# How is the 'cluster decomposition principle' implemented in holographic theories?

Since holographic theories are non-local by definition, how is this principle implemented?

Naively, it seems to me it is not, at least, in some sense.

I would appreciate an explanation as simple as possible.

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It is implemented on the boundary, but this is too glib. There is an S-matrix formulation, although this is for flat space. –  Ron Maimon Aug 19 '12 at 6:20

I wouldn't say that "holographic theories are non-local by definition". On the contrary, in AdS/CFT the CFT is completely local and satisfies cluster decomposition.

The cluster decomposition property in AdS can be proved using the CFT bootstrap for all CFTs in $d > 2$ (see http://arxiv.org/abs/arXiv:1212.3616, the proof only requires CFT `axioms', i.e. unitarity and crossing symmetry). In the CFT it amounts to a statement about the large angular momentum limit of the OPE of pairs of operators. This translates into the statement that any two "blobs" in AdS can be set to orbit each other, and at large angular momentum ~ large separation, the blobs become independent. So at distances large compared to the AdS scale, all theories of quantum gravity in AdS$_D$ with $D > 3$ are local.

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Thanks. I did know the CFT is local and verifies cluster decomposition. However, a duality between a theory in the bulk a theory in the boundary is nonlocal. I infer from what you claim that at distances not larger than the AdS scale, the theory is nonlocal and there is no cluster decomposition, right? –  drake May 25 '13 at 22:52