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The dilatation operator is given by

$$D=x^{a}\frac{\partial}{\partial x^{a}}+z\frac{\partial}{\partial z}$$

How the norm can be $$D^{2}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}x^{\mu}x^{\nu}+z^{2})$$ where the metric of $AdS_{d+1}$ in Poincare patch is

$$ds^{2}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2})$$

Explicit calculation will highly be appreciated.

The Killing vector that generates dilatations is $$ \xi^a = \left( x^\mu , z \right) $$ The norm of this is $$ \| \xi \| = g_{ab} \xi^a \xi^b = g_{\mu\nu} \xi^\mu \xi^\nu + g_{zz} \xi^z \xi^z = \frac{L^2}{z^2} \left( \eta_{\mu\nu} x^\mu x^\nu + z^2 \right) $$

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