# From quantization under external classical gauge field to a fully quantized theory

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.

-
I noticed a possibly related comment by Xiao-Gang Wen from this post, at the bottom of the page: –  Jia Yiyang Jul 8 '12 at 12:24
"There are three kinds of gauge theories: (1) Classical gauge theory where both gauge field and charged matter are treated classically. (2) fake quantum gauge theory where gauge field is treated classically and charged matter is treated quantum mechanically. (3) real quantum gauge theory where both gauge field and charged matter are treated quantum mechanically. Most papers and books deal with the fake quantum gauge theory..." –  Jia Yiyang Jul 8 '12 at 12:25
Does the background field method go towards answering your question? en.wikipedia.org/wiki/Background_field_method –  Michael Brown Jan 3 '13 at 15:15
I just want to say that do quantization like that, you have to be able to solve the Dirac equation for arbitrary gauge field configuration, and slightly even more than that. I guess the reason people usually go the other way is the computational feasibility. However, in some cases, it is possible to follow you approach. For example, one of the infinite number of solutions of the Schwinger model goes precisely that way. If it is of any interest, I can write up an answer in some time. –  Peter Kravchuk Jun 3 '13 at 8:41
@PeterKravchuk: Yes please, I'm quite interested. –  Jia Yiyang Jun 3 '13 at 9:47

This may not be a satisfactory answer to your question but hopefully it will be of some help : In order to quantize a classical field theory general procedure (apart from path integral method ) is to first solve the classical equations of motion, and define a Fock space$^1$ out of these (Just as you mentioned). Now for a free theory, i.e. a classical theory for which solutions can be explicitly found this procedure certainly gives a nonperturbative definition of QFT; but if your classical equations are nonlinear then (in general) it in not possible to solve them exactly and you have to resort to perturbative methods. The same thing happens in case of Dirac equation when you couple it with electromagnetic field. In principle you can still apply the same procedure as you used for the case of free Dirac field; all you need to do is to find space of exact solutions and use some quantization method like geometric quantization. But that is at least very difficult, if not impossible. So one makes use of perturbative methods$^2$.
$1$ : When the classical configuration space is $Q$ (so that phase space is $T^*Q$) then space of states is space of functions on $Q$. In the case when $Q$ is a linear space (or can be made linear through choice of some linear structure) then space of (polynomial) functions on $Q$ can be written as a Fock space.
$2$. See e.g. Peskin-Schroeder