Let me try this coarse overview. Corrections always welcome.dextercioby said:I dunno what you mean by "representation",but i'm sure there's no simple explanation of this issue.Heavy mathematics is needed.
I infer you to chapters 13 & 14 from Bogolubov's book  for a serious treatment.
N.N.Bogolubov et al.,"Introduction to Axiomatic Quantum Field Theory",Benjamin/Cummings,1975.
This is the reduction formalism. I am more interested in the spectrum- why the masses are the poles of the propagator, and why the spin is the number of degrees of freedom of the propagator?selfAdjoint said:Let me try this coarse overview. Corrections always welcome.
First of all interactive quantum field theory is not well-defined. It has as many Hilbert spaces as there are points on a line, which is not physically meaningful. This is Haag's theorem. This problem shows up as a shaky vacuum state condition for interactive QFT; field states that should be over and gone continue to affect the vacuum. The LSZ formalism intends to work around this by going into the remote past and seeking "asymptotically localized" solutions of the theory. These will than be used as the incoming particles to the intreractions defined, e.g., by Feynman diagrams. As Daniel said, the math of doing this is very non-trivial, and Bogoliubov's book is a good intro to it. See also R. Haag, Local Quantum Physics, p. 81 ff.
The masses are the poles of the propagators because in free field theory, this is the case. We somehow want to consider our particles "free" in the far past and future. Of course they aren't, really, and there are serious problems in doing so. But even if interacting field theory is not free field theory, "it plays free field theory on TV, early in the morning, and late in the evening" And so you try to identify what aspects of the interacting theory mimick a free field theory: the pole of the propagator (the full, interacting one) looks a lot like the pole of a free field theory propagator ; so that pole is then identified as the mass. At least that's how I understand this correspondence.Itzhak the cat said:This is the reduction formalism. I am more interested in the spectrum- why the masses are the poles of the propagator