Dirac Operators for Kinetic and Potential Energy

It has long been known that energy can be calculated from the time derivative of Dirac bispinor wave functions. However, the physical origin of that energy has been unclear. Using a simple model of incompressible shear plane waves propagating in an elastic solid, we derive the Dirac operators for rotational kinetic and potential energy density. The key field variable is spin density, which is the axial vector field whose curl is equal to twice the incompressible part of momentum density. The Dirac equation is obtained by factoring a nonlinear second-order wave equation into a first-order bispinor equation. Rotational kinetic energy density is computed from the term representing local rotations of the solid medium. Rotational potential energy density is computed from two terms representing wave propagation and rotation of wave velocity relative to the medium. These “rotational energies” are in quadrature to their conventional counterparts, but with the same integrals over space. The mass term in relativistic quantum mechanics arises from rotation of wave velocity and corresponds to twice the conventional potential energy.