# Do virtual photons Doppler-shift?

Just a casual thought I had, with no answer since I'm not well-versed in QFT:

Given two point charges, the force between them is given by Coulomb's law, $$\frac{1}{r^2}$$. I know this is only fully accurate when the particles are stationary. What I'm wondering, is that since the force is mediated by (virtual) photons, whether the effect of motion could be approximated as a Doppler-shift.

In other words, could you then approximate the Coulomb potential between two moving particles as:
$$V(r) = \frac{1}{r}\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}\quad, v = \frac{dr}{dt}$$

I haven't put a lot of thought into this, so I might be completely wrong :) But it does feel intuitively correct to assume a stationary charge would 'feel' a greater force from a particle headed towards it than away from it.

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## Answers and Replies

Matterwave
Gold Member
I don't know much about QFT so I can't answer your question, but Coulomb's law is a $$\frac{1}{r^2}$$ relation not $$\frac{1}{r}$$

The force is 1/r^2 the potential is 1/r. The first equation was a typo.

Matterwave
Gold Member
Oh, but Coulomb's law concerns forces, not potentials. :P Either way, it's not a big deal.

This is an interesting question. Perhaps the following will help. In Compton scattering, a photon scatters off an electron at rest. The cross section at low energies is roughly 2/3 barn. But what happens when the electron is relativistic? In this case in order to calculate the scattering kinematics, the incident photon is Lorentz transformed from the lab to the electron rest frame, and the recoil photon is then Lorentz transformed back into the lab rest frame. If the incident photon hits the electron head on, the recoil photon is gamma-boosted to an energy that is roughly gamma-squared times the incident photon energy.

The Lorentz transfom kinematics are found at

http://pdg.lbl.gov/2008/reviews/contents_sports.html

Click on the Kinematics, cross sections, etc. category and go to paragraph 38.1 Lorentz transforms.