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.

Hello,

Yes, not difficult to guess from my former comments on your website that I’m the person who asked the question regarding yours, and mine, “favorite approach.” I think Witten’s reply was somewhat disappointing since I see no reason to constrain oneself to Kähler’s structure.

I always felt that the inverse problem was at least as important; what is the corresponding {\em classical} model corresponding to some quantum system?

Quantization is not unique, of course, but the classical limit of a quantum system is unique (although it may have different descriptions).

My impression is that mathematicians and people doing physical mathematics don’t spend much effort on the inverse problem. It has a lot of relevance to real physics, though. For example, there are interesting condensed-matter systems which should correspond to field theories with Lagrangians (hence the classical limit), but where the connection is not proven.

Peter Orland,

unfortunately, the classical limit of a quantum system is not unique either; it depends on the choice of a family of (generalized) coherent states in the Hilbert space of the quantum system, which is not unique. For the classical limit in terms of coherent states, see, e.g.,

Yaffe, L. G. (1982). Large N limits as classical mechanics. Reviews of Modern Physics, 54(2), 407.

Arnold Neumaier,

That is not really the issue. It depends upon what you mean by classical, i.e., which parameter you take to zero. If that parameter is $\hbar$, there is no ambiguity. If you decide that the parameter is 1/N, as in the example you give, what you obtain can be entirely different.

On the other hand, the problem I raised includes many large-N models, which have not been successfully been written as classical systems. The result should be unique, but we don’t usually know what it is.

What I was referring to above, was that the classical limit of some systems can’t yet be directly produced, only conjectured. An example is the XXX spin chain, at large spin s. This is supposed to be the O(3) sigma model (Haldane’s Nobel Prize was at least partly due to this identification), possibly with a $\theta=\pi$ term (for half-integer s). There is a lot of evidence for this conjecture, but it is not proved. There are other such conjectures around.

Personally, I think the case for the finite dimensional case is pretty close to solved in many aspects, and now fully streamlined in the work of Landsman:

-Lie Groupoids and Lie algebroids in physics and noncommutative geometry (https://arxiv.org/abs/math-ph/0506024)

-Between classical and quantum (https://arxiv.org/abs/quant-ph/0506082)

-Quantization commutes with singular reduction: cotangent bundles of compact Lie groups (https://arxiv.org/abs/1508.06763)

Field theories, of course, are a different beast.

“To get physically sensible results one must choose a polarization that distinguishes between positive and negative energy”.

That’s true, although I would say the problem is not so much in the quantization step (an abstract Weyl C*-algebra can be defined without that choice), but in obtaining a concrete representation of that algebra, where you indeed need to make that choice (which amounts to select a particular quasi-free/Gaussian algebraic state on which to base the GNS representation of the algebra; if there’s a timelike Killing vector field on the spacetime, you can make the interpretation of that step in terms of “positive/negative energy Fourier modes”) And, still, all of this is in the case of free/non-interacting field theories!

Peter Orland,

in quantum theories not derived from an action principle there may not even an $\hbar$ in the theory!

Of course if you have a Hamiltonian where you know (or assume) the precise $\hbar$ dependence and know (or assume) which quantities should survive the limit $\hbar\to 0$ then there is no ambiguity about the classical limit, though finding it may still be nontrivial.

Can you provide references for ”There is a lot of evidence for this conjecture”?

I did not mean specifically theories with known S matrices, e.g. satisfying Yang-Baxter equations.

The literature on the Haldane conjecture is substantial, and the best place to find more discussion is in Haldane’s paper and those citing it.

The same problem occurs for pretty much any spin chain or vertex model which is argued to be Bosonic QFT. Some degrees of freedom are relevant, hence should be kept, but others should be thrown away.

Higher-dimensional examples include quantum link models of non-Abelian gauge theories. The Yang-Mills Lagrangian has not been derived from these. Long ago, Daniel Rohrlich and I argued that some of these are YM, but some are not even relativistic.

Alex,

One of the main points Witten was making is that quantization is not just producing an abstract algebra, but also a state space for them to act on. This is where you really need to introduce extra structure.

He doesn’t describe this in general, but he’s referring to the fundamental problem of Lie algebra representation theory: given a Lie algebra you have an abstract associative algebra (the universal enveloping algebra), which is a quantization of a symmetric algebra, but producing irreducible representations is very tricky. The orbit methods says they should be quantizations of co-adjoint orbits, but how to do this in general runs into interesting problems (which he hopes brane quantization might resolve).

Is there some qualifier for this remark? Or is quantization really only something that can be trusted to 1st order?

ARoxdale,

It’s not a question of being trusted, the problem is more one of ambiguity: there are lots of possible quantizations of a given phase space, all of which differ by terms which vanish in the classical limit hbar goes to zero. This is mentioned in some textbooks as the “operator ordering problem”.

If you impose various conditions you want your quantization to satisfy, you can sometimes get a unique quantization, but only for some special class of functions on phase space.

My QM textbook

https://www.math.columbia.edu/~woit/QMbook/qmbook-latest.pdf

does emphasize this problem, explains in great what happens for phase space R^2n

Witten is interested in more general symplectic manifolds, where the situation is much more complicated, with “geometric quantization a partially successful way to deal with it (while requiring the specification of a polarization, as Witten discusses).

I have no idea why Witten is concerned with this problem.

Quantum mechanics is the fundamental theory.

So one could, perhaps, be concerned with the classical limit of quantum mechanics, not the other way round.

Physics is concerned with computing experimental results. So you do this computation and see if this class of experiments has a macroscopic limit.