= Nullity of real N=1 supercharges
{title2=$\dim_{\mathbb R}\{u:u\cdot Q\,\psi=0\}\in\{0,2,4\}$}
For a normalized state in a <unitary representation> of the ordinary four-dimensional <Super-Poincaré algebra>, define $M_{ab}=\langle\{Q_a,Q_b\}\rangle/2$ in a real Hermitian charge basis. Then $\|u\cdot Q\,\psi\|^2=u^TMu$, so its annihilator is precisely the <kernel of a linear map> represented by the positive matrix $M$. In a suitable real basis its <eigenvalues> are the two <eigenvalues> of $\langle P^0\rangle I+\langle\mathbf P\rangle\cdot\boldsymbol\sigma$, each repeated twice. A nonzero kernel therefore has dimension two at nonzero null momentum expectation or four at zero expectation. Complex non-Hermitian lowering operators and algebras with domain-wall extensions are not covered by this argument.
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