Spinorial Div-Curl Structure and Bilinear Null-Form Estimates for Dirac Equations

Abstract

We identify a physical-space balance-law mechanism underlying bilinear null forms for the free Dirac equation in three space dimensions. For each spatial direction, we decompose a spinor into the two eigenspaces of the directional Dirac symbol. The principal parts of the corresponding modes propagate in opposite directions, while the transverse derivatives and the mass term couple them. After integration over the transverse variables, their charge densities satisfy a pair of one-dimensional balance laws, and a div--curl interaction estimate controls the mixed product of these densities. The algebraic anticommutation condition defining the spinorial null form exchanges exactly the same two eigenspaces. This identifies the algebraic cancellation with the interaction selected by the balance laws. Combined with angular localization, the argument yields frequency-localized $L^2$ spacetime estimates while preserving the natural first-order formulation of the Dirac equation. In the massless case, the estimate has the same lower-frequency scaling as the three-dimensional wave null-form estimate and gains half a derivative over the direct product bound. In the massive case, we obtain a channel-dependent refinement. For the pseudoscalar interaction, anticommutation with the full massive Dirac Hamiltonian gives an additional frequency-to-mass factor $K/m$ for low-frequency interactions within the same energy branch, where $K$ is the frequency scale and $m$ is the mass. This gain is absent in the scalar channel. The bilinear estimates also yield factorized bounds for cubic spinorial null forms arising in nonlinear Dirac models.