= Radial-ordering sign in commuting beta-gamma contractions
{title2=$\beta(z)\gamma(w)\sim-(z-w)^{-1},\quad\gamma(z)\beta(w)\sim+(z-w)^{-1}$}
Commutation allows exchanging the factors at their existing arguments; it does not allow exchanging their arguments while retaining the same pole sign. With the displayed convention, differentiation produces $\partial\beta(z)\gamma(w)\sim(z-w)^{-2}$ and $\partial\gamma(z)\beta(w)\sim-(z-w)^{-2}$. These signs are essential for the <conformal weights> and <central charge> of a commuting <beta-gamma system>.
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