Ghost oscillator vacuum ($b_{n\ge 0}|0⟩=c_{n>0}|0⟩=0$)

For the anticommuting bc system, $\{b_m,c_n\}={\delta }_{m+n,0}$. A useful oscillator vacuum obeys $b_n|0⟩=c_n|0⟩=0$ for $n>0$ and $b_0|0⟩=0$. Its partner $c_0|0⟩$ is nonzero. This specifies a ghost zero-mode doublet and differs from imposing annihilation by both zero modes.

With Hermitian $b_0,c_0$ and $\{b_0,c_0\}=1$, a ket annihilated by $b_0$ has zero bare full-ghost norm. Nonzero amplitudes need FP ghost zero-mode saturation. Low-level Virasoro descendant calculations can instead normalize the highest-weight coefficient and compute $L_n{L}_{-n}|\Omega ⟩$ algebraically. Such normalized contravariant coefficients are not positive-definite Hilbert norms. The pairing convention must be stated before using FP ghost descendants in a norm argument.