In Part I, we discovered that tensorial propagators in quantum field theory are often non-invertible which poses an issue when one is trying to determine the form of the propagator. Then, in Part II, we discussed possible techniques involving both symmetries and invariants of the Lagrangian. Here, we aim to reveal how invariant quantities may be useful for propagators, and gauge fields specifically, and we will ultimately derive the form of the photon propagator.
In Part I, we found the central issue when dealing with tensorial propagators $latex -$ they are usually non-invertible. So then how do physicists obtain these propagators? In order to obtain these propagators, they usually exploit gauge invariance. Here, we will discuss what gauge invariance is and continue with our analysis of the photon propagator.
Here we study some issues with tensorial propagators that are often encountered in the study of quantum field theory. We will use the photon propagator from electromagnetism as an example to guide us through the troublesome calculations.