= 0$, one can introduce an odd generator $\Omega$ (“the BRST generator”) in an extended phase space containing further fermionic conjugate pairs $(\eta^a, {{\cal P}}_a)$ (the “ghost pairs”) which has the following properties : $$\begin{aligned}
[\Omega, \Omega] &=& 0 \label{nilpotency} \\
\Omega &=& G_a \eta^a + \mbox{``more''}.\end{aligned}$$ Here, “more” stands for terms containing at least one ghost momentum ${{\cal P}}_a$. We take the ghosts $\eta^a$ to be real and their momenta imaginary, with graded Poisson bracket $$[{{\cal P}}_a, \eta^b] = - \delta_a{}^b$$

The BRST derivation $s$ in the extended phase space is generated by $\Omega$, $$s \bullet = [ \bullet, \Omega]$$ and is a differential ($s^2 = 0$) because of (\[nilpotency\]). One also introduces a grading, the “ghost number” by setting $${\mbox{gh}}\eta^a = - {\mbox{gh}}{{\cal P}}_a = 1, \quad {\mbox{gh}}q^i = {\mbox{gh}}p_i = 0.$$ The ghost number of the BRST generator is equal to 1.

The BRST generator $\Omega$ is constructed recursively as follows. One sets $$\Omega = {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} + {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}} + \cdots$$ where ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ contains $k$ ghost momenta. One has ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} = G_a \eta^a$. The 