## Tuesday, October 30, 2018

### New papers by Maldacena, Susskind, and Witten

I am confident that for most people who follow the hep-th listings, most of the new papers are as boring as they are for me. Most papers look fine but they're either too far from one's narrower interests or they look trivial or there is something straightforward about them. In most cases, if I understand the paper at all, I can say "what a big deal", I could probably do the same thing as well – if I were motivated.

Sometimes, there are very exciting papers. During the explosive periods, physicists may eagerly await new exciting papers every day: there is a significant probability that some paper or papers will make us breathless. Papers by big shot authors are more likely to do so – up to some moment, every paper by a sufficiently worshiped author makes enthusiastic students thrilled. I sincerely hope that some students like that still exist somewhere on this blue, not green planet. And they are looking forward to another paper by Witten or someone like that.

But I am not sure. Hasn't the constant influx of anti-physics hatred by worthless, toxic, pseudoscientific NPC demagogues such as Ms Hossenfelder – often echoed by shameless "science" (in practice, anti-science) media – made all students who love physics to be afraid to express their excitement, even in front of themselves? Hasn't the constant imported frustration and the "politically incorrect status of physics discovered by actual brilliant minds" suppressed the glorious feelings of actual brilliant people who are still survivors?

Today, three papers have pretty famous authors. One is Maldacena, another one is Susskind, and the third one is Witten. Not bad:
Black hole entropy and quantum mechanics (JM)

Three Lectures on Complexity and Black Holes (LS)

Open Strings On The Rindler Horizon (EW)
I don't know whether this Holy Trinity needed to synchronize their publication.

OK, Maldacena just contributed 11 modest pages reviewing the fate of black holes within quantum mechanics. The pages will be included in some Jacob Bekenstein Memorial Volume and I doubt too many people will read it. Juan starts with some thermodynamic aspects of the black hole entropy, then he revisits the search for microstates, and then he ends up with some hints of holography and entanglement as seen in Penrose diagrams. Clearly, Juan's contribution to most of these things has been profound. I guess you can imagine that a revolution is unlikely to appear in a 11-page review.

In his much longer, 84-page-long document, Susskind discusses the concept of complexity – close to "computational complexity" – that many authors tried to make relevant for the black hole physics. In these considerations, the "complexity" is meant to play an analogous physical role as "entropy" – in particular, there is a "second law of quantum complexity". But complexity isn't quite the same thing as entropy, not even after some averaging.

Folks like Susskind clearly think that "complexity" is more relevant and smarter than "entropy" and the black hole phenomena should be rephrased in terms of complexity. Well, I am rather skeptical about this whole philosophy. "Complexity" is a notion that "knows" about clever ways to compress the data and do other things. But I think it is exactly this cleverness that simply cannot be relevant for the formulation of the fundamental laws of Nature – and laws of black holes.

The talk about complexity is analogous e.g. to compression algorithms. You know, computer files of various kinds may be compressed. Is there some canonical method to determine the shortest possible file that may be considered the compressed version of the original, longer file? When you talk about some important but standard enough types of compression, the operation may make sense, especially if the "amount of raw data" is vastly greater than the length of the algorithm needed to compress or decompress the data. But if you really want to apply these principles in generality, you will find out that the compressed file size depends on all the details how the compression algorithm is described in and incorporated into the compressed file, what is allowed, what is not, and many other things.

What I want to say is that compression depends on some human-like cleverness and human-like cleverness cannot be canonically quantified. There can't be truly unique formulae that quantify the compressed file size – and analogously complexity. For this reason, all these concepts have an irreducible dependence on conventions, details in arbitrarily chosen procedures followed by humans. In this sense, the complexity is always partly a social science if not a pseudoscience from the humanities.

In other words, all questions about the complexity seem to be self-evidently derived to me. You first need some rules, and then you may ask "what is the minimum file size that can do something" or similar questions. If you don't know the underlying fundamental rules, you can't talk about complexity – which is another level that depends on the lower floor. That's why I think that the feeling that the black hole complexity considerations tell us something fundamental about black holes will always turn out to be illusions.

Clever algorithms and compressions are useful for humans to calculate various things but if they're ignorant about a clever trick that allows to calculate something hard, it cannot be interpreted as a violation of a law of physics. The knowledge of all derived facts from complexity theory cannot be understood as a necessary condition to formulate the laws of physics, I think. Only specific clever tricks, algorithms, and compression may be relevant in specific situations.

On top of that, Susskind uses the term "qubit" 114 times and as far as I can see, he really means that there is something special in encoding information in separate base-two pieces of data. I think it's absolutely and ludicrously wrong. Base-two encoding is actually very unnatural in physics – the Hilbert spaces in CFTs etc. really have dimensions that are $$\exp(S)$$ for some "simple" $$S$$ rather than $$2^S$$ – and if Susskind sees evidence for the opposite claim, I think he has hallucinations.

Witten and continued orbifolds

Today, it's clearly Witten's paper that is closest to my own research and the intuition about what I consider a promising path. It is much more specific when it comes to the technical content – but it may still be relevant for some truly fundamental insights in quantum gravity.

Witten elaborates on the idea – that appeared in a paper by Dabholkar but there is some pre-history – that physics of the Rindler space (a quarter of the 2D Minkowski space, perhaps with some extra "dull" flat $$D-2$$ dimensions, that is parameterized by the radial coordinate and the hyperbolic angle) may be calculated from orbifolds.

Well, one may try to start with compactifications on the "multiple cover of the plane" and Witten says that there's no way to achieve such states in string theory. I think there actually is a way, one that effectively composes the regular orbifolds with some T-duality – but I think it's sufficient to follow his calculation because its interesting results are probably equivalent.

OK, he wants to calculate the Rindler space entropy. The Rindler density matrix is $$\rho$$ and the entropy is$S = -{\rm Tr} (\rho \log \rho)$ as you know. Can this quantity be calculated from a path integral? The logarithm of a density matrix is a complicated function, isn't it? However, there's a nice way to get a logarithm. If you differentiate$f({\mathcal N}) = {\rm Tr} (\rho^{\mathcal N})$ with respect to $${\mathcal N}$$, you get $${\mathcal N}$$ times $$\log \rho$$ somewhere, don't you? If you substitute $${\mathcal N}=1$$ at the end, the unwanted power factors drop. So you will be capable of computing things like entropy – which depend on the logarithm of the density matrix – if you can compute the partition sum that involves a general power of the density matrix, with a continuous exponent.

In other words, for this calculation of the Rindler entropy, you need to compute the trace of a general power of the density matrix – where the exponent is continuous, perhaps general complex. Well, there seems to be a way to do it. If the partition sum is understood as the path integral inside $$\RR^2$$ where the (Euclideanized) time is the angle $$\phi$$ going around the origin, the replacement of $$\RR^2$$ with $$\RR^2 / \ZZ_N$$ has the effect of replacing $$\rho$$ with the $$N$$th root, $$\rho^{1/N}$$. Why? Well, $$\rho$$ is basically the unitary evolution operator over some imaginary time and you may divide that time by $$N$$ by taking the $$N$$th root.

To complete this calculation, you need to compute the partition sum on orbifolds – but be able to analytically continue the integer $$N$$ from the $$\RR^2/ \ZZ_N$$ orbifold to a general, continuous (and perhaps complex) value of $$N$$. I am writing these things as a bloc because I've thought about these very things in the past, too.

Now, there are some hypotheses here whose validity is not guaranteed. Does any "theory" exist for non-integer $$N$$ at all? Does it have a well-defined set of CFT operators? And independently of that, does it have at least the partition sum?

Witten suggests that at least the answer to the last question – which is one that we really need – is affirmative. How does he do it? He computes the partition sum in the open string sector. And on the simple orbifold $$\RR^2 / \ZZ_2$$, even superstrings break supersymmetry and you get tachyonic states in the twisted sector. When it comes to the partition sum, this closed string tachyon makes it infrared-divergent.

But these tachyonic states only exist for the integer values $$N=2,3,\dots$$, those that you can imagine. But Witten boldly continues the orbifold's results to $$N=1+\epsilon$$ where the tachyon basically disappears and the partition sum becomes much more well-defined. So it's good news, maybe the Rindler entropy and other things could be calculable in this straightforward way. Witten says that the whole resulting CFT describing the continued orbifold, if one exists, is probably a non-unitary logarithmic CFT.

(Well, I can imagine that the correct answer to the first question actually says: No, there is no full-blown CFT for the unphysical values of $$N$$ because the number of primary operators etc. is formally non-integer or complex – and CFTs etc. with a non-integer number of "objects" may exist at most formally.)

Much of the paper computes some partition sums for the world sheet cylinder – that can be read in two different ways, as the closed-string channel and the open-string channel. His expressions are full of eta-functions and ratios of $$\sinh x$$ and $$\cosh x$$ of various types.

The strategy is that many things – including the general behavior of he vicinity of the event horizon, among other things – could be encoded in some analytic continuation of physics and expressions that we know very well. Certain things about the Rindler space may look new but they could be just the well-known old things with unusual and seemingly unphysical values of parameters.

I have had great sympathies for this kind of reasoning since the age of 8 or so – when I was literally obsessed with continuing everything to complex values of every parameter. I didn't really understand complex analysis well so I thought that $$x^y$$ had to be very complicated and store some deep mystery, before I learned that it's just $$\exp(y\log x)$$ which is straightforward – and has multiple values, due to the many values of the complex logarithm.

But string theory allows us to go further. It is at least morally true that this is the smart way of calculating things (I guess that Witten shares the "morally true" adjective as well – and I am not sure whether I was using it independently before I heard it from him, I think that I did use it). There are some clever continuations like that which allow you to address the Rindler space – and possibly also wormholes and other things that are a bit more complicated than the Rindler space – and you can try to calculate their partition sums etc. in quantum gravity.

You can try to calculate them in much more specific realizations of quantum gravity, namely string theory. Well, in individual and special string vacua.

Why do I think this approach is morally true? In many cases, people could be tempted to say that "we don't know how to compute some partition sum or other quantities on some spacetimes of unusual topologies" and similar things. But this negative statement isn't really backed by any proof – and many people believe it because they fail to see that certain quantities might be obtained by the analytic continuation (and gluing and otherwise combining) well-known theories and expressions, and perhaps the differentiation of such constructions.

We already know a lot – in some cases, we just don't realize that we have the tools to quantify many things that look hard. I personally believe that any continuation-or-gluing procedure that addresses quantities in the spacetime and seems to work "formally", according to a geometric picture, is ultimately correct even when all the quantities are made explicit.

Susskind vs Witten, social science vs hard science

Juan mostly reviewed things but I want to compare Susskind's and Witten's papers. You see that both of them are really trying to demystify some issues in quantum gravity – the Rindler space in Witten's paper is clearly a local toy model of the event horizons for black holes, among other things. But they're betting on very different "hot proposals" how to make new dramatic progress – complexity vs analytical continuations of stringy orbifolds.

I think you can see the personalities of the two people in the two topics. Complexity is a somewhat "social science" as I argued, it is energized by some of the "interdisciplinary" hype, and so on. On the other hand, substitutions of unusual numbers to stringy orbifolds in Witten's paper looks like a very technical procedure that no popular science writer has ever written about. She wouldn't even want to write about it because it doesn't really sound like the "mass culture garbage" and the "interdisciplinary hype" that pop science so existentially depends upon.

Well, yes, I think that these relatively technical – yet unconstrained and free – manipulations with particular string vacua, such as the continuations of orbifolds to unusual values, are much more likely to teach us something really solid about the Rindler space, black hole information puzzle, wormholes, and the general formulation of string/M-theory or quantum gravity.