Experimental verification of quantization

Question

I understand quantization as a map from Symplectic Manifolds $M$ (either finite dimensional or not) to Hilbert Spaces $H$, along with a rule that attach to every function $F$ in $M$ a hermitian operator $A_F$ in $H$. Dirac proposed that the map should satisfy the rule $-i\hbar[A_F,A_G] = A_{\{F,G\}}$, where $[A,B]:=AB-BA$ is the commutator and $\{F,G\}$ is the poisson bracket.

It is known that this dream cannot be exactly realized, and this vagueness casts a shroud of mystery over quantization to outsiders, who see how physicists quantize without any apparent order. A relevant question here.

I'm aware of some methods of quantization. Pseudo-differential operators and Weyl quantization are widely used in mathematical analysis. There is a geometrical quantization well explained here.

I think that the only way to settle the problem is through experiments, or at least to clarify the meaning of quantization for some restricted families of operators, say polynomials of degree less than four in $p$ (momentum) and $q$ (position). It's been many years after the foundation of quantum mechanics, so what are the obstructions to this kind of verification?

For example, I suppose that experiments with nanotubes should help to understand the quantum realization of observables when the manifold $M$ is not flat.

Answer 1

Agree absolutely. Quantization is not a functor. This is exactly why physics requires experiment to see what nature actually does as opposed to what we might think it does. After saying all this there are systems where the experiments are quite clear that what we are doing are correct. This includes all of chemistry and solid state physics where we know the underlying ground rules, and the remaining problems are that exact/controlled solutions of the Schroedinger equation for many body systems are hard.

Perhaps life becomes harder when we do relativistic field theories? There was a time when it was thought that if opertor ordering ambiguities (Groenewold-Van Hove) made ordinary quantum mechanics complicated, how much harder things were going to be in four dimensional field theories. Thanks to Ken Wilson who told us what field theories actually are, we know that that that this fear is unjustified. By connecting the removal of the high-momentum cutoff in the renormalization process to the concept of critical-point universality in the statistical mechanics theory of second order phase transitions he showed that most operator-ordering ambiguities are irrelevent in the technical sense of the word --- that they are washed out as we approach the continuum limit. This makes physicists much more confident in what they are doing when working with QCD and otherfield theories.

The Wilsonian view of quantum field theory also meant that for issues such as quark confinement the relevent rigorous mathematics become more probability theory (Markov fields and a search for stable distributions as a higher-dimensional version of the central limit theorem) than quantization as in Poisson brackets $\to$ commutators. The latter game is still very much in play though as in BRST quantization, and continues to enrich the geometry/physics cross fertilization.

Answer 2

There cannot be any experimental evidence for quantization, because quantization is not a theory of nature. It is just a recipe for constructing theories of nature that have some desired set of properties.

Example: We can define quantum electrodynamics (QED) without any reference to "quantization" whatsoever. (And by the way, we can define it mathematically rigorously by treating spacetime as an extremely fine discrete lattice, so fine that no practical experiment would ever notice.) The role of "quantization" is only to motivate the structure of QED based on what we already know about classical electrodynamics. As a testable theory, QED stands on its own without any dependence on the chain of reasoning that led us to consider it, and without any dependence on classical electrodynamics. After all, classical dynamics is an approximation to quantum electrodynamics! Quantization is just a (formalized) chain of reasoning that leads us to consider testable theories like QED.