# The open problem of nth quantization

In trying to explain a quote by E. Nelson, "First quantization is a mystery, but second quantization is a functor!" Baez points out what follows (full text available in this week find; I'm also reading this other -and mathematically clearer- week find)

First quantization is a mystery. It is the attempt to get from a classical description of a physical system to a quantum description of the "same" system. Now it doesn't seem to be true that God created a classical universe on the first day and then quantized it on the second day. So it's unnatural to try to get from classical to quantum mechanics. Nonetheless we are inclined to do so since we understand classical mechanics better. So we'd like to find a way to start with a classical mechanics problem - that is, a phase space and a Hamiltonian function on it - and cook up a quantum mechanics problem - that is, a Hilbert space with a Hamiltonian operator on it. It has become clear that there is no utterly general systematic procedure for doing so.

Mathematically, if quantization were "natural" it would be a functor from the category whose objects are symplectic manifolds (= phase spaces) and whose morphisms are symplectic maps (= canonical transformations) to the category whose objects are Hilbert spaces and whose morphisms are unitary operators. Alas, there is no such nice functor. So quantization is always an ad hoc and problematic thing to attempt. A lot is known about it, but more isn't. That's why first quantization is a mystery.

(By the way, I have seen many "no-go" theorems concerning quantization but I have never seen one phrased quite like the above. "There is no functor from the symplectic category to the Hilbert category such that ... holds." Is anyone up to the challenge?? If this hasn't been done yet it would clarify the situation.)

I find quite interesting the highlighted part: is anyone of you more acquainted with this topic? Is anyone really up to this challenge?

From a really naive point of view, and with my really narrow knowledge about QM, I think that if we call $CS$ (resp., $QS$) the category of classical (resp., quantum) physical systems (deliberately avoiding to give a geometrical shape to these categories), then any "reasonable" functor $CS\to QS$ must encode the "canonical quantization" procedure, sending the phase-space of a classical system in a suitable Hilbert space, and "deforming" the canonical Poisson structure on the former space into a noncommutative one in the latter (even more naively, exchanging Poisson brackets with commutators of operators).

I likely believed that any no-go theorem in this setting should be formulated in this way ("$\not\exists$ canonical quantization functors") even before reading Baez's note: is anyone out there trying to walk this path?

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Isn't the problem that the functoriality is backwards? The universe (as far as we understand it) is fundamentally quantum-mechanical, but passing to macroscopic/high-temperature scales yields behavior that can be modeled with classical-mechanical approximations. One would expect that taking such a semi-classical limit yields a well-defined functor for a large class of quantum systems, but there doesn't seem to be a good reason to expect a canonical inverse to exist without imposing additional constraints. –  S. Carnahan Jan 13 '12 at 2:48
Probably you are looking for Harold's answer to mathoverflow.net/questions/8606/…. I'm sure that there is more to discuss in this direction, though. At present, I think your question could be closed as duplicate — perhaps you could provide more direction so that answerers know whether to complement, supplement, ... the earlier discussion. –  Theo Johnson-Freyd Jan 13 '12 at 4:05
Sure, I noticed the other question too late. Thanks! –  tetrapharmakon Jan 13 '12 at 10:33