How Do On-shell and Off-shell Solutions Relate in Feynman Diagrams?

[Lapidus]

https://perimeterinstitute.ca/news/new-face-feynman-diagrams/deeper-dive-shell-and-shell that if you add two momentum vectors of two on-shell particles, you get an off-shell particle.

Two questions:

1. Since on-shell solutions are solutions to the free equation of motion, should no adding two solutions also give a solution?

2. Why should adding two on-shell particles represent an interaction?

thank you

 

[mfb]

https://perimeterinstitute.ca/news/new-face-feynman-diagrams/deeper-dive-shell-and-shell that if you add two momentum vectors of two on-shell particles, you get an off-shell particle.

Not necessarily (you can produce a heavier particle if the energy matches), but this is a typical result.

2. Why should adding two on-shell particles represent an interaction?

The opposite direction: If they interact to form a new particle, then you have to add the momenta. There are other interactions where the sum of momenta is irrelevant.