There is a well known duality between quantum mechanics in D dimensions and classical statistical mechanics in D+1 dimensions. Specifically, the Euclidean version of the path integral in D dimensions is isomorphic to a classical partition function in D+1 dimension, if you replace kT with h. So if there is a physical quantity that's difficult to compute in one of these types of systems, but easy to compute in the other, you can just switch over to the other one and compute it, then translate it back.

Somewhat similarly, there is a conjectured duality between quantum field theory in D dimensions and quantum gravity in D+1 dimensions (the holographic principle). The most concrete example of this being AdS/CFT. Similarly, there have been many instances where when something is hard to compute on one side of the duality and easy on the other, you can use the easier theory to compute it.

What I'm wondering is if by some kind of weird transitivity, you could find some quantities that are easy to compute in D+1 dimensional classical stat mech and get out answers having to do with quantities in D+1 dimensional quantum gravity. Since the dimensions match, we can even drop the +1, and say for instance... use 4 dimensional classical stat mech to say something about 4 dimensional quantum gravity. Has anyone tried this? Or is it just nuts?

  • $\begingroup$ I would think that this has been explored to death, but I don't see how a Wick rotation or similar approach makes the computation easier. The only "out" from the naive computational burden of the path integral is a completely new formalism that avoids the trap the path integral approach leads to. In the general case, rather than on some more or less physical limit, no such identity will save the bacon, anyway. The "solution" to these equations is known: it's the universe all around you. Does it look like it came from a trivial formula? $\endgroup$ – CuriousOne Apr 28 '16 at 20:18

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