Overvation 1: Whatever quantization process is used, it is common to define a QFT from a classical field theory.
Observation 2: On the other hand, given a lagrangian QFT, one could try to "classicalize" the theory by transforming operator valued fields into classical field having the right Poisson bracket relations.
My questions are the following:
- A priori, the "classifying process" for lagrangian QFTs described in the second observation is not possible for any field theory, is it ? Is there an obvious class of lagrangian QFTs for which it is not possible ?
- Now if we drop down the lagrangian requirement one could ask first, are there some QFTs that are defined in other ways than with their lagrangian ? If so, I suppose that finding if they are lagrangian or not can be troublesome.
- Are there some QFTs that are proved to be non-lagrangian QFTs ? How are they then defined ?
