

The main theoretical weapons that allowed us to learn that the information is preserved were new approaches to string theory, especially the AdS/CFT correspondence and Matrix theory (that I plan to write about soon). These descriptions of string theory are manifestly unitary – they evolve quantum states in a one-to-one way from the past to the future just like your textbook models of quantum mechanics – but they may also be shown to incorporate (evaporating) black holes.

Stephen Hawking has admitted that he was wrong and he surrendered a famous bet against John Preskill. If quantum gravity is treated properly, the information comes out.

• Do black holes leave remnants with a lot of information after they evaporate?

Because the Hawking radiation depends on the initial state – subtle correlations in this seemingly thermal radiation betray what the black hole was made of – there is no need for a remnant that would carry the information after the bulk of the black hole mass is evaporated away.

In fact, remnants – essentially point-like objects that may carry an arbitrarily high amount of information – would yield the theory (or Nature, if she suffered from the remnant illness) inconsistent. Remnants would violate the holographic entropy bounds: one can't really squeeze too much information to too small a volume. Also, there would be infinitely many types of remnants and their pair production would be infinitely frequent and it would correct many ordinary physical quantities by infinite amounts, and so on.

So yes, the information is getting out of the black hole as it Hawking-radiates. When the black hole gets very small, the higher-derivative terms in quantum gravity and various stringy/M-corrections become very important. In the very final stages of the evaporation, the evaporating black hole behaves just like an unstable elementary particle. The transition from a large black hole microstate to a particular elementary particle species is gradual; there is no qualitative difference between them.

These "unified objects" may be described as black holes (whose corrections to Einstein's equations are small) if they're much heavier than the Planck mass; and they may be described as elementary particles of different species (whose gravitational force may be neglected) if they're much lighter than the Planck mass.

The only region where the full quantum theory of gravity i.e. string/M-theory fully exposes its muscles (and the aforementioned approximations are not enough) is the regime in which the mass of the objects is comparable to the Planck mass. String/M-theory is the peacemaker that is needed for the smooth interpolation between the elementary particles propagating on a mildly curved, nearly flat background; and the general relativity that is needed for heavy black holes. Macroscopic physical phenomena in these two opposite extremes are captured by two copies of an effective quantum field theory (and in fact, the classical field theory limit is the most important part of them, especially on the black hole side); however, the interpolation required by consistency is inevitably a slightly more general theory than quantum field theory, namely string/M-theory.

• Does the Hawking radiation carry nonlocal correlations that contain some information?

Yes, it does. Several researchers have offered their toy models of the code by which the correlations are stored and none of them has been convincing enough for everyone else so far. However, despite the absence of an easily calculable toy model, we know that the Hawking radiation does carry correlations of the most general type – i.e. non-local entanglement between all the Hawking particles – that stores the information about the initial (or later) state.

It shouldn't be shocking that the black hole is able to produce this subtle entanglement. The Hawking radiation itself may be interpreted as quantum tunneling. The information tunnels out of the black hole interior, too. In other words, the causal diagram that seemingly strictly tells us that the black hole interior is separated from the exterior shouldn't be taken too seriously because the metric tensor is a fluctuating observable. Much like the alpha particle can't be guaranteed to remain inside a nucleus – that's why we occasionally observe alpha-decay – Hawking particles can't be "totally confined" within the black hole interior and may appear in random directions outside the black hole. They're still coming from the same origin and may be entangled.

• How to make the semiclassical limit nonlocal?

This question appears in Sabine Hossenfelder's chart but it is completely misguided. You can't change the properties of objects in Nature. It's up to Nature to decide whether some things are local or nonlocal; we can't "redesign" Nature. And while Nature produces nonlocal correlations that may be found in the exact treatment of an evaporating black hole, e.g. in AdS/CFT or Matrix theory if you want to be really explicit, the nonlocal correlations inevitably disappear if we reduce our description to the semiclassical limit.

The semiclassical limit is what Stephen Hawking calculated in the 1970s and he correctly determined that the radiation couldn't carry the information away in the form of (nonlocal) correlations because that would violate the locality. In fact, even if he calculated the exact answer to all orders in the perturbation theory in \(\hbar\), it would still be right that the information can't get away.

The preservation of the information may only be seen if one goes beyond the perturbative expansion – beyond all orders in a Taylor expansion in Planck's constant. Because the question above appeared in Sabine Hossenfelder's chart, one must say that the right answer to the questions about the fate of the information in black holes isn't included in the chart, despite the plethora of wrong answers that are included.

• Do black holes have hair, after all?

For general relativity in low enough dimensions and its simple enough extensions, one may rigorously prove that the black holes can't have hair. However, the black hole has many microstates so it surely does remember the information in some form of "hair". The only question is whether the hair may be visualized in some geometric way – whether the information carried by a black hole microstate gives the black hole some detailed complicated "shape".

Samir Mathur is behind the most convincing proposals that would yield a "Yes" answer to this question. His fuzz balls literally look like very complicated objects that totally change the character of the black hole interior – fill it with complicated fuzz that carries a huge amount of information. It should still be true from locality (validity of general relativity at long distances, for fat enough probes) that an infalling observer generically experiences empty space. This emptiness must result from some averaging over all the complicated types of fuzz.

In some simple enough contexts, Mathur and collaborators have constructed a literal "local" representation of hair – the microstates are genuine solutions of ordinary extended Einstein's equations. In more generic situations, however, the degrees of freedom needed to describe the hair probably need to be more nonlocal themselves.

These are cute constructions that didn't have to exist and many people still believe that they either don't exist or they're wrong. But whether such "visualizations" of the information carried by black holes (and by their Hawking radiation) may be found or not, the answers to the remaining questions listed in this blog entry are almost certainly irreversible at this point. What would many people love to see is some readable answer to the question "where in space and how" the information is stored on the black hole horizon or in the radiation. However, the exact description of these objects doesn't seem to have a form of a local field theory in the spacetime so the questions of the type "where do things live" may be misplaced.

On the other hand, it's conceivable that they're not misplaced. There could exist some "much more spacetime-local" description of the black hole evaporation than what you can get from the AdS/CFT correspondence or Matrix theory. It wouldn't be the first time when an "ordinary description" of a structure that was believed by many experts to inevitably transcend field theory was found: the membrane minirevolution found some explicit field-theoretical Lagrangians for theories that many people had believed to be inequivalent to Lagrangian field theories.

Bribed, stealing officials enjoy their $500k before they're caught

David Rath, the main symbol of corruption of the Czech social democratic party, took most of the $500,000+ bribes in a major Czech corruption scandal, the grandest one at least in the last 20 years.

(He has obviously stolen much more money throughout his life, some of those millions of dollars are being investigated at this very moment. Just in his house, they found additional $1.5 million and then extra $0.5 million. Rath's father claims that those latter $0.5 million are just his – the father's – savings he earned somewhere in Emirates 20 years ago.)



Ex-governor of Central Bohemia Dr David Rath and Ms Kateřina Pancová, an ex-director of a hospital in the region.

He defended himself by saying he thought that the shoebox contained wine. This defense is truly ludicrous given the fact that every single place where he and his key collaborators – Ms Pancová, a director of a hospital, and Mr Rott, her partner who left the Czech Parliament after being totally drunk during an important vote – were meeting was eavesdropped for half a year.

Some of these impressive police's eavesdropping skills may have been inherited from socialism when it was normal to monitor the (suspicious and inconvenient) citizens.

So the police knows about – and can easily expose – his (and their) feeling about every penny and every motion of a penny during the last 6 months or so. Their discussion minutes before Dr Rath was arrested is kind of amusing. Well, at least now it seems amusing when we already know that they couldn't laugh to justice for too long (at least so far it looks so).




It took place in Ms Pancová's house in Rudná near Prague on Monday, May 14th, 2012. They're dividing over $500,000, most of which went to Dr Rath.

Via novinky.cz.

Pancová: ... what he has at his disposal. It would be nice of you. I would even love to have such nice cute money.

Rath: So nice little money. Shiny, so pretty yellow, yellowly glossy. So pretty, so goldish.

Pancová: Yup, yup, I would like to have them. It doesn't get spoiled or mouldy. One may hide them well.

Rath: Yup, exactly. Yeah, yeah. Kha kha kha kha (laughter).

Pancová: It's enough to hide them somewhere.

Rath: Where, into your little pocket?

Pancová: Into my little pocket, khi khi khi, so that I have them for the bad times.

Rath: Kha kha kha.



Pancová: I would be like that Tsar Nicolai II [of Russia]. His family was being shot at but they were not dying at all because they were covered by gold and those diamonds. They thought that the tsar family was protected by tsar, I mean God. But they had gold all over their bodies. Haven't you read it? [More than 1.3 kg of diamonds on Olga and Maria.]

Rath: No, I haven't read it.

Pancová: They were shooting into them all the time but they were still alive. Now, the chaps from the Red Army [sic] were already completely stunned by it, thinking: what's going on? What's going on? This tsar must really come from God. And only later, they found out that they were wrapped in gold.

Pancová: Look, you have one more suitcase here.

(Rath and Kott are leaving the room in the upper floor.)

Rath: Won't you kindly plaster it once more for me?

Pancová: Don't worry. Wrap it. Do you want one more beer?

Rath: Please, no, I must already leave because one more person is visiting me later [it turned out it was Mr Michal Pohanka, a socialist ex-lawmaker who switched and supported a center-right government in 2007; some people accused him of having been bought; he's a candidate for a mysterious person who may have been scheduled to receive 1/2 of Rath's $350,000 in the shoebox]. Ouch, so what do you say, Peter?

Kott: Well, I know one answer to this question.

Pancová: The world is a shitty place, my beloved little sister used to say.

[Informal conversations continue up to Rath's departure.]

Rath: On Friday morning, we will meet at your place. [The debate took place on Monday evening.]

Pancová: Yes.

Rath: We will agree about the Saturday plans later. So far, our plan is that you will pick me up.

Pancová: Right, at 8:15.

Rath: Sounds good, bye bye.

Kott: See you.

Pancová: Jesus Christ, what a chilly weather is over there... Don't leave! [For a few minutes, Ms Pancová believes that the climate is her and Dr Rath's greatest problem.]



The conversations took place in this house of Ms Pancová and Mr Kott in Rudná near Prague. Of course, all the three "top people" in this scandal have lots of other real estate. Most of them have already been sealed.

Rath is leaving. In two minutes, Pancová and Kott find out that something is going on in front of the house.

Kott: An urn [a special police car, URNA]! Don't do anything! Don't do anything!

Pancová: We're in deep shit, we're in deep shit.

Kott: Calm down, calm down.

Pancová: So we're in deep shit!

Kott: Perhaps. Calm down, calm down!

:-)) Sounds just like from some parodies and comedies except that this conversation is real. The degree of happiness that these folks experienced when they stole half a million dollars is high, indeed.

A few minutes later, Ms Pancová and her sex toy, Mr Kott, had to go to the cold weather, too.



So far, David Rath enjoys one of the toughest Czech prisons in Litoměřice. He decided to become a marathon runner to stay in a good shape. Photo via Blesk.cz

Recent JS-Kit comments

This widget used to be in the right sidebar. I removed it on June 21st, 2012, because Echo is being decommissioned, the "recent comments" in the widget are no longer recent, and the Echo comments will ultimately be imported into DISQUS.

Recent Echo comments

New Echo comments aren't being accepted anymore and Echo will disappear on October 1st. Use DISQUS instead – where old Echo comments will be imported.

Sorry for the wrong spacing. It looked correct in the sidebar.






That's it.

Recent slow Blogger.com and JS-Kit Echo comments

Recent Blogger.com slow comments

This used to be inside the right sidebar instead of the DISQUS recent comments combination widget:

You see that it includes the fresh DISQUS comments because I turned on synchronization of DISQUS and Blogger.




But it's no longer there.

Recent Echo comments ☼

In the sidebar, there were also old Echo fast comments.

This widget used to be in the right sidebar. I removed it on June 21st, 2012, because Echo is being decommissioned, the "recent comments" in the widget are no longer recent, and the Echo comments will ultimately be imported into DISQUS.

Recent Echo comments

New Echo comments aren't being accepted anymore and Echo will disappear on October 1st. Use DISQUS instead – where old Echo comments will be imported.

Sorry for the wrong spacing. It looked correct in the sidebar.

That's it.

The latest version of the gadget, before it was erased (moved here), was the following:

The first Echo (well, Haloscan) comments were posted on March 12th, 2005, when the blog was 5 months old. See some statistics.

New Echo comments aren't being accepted anymore, old Echo comments are still available via the pirate icon beneath each blog entry, but Echo will disappear completely on October 1st. Use DISQUS instead – where old Echo comments have been imported.

Another JS-KIT feed I have removed...

Knud Rasmussen pictures: Greenland is melting less quickly than 80 years ago

Knud Rasmussen (1879-1933) was a Danish polar explorer and the father of Eskimology. By an accident, 80-year-old pictures of the Greenland taken during his expeditions were just found in a Danish basement:

Daily Mail, TG Daily, Nature, Live Science, Newswise, Science Codex

The pre-satellite pictures of ice shelves are rare. One may evaluate them in various ways.

For example, 55% of the retreating glaciers were retreating at a faster rate 80 years than in recent years. On the other hand, the average rate of retreat (in meters per year) is higher today than it used to be because there are some isolated very quickly retreating glaciers today.



So whether the retreat of Greenland's glaciers was faster 80 years ago than today may depend on the detailed specification of the "contest". However, one thing is clear.




And the clear fact is following one: the evidence convincingly shows that the dynamics of the Greenland's ice is qualitatively similar to what it was 80 years ago when the carbon dioxide emissions per year were 4 times lower than today.

Via helvio

Why \(\NNN=8\) supergravity is probably divergent at 7 loops

Arguments in favor of finiteness are much more sloppy

I have been discussing the maximally supersymmetric supergravity and its conjectured perturbative finiteness many times on this blog. But the immediate reason for a new entry is the following paper by my ex-adviser Tom Banks,

Arguments Against a Finite \(\NNN=8\) Supergravity

In this text, I want to review the \(\NNN=8\) supergravity in \(d=4\), its field content and "other objects content", the arguments that have been raised for its perturbative finiteness, the explicit 7-loop counterterm that is allowed by all the symmetries, and the argument why its coefficient is probably nonzero.




As I discussed in the article about the royal status of 11-dimensional supergravity, 11 dimensions is the maximum number of large dimensions in which one may have a theory with unbroken supersymmetry and a nontrivial particle content at long distances. The theory was discovered in the late 1970s and is known as the 11-dimensional supergravity.

In such a high spacetime dimension, it's of course non-renormalizable as a field theory but we have known since 1995 that it is the right long-distance approximation to a theory that is fully consistent and finite, namely M-theory which is an important cousin of string theory. More precisely, M-theory is another vacuum or limit of string theory that, unlike the 10-dimensional vacua, has 11 spacetime dimensions. The 11-dimensional dynamics may be obtained by sending the string coupling \(g_s\) to infinity either in the \(E_8\times E_8\) heterotic string theory or (which is easier) in type IIA string theory. When we do so, a new, eleventh dimension of spacetime emerges and grows larger.



The elf-dimensional supergravity

This 11-dimensional M-theory may be compactified down to various lower spacetime dimensions. From the viewpoint of the bread-and-butter physics, the most important ones among these compactifications are compactifications on singular 7-dimensional manifolds of \(G_2\) holonomy: it's one of the several major descriptions by which string/M-theory may describe the Universe around us and which has all the desired qualitative properties to agree with all the particles and forces we have observed in our world. These compactifications were recently promoted as the most canonical example of "generic string/M-theoretical compactifications" by Gordon Kane, Bobby Acharya, and their colleagues in their phenomenological papers. Edward Witten has played a key role in the discovery of this new "stringy scenario" a decade ago, too.

However, those compactifications are complicated and only preserve four real supercharges which is equivalent to the \(\NNN=1\) supersymmetry in \(d=4\), the usual extent of supersymmetry that is employed in the realistic model building. From a top-down viewpoint, the more fundamental are the more supersymmetric compactifications, especially the maximal i.e. \(\NNN=8\) supergravity with 32 real supercharges, the same number as the original 11-dimensional theory. This theory is pretty much unique and so is its consistent ultraviolet completion, M-theory on a 7-torus.

The maximal supergravity: symmetries, field content

As we have recalled in the royal article, the 11-dimensional supergravity contains one supermultiplet of fields, the gravitational supermultiplet, which boasts \(2^8=256\) physical polarizations for each (light-like) momentum. One-half of them are fermionic, they're the components of the spin-3/2 gravitino (a maximally constrained tensor product of a spinor and a vector), while the remaining 128 polarizations are bosonic ones. 44 of them are components of the metric tensor; 84 of them are components of the \(C_{\lambda\mu\nu}\) three-form potential that couples to membranes and fivebranes.

What happens if you compactify the theory to 4 dimensions i.e. dimensionally reduce it over 7 dimensions? Some values of the vector-like Lorentz indices will become scalar-like. The spin as understood in \(d=4\) will be smaller than the spin as extracted from \(d=11\). How many scalar fields will we get?

We only get scalar fields from the bosons; the spin-statistics relation always holds in these and other theories. From the metric tensor, we get the components \(g_{\mu\nu}\) where \(\mu,\nu\) go over the 7 dimensions we have compactified on a tiny 7-torus. The number of such components is \(7\times 8/2\times 1 = 28\). These components remember the radii and angles in the 7-torus defining the compactification. The gravitational excitations should be traceless but we will assign the task of keeping the tracelessness to the components in the uncompactified dimensions so there's no elimination of components here.

There are also now-scalar components of the three-form potential, \(C_{\lambda\mu\nu}\). If all the indices are set to some of the numbers in the list of 7 compact dimensions, we get \((7\times 6\times 5)/(3\times 2 \times 1) = 35\). I would have to spend a lot of time to explain the complete justification but we must also count the components of the dual 6-form potential \(\tilde C_{\lambda\mu\nu\pi\rho\sigma}\) that couples to the fivebranes. These six indices may be set to the seven compactified dimensions with one missing exception; so there are seven ways to do so. We get 7 new scalar physical polarizations.

In total, we have \(28+35+7=70\) scalars in the four-dimensional \(\NNN=8\) supergravity. If you were a very good numerologist, you would be able to realize that \(70=133-63\) and because \(133\) and \(63=8^2-1\) are dimensions of the \(E_{7(7)}\equiv E_{7(7)}(\RR)\) and \(SU(8)\) groups, those 70 scalars could be parameterizing the coset or quotient \(E_{7(7)}/SU(8)\) of a noncompact exceptional group and its largest noncompact subgroup.

The arithmetic check above is of course very far from being a full proof that the symmetries are relevant for the theory; but the conclusion is right. The supergravity theory has an exact, internal, noncompact \(E_{7(7)}\) global symmetry. Its \(SU(8)\) subgroup's action on the scalar fields is trivial; so the scalar fields inevitably parameterize the coset. There are various independent ways to see why the noncompact exceptional symmetry exists – a brute force algebraic calculation; arguments that the high degree of SUSY implies that the moduli spaces of scalars have to be cosets; reconciliation of partial stringy symmetries such as T-dualities – but just believe me that the symmetry is there.

Charged objects in the theory (its stringy completion)

To have some fun, let me show you one more numerological check that the \(E_{7(7)}\) exceptional symmetry exists. Consider the stringy completion of our \(d=4\) \(\NNN=4\) SUGRA, namely M-theory on a seven-torus, and let's count the number of individual \(U(1)\) gauge symmetries (each of them has a new kind of a charge and the corresponding charged objects, too).

First, remembering Kaluza and Klein 90+ years ago, we know that general relativity compactified on a torus produces \(U(1)\) groups. We get seven of them; the gauge fields are \(A_{\mu}^{(n)}=g_{\mu n}\) where \(\mu\) is the 4-dimensional Lorentz index and \(n\) goes over the seven compact dimensions and labels the seven different \(U(1)\) groups. The charged objects are any particles that are moving in the directions of the compactified seven-torus; recall that quantum mechanics guarantees that the momentum on a compact space is quantized (and, consequently, so is the resulting electric charge).

However, we also have \(A^{(m,n)}_\mu=C_{\mu mn}\). The indices \(m,n\) take 7 possibilities each and due to the antisymmetry, they produce \(7\times 6/2\times 1=21\) different \(U(1)\) fields. The corresponding charged objects are M2-branes with both dimensions wrapped on a two-torus within the spacetime seven-torus.

Similarly, there are five-branes wrapped on five-tori that are charged under similar \(U(1)\) symmetries whose gauge fields are \(\tilde C_{\mu mnopq}\) where \(mnopq\) form a quintuplet of indices chosen from the list of seven compact directions. We get "7 choose 5" which is 21 different \(U(1)\) fields again.

There are some extra charged objects (and their corresponding \(U(1)\) groups) I haven't counted yet. The charged objects are the Kaluza-Klein monopoles which are really 6-branes (we want fully wrapped ones) with 1 additional "systemic" compact spatial dimension. Any of the 7 spacetime toroidal dimensions may play the role of this "systemic" Kaluza-Klein direction so we get 7 independent Kaluza-Klein monopole charges.

In total, we may find \(7+21+21+7=56\) different \(U(1)\) groups. It's only 28 times and not 56 times more electromagnetic fields than in Maxwell's theory because in the 56-based counting, Maxwell's theory has two types of charges, the electric ones and the magnetic monopoles. Now, it is not a coincidence that 56 is the dimension of the fundamental representation of \(E_{7(7)}\) – which is just a continuation of the compact \(E_7\) Lie group. In fact, these charges do transform under the \(E_{7(7)}\) non-compact global symmetry that I have already discussed.

The supergravity itself, i.e. the long-distance limit of M-theory on a seven-torus, doesn't admit any charged objects. The torus is infinitely tiny in this limit so the Kaluza-Klein particles are infinitely heavy and as long as we are at a generic point of the moduli spaces, the other charged objects are related to the Kaluza-Klein particles by the symmetry so they must be equally heavy. And in fact, you could think that in the supergravity, the charged objects and their precise charge quantization conditions don't make any difference at all.

However, in the full stringy theory, namely M-theory on a seven-torus, these charged objects – Kaluza-Klein particles (moving along the new dimensions), wrapped membranes and fivebranes, and wrapped Kaluza-Klein monopoles – are allowed, real, and they make a difference. Because the Kaluza-Klein particles are electromagnetic duals of the wrapped Kaluza-Klein monopoles and the wrapped M2-branes are electromagnetic duals of the wrapped M5-branes – it is not an accident that the list 7,21,21,7 was symmetric – we must remember the Dirac quantization rule for the magnetic charges.

When we impose this rule on all the charges we have, we find out that these charges must belong to a particular lattice. The lattice isn't quite unique but by \(E_{7(7)}\) transformations, you may get all the solutions (all the lattices) from a particular one. It means that the 70-dimensional (real) moduli space labeled by the 70 scalars we started with\[

{\mathcal M} \approx E_{7(7)} / SU(8)

\] is nothing else than the space of all possible lattices which may be chosen as allowed charges of the 56 independent types of electromagnetically charged objects. In fact, the actual moduli space is locally equal to what I just wrote but its exact global structure has some identifications. The moduli space is\[

{\mathcal M} = E_{7(7)}(\ZZ)\backslash E_{7(7)} / SU(8)

\] It's a quotient taken from both sides. The extra discrete group at the beginning of the right hand side is the U-duality group; it's a group of the noncompact symmetry transformations that don't belong to the compact \(SU(8)\) subgroup but that still preserve the lattice of allowed charges.

In other words, the full stringy theory of quantum gravity – which takes things such as the Dirac quantization rules into account – breaks the continuous noncompact \(E_{7(7)}\equiv E_{7(7)}(\RR)\) symmetry down to its \(E_{7(7)}(\ZZ)\) discrete subgroup. It's a good thing: quantum gravity morally prohibits continuous global symmetries so it's good that string theory only keeps a discrete subgroup of it – and this subgroup is actually a group of local symmetries if you look a bit more carefully. (The difference between global and local symmetries is subtle for discrete groups; the existence of monodromies and cosmic strings is what operationally decides about the right adjective.)

So string theory has a moduli space \({\mathcal M}\) of inequivalent vacua (in the old, quantum field theory era, we would probably talk about inequivalent theories) i.e. of different superselection sectors.

Now you should read two roads from \(\NNN=8\) supergravity to string theory if you haven't read it previously. In that article, I argued that this supergravity theory has two basic bugs from a phenomenological viewpoint. It's too supersymmetric so that it can't produce a realistic spectrum; and it's divergent and inconsistent, at least at the nonperturbative level.

If you work hard to fix either of the two glitches, you are led to the full string/M-theory in both cases. To fix the excessive supersymmetry that forbids realistic quarks and leptons, among other things, you have to find out that the supergravity theory is really a compactification of a master \(d=11\) theory and you have to consider more sophisticated compactifications on the \(G_2\) holonomy manifolds.

To fix the problems with the divergences, you must include all the extra objects and excitations – objects moving or monopole-charged in new Kaluza-Klein dimensions; wrapped branes – which make the theory finite and consistent at very short distances in the same sense in which the W-bosons regulate all the divergences we know from Fermi's non-renormalizable four-fermion theory of the weak nuclear interaction.

There is no doubt that a fully consistent completion of this \(\NNN=8\) supergravity theory has to include black holes, including the charged ones under the \(2\times 28\) \(U(1)\) groups (it's always possible to create charged black holes in electromagnetic fields, even classically or "astrophysically") and quantum mechanics dictates that the charges have to be quantized (Dirac quantization condition). If you think for a while, you will realize that consistency requires the small quantized charged objects such as the Kaluza-Klein stuff and M2-branes and M5-branes and all their interactions and properties are pretty much dictated by consistency, too.

Even if you assume nothing else than consistency, you are led to the full string/M-theory as the only "complete cure" for the problems of all the older theories that approximate string/M-theory.

Perturbative finiteness

The non-perturbative inconsistency of the \(\NNN=8\) SUGRA without the stringy objects is indisputable: it either violates the rules of general relativity that give every region the human right to create charged black holes; or it violates the Dirac quantization rule for such charged objects. Or something else.

However, it has seemed possible to many people including your humble correspondent that the \(\NNN=8\) theory could be perturbatively finite. You could calculate the scattering amplitudes of the gravitons (and their superpartners) in the perturbative expansion, adding one order after another, and you could never experience a divergence. All of them would cancel.

I no longer think it is too likely but let me mention two main arguments – which I consider too sloppy to be given too much weight at this moment. One set of these arguments are papers by Renata Kallosh such as this March 2011 paper. She has argued that the theory is perturbatively finite and has used various methods – including the light cone gauge (which I kind of like) and some "deformed" versions of the noncompact symmetry – to prove her point.

A more general reason why I was very open to the possibility that the theory was finite were the KLT relations: the \(\NNN=8\) supergravity looks like the \(\NNN=4\) gauge theory "squared". Because the gauge theory – coming from open strings – is (not only) perturbatively finite, it seemed plausible to me that the supergravity theory – arising from closed strings (in some moral sense, open strings tensor squared) – may be perturbatively finite as well. Various objects relevant for the supergravity case – including the Riemann surface moduli spaces in string theory – looked like the "squared cousins" of similar objects in the open string theory. So the finiteness of the gauge theory could have been squared and preserved, too.

But the relationship between the amplitudes at the loop level isn't exact so the argument is really sloppy, I think today. So I joined those who find it much more likely that the \(\NNN=8\) supergravity is divergent even perturbatively. Simeon H. was among those who have been telling me it's the likely right answer for years.

Now, in the 2006 article about the finiteness of supergravity theories, I explained that at one-loop, the problems of the quantized Eistein's theory cancel. It's really because the would-be counterterms are of the form \(R^2\), the squared Riemann tensor. There are just three independent ways how to contract the indices (an easy-to-see basis is the Riemann tensor squared, Ricci tensor squared, Ricci scalar squared). One combination is topological (the Euler density) which means that it doesn't influence the local physics (or equations of motion: it is a total derivative) and the other two combinations may be eliminated by a field redefinition that adds a multiple of the Ricci tensor (or Ricci scalar times the metric) to the metric tensor.

However, at two loops, there is a well-known divergence with a complicated coefficient. It must be cancelled by a counterterm. The divergent coefficient of the counterterm arrives along with a new residual finite coefficient that may be adjusted (and must be adjusted to make the theory specific). At higher loops, the vertices combine in infinitely many ways to produce infinitely many different types of counterterms whose values have to be determined. It's not possible using a finite number of experiments. So the quantized general relativity is totally unpredictive; that's the right description why its non-renormalizability is a problem.

If you consider supergravity with some supercharges, they may cancel some leading divergences but they usually just "delay" where the first divergences occur. In combination with other known symmetries, the \(\NNN=8\) supergravity shifts the "leading" possible divergence to seven-loop diagrams. Be sure that these are complicated mathematical objects – one could say that they are sums of millions of complicated 28-dimensional integrals – but many people became very good in dealing with these objects.

The leading candidate 7-loop divergence

Although the supergravity seems to have many symmetries (noncompact symmetries, many supercharges), they don't seem enough to eliminate infinitely many types of candidate divergences. After all, there's just a finite number of such symmetry generators so it would be surprising if they could cure infinitely many diseases.

In this respect, the situation is different from the \(\NNN=4\) gauge theory which celebrated its 35th birthday some months ago. This gauge theory has an infinite-dimensional symmetry that underlies it, the Yangian, and other structures. Although the supergravity theory has a doubled number of supercharges, it seems less constrained than the gauge theory.

At any rate, even when you impose all the symmetries, you see no reason why the theory shouldn't generate a particular leading 7-loop counterterm which may be written in superspace approaches in an extremely simple way: \(\int d^{32}\theta\). It's just the integral over (i.e. total generalized volume of) the full 32-dimensional superspace!

Superspaces that are this huge suck for the other terms so you want to translate this schematic form into components. It turns out that in the language of the component fields, such a counterterm is equivalent to something like \(D^8 R^4\), schematically speaking. So it's the eighth power of the superderivative acting on the fourth power of the Riemann tensor – with various orderings and contractions of indices.

See e.g. this 2010 paper by Elvang, Beisert, Freedman, and others that discusses these counterterms.

Dimensionally speaking, the superderivative \(D\) has the same units as the square root of the curvature tensor (which is bilinear in the ordinary derivatives) so \(D^8\) is dimensionally equivalent to \(R^4\) and \(D^8 R^4\) has the units of \(R^8\). That's OK – thanks to kristan for corrections – because a \(k\)-loop expression should produce terms with units just like \(R^{k+1}\). Recall that the tree-level (0-loop) Einstein-Hilbert action is proportional to \(R\); we canceled the \(R^2\) one-loop corrections; but the \(R^3\) two-loop terms gave us the first lethal divergences in the quantized general relativity. Extrapolate this process to 7 loops and you will get the right units.

The only "hope for perturbative finiteness" could be that this candidate counterterm is produced with a coefficient that is zero. All the contributions to it should cancel. However, there's no known convincing reason why it should be the case; all the known "symmetry weapons" have already been used to cancel other terms.

Now, Tom argues that a problem must arrive at a finite order in the perturbative expansion of the supergravity because it's important for the consistency of black hole physics to break the \(SU(8)\) symmetry, the maximum compact subgroup of the noncompact symmetry. I am confused by his argument because I have believed that the \(SU(8)\) symmetry is exact even in the full string/M-theory.

Tom also writes, much like your humble correspondent, that the classicalization papers are wrong although I haven't studied his reasoning sufficiently carefully to know whether he has the same reasons, less correct reasons, or more correct reasons for the dismissal of these Dvali et al. paper as/than I do (there are surely some similarities in the strategies how to think about these matters).

Tom Banks mentions that "SUGRA has divergences at the 7-loop level" is a "consensus" of the experts. I am a bit surprised by his using this language because while Tom is left-wing, he is not a guy who would promote the "consensus science". After all, he refused to belong to various consensuses himself. ;-) But OK, he is making a sociological point, one that e.g. Renata Kallosh could disagree with.

Shmoits, Shwolins, and related feces

Finally, let me mention that some of the world's most notorious crackpot discussion forums such as "Not Even Wrong" have been running the campaign forcing everyone to believe that the \(\NNN=8\) supergravity had to be finite (they don't understand the difference between perturbatively finite and finite at all so they couldn't discuss this subtlety). Of course, they have never had any arguments – and it seems to disagree with the evidence at this point. It was just decided by some crackpots-in-chief that such a conclusion is more convenient for the dishonest propaganda addressed to the human trash that reads "Not Even Wrong" – that it could be more helpful to invent stories that string theory is not needed.

This approach was not only dishonest but totally irrational at every conceivable level. First, these would-be folks have been irrationally attacking supersymmetry (and, therefore, supergravity as well) almost as intensely as string theory so it's nonsensical to suddenly worship its miraculous (and probably non-existing) properties of a supergravity theory. Second, it's obvious and it's been proven above that supergravity can't be nonperturbatively consistent: it would either prohibit the production of charged black holes (which must be allowed) or it would violate the Dirac quantization rules (making their wave functions multiply-valued etc.).

Third, and it's been also discussed above, the maximal supergravity can't lead to a realistic particle content and the less-supersymmetric versions of it would surely violate the finiteness properties, anyway. Fourth, it's nonsensical to try to put a less complete theory, supergravity, against string theory because the research of supergravity has been incorporated into the research of string/M-theory and everyone realizes that the former is an approximation of the latter. The latter is more complete, more comprehensive, and for many questions, it's totally needed for the consistency.

One must really be a very stupid, brainwashed pile of scum to take any of this dishonest Shmoit-like idiocy seriously.

And that's the memo.