Electron and Positron beams collide at SLAC and CERN for high energy experiments. Electron beam is produced by thermionic emission from tungsten filament and then launched in the Electric field to gain desired energy. So up to the point before the collision this beam is classical as we are using classical mechanics to control its position and momentum (hence energy). At the point of collision, however, the beam becomes a Quantum Field, as it is participating in the Feynman Diagram. Essentially we are accelerating a quantized Field in the classical lab using classical apparatus. We are using PURE classical mechanics to do calculations before the collision and PURE quantum calculation during and after the collision.
My question is how is that possible??