Relativistic Wave Equations

We derive the most general relativistically covariant linear differential equations, having at most two derivatives, for scalar, spinor and vector fields. We introduce the corresponding Lagrangian and Hamiltonian formalisms and present the expansion of the solutions in terms of plane waves. In each case, we study the propagation properties of the corresponding Green functions. We start with the simplest example of the Klein–Gordon equation for a real field and generalise it to that of N real, or complex fields. As a next step we derive the Weyl, Majorana and Dirac equations for spinor fields. They are first order differential equations and we show how to adapt to them the canonical formalism. We end with the Proca and Maxwell equations for massive and massless spin-one fields and, in each case, we determine the physical degrees of freedom.