Let me take QED for example to clarify my question: The textbook-approach(at least for Peskin&Schroeder) to quantize ED is to first quantize EM field and Dirac field as free fields respectively, and then couple them together perturbatively to represent the interaction, and we will have a fully quantized theory. Here by "fully quantized" I mean both EM field and Dirac field are quantized.
On the other hand, we may quantize the Dirac field under an given external classical EM field: briefly speaking, one solves minimally coupled Dirac equation and take the solution space as 1-particle Hilbert space, and then second quantize the theory by building a Fock space based on this 1-particle space.(A more detailed description can be found in chap 10 of "the Dirac equation" by Thaller.B) Such a theory is defined non-perturbatively, and describes many-electron systems interacting with an external classical EM field but non-interacting among themselves, so in this sense it's a "better" theory than a quantized free Dirac field. However I cannot see a way to go from this theory to a fully quantized one, if there is, do we get exactly the same theory as given by the text-book approach, or possibly a better one? Any answer, comment or reference will be appreciated.
This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Jia Yiyang