I Why are 'virtual particles' allowed to be 'off-shell'?

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?

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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Did you read up on virtual particles in the excellent insight series by Arnold Neumaier ?
I read it briefly last night.

I appreciate the vacuum fluctuation myth portion - it meshes well with the impression I've formed so far beating my head against QFT. (PS - why are the popular mental pictures of what the math is supposed to be doing in QM so especially obscurantist and misleading? I've never encountered another area of engineering or physics where I've had to revise and re-revise my mental picture of what is going on so often due to misleading explanations.)

I still don't have a good answer for how the propagator (what I was calling a Green's function) for a real vs. virtual particle can be different. Also, insisting that virtual particles aren't real is a little silly - it's like insisting that static electric fields aren't real in the context of classical electrodynamics since they aren't plane wave solutions to homogenous Maxwell's equations - they still have to obey the equations!

In the context of a physical theory, any element that effects the outcome is "real" and better obey the fundamental laws!

Where does the propagator for a virtual particle come from? Why can you get away with using a different propagator than for a real particle? (Why, for example, in Feynman in Elementary Particles and the Laws of Physics) blithely integrate over invalid worldlines for virtual photons without any sort of weighting?) (Or a weighting that is nonzero outside the lightcone and inside it as well.)

PPS - is lattice QFT similar in any way to the mattress picture model of QFT? (something which I think I understand.)
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Vanadium 50

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hy are the popular mental pictures of what the math is supposed to be doing in QM so especially obscurantist and misleading
One might as well ask, "how come QFT requires years of study and can't be easily understood by those who don't put the effort in?" Why should popularizations be accurate?

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