Gauge-boson propagator

For the Euclidean covariant gauge term $({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$, the free gauge-boson propagator is

At $\xi =1$, the Euclidean photon propagator is $D_{\mu \nu }(k)={\delta }_{\mu \nu }/k^2$. Its Lorentz-index contraction simplifies loop numerators in quantum electrodynamics.

For nonzero momentum $k$, the transverse projector isIt annihilates the component parallel to $k$.

The longitudinal projector is $P_L^{\mu \nu }=k^{\mu }{k}^{\nu }/k^2$. It obeys $P_T+P_L=I$, $P_T{P}_L=0$, and $P_T^2=P_T$, $P_L^2=P_L$.