The Problem of Quantization

I’ve been watching Witten’s ongoing talks about geometric Langlands mentioned here, and wanted to recommend to everyone, mathematician or physicist, the first of them, on The Problem of Quantization (pdf here, video here, the question session is very worthwhile). For those very sensibly not interested in the intricacies of geometric Langlands, this talk is about the fundamental issue of “quantization”.

Hamiltonian mechanics gives a beautiful geometrical formulation of classical mechanics in terms of the Poisson bracket on functions, while quantum mechanics involves operators with non-trivial commutators. It was Dirac’s great insight that “quantization” takes functions to operators, taking the Poisson bracket to the commutator. In mathematician’s language, it’s supposed to be a unitary representation of the Lie algebra of the infinite dimensional group of canonical transformations of a symplectic manifold, so a homomorphism from functions with Poisson bracket to the Lie algebra of skew-adjoint operators on a complex vector space.

The problem with this is that you’d like to have an irreducible representation, but the only way to get this is to pick some extra structure on the symplectic manifold. The standard example is the phase space $\mathbf R^{2n}$, where you have to pick a decomposition into position and momentum coordinates. The state space will then be functions of just position, or just momentum. A different choice is to complexify, and look at functions of either holomorphic or anti-holomorphic coordinates. This choice is called a “polarization”. One aspect of the “problem” of quantization is that, given a phase space (symplectic manifold), there may not be an appropriate polarization. Or, there may be many different ones, with no obvious reason why they should give the same quantum theory.

Witten doesn’t mention one aspect of this that I find most fascinating. For relativistic quantum field theories the phase space is a space of solutions of a relativistic wave-equation. To get physically sensible results one must choose a polarization that distinguishes between positive and negative energy (or between functions which extend holomorphically in the positive or negative imaginary time direction).

In these lectures, Witten advertises a rather exotic quantization contruction, using (even for a finite dimensional symplectic manifold ) conformally invariant boundary conditions in a two-dimensional QFT. I’m not convinced that this is really a good way to deal with the case where what you’re doing is looking for representations of a finite-dimensional Lie algebra, but it’s plausible this is the right way to think about the geometric Langlands situation, where you’re trying to quantize a moduli space of Higgs bundles.

In the question section, someone asked about my favorite approach to this problem, essentially using fermionic variables and cohomology. This can be thought of in general as using spinors and the Dirac operator, with the Dolbeault operator a special case when the symplectic manifold is Kähler. Witten responded that he had only really looked at this in the Kähler special case.

Deterioration of the World’s Thinking About the Deepest Stringy Ideas

For quite a few years now, I’ve been mystified about what is going on in string theory, as the subject has become dominated by AdS/CFT inspired work which has nothing to do with either strings or any visible idea about a possible route to a unified fundamental theory. This work is very much dependent on choosing a special background, in tension with the idea that, whatever string theory is, it’s supposed to be a unique theory that relates all possible backgrounds. This issue came up in a discussion session at Strings 2021, and it turns out that others are wondering about this too. There’s this today from Lubos Motl:

Aside from more amazing things, the AdS/CFT correspondence became just a recipe for people to do rather uninspiring copies of the same work, in some AdS5/CFT4 map, and what they were actually thinking was always a quantum field theory, typically in D=4 (and it was likely to be lower, not higher, if it were a different dimension!) whose final answers admit some interpretation organized as a calculation in AdS5. But as Vafa correctly emphasized, this is just a tiny portion of the miracle of string/M-theory – and even the whole AdS/CFT correspondence is a tiny fraction of the string dualities.

This superficial approach – in which people reduced their understanding of string theory and its amazing properties to some mundane, constantly repetitive ideas about AdS/CFT, especially those that are just small superconstructions added on top of 4D quantum field theories – got even worse in the recent decade when the “quantum information” began to be treated as a part of “our field”. Quantum information is a legitimate set of ideas and laws but I think that in general, this field adds nothing to the fundamental physics so far which would go beyond the basic postulates of quantum mechanics…

When Cumrun correctly mentioned that the real depth of string theory is really being abandoned, Harlow responded by saying that there were some links of quantum information to AdS/CFT, the latter was a duality, and that was important. But that is a completely idiotic way of thinking, as Vafa politely pointed out, because string theory (and even string duality) is so much more than the AdS/CFT. In fact, even AdS/CFT is much more than the repetitive rituals that most people are doing 99% of their time when they are combining the methods and buzzwords of “AdS/CFT” and “quantum information”. Many people are really not getting deeper under the surface; they are remaining on the surface and I would say that they are getting more superficial every day.

According to Lubos, he’s not the only one who feels this way, with an “anonymous Princeton big shot” agreeing with him (hard to think of anyone else this could be other than Nima Arkani-Hamed):

There is a sociological problem – coming from the terrifying ideological developments in the whole society – that is responsible for this evolution. I have been saying this for a decade or two as well – and now some key folks at Princeton and elsewhere told me that they agreed. The new generation that entered the field remains on the surface because it really lacks the desire to arrive with new, deep, stunning, revolutionary ideas that will show that everyone else was blind. Instead, the Millennials are a generation that prefers to hide in a herd of stupid sheep and remain at the surface that is increasingly superficial…

So most of the stuff that is done in “quantum information within quantum gravity” is just the work of mediocre people who want to keep their entitlements but who don’t really have any more profound ambitions. As the aforementioned anonymous Princeton big shot told me, their standards have simply dropped significantly. The toy models in the “quantum information” only display a very superficial resemblance to the theories describing Nature. That big shot correctly told me that in the early 1980s, Witten was ready to abandon string theory because it had some technical problems with getting chiral fermions and their interactions correctly.

Harlow says that many of the people – who may be speakers at the annual Strings conference and who may call themselves “string theorists” when they are asked – don’t really know even the basics of string theory. And they can get away with it. Just like there is the “grade inflation” and the “inflation of degrees”, there is “inflation in the usage of the term string theorist”. Tons of people are using it who just shouldn’t because they are not experts in the field at all. Harlow said that many of those don’t understand supersymmetry, string theory etc. but it’s worse. I think that many of them don’t really understand things like chiral fermions, either. It’s implicitly clear from the direction of the “quantum information in quantum gravity” papers and their progress, or the absence of this progress to be more precise. They just don’t think it’s important to get their models to a level that would be competitive with the previous candidates for a theory of everything – like the perturbative heterotic string theory, M-theory on G2 manifolds, braneworlds, and a few more. They are OK with writing a toy model having “something that superficially resembles a spacetime” and they want to be satisfied with that forever.

I don’t want to start here an ad hominem discussion of Lubos and his often extreme and eccentric views. On the topic though of the devolution of string theory as a TOE to playing with toy models of AdS/CFT using quantum information, it seems quite plausible that not only the “anonymous Princeton big shot” but quite a few other theoretical physicists see the current situation as problematic.

Even More Langlands

Various news at least tangentially related to the Langlands program: