Field equations for massless bosons from a Dirac–Weinberg formalism

Dirac’s first-order differential equation for relativistic spin 1/2 fermions can be extended to massless bosons using a formalism developed by Steven Weinberg. Weinberg’s theory is based heavily on details of the Lorentz and Poincaré groups and may not be accessible to the average reader. Although Weinberg’s theory was formulated over 40 years ago and his equations are easy to write down, few physicists seem to be familiar with them, and they are not discussed in standard quantum texts. In this paper we demonstrate the utility of his methods. Weinberg’s equations, which are valid for arbitrary spin, can be put in a form analogous to the Dirac equation for massless particles. Due to transversality conditions imposed on the theory so that it satisfies the second-order Klein–Gordon equation, we recast the theory in terms of a two-component Dirac–Pauli spinor. The theory is presented in some detail for $j=1$⁠, corresponding to the photon and the vector Maxwell equations, and for $j=2$⁠, corresponding to a graviton-like particle and a tensor version of Maxwell’s equations. These latter equations determine the polarization of gravitational waves in the weak field approximation. Our results correspond to those from the traceless-transverse gauge of general relativity.