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Nima, singularities, and the fall of Wilsonianism

MmanuF at Twitter has kindly recommended me this recent, 2.5-hour-long video by Nima Arkani-Hamed:

Because he suspected that I may find it a bit long, being occupied with things like the insane anti-white and anti-Slavic racism thriving in the United Kingdom that has assaulted my homeland through soccer, MmanuF has also helpfully recommended a 10-minute-long excerpt to focus on, 0:20:00-0:30:00, and I did watch that.

Among other things, Nima has declared the end of Wilsonianism. The Wilson paradigm is just wrong, he stated at one point, and you can't organize physics according to the distance scales so that the theory at the distance scale \(L\) is always determined from the theory at a shorter length scale \(L(1-\epsilon)\). Instead, in a theory of quantum gravity (and he may apply it even to some other related theories), there are numerous UV-IR connections and communication in between the scales – aside from some role played by numerical coincidences.

Because Nima has been a hep-ph person and Wilsonianism defines it, while string-trained hep-th folks may be eager to throw away Wilsonianism, you might think that it would be bizarre if Nima abandoned Wilsonianism and I would defend it. But things have many extra layers of subtlety. In particular, my (former or forever) PhD adviser Tom Banks has been a great Ken Wilson worshiper. He admired that paradigm shift of the 1970s and I had to read some original papers about it. Note that Ken Wilson died in 2013.

Aside from that, Nima lives at Princeton where Wilson is being removed from all places because "he was racist" according to the currently omnipotent unhinged racists there. In my city of Pilsen, we have bridges and many other things named after Wilson – who was a de facto co-father of Czechoslovakia. It was Woodrow Wilson, a Democratic president hated by most of the real Republicans, and a different Wilsonianism but who cares. ;-)

A substantial part of the aforementioned 10-minute segment was dedicated to a "warning" not to make a certain kind of an error. These insights may induce many reactions. Some people may fight them and claim that Nima must be wrong. Others may claim that he is saying obvious things that they have known for a long time. Others may feel illuminated.

I tend to choose "Nima is saying things that I have known for decades" but there is a problem. Just a few days ago, I basically defended Wilsonianism (although I haven't used these precise words) when I emphasized my view that string/M-theory may still be viewed as a framework to derive the (field and particle) spectrum of effective field theories and their parameters. OK, what is going on?

Nima said a nice and important thing that when we say that the Wilsonian effective theory fails at distances shorter than \(\ell\), we must interpret \(\ell\) as a proper Euclidean length... in the Euclidean spacetime. Why? Well, in our real world, we have both space and time and nearly light-like (null) intervals may be boosted to very short (timelike or spacelike) intervals. In the Minkowski spacetime, separations may look "very long" in the coordinate space but the proper distance or the proper time may be very short.

That is why the "balls" really look like "seemingly noncompact" solid hyperboloids in the Minkowski spacetime and the work with this topology is problematic. For this reason, the only "topological natural" discussion takes place in the Euclideanized spacetime where the short distances are confined in a ball and the proper distances inside the ball are shorter than \(\ell\) and equivalent to spacelike distances. Great. I agree with that.

However, as his discussion immediately realizes, we are not really behaving like that in most cases when we pretend to use the Wilsonian paradigm. In fact, by "long distance scales", we often mean the behavior of functions like Green's functions for a very long real time in the Minkowski space. But that long time is equivalent to a long imaginary spatial distance which isn't quite the same thing as the "small but real and positive" distances that define "short distances" in the Wilsonian paradigm. But in most cases, we assume that this "detail" (that we look at long times and not long spatial distances) doesn't matter because the relevant functions behave as Taylor series and those are qualitatively the same in all directions of the complex plane.

I have added this encapsulation of the discussion. I haven't heard it and it's possible that Nima has encapsulated the discussion unequivalently but I tend to believe that the disagreement wouldn't be substantial. Fine. At any rate, the Taylor expansion assumption may be incorrect and Nima states that it is incorrect in very typical Green's functions, indeed. Imagine that the experimenters carefully measure some Green-like function in the real Minkowski space, as a function of time, and they get\[ f(t) = \frac{2021}{t}. \] I added the numerator to prevent the future generations from repeating my blog posts and to encourage them to invent something new instead. That is great. It is the inverse function and you may calculate the counter-clockwise integral\[ \frac{1}{2\pi i} \oint f(t) dt = 2021. \] There is a nice singularity at \(t=0\), we may deduce, and the contour integral around that gives you \(2\pi i\) times the current year. The function is helpful because you may use it as a calendar but only up to the next New Year's Eve. Fine, it is getting time-consuming and I will switch to the units where \(1=43\cdot 47\) right now. :-) Do we know from the \(1/t\) asymptotic measurements that there is a singularity at \(t=0\)? No, we don't. The function could also be\[ f_2(t) = \frac{1}{t-1} = \frac{1}{t} \cdot \frac {1}{1-1/t} = \frac{1}{t} \zav{1 + \frac{1}{t}+\frac{1}{t^2}+\dots } \] The second factor comes from the geometric series. You may see that the original \(1/t\) is corrected by a power law expansion. This is quite a "textbook Wilsonian" expectation because such power law expansions in \(1/t\) are almost equivalent to the power law expansions in \(E\), i.e. the expansion according to the number of derivatives in the operators included in an effective theory. If your undergraduate Fourier knowledge is even deeper, you know that \(E\) doesn't quite get Fourier-translated to \(1/t\) but rather to \(\partial / \partial t\) but the effect is rather similar if these operators act on functions that admit a Taylor expansion and behave as a power law for \(|t|\to\infty\) in all directions of the complex plane \(t\in\CC\).

But again, as Nima says, it doesn't have to be the case. Consider the function\[ f_3(t) = \frac{1 - \exp(-kt)}{t} \] We have added an exponentially dropping correction, for \(t\to + \infty\), in the numerator. The effect may be unmeasurably small for a large \(t\) that the experimenters see. But we have changed the behavior in other parts of the complex plane. In particular, where are the singularities of \(f_3(t)\)? There are none. The value \(t=0\) used to be a singularity due to the denominator but that singularity and the \(t\) in the denominator is cancelled against \(1-\exp(-kt)\approx kt\) for \(t\to 0\) which appears in the expansion of the numerator. The exponential is smooth everywhere and doesn't introduce any new singularities. Well, more precisely, it doesn't introduce any singularities for a finite \(t\in\CC\). If you have listened to the right classification of singularities, you know that you mustn't hide the terrifying (or what is the right adjective LOL) singularity for \(|t|\to \infty\). OK, I think that it is called the essential singularity. Is it one?

While \(f_2(t)\) only moved the \(t=0\) singularity to \(t=1\), the new \(f_3(t)\) has erased it from everywhere. The flip side of this "beautification" is that \(f_3(t)\) behaves "terribly" far away from the real positive \(t\) axis. It oscillates on the imaginary \(t\) axis (which is the real Euclideanized time) and it exponentially grows for a negative \(t\).

Excellent. Functions may be messy and this may look artificial but as Arkani-Hamed assures you, this behavior has actually been seen everywhere in the AdS/CFT correspondence. The corrections similar to the exponential one are actually there, they are everywhere, and they are why the full theory of quantum gravity actually "erases" the singularities that seem unavoidable according to the Wilsonian or Taylor expansion.

Over 18 years ago, with Andy Neitzke, and Nima was watching us, we saw some similar and perhaps even more subtle kinds of behavior in the "asymptotic circle of the complex plane" spanned by \(r\), the tortoise coordinate, which isn't quite the same thing as the time \(t\) but it is geometrically equivalent, up to some Wick rotation. In our paper, we needed to compute some highly-damped quasinormal frequencies. To do so, we described some solutions as Bessel's function in the asymptotic region but these annoyingly advanced (but not too advanced) functions have some counterintuitive behavior such as the Stokes phenomenon. That is the trait which makes the asymptotic behavior look different in different directions of the complex plane. Also, the two-dimensional space of solutions to a second-order differential equation has exhibited a monodromy after you made a circle around the complex plane.

So a similar behavior which just prevents you from using a simple power law expansion for \(|r|\to\infty\) or \(|t|\to\infty\) is almost certainly omnipresent in a theory of quantum gravity – and our calculation with Andy was really some radiation from a black hole in the classical gravity although the purpose of this solution was to discuss radiation modes in a semiclassical approximation of quantum gravity and to say something about the full quantum gravity.

I agree with Nima that one has to be careful and the assumption about a universal \(1/t^M\) in all directions of the complex \(t\)-plane is broken at very many places. I only agree with him partially concerning another statement – but this partially problematic insight may be and probably should be illuminating for many – that this deviation from a simple, power-law-like behavior in all the directions of the complex plane is basically equivalent to the breakdown of the Wilson paradigm i.e. to some unavoidable deviations from the local effective quantum field theory.

Why do I say that this proposition is problematic? Well, it is only problematic if you actually did use functions like \(f_j(t)\) in all directions of \(t\in\CC\) in the complex plane. You could have taken a more experimental approach and talk about the behavior for \(t\) which is at (or very close to) the real positive \(t\) semi-axis. If that is how you approach it, then the power law expansions are just some pragmatic expansions of a function of the real variable and the behavior in the faraway regions of the complex plane (or even the possible non-existence of such analytic continuations) may be ignored.

Also, while the Wilsonian approach normally generates functions that are represented by power law expansion, we know that some other terms may hypothetically appear – such as the exponentials of field operators. Similar non-Taylor terms may be assumed to be non-perturbative corrections. Some of them may even be explicitly constructed in terms of instantons. You should be careful here, however. These two things are not equivalent, at least not self-evidently. The instantons scale like \(\exp(-C/g^2)\) as a function of a coupling constant while the non-Taylor corrections to the Green's functions require the decreasing exponentials in the field operators. They are a priori different things. But by this moment, you are probably open-minded enough to realize that both of these non-perturbative things, exponentials of inverse fields and exponentials of inverse coupling constants, may appear in the "full form" of Green's functions, along with some terms or the behavior that is hard to be expressed explicitly.

Whether these subtleties look like a minefield to you therefore primarily depends on the question whether your power law expansions are thought of as expansions of a function of a real variable or a function of a complex variable. Many functions may be extended to real or complex variables but the two methodologies are not quite the sime. In particular, you may have functions of a real variable that are zero to all orders but they are not zero. Well, it is true even in the complex plane but on the real axis, it looks "healthy" to connect such a function, like \(\exp(-1/g^2)\) for \(g\gt 0\), to the function \(0\) for \(g\lt 0\).

Many such subtleties, monodromies, divergent terms, perhaps singularities of slightly unexpected types may (and do) modify our expectations about Green's functions and reduce the direct relevance of the Wilson paradigm. I would still insist, as in my de Sitter post, that an effective field theory may be considered a language into which an arbitrarily subtle or nonlocal theory of quantum gravity may be translated. The nonlocal interactions over roughly Planckian distances are unavoidable but the real question is whether they must be considered unavoidable even at very long distances. I still believe that in the Minkowski spacetime, the nonlocality should be just a Planckian subtlety.

However, if our world is close to a de Sitter space, I do find it possible that the expansion for \(|t|\to\infty\) may be confusing even at cosmological distances because the physically relevant region isn't \(|t|\to\infty\) but \(t\sim R_{dS}\) which is a finite cosmological infrared cutoff and the cosmological constant could be tiny due to some numerical coincidence or a "miracle". Subtleties similar to those above could be relevant for the cosmological constant problem and I guess that much of Nima's talk must be about their possible relevance for the hierarchy problem (why the Higgs is so much lighter than the Planck mass). I have been informed about a sketch of Nima's thinking and I've had some independent and overlapping thoughts. It may be very interesting to watch but 35 minutes from now, I must check that the best Czech soccer team won't be murdered by some visitors from Arsenal FC – let's hope that they won't be the same kind of unhinged fanatics and anti-Czech racist thugs as those in Rangers Glasgow.

No proofreading is planned, sorry.

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