QED sum over polarization states

[Silviu]

Hello! In the calculation of the QED matrix element, it says in the book I read that we have to sum over the polarization states of the photon: $$\sum_\lambda \epsilon_\mu^\lambda\epsilon_\nu^{\lambda *}=-g_{\mu\nu}$$ I am a bit confused why do we do a summation over the orthonormal basis describing the polarization and we don't do an integral over all possible polarization states? Is it any reason why the photon would leave in an eigenstate and not in a linear combination of polarization basis?

 

[LightHero]

The reason why we sum over the polarization states of the photon instead of integrating over all possible polarization states is because we are dealing with a quantum system. In a quantum system, the wavefunction of the system can be represented as a linear combination of the eigenstates of the system. In the case of the photon, the eigenstates are the polarization states of the photon. Therefore, when we calculate the matrix element of a QED process involving photons, we sum over the polarization states of the photon instead of integrating over all possible polarization states.