Photon effective mass in plasma

Question

How can we compute the effective mass of the photon in a plasma in QED? I assume there must be a way of computing it from QED Lagrangian.

Answer 1

Light wave in plasma obeys the dispersion relation $$\omega^2=\omega_p^2+k^2c^2,$$ where $\omega_p$ is the plasma frequency. Multiplying this equation by $\hbar$ we have $$\hbar^2\omega^2=\hbar^2\omega_p^2+\hbar^2k^2c^2 \leftrightarrow E^2=m^2c^4+p^2c^2.$$ We now readily identify $$ m^2c^4=\hbar^2\omega_p^2\Rightarrow m=\frac{\hbar\omega_p}{c^2}.$$

Sanity check: here is a link that comes up in google where similar derivation is done.