What would be an experimental test of AdS/CFT correspondence? Or it's extensions?
I've heard that people are studying AdS/CMT (condensed matter) correspondence, but I don't know the details of it?
But in general, how would you test it?
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It is hard to prove something experimentally that is actually an equivalence of two theories on a mathematical level. The AdS/CFT correspondence is supposed to be an exact correspondence between a quantum field theory and a string theory.
However, it is in principle possible to find support for the correspondence in the following way: construct the holographic dual (or a suitable approximation to it) of a theory that we know describes physics as we know it (e.g. QCD) and use this dual to calculate measurable quantities. If the results agree (if applicable, within the bounds of the approximation) with the observations, you have found experimental support for the correspondence.