Path integral quantum field theory, perhaps single handedly constructed by Richard Feynman, remains both elusive to undergraduates that wish to study the subject and immensely useful for performing the calculations found in quantum field theory due to the path integral formulation being manifestly symmetric between space and time. Here, we will enter the world of path integral quantum field theory by meticulously performing the calculations so that it is accessible to an advanced undergraduate.
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.
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.
In quantum field theory, the propagator gives a probability amplitude for a particle traveling from some point $latex (t_i,x_i)$ to $latex (t_f,x_f)$ with a certain energy and momentum. These propagators are the first steps into quantum field theory, I aim to bring these to the masses. The one that is most interesting and easy to grasp is the free propagator for a free field theory. The following derivation is inspired by Anthony Zee’s Quantum Field Theory in a Nutshell of UC Santa Barbara.