Dirac Equation for Spin Density in an Ideal Elastic Solid

There are two types of momentum associated with shear waves in an elastic solid: the “canonical” momentum that is proportional to the velocity of the medium and the “field” or “wave” momentum associated with force and propagation of energy. It logically follows that there are also two corresponding types of angular momentum in an elastic solid: “intrinsic” or “spin” angular momentum associated with rotations of the medium and “wave” or “orbital” angular momentum associated with wave propagation and torque. Spin density is uniquely defined from a Helmholtz decomposition as the field whose curl is equal to twice the incompressible part of momentum density. It is related to the usual “moment of momentum density” through integration by parts. A nonlinear wave equation is derived for spin density in an elastic solid, and the corresponding Dirac equation is derived by factoring the second-order wave equation to obtain a first-order bispinor equation. Plane wave solutions are analyzed to derive Dirac operators for kinetic and potential energy density. Wave interactions yield the Pauli exclusion principle and interaction potentials, with proposed interpretations of magnetic flux and electric charge.