BRST cohomology

BRST cohomology identifies physical states and observables with BRST-closed objects modulo BRST-exact ones. A change of gauge-fixing fermion changes the action by a BRST-exact term and therefore leaves BRST-cohomology classes unchanged when the measure has no BRST anomaly.

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.

A BRST-closed operator $O$ satisfies $QO=0$. It represents a physical observable when it is not also BRST exact.

A BRST-exact operator has the form $O=QX$. It is BRST closed by nilpotence and represents the zero class in BRST cohomology.

When separated operators $G_i$ are BRST-closed, the graded BRST Ward identity makes an exact insertion vanish. This is the correlator form of identifying closed representatives that differ by exact terms in BRST cohomology. The statement presupposes an invariant measure and suitable boundary conditions.