This is a question not about the Physics (or Math) but about the convention. One usually defines (as in the yellow book Di Francesco) the special conformal transformation (SCT) as \begin{equation} x^{'\mu} = \frac{x^\mu - x^2 b^\mu}{1 - 2x\cdot b + x^2 b^2} \end{equation} or \begin{equation} \frac{x^{'\mu}}{x^{'2}} = \frac{x^{\mu}}{x^{2}} - b^\mu \end{equation} It's essentially a translation conjugated by an inversion, but with a minus sign compared to the usually definition of translation. To be more precise we usually define $P_\mu=\partial_\mu$ as the Killing vector for translation and $K_\mu=2x_\mu x^\nu\partial_\nu - x^2\partial_\mu$ (not $-2x_\mu x^\nu\partial_\nu + x^2\partial_\mu$) as the Killing vector for SCT. So under the conjugation by inversion we have \begin{equation} P_\mu \to -K_\mu, \; K_\mu \to - P_\mu \end{equation} It seems not as simple as it can be with the extra minus sign. I am wondering if there is any reason we stick to the current convention.



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