Notes on basis-independent computations with the Dirac algebra: https://arxiv.org/abs/2201.01802
Walter Grimus
In these notes we review Pauli's proof of his `fundamental theorem' that states the equivalence of any two sets of Dirac matrices. Due to this theorem not only all physical results in the context of the Dirac equation have to be independent of the basis chosen for the Dirac matrices, but it should also be possible to obtain the results without resorting to a specific basis in the course of the computation. Indeed, we demonstrate this in the case of the behaviour of Dirac spinors under Lorentz transformations, the quantization of the Dirac field, the expectation value of the spin operator and several other topics. Finally, we compare the basis-independent manipulations with those performed in the Weyl basis of Dirac matrices. The present notes serve as a supplement to an introductory lecture on quantum field theory, for students who want to delve deeper into mathematical and computational aspects of the Dirac theory.
Comments: 54 pages, no figures