Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

(propose a free ad)

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 $$ \| \xi \| = g_{ab} \xi^a \xi^b = g_{\mu\nu} \xi^\mu \xi^\nu + g_{zz} \xi^z \xi^z = \frac{L^2}{z^2} \left( \eta_{\mu\nu} x^\mu x^\nu + z^2 \right) $$

user contributions licensed under cc by-sa 3.0 with attribution required