The Klein–Gordon Equation and Its First-Order Roots in Spacetime Algebra

This note classifies all Lorentz-covariant first-order square roots of the Klein-Gordon operator when fields are valued in the full spacetime algebra. Two algebraically consistent roots appear: a trivector root corresponding to the Dirac equation, and a spacelike vector root. The vector branch yields only static spatial exponentials and fails to provide finite-energy, propagating solutions. Imposing physical boundary conditions therefore leaves the Dirac root as the unique physically admissible factorisation. The result offers a clear, representation-independent view of how the Dirac equation arises directly from the structure already present in the Klein-Gordon operator.