What does a QFT particle state have to do with a classical point particle? In the question Can one define a “particle” as space-localized object in quantum field theory? it is said that in quantum field theory, a particle state is a state with well defined energy and momentum, related with dispersion relation $E^2=p^2+m^2$. This thing is localized in momentum space, which means it must be delocalized in coordinate space.
On the other hand, in classical mechanics, the most striking feature of a point particle is being localized in coordinate space.
On the first glance, these two objects may seem very different with no obvious link between them. It seems that it is usually just postulated that the above defined QFT states are particles, without any clear justification why should they be related to the point-like classical objects. Satisfying the same dispersion relation isn't good enough justification, as it is not obvious that e.g. some classical field configurations can't satisfy it.
So my question is: Why do we call them both particles, how do we see that they behave similarly (in some appropriate limit)?
 A: This is largely a question of definitions so there's no single right answer, but I personally would not call a momentum eigenstate a particle, precisely because it is delocalized.  I think of a particle as a wave packet which is narrowly, but not perfectly, peaked around some value in both real and momentum space.  So macroscopically it looks localized, but if you zoom in enough you see it has spatial extent.  Momentum eigenstates are certainly convenient to work with mathematically, but in practice every particle we've ever observed has been very small - certainly not spread over an entire laboratory, let alone the whole universe, as a momentum eigenstate would be.
For example, the beginning of Srednicki's chapter 5 (on the LSZ reduction formula) explicitly assumes that all particles are localized by a wave-packet envelope, which is necessary so that one can freely integrate by parts without having to worry about surface terms.
A: In quantum field theory, particles are asymptotic states; they can be represented both in the momentum representation and in the position representation. In theoretical scattering theory, the momentum representations is the one usually employed for the derivation of the formulas. However, the states actually prepared in beams are essentially coherent states with a fairly well-defined position (especially in the directions orthogonal to the beam) and momentum (especially in the beam direction), limited only by the uncertainty principle. 
The possible pure states of a particle of spin $s$ are the unit vectors in a Hilbert space $L_2(R^3)^{2s+1}$. It contains neither position nor momentum eigenstates. 
For heavy particles, the coherent states behave essentially like classical point particles, if one does not look at them at short distance or short times. 
