Re: electromagnetic interaction of charged particles - virtual photons

Dear Barry!

The first thing you have to understand when dealing with virtual particles (photons and others) is that their emission and absorption is not a `simple repetitive act', as you state. Virtual particles are different from normal particles in that they do not have the usual relation between energy and momentum, and that's why they are, in principle, unobservable. The energy connected with virtual particles is partly `borrowed' from the vacuum, and has to be returned after a process in order to fulfill energy-momentum conservation.

Second, virtual particles are a concept that has emerged from a perturbative treatment of quantum theory. The process of repeated emission and absorption of virtual photons by which we describe the interaction between electrons is a convenient picture that allows us to split the very difficult problem of quantum scattering into pieces that can be ordered according to their importance: You calculate just as many pieces as you need to get the desired accuracy. In general, the more virtual particles the process that you wish to look at contains (or, to be more accurate, the more electron-photon vertices the feynman diagram has), the smaller is its contribution to the result. In reality, however, the scattering process is a complicated interaction of the electron and photon fields, and one would have to take into account all orders of perturbation theory to get an exact answer.

Third, you assume that a single, isolated electron would not emit any virtual photons at all. This is wrong: Charged particles emit and absorb virtual photons continuously, all the time. This is an important contribution to their overall charge and energy and thus their mass. An electron has a `cloud' of virtual photons and virtual electron-positron pairs around it (the pairs emerge from virtual photons). Looking very closely at an electron by means of high-energy accelerators reveals this structure and shows that the charge changes depending on how close you look, i.e. how much of the `virtual cloud' your apparatus is capable to see trough.

That said, there is still the unresolved issue of the frequency of virtual photons. Well, even the production of virtual particles has to obey the law of energy-momentum conservation. To understand the answer you first have to learn something about what one really calculates in that case. The `ultimate' answer is in most cases the scattering cross section, i.e. the probability for scattering particles (e.g. electrons) into certain angles. As one nearly always performs the calculations in the centre of mass frame, the energy of an ingoing electron that scatters off another electron is the same before and after the event - only the direction of its momentum vector changes. In that case, the so-called Feynman amplitude which is the basis for the calculation of the scattering cross section is proportional to the inverse squared of the mentioned momentum difference, which means that low-frequency virtual photons are much favoured, because the momentum difference is exactly the momentum of the virtual photon, by simple momentum conservation.

Now we know that the momentum of the virtual photon in the centre of mass frame is the difference of ingoing and outgoing momentum for an electron. But the energy of the virtual photon is zero! We can nevertheless calculate an energy from the momentum just by assuming the usual energy-momentum relation for photons (which does not hold for virtual photons, as mentioned above). It is this energy which is `borrowed' from the vaccum to provide the interaction between the two electrons. In the case of photons this energy is directly proportional to the modulus of the momentum vector.

Summing up, one can say that the naive picture of virtual particles which can often be found in popularizations is very incomplete. It disregards the fact that the perturbative treatment from which the virtual particle picture emerges is just a computational tool which helps us to visualize interactions in a convenient way. Reality is more complicated and doesn't care about our ability to carry out calculations.

Hope that helps,
Georg.

PS: Check out this Physics FAQ article about virtual particles. It is really worth reading!