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I am trying to understand the argument for why internal lines in a Feynman diagram are virtual particles that do not obey the energy-momentum relation.

For the specific case of the following Feynman diagram, I was given the argument that the energy and momentum of the photon are over-constrained, hence the photon violates one constraint ie. the energy-momentum relation.

Electron Scattering Feynman diagram

However I fail to see why this is the case. This is my understanding: $E_1$, $E_2$, $p_1$, and $p_2$ are known as they are experimental parameters, so this leaves 6 unknowns (energy and momentum of the final states and of the photon). If I assume all of them follow the energy-momentum relation then this is reduced to 3 unknowns. Energy is conserved in each vertex which gives the relation between the remaining unknowns leaving 1 degree of freedom.

Is there any constraint I am missing? The only other reason I can think of is that the energies of the two final electrons are experimentally found not to depend on each other, meaning that this requires 2 degrees of freedom instead of one. Is this maybe the case?

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Energy and momentum are conserved on internal lines, which means that the mass shell condition cannot hold (as you can establish with a simple calculation). $$ m^2\ne p^2 = E^2 - \mathbf p^2$$ The question of whether internal lines represent "virtual" particles is primarily philosophical. Feynman for one regarded them as real particles which are not directly observed.

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  • $\begingroup$ Maybe you mean are not conserved ? Else, your answer is impossible ot understand. $\endgroup$
    – Mathieu Krisztian
    Commented Mar 20, 2022 at 13:50

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