Path Integral Free Propagator

In quantum field theory, the propagator gives a probability amplitude for a particle traveling from some point $(t_i,x_i)$ to $(t_f,x_f)$ 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.