The interpretation of the quantum field In QM we have always been told that for each quantum mechanical field there is an associated particle. This works in the free theory where from canonical quantisation we promote a field to a field operator using ladder operators with the ladder operators generating single particle states. We say the field operator couples the state to the vacuum
$$\langle  0|\phi|p \rangle = e^{-ip \cdot x}$$
In the interacting theory we don't know how to solve exactly, we resort to perturbation theory and the interaction picture. For small perturbations there is a similar interpretation but this can't really hold for strong coupling regimes. In general the field operator couples many states to the vacuum, now we have
$$\langle  0|\phi|p \rangle = \sqrt{Z}e^{-ip \cdot x}$$ for $|Z|<1$. How do we interpret the field operator in this case?
 A: (This answer is written from the perspective of lattice QFT, where everything is mathematically well-defined. Lattice QFT is not fundamental, of course, but most of the QFTs we use in physics are not fundamental anyway.)
In general, quantum fields are the mathematical ingredients from which the theory is constructed, and particles (if any) are phenomena that the theory predicts. Observables (represented by operators on the Hilbert space) are expressed in terms of quantum fields. The fields themselves are usually not observables, but using fields as the basic ingredients often allows the theory to be specified in a compact way, using a lagrangian, where locality of interactions is manifest. When a simple correspondence between fields and particles happens to exist, it's something we derive, not something we assume.
A simple field-particle correspondence is characteristic of free fields. For fields with weak couplings, applying a field operator to the vacuum state no longer gives a pure single-particle state, but the LSZ reduction formula is a way of isolating the single-particle terms for use in calculating scattering amplitudes. For fields with strong couplings, such a simple field-particle correspondence is not generally expected. Topological quantum field theory gives a rich supply of examples where a traditional field-particle correspondence does not exist at all.
