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How is the black hole complementarity version of the holographic principle derived from path integrals and/or string theory? That has never been obvious to me. Can someone show me how to do it step by step?

I know string theorists have shown extremal and near-extremal BPS black holes have stringey entropies which match the holographic bound, but not only does this not apply to black holes in general, there are no signs of complementarity in the derivation. The AdS/CFT correspondence is also conjectural and based upon nontrivial consistency checks, but no derivation?

Thanks a lot.

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Path integrals don't make sense for both interior and exterior, since by complementarity, the interior sum is determined by the exterior, and by holography, both are really determined by a boundary sum. So if you start by using a path integral, you are overcounting.

There is no derivation from any formalism. This is not something that people have derived by woring backwards from an existing formalism, but a new principle which allows you to figure out new regimes.

What you want to know is how you arrive at black hole complemetarity in a mathematically precise way. There is no way to do that right now, because the map between interior states and exterior states is not known even in the most basic of model black holes. Complementarity was understood by Susskind by making a reconciliation between black holes and unitarity.

If you take a model black hole, like IIB 3 branes, and look near the horizon, the description of the model black hole near-horizon dynamics is by a field theory on the horizon. These black holes are infinitely cold, and it is difficult to think of the horizon in a near horizon description as being at any one spot, because space there is very symmetric. But the interior and the exterior are described by the same sort of thing, and it is nonlocal, so that the interior coordinates and the exterior coordinates are never commuting observables.

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