This question builds off of this previous question particularly the excellent answer by @Cosmas Zachos and the this document which he attached.
Quantization whatever form it takes always seeks to conceptually, if I understand it correctly, take inspiration from a classical theory and construct a quantum theory from it. Certainly this was Dirac's grand vision and perhaps even the most rigorously defined processes of quantization (Moyal etc) seek to take this idea to another level by literally forming a function between classical and quantum observables.
Is this the only way we are able to consturct a Quantum Theory? Is it possible to construct quantum mechanical hamiltonians outside of the notion of quantization, without any reference at all to the classical world.
In saying this I appreciate that any quantum theories built through quantization hold on their own merits and I am in no way saying it is a flawed approach.Also it is good to note that philosophically one might say that of course all must take inspiration from classical mechanics given that we live in a classical world, but I encourage the reader to think of how classical mechanics should be a subset of the more general quantum mechanics and how with this thought in mind it seems to me that there should be quantum processes which fundamentally cannot be attempted to be described through quantization.
If so, what process would this follow and what examples of this are there within the coloured history of this great field?
