fxdung said:
Is there any concept relationship between quantum field and wave function of particle?
This is a subtle question. First of all one has to emphasize that the short answer to the question in the subject line is that the theoretical definition of an elementary particle is that it can be described by a local microcausal quantum field theory that induces an irreducible representation of the proper orthochronous Poincare group (see also #15).
This is a strict definition from a theoretical point of view, but physically it's quite useless. From an empirical point of view, using the abstract definition, it depends on the collision energies of the particles you look at. At low energies, e.g., a proton simply looks as if it were an elementary particle. At very low energies, in atomic physics, it's even accurate enough for a quite good description to just treat it non-relativistic and treat it as a point-particle electric Coulomb field, leading to the binding of an electron to form a hydrogen atom. At higher energies, you'll resolve a bit more of the structure of the proton and realize that it is not elementary, but that it has a non-trivial charge distribution. Then you introduce form factors. At very high energies ("deep inelastic electron scattering"), you won't find this accurate enough anymore, but you find socalled scaling relations, leading to the conclusion that a proton consists of three quarks. With today's awailable energies in particle colliders, there's no hint yet that the quarks are not elementary. So we assume that they are, but it's not said that one day one finds that also the quarks are composed of other constituents, yet not known.
Now to your question about the wave function. A wave-function description is always applicable, when you deal with a problem of fixed particle number, i.e., when the interactions involved do not create or destroy particles. This is the case for low energies, and then you can often treat the problem in the non-relativistic approximation of a potential problem like in atomic physics. Of course, you can make stystematic improvements of the non-relativistic approximation to take into account relativistic effects as perturbations and also radiative corrections, leading to phenomena like the Lamb shift of spectral lines, which is one of the most accurate results in particle physics (QED).
In the relativistic realm, i.e., at high energies, the description of particles with wave functions is quite difficult and not very consistent, because one can create and destroy particles, e.g., an electron and a positron can annihilate to two photons or an electron scattering at a nucleus irradiates a photon (bremsstrahlung) or additional electron-positron pairs (pair creation), etc. This is why the most natural description of relativistic quantum theory is relativistic quantum field theory, and here the local microcausal ones provide the successful models.
This brings me back to the beginning of this posting: Now you have a very clear recipe to guess particle models. You start from Poincare invariance and look for the irreducible ray representations (which all are induced by unitary representations of the covering group) and pick out the local and microcausal models which have a ground state (i.e., the Hamiltonian is bounded from below). Than you take into account additional symmetries (e.g., the approximate chiral symmetry to describe hadrons) and so on.
