#### marcus

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what do you think is usually meant by quantizing a classical (non-quantum) theory? And how does this connect to these axioms of what a quantum theory ought to be

Here is a mainstream summary description of what quantizing means:

(just quoting from a June 2002 paper by Bojowald)

"Quantization consists in turning functions on the phase space of a given classical system into operators acting on a Hilbert space associated with the quantized system.

To construct this map one selects a set of 'elementary' observables, like (q,p) in quantum mechanics, which

generate all functions on the phase space and form a subalgebra

of the classical Poisson algebra. This subalgebra has to be

mapped homomorphically into the quantum operator algebra, turning real observables into selfadjoint operators..."