## QED

### QED Lagrangian

1. Full form of the QED Lagrangian
2. Mass of the Photon field
3. Intermezzo 1: P and C of the photon field
4. Intermezzo 2: Polarisation vectors of the photon field
Solutions of the wave equation for the electromagnetic four-potential in the Lorentz gauge
are given by
where the four-vector εμ is the polarisation vector of the photon and only depends on the three-momentum of the photon since q2=0.

The polarisation vector describes a spin-1 boson but has four components!
How is this possible?

First, the Lorentz gauge results in
reducing the number of independent components to three.

There is still the freedom to make an additional gauge transformation:
with Λ being an arbitrary scalar function satisfying
so that the Lorentz condition is still valid.

Chosing
the physics remains unchanged by the transformation
This gives the freedom to set ε0=0 so that the Lorentz condition reads
This noncovariant gauge is called the Coulomb gauge.

As a consequence, there are only two independent photon polarisation vectors, both being transverse to the three-momentum vector of the photon.
For a photon travelling in the z direction one may choose

The following linear combinations
describe photons of helicity +1 or -1, respectively.
This can be seen by considering a rotation around the z axis.
These polarisation vectors are also called circular polarisation vectors.