It is an error to sum probability amplitudes from Feynman diagrams which produce a different number of particles

In the Matthew Schwartz calculation of the vertex function we encounter two "corrections" two elastic Coulomb scattering.

https://canvas.harvard.edu/files/936402/download%3Fdownload_frd%3D1%26verifier%3DF760LwWdmCja7JqcNpFj2C8vROjlRSTc8ZhpJkRA

One of them is the claimed correction due to soft photons in bremsstrahlung.

         e- -----------------------------------------

                                  |  virtual

                                  |  photon

                                  |  p

        Z+ -----------------------------------------

Above we have elastic Coulomb scattering.

         

       

                                             soft real photon

                                             ~~~~~~~

                                           /

         e- ------------------------------------------

                                  |  virtual

                                  |  photon

                                  |  p

        Z+ ------------------------------------------

Above we have a diagram of the electron sending a photon of extremely low energy.

There cannot be any interference between the e- and Z+ in the upper diagram versus the e- and Z+ in the lower diagram. We can, in principle, observe the soft photon: that fact destroys any interference.

How to sum Feynman diagrams which produce a different number of particles?

How can we sum Feynman diagrams which produce different particles? We cannot.

Literature is silent about this problem. How do we combine elastic Coulomb scattering with bremsstrahlung of soft photons? How to calculate probabilities?

We have presented the hypothesis that elastic Coulomb scattering really does not exist. The electron always sends soft photons. However, the photons in most cases are infinitesimal and their effect is negligible.

We want to model the fly-by of an electron. How to do that?

Suggestion. If we only measure large photons of bremsstrahlung, then the emission of a photon is quite rare. Its probability might be P = 0.001. Such a photon requires that the momentum transfer |p| is relatively large. We should subtract the probability P from the probability of elastic scattering with a high momentum transfer |p|. We ignore soft photons and assume that the rest of fly-bys are elastic.

Classically the effect of soft photons is negligible

We need to find out why literature claims that soft photons could have a significant effect on the trajectory of the electron. Classically, the energy in low-frequency bremsstrahlung is very small. Classically it cannot have a significant effect on the electron.

We calculated that classical bremsstrahlung is only 0.05 eV for a mildly relativistic electron which passes a proton at the distance 2.4 * 10⁻¹² m. The photon spectrum of classical bremsstrahlung is from 0 to roughly 250 keV, and the power density is approximately constant for all frequencies f in the spectrum. The total energy of soft photons of energy, say, < 1 eV, is extremely small in the classical process. Soft photons have a negligible effect on the classical process. Why would they have a large effect on the quantum process?