Quantum-classical correspondence of dynamical observables, quantization, and the time of arrival correspondence problem

We raise the problem of constructing quantum observables that have classical counterparts without quantization. Specifically, we seek to define and motivate a solution to the quantum-classical correspondence problem independent from quantization and discuss the general insufficiency of prescriptive quantization, particularly, the Weyl quantization. We demonstrate our points by constructing time of arrival operators without quantization and from these recover their classical counterparts.