This chapter presents a short introduction to relativistic quantum mechanics that conforms to the style and methods of this book. In this context, relativity is a special kind of symmetry. It turns out that it is a fundamental symmetry of our world, and any understanding of elementary particles and fields should be necessarily relativistic – even if the particles do not move with velocities close to the speed of light. The symmetry constraints imposed by relativity are so overwhelming that one is able to construct, or to put it better, guess the correct theories just by using the criteria of beauty and simplicity. This method is certainly of value although not that frequently used nowadays. Examples of the method are worth seeing, and we provide some.

We start this chapter with a brief overview of the basics of relativity, refresh your knowledge of Lorentz transformations, and present the framework of Minkowski spacetime. We apply these concepts first to relativistic classical mechanics and see how Lorentz covariance allows us to see correspondences between quantities that are absent in non-relativistic mechanics, making the basic equations way more elegant. The Schrödinger equation is not Lorentz covariant, and the search for its relativistic form leads us to the Dirac equation for the electron. We find its solutions for the case of free electrons, plane waves, and understand how it predicted the existence of positrons. This naturally brings us to the second quantization of the Dirac equation. We combine the Dirac and Maxwell equations and derive a Hamiltonian model of quantum electrodynamics that encompasses the interaction between electrons and photons. The scale of the interaction is given by a small dimensionless parameter which we have encountered before: the fine structure constant α ≈ 1/137 ≪ 1. One thus expects perturbation theory to work well. However, the perturbation series suffer from ultraviolet divergences of fundamental significance. To show the ways to handle those, we provide a short introduction to renormalization. The chapter is summarized in Table 13.1.