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It is my understanding that virtual particles are those that are internal to a particular interaction. They do not fulfill the requirements of a freely-travelling particle (ie a plane wave) and as such do not have the usual rest mass of "real" particles. In other words, they are "off mass shell". Many sources mention that virtual particles should not be treated as particles at all, and instead describe them as strictly mathematical constructs to calculate the integrals represented by Feynman diagrams.
My confusion with this description is, what really makes something internal to an interaction? For any given "external" line in a Feynman diagram, why couldn't we simply expand the diagram to include further interactions of the outgoing particle, such that it becomes "internal" to our new diagram, and therefore a virtual particle? Unless we're talking about a universe of only a couple of particles that really do interact once and never again, I don't see a fundamental distinction between these two types of particles. People talk about how only non-virtual particles are measurable, but isn't any measurement just another long series of interactions, of which our "measured" particle is some internal part?
To give an extreme example, take a photon emitted from the sun by some process, which travels all the way to Earth before interacting with some other particle on the surface. Couldn't I construct a Feynman diagram of this interaction, where the photon is just an internal line exchanged between two other particles, making it "virtual"?
Have I made a fundamental misunderstanding of virtual particles here, or are all particles in QFT, real and virtual, simply excitations of the field that asymptotically approximate on-shell particles the further away the two interaction nodes are?
Edit: There seems to be some lack of consensus on this question. Some answers to similar questions are saying that the notion of being "on-shell" is simply an asymptotic approximation for very long-lived particles, and that fully on-shell states are unphysical as they correspond to a completely non-interacting particle. In other words, the difference between "real" and "virtual" particles is not a discrete distinction:
Other answers seem to say that virtual particles are fundamentally different to real particles. They only make sense within the context of a Feynman diagram, and a Feynman diagram is strictly a useful tool for perturbative calculations. A single Feynman diagram does not make physical sense on its own, and does not represent the actual, physical world lines of particles:
Are W and Z bosons virtual or not? (see "Can one distinguish real and virtual photons?")