Types of quantum field thories
An interesting subclass of quantum field theories is thought to arise from some kind of process that reads in certain – usually differential geometric – data, interprets this data as specifying the dynamics of some physical system, and spits out the quantum field theory that encodes the time evolution of this system.
Historically, it was an approximation to the true time evolution that was originally found and studied, by Newton, Maxwell, Einstein and others. This is now known as “classical physics”. The true dynamics in turn is “quantum physics”.
In view of this, quantization is often understood as a right inverse to the procedure that sends the full quantum dynamics to its classical limit. As such it is not well defined, i.e. unique, when it exists. Additional structures sometimes make it unique.
A general geometrically inclined introduction can be found in
Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf
N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Math. 1998. xx+529 pp.
A proposal for a full formalization of the notion of quantization for “finite” theories such as Dijkgraaf-Witten theory is in
A historical discussion by one of the labizants is here: mathlight:quantization. See also Urs’s manifesto at Mathematical Foundations of Quantum Field and Perturbative String Theory.