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In this paper by Sean Carroll (What if Time Really Exists), there's a section "Lessons from Duality" where he says that the holographic principle (and in particular, that a lower dimensional non-gravitational quantum theory can be dual to a higher dimensional theory of quantum gravity) suggests that the Wheeler-deWitt equation need not hold.

As I understand this, the issue is that the total energy in classical GR is zero, while the total energy in ordinary quantum field theory is typically non-zero.

But if there really is any valid argument that the total energy is zero in quantum gravity (presumably some people think this follows from the total energy being zero in classical gravity with no cosmological constant?) then the implication would seem to be that whatever conformal field theory lives on the boundary has to also have zero total energy. Instead, he seems to be drawing the opposite conclusion--that since the total energy can be anything in ordinary QFT, this duality suggests that it can be anything in quantum gravity. Is either of these a valid conclusion to draw?

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  • $\begingroup$ The "Wheeler-deWitt equation" is just the statement that the Hamiltonian generically vanishes on the physical states of a time-reparametrization invariant system. If you are able to write the system in a non-time-reparametrization invariant way (by duality, holography or whatever, the procedure itself doesn't matter), then the Hamiltonian constraint, i.e. the "Wheeler-deWitt equation", is gone. I'm not sure what the precise question is here. $\endgroup$ – ACuriousMind Aug 23 '15 at 16:44
  • $\begingroup$ See e.g. my answer here for why identifying the Hamiltonian with "energy", as this question seems to assume, is, in general, a bad idea if the system contains gauge degrees of freedom, in particular time-reparametrization. $\endgroup$ – ACuriousMind Aug 23 '15 at 16:47
  • $\begingroup$ @ACuriousMind Ah, looking at your answer to the question about examples where the Hamiltonian does not correspond to the total energy gives me a hint as to what I'm missing. Maybe all that Carroll meant was that if there is a dual description of the theory of quantum gravity then the Hamiltonian constraint would need not apply in that description. Even though it <i>would</i> still apply in the original description. I was assuming if it applied in one it had to apply on both sides of the duality, but perhaps not if the Hamiltonian represents different things in each description? $\endgroup$ – reductionista Aug 23 '15 at 17:51
  • $\begingroup$ I suspect that that is what he meant. However, since I don't really know about holography, it could be something else (which is why I didn't make that an answer). $\endgroup$ – ACuriousMind Aug 23 '15 at 18:58

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