Conserved quantity partition for Dirac’s equation

Let $M$ be the $(n + 1)$-dimensional Minkowski space, $n \ge 3$. The energy of a solution $\psi$ to Dirac’s equation in $M$ is a sum of $n$ terms, the $j$ th term depending on $\psi$ and the space derivative $\partial \psi /\partial {x_j}$. We show that if the Cauchy datum for $\psi$ is compactly supported, then each of these terms is eventually constant. Specifically, if $\psi$ is initially supported in the closed ball of radius $b$ about the origin in space $\left ( {{R^n}} \right )$, then for times $\left | t \right | \ge b$, the $j$th term is equal to the energy of the $j$th Riesz transform ${( - \Delta )^{ - 1/2}}(\partial /\partial {x_j})\psi$, which also solves Dirac’s equation.