kmm stated that "we have our quantum field and we write it as the sum of our elementary wavefunctions", so I assumed that any quantum field can be described by a superposition of the elementary wavefunctions convolved with some sort of particle-position function P(p, x) (where, roughly, P(p, x) = E if particle p exists at position x with energy E, otherwise P(p, x) = 0). Am I wrong to assume this?
Edit: from what I can understand about the formation of the Standard Model from the Lagrangian, the domain restriction which makes the elementary wavefunctions elementary is wavefunctions which evolve over time (according to EM theory) such that when viewed as particles, their motion is compatible with QM (or relativity?), and that there are only a finite number of distinct wavefunctions which are needed to span this space. Is this roughly correct?