# Coulomb law and photons

When we consider process like

$e^- e^- \to e^- e^-$

in QED, we see that from exchanges of one photon (tree-level diagrams) one can obtain Coulomb's law, while loop-diagrams give quantum corrections to this law. I know that virtual particles, which appear in loop-diagrams, are not observable. But real particles can be observed.

So, I wonder, are such photons that contribute to classical Coulomb's law (tree-level diagrams) observed? For example, we can increase distance between two electrons, so that virtual photon becomes almost real.

Or is there any way to experimentally show that electromagnetic forces are caused by discrete particles - photons?

• Are you asking whether the quantum correction to Coulomb's law have been observed? You will not observe the "photons" because they are virtual - there are no actual particle states corresponding to them, so the language "are such photons observed" is misleading. – ACuriousMind Apr 27 '16 at 19:02
• Even the electrons are not "discrete particles". All of these entities are quanta of fields. Admittedly, it's easier to think in small balls than in field quanta, so we keep doing it, even when we know that it's wrong. If you need to remind yourself about the difference: the multi-particle wave function in that scattering example has to be anti-symmetric because you are dealing with fermions, something which wouldn't apply to balls. But what would it mean to anti-symmetrize two balls, anyway? – CuriousOne Apr 27 '16 at 19:50
• That they can come arbitrarily close to becoming real means that they can actually become real and appear as soft Bremsstrahlung photons that are emitted but not detected. This needs to be taken into account together with the soft virtual photons that start and end on the external lines to get as finite result for the scattering cross section. – Count Iblis Apr 27 '16 at 20:49

## 1 Answer

It does't really make much sense to talk about a tree-level truncation (it helps for calculations, but that's it) or to take the first Feynman diagram as a true representation of reality. By the way, in your $e^- e^- \to e^- e^-$ example, the whole notion of spatial separation is ill-defined since this is a t-channel process.

If going from virtual to real photon were possible here, you'd be able to create a single photon out of the annihilation of an electron-positron pair ... however, energy-momentum conservation forbids such a thing to happen (see this discussion for instance).

• "The classical law is the result of the integration over all the possible quantum contributions. So it does't really make much sense to talk about a tree-level truncation (it helps for calculations, but that's it)" is false, the Coulomb force arises exactly from the non-relativistic limit of the tree-level amplitude of an electron-electron interaction, see, for instance this question. – ACuriousMind Apr 27 '16 at 21:15
• Yes, you're right. The point I wanted to make is that truncating the series doesn't make much sense in QFT. I was sloppy, I'll fix this. – xi45 Apr 27 '16 at 21:31