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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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There are various ways of arguing that $z$ should be an inverse energy scale, none of them to my knowledge very precise. In fact, it's not clear to me that it is possible to make a precise relationship between $z$ and energy scale. That said, perhaps the simplest way of arguing for the relationship is to note that for massive particles, AdS is like a well: ...

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Let me work in the usual limit where a classical theory of gravity maps to a strongly interacting conformal field theory (CFT) with some color like parameter $N$ that is taken to be very large. In this limit, a central statement of the correspondence is that a generating function for correlation functions on the CFT side is given by the on-shell ...

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I usually work in the Poincare patch for which the line element is $$ds^2 = {{d {\vec x}^2 + dz^2} \over {z^2}} \ .$$ Given translation invariance in $x$, one can then assume a solution of the form $\phi(\vec x,z) = e^{i \vec k \cdot \vec x } f(z)$. The wave equation for a massive scalar in $d+1$ dimensional AdS space then reduces to  z^{d+1} \partial_z ...

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Independently of the AdS/CFT correspondence, string field theory becomes problematic for closed strings. Closed string dynamics is the stringy extension of quantum gravity which is really different from a local quantum field theory. On the other hand, open string dynamics is a stringy tower extension of Yang-Mills theory, so it preserves its proximity to ...

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