Anti-Quill Part 8: Quantization-1

  In physics a very natural construction is that of quantization, where we take a classical system and quantize it. This involves, to be very naive, taking a symplectic pair $(M, \omega )$ and turning the functions in $L^{2}(M)$ into operators, and the Poisson brackets $\{\cdot, \cdot \}$ into commutators $[\cdot, \cdot ]$. But getting to $L^{2}(M)$ is hard. In fact, prequantization is as far as we've gotten to here. The objective at the end of the day is to take this symplectic manifold we have and obtain a Hilbert space. Really, what you would work with would be subspaces and not $L^{2}$ spaces, etc etc. There's many sides to how you do this though. In fact, this is a very crucial part of physics -- or in broad terms mathematical physics, since a lot of the developments in this field come not from physicists but from mathematicians. E.g. deformation quantization was formulated by Maxim Kontsevich (the "t" comes before the "s", this comment is directed at my collaborator), a lot of geometric quantization was done by mathematicians just trying to do symplectic geometry and brane quantization, while initiated by Gukov and Witten and is of interest to string theorists primarily, is also of interest for differential geometry and representation theorists. The case with brane quantization is that there's boundary conditions involving Lagrangian and coisotropic A-branes, and stuff I don't fully understand yet. See Gukov-Witten and Gaiotto-Witten for more about this. Interestingly, the ordinary sort of quantization that physicists do is not fully meaningful either. A good example is the work by Suvrat et al on the Hilbert space of de Sitter quantum gravity, where you take the ordinary Hilbert space (obtained from Dirac quantization) and you have to solve the constraints next, for which you define the rigging map and do group-averaging. This forms the so-called refined algebraic quantization scheme, which is really just Dirac quantization extended. So even though quantization as a term seems trivial from undergrad physics a la canonical quantization, the actual mathematically clear way to do it is very nontrivial and they branch out into several formalisms. In the words of Ludwig Faddeev, "quantization is an art, not a science". More on this soon.