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If virtual particles are supposed to be some sort of Green's function excitation of a field following a particular Lagrangian or PDE (induced by the presence of another particle, free or virtual), then why are they allowed to be "off shell"? (Especially: Why are they allowed to be nonzero outside the light cone). This doesn't seem right: First of all, if it doesn't evaluate to a delta-function at the origin, it isn't a Green's function to begin with. The behavior claimed for Feynman's propagator doesn't appear to fit. Second of all, why are influences allowed to propagate in ways that free particles are not if where you draw the boundaries of the diagram are arbitrary?

Why are "negative energy states" (negative time frequency components of a Green's function) forbidden for quantum field theory when we use them all the time for classical fields? (Water waves or classical radio waves don't seem to have any 'negative energy' instability problems. Why do quantum fields?)

If they're allowed to violate the fundamental equation they are supposed to represent, what rules *do* virtual particles follow?

Edit:

Another way of phrasing my question: If the dispersion relation for the motion of free and virtual particles comes from the same lagrangian leading to the same PDE, then how can the support for the Green's function for a virtual particle be any different than for a free particle?

I've heard the "virtual particle's aren't real, so we can do whatever we want" excuse before, but I don't buy it. There's still some way of weighting all (k,w)-vectors in your plane-wave decomposition that comes from somewhere - why is it different if it's virtual or free? If I were trying to invent semi-classical field theory, a virtual photon couldn't do anything an actual photon couldn't do since they would both be derived from Maxwell's equation (and fundamentally the same thing) - a disturbance in an EM field firmly nailed to a light-cone with zero support outside.

Why are "negative energy states" (negative time frequency components of a Green's function) forbidden for quantum field theory when we use them all the time for classical fields? (Water waves or classical radio waves don't seem to have any 'negative energy' instability problems. Why do quantum fields?)

If they're allowed to violate the fundamental equation they are supposed to represent, what rules *do* virtual particles follow?

Edit:

Another way of phrasing my question: If the dispersion relation for the motion of free and virtual particles comes from the same lagrangian leading to the same PDE, then how can the support for the Green's function for a virtual particle be any different than for a free particle?

I've heard the "virtual particle's aren't real, so we can do whatever we want" excuse before, but I don't buy it. There's still some way of weighting all (k,w)-vectors in your plane-wave decomposition that comes from somewhere - why is it different if it's virtual or free? If I were trying to invent semi-classical field theory, a virtual photon couldn't do anything an actual photon couldn't do since they would both be derived from Maxwell's equation (and fundamentally the same thing) - a disturbance in an EM field firmly nailed to a light-cone with zero support outside.

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