= Positive one-particle BRST cohomology
{title2=$\{\varepsilon:k\cdot\varepsilon=0\}/\operatorname{span}\{k\}$}
For a nonzero null momentum and a nontrivial <BRST charge> on the gauge/ghost one-particle sector, closed vector states obey $k\cdot\varepsilon=0$ 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 $\varepsilon^3=\varepsilon^0$, so the norm is $|\varepsilon^1|^2+|\varepsilon^2|^2$. 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.
Back to article page
