Physics Monkey said:It is a fact of experience that we observe experimentally something we call an electron that has a pretty well defined mass and charge. Any theory that is useful for predicting phenomenon involving our observed electron must be able to be viewed as 'containing' the electron we see. As vanesch said, for non-interacting field theories and for interacting field theories treated perturbatively, we can make sense of the theory in terms of particles we observe experimentally (using renormalization). In free field theories, the electron is defined as an eigenstate of the free Hamiltonian, but no one knows any exact eigenstates of the interacting Hamiltonian. It isn't totally clear what the exact energy eigenstates of QED look like. Note for instance that the number operator (defined in terms of the old 'electron' and 'photon' operators) doesn't commute with the interacting Hamiltonian, so it would seem that the exact energy eigenstates of the theory don't contain definite numbers of the quanta described the 'electron' and 'photon' operators.
The question of the equivalence between fields and particles (and what we even mean by particles) in QFT has been attacked in a rigorous manner using the tools of so called general or axiomatic quantum field theory. I would recommend the book by Rudulf Haag, "Local Quantum Physics: Fields, Particles, Algebras" as good place to start if you are interested in this approach.
Thanks! I am now talking of memory but i think that Lubos Motl wrote an Amazon review on that book, and if i remember correctly Motl said (in his well know style) that Haag was "nonsense".
Effectively, in Motl words
Axiomatic Field Theory has given no physical predictions and it has led to no conceptual developments. Today, Axiomatic Field Theory is not an active field of physics anymore. Moreover, most of its conclusions are believed to be incorrect.
The discovery of the Renormalization Group (RG) showed that many exact - and seemingly rigorous - ideas about the operator algebras were too naive to be true... There exists almost no useful quantum field theory that would satisfy the axioms of Axiomatic Field Theory, and therefore the "theorems" derived within the framework of Axiomatic Field Theory have almost no physical impact. Although there are many correct and useful statements in the book, the number of incorrect and misleading sections is too large and it makes the book useless.