Exploring On-shell and Off-shell Solutions in Feynman Diagrams

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In summary, adding two momentum vectors of two on-shell particles can result in an off-shell particle, but not always. This is because the energy of the particles can also play a role in creating a new particle. Additionally, the addition of two on-shell particles can represent an interaction, but it is not the only way interactions can occur.
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Lapidus
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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
 
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Lapidus said:
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.
Lapidus said:
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.
 
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FAQ: Exploring On-shell and Off-shell Solutions in Feynman Diagrams

What is the purpose of exploring on-shell and off-shell solutions in Feynman diagrams?

The purpose of exploring on-shell and off-shell solutions in Feynman diagrams is to understand the behavior of particles and their interactions in quantum field theory. These diagrams are used to calculate the probability of particle interactions and to make predictions about the outcomes of experiments.

What is the difference between on-shell and off-shell solutions?

In on-shell solutions, all particles involved in the interaction are on their mass shells, meaning they have their physical masses and energies. Off-shell solutions, on the other hand, involve virtual particles that do not satisfy the on-shell condition, meaning they do not have their physical masses or energies.

Why are off-shell solutions important in Feynman diagrams?

Off-shell solutions are important because they allow for the consideration of all possible interactions and processes, including those that cannot be observed directly. By including these virtual particles, Feynman diagrams can provide a more complete understanding of the behavior of particles and their interactions.

What are the limitations of using off-shell solutions in Feynman diagrams?

One limitation of using off-shell solutions is that they do not correspond to physical observables, and therefore cannot be directly measured. Additionally, the inclusion of off-shell solutions can lead to divergent calculations, which must be dealt with using mathematical techniques such as renormalization.

How do on-shell and off-shell solutions affect the conservation of energy and momentum in Feynman diagrams?

In on-shell solutions, energy and momentum are conserved at each vertex of the diagram. However, in off-shell solutions, momentum and energy are not conserved at individual vertices, but are still conserved overall in the entire diagram. This is due to the uncertainty principle, which allows for the temporary violation of energy and momentum conservation in virtual particle interactions.

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