Place an electron and positron near each other, and they will attract, meet, and release a photon or photons as they annihilate each other.
If they are point particles, how do they manage to meet? How do they shed their excess angular momentum in order to meet? What happens to the energy released in descending the infinitely deep potential well? (There's infinite energy there, so it's surprising a electron-positron accelerator is even needed to provide the energy to create new particles.)
The answers are probably "uncertainty principle".