Is it possible to minimize the number of axioms/rules of the canonical quantization?

Question

In the standard canonical quantization procedure there are two rules.

Transform all quantities to operators.
Transform the Poisson bracket to a commutator.

Of course it will be nicer to minimize the number of axioms, so my question is very simple: Is there a way to derive the second rule from the first one (or the opposite)?

mike stone · Answer

I'm not sure that the rules are axioms as they can only be followed strictly for operators that are no more than quadratic in $p$'s and  and $q$'s (Groenewold's theorem).  The wikipedia article on canonical quantization has sets of axioms, and shows that they are always inconsistent.  Thus quantization is an art and not a functor.

Is it possible to minimize the number of axioms/rules of the canonical quantization?

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