Calculating Transition Amplitudes from Feynman Diagrams

When one thinks of quantum field theory, one’s mind is undoubtedly drawn to Feynman diagrams. The näıve view these diagrams as merely a concise and standardized way of describing collisions and processes. Feynman diagrams, however, encode a great deal of information, for each one corresponds exactly to the probability amplitude of the transition it describes. In other words, it is easily possible to write and calculate the transition amplitude of a given process using only the process’ Feynman diagram. Such monumental utility arises thanks to the efforts of Dyson, Wick, and, of course, Feynman. In this short paper, we will briefly provide some background information on concepts required to understand Feynman diagrams. In particular, we will give an overview of Feynman diagram construction from terms in the S-matrix as expanded by Wick’s theorem. We will then calculate a simple transition amplitude by hand. Afterward, we will present the rules that can be used to derive the transition amplitude much more simply from Feynman diagrams. Throughout this paper, we will refer to equations in the book by Mandl and Shaw and frequently use its notation. Where convenient or deemed sufficiently important, certain equations will be rewritten here. This paper comprises a direct adaptation from the lecture notes I used for Chapter 7, and is in most cases verbatim. Hence the paper assumes a working familiarity with concepts of previous chapters and should not be considered comprehensive.