How Do Virtual Photons Influence the Range of Electromagnetic Interactions?

[lilphil1989]

In a particle physics formalism, electromagnetism can be described in terms of interchange of virtual photons.

The range is then c.t = hbar / E

with t the lifetime, and E the energy of the virtual particle.

What is the argument then, for long range EM interactions, in terms of E?
Is it that the photon is simply produced with low enough frequency to cover the necessary distance? To me, this implies that the virtual photon "knows" how far it must travel, before it does so.

 

[tom.stoer]

In a particle physics formalism, electromagnetism can be described in terms of interchange of virtual photons.

The range is then c.t = hbar / E

with t the lifetime, and E the energy of the virtual particle.

What is the argument then, for long range EM interactions, in terms of E?
Is it that the photon is simply produced with low enough frequency to cover the necessary distance? To me, this implies that the virtual photon "knows" how far it must travel, before it does so.

The virtual photon is not a physical particle that is emitted in order to be absorbed one light year away; it is a purely mathematical concept. In terms of Feynman diagrams it is a rule how to integrate over all allowed intermediate particle states including states off-mass-shell.

In the language of Feynman diagrams a propagator of a massless particle ~1/k² has a Fourier transform 1/r^D-2 for spatial dimension D=3, 4, ...; this is the mathematical reason for the long-range Coulom potential.

If you chose a different gauge in QED (a physical gauge like the Coulomb gauge or the axial gauge) you find the Coulomb potential ~1/r directly in the interaction Hamiltonian; the virtual particles are fluctuations on top of a static potential; this shows that a "virtual particle" is by no means uniquely defined.

 

[Patrus89]

This is a good reason to not call these elements of path-integral formulation virtual particles. The confusion which often results is so contrary to scientific endeavor.

 

[tom.stoer]

In my course in QFT we first derived very simple Feynman diagrams in QED in the Coulomb gauge. The benefit is that you see both, the Coulomb potential and the propagators close to each other. It's more difficult than the radiation gauge but it makes very clear that the propagator is only a mathematical artefact.

If you simply take Feynman rules and diagrams w/o doing the calculation you never see the difference between an internal line and an external line; both look identical, but mathematical they are totally different entities. The external lines are somehow gauge-invariant physical entities, whereas an internal line is a gauge-dependent and therefore unphysical rule how to calculate things. As soon as you do the calculation you immediately see that external lines are something like asymptotic states (= particles) whereas internal lines are just mathematical expressions, namely certain integrals over 4d p-space.

Feynman invented his diagrams just for book-keeping; afaik the integrals existed prior to the invetion of the diagrams. Today it's the other way round; people first see the diagrams w/ knowing the math behind them.