Positive one-particle BRST cohomology ($\{\epsilon :k\cdot \epsilon =0\}/\mathrm{span}\{k\}$)

For a nonzero null momentum and a nontrivial BRST charge on the gauge/ghost one-particle sector, closed vector states obey $k\cdot \epsilon =0$ and exact vector states have polarization proportional to $k$. The Faddeev-Popov ghost field state is exact and the antighost field state is not closed. With mostly-plus Minkowski metric, choose $k=(\omega ,0,0,\omega )$; closure sets ${\epsilon }^3={\epsilon }^0$, so the norm is $|{\epsilon }^1{|}^2+|{\epsilon }^2{|}^2$. Quotienting by the null longitudinal polarization leaves two positive-norm transverse polarizations per group index. Self-adjointness and nilpotence alone do not establish positivity on an arbitrary state space: the nonzero momentum, nontrivial charge and stated indefinite inner products are essential here.