In the standard canonical quantization procedure there are two rules.
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)?
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.
what if i wraps around itself, in a (operator based) number system where something like i=0. pq=qp+i. If the number system wraps around on itself in the complex direction then they would commute. It may have a factor of 2pi if needed. If they commute, then one can define a insanely large fourier transform (with exponential kernels). One simple way is to transform K(p,q) (leaving p constant), to R(p). Then one can transform it back, to result in W(q). Combining these ideas, one can have a foundation for a consistent quantum mechanics. If we have enough quantities, which I wonder if mass (based on the fact the p hat is the jacobian and p is the mass times the velocity, which creates the derivative (possibly proper time)), transforms to iUp^2, then transforms through (t_proper->U) (U->t_proper). Then that could incorporate mass into the translation of classical theories to quantum theories. Don't forget that the dot product should be based on the metric tensor. There are many possibilities of numerous integrals, for a leaky-fourier i-truncated transform based theory of quantum mechanics that needs to be explored. Instead of writing the axioms, find an equation or system that satifies the axioms automatically. One can write a list (Maybe for santa, who knows) for these transforms that match up with the axioms with new complex systems for commutivity that makes anti-commutivity meaningless.
