Action principle for spin-12wave equations

Abstract

An action principle is presented whose variation yields both Dirac's spin-1/2 equation and Staunton's positive-energy spin-1/2 equation. The Lagrangian used is a function not only of the matter, electromagnetic, and gravitational fields, but also of the structure constants of the internal group. Variation with respect to the matter field gives the dynamical equations of motion. Variation with respect to the electromagnetic and gravitational fields yields Maxwell's and Einstein's equations, respectively. These equations include source terms that are the electromagnetic current and the stress-energy tensors for the matter and electromagnetic fields. Variation with respect to the structure constants determines the internal group, thereby projecting out either Dirac's or Staunton's equation. The matter stress-energy tensor herein determined is used to construct the energy operator for the second-quantized free fields. The results obtained in the case of Dirac's equation are the standard ones. In the case of Staunton's equation, the matter stress-energy tensor and energy operator are found for the first time. (AIP)