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Are timelike diffeomorphisms really redundancies in description in quantum gravity? Certainly Yang-Mills gauge transformations can be considered redundancies in description. Ditto for p-form electrodynamics. Even spatial diffeomorphisms too. What about timelike diffeomorphisms? If they are, do the only real degrees of freedom in quantum gravity lie on the conformal boundaries, which are invariant under diffeomorphisms? This would be an extreme version of holography.

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By timelike diffeomorphism, do you mean a diffeomorphism generated by an everywhere timelike vector? – Ron Maimon Jul 22 '12 at 9:48
I think that temporal diffeomorphisms are also gauge redundancies as long as they do not change the "initial" and "final" hypersurfaces in the normal direction. – drake Jul 27 '12 at 0:18

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In classical general relativity in the ADM formalism, it is possible to define a well-defined equivalence relation between ADM configurations with two configurations being equivalent if and only if there is a diffeomorphism relating both of them. The idea is, there is only one block spacetime, and different global spatial slices of it describe the one and same block spacetime. In this sense, timelike diffeomorphisms are indeed redundancies in the state description.

Now, it's possible to carry over the same idea to quantum gravity. The problem is, evolving a quantum state forward in time typically leads to a state with a superposition. If the equivalence relation is also applied here, one has to subscribe to the many worlds interpretation with no collapses allowed. Equivalences aren't between branches, but the whole wavefunctional. This does not occur with Yang-Mills gauge transformations, or even spatial diffeomorphisms. There, a state with no superposition in some "preferred basis" maps to another state with no superposition under the gauge transformation, and one can imagine the equivalence relation is between branches.

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Temporal diffeomorphisms are also gauge redundancies as long as they do not change the "initial" and "final" hypersurfaces in the normal direction so that they do not change the hyper-volume in which the Lagrangian is integrated to give the action.

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