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In some QFT textbooks, an external line which is off mass shell also concerns us. But according to the motion equation, shouldn't the single external line be on the mass shell? Especially when we compute some Feynman diagram, we usually use the equation of motion, e.g. $$\require{cancel}(\cancel{p}-m)u(p)=0.$$ So how to understand this problem and what is the significant of such research?

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One possible scenario is what Vladimir Kalitvianski said. However from my experience, it is more often the case that "external" refers to the fact that some lines are diagramatically external, but not necessarily external in the sense that they correspond to any S-matrix states, and the latter is a subset of the former. Once we adopt the former and more general definition of "external", it becomes clear that external lines can sometimes represent propagators, and we call this kind of external line off-shell, where on-shell external line is reserved for polarization vectors, i.e. S-matrix states. The significance is that we frequently need to study a component of a Feynman diagram rather than the whole, e.g. 1-particle irreducible component.

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Normally it should be a free (on mass shell) line, but if there is an external field, it is not so any more. The context of your question is unclear.

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