Why do physicists say that elementary particles are point particles? For example, an electron, it has mass and charge, but is considered to have point mass and point charge, but why? Why are they assumed to have charge and mass in a single infinitely small point in space? Doesn't QFT show us that point like particles aren't really points rather extended excitations of fields?
 A: Scattering experiments can be used to determine the size of a particle.  The results for an extended object are different than that of a point particle.  But all of these scattering experiments depend on getting the probe particle "close" to the scattering object.  In the case of electrons, that means launching the probe with enough energy to overcome the Coulomb repulsion ... and get "close".  How close depends on the energy of the probe particle.  But there's a limit on the energy that can be given to the probe, so there's a limit to how "close" the probe can get.   Consequently, we can't know what happens at distance shorter than some value.
The best scattering experiments done on electrons to date show a scattering pattern identical to that of a point particle.  
That doesn't mean that it actually is a point particle, it just means that to the best of our knowledge it is.   No experiment to date, scattering or otherwise, has shown any sign of a size for the electron.   Our theory and our experiments are both happy ... for now ... with the notion that the electron is (more accurately: behaves exactly like) a point particle.
