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In loop quantum gravity, the volume operator for a given region is not a diffeomorphism invariant. But classically we know that volume is a scalar quantity under a diffeomorphism even if we take the full manifold or any region.

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    $\begingroup$ Volume is not invariant under a diffeomorphism, only under an isometry. $\endgroup$ – Javier Jul 13 '18 at 21:34
  • $\begingroup$ But the volume element in n-manifold is n-form which is a scalar quantity under any (passive or active diffeomorphism) $\endgroup$ – o.nemoul Jul 14 '18 at 10:51
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    $\begingroup$ @ravjotsk n-forms are scalar quantities, but the components of the n-form who transform as a tensor. The volume element in a 4d- metric space (M,g) is: sqrt( det(g) )*d^4x which is a scalar quantity. $\endgroup$ – o.nemoul Jul 14 '18 at 19:44
  • $\begingroup$ @o.nemoul My bad. I’ll remove my comment since it’s not useful. However I thought that the term scalars was used for tensors of rank zero and an n-form is an alternating tensor of covariant rank-n. Although the tensor itself as an object remains invariant under coordinate transformations, the fact that its components change means its not a scalar (which should have the same value no matter what coordinates you use) $\endgroup$ – ravjotsk Jul 14 '18 at 20:12
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Why would you think that volume is not diffeo-invariant?

If the region moves along with the diffemorphisms (aka passive diffeomorphisms), the volume is invariant. In LQG, the volume is also invariant (if you disagree, please explain why).

If the region doesn't move with diffeomorphisms (aka active diffeomorphisms), the volume changes. That is because the region here is only a region in the coordinate space, which doesn't have a well-defined notion of volume. The same is also true in LQG.

LQG is really not different from General Relavitity when it comes to diffeomorphism invariance.

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  • $\begingroup$ Volume is not invariant under Lorentz transformations (through the contractrion factor). If the diffeomorphism invariance is meant only for the 3d spacial submanifold(then volume would be invariant) of full diffeomorphism invariant for the 4d pseudo riemannanin manifold (then 3 volume.isnt invariantn): I have.no.idea. $\endgroup$ – lalala Jul 15 '18 at 16:14
  • $\begingroup$ @lalala Lorentz transformations are active diffeomorphisms. Volume is not invariant under those neither in GR or in LQG, as I said in my answer. $\endgroup$ – Solenodon Paradoxus Jul 15 '18 at 16:18
  • $\begingroup$ Ah now I see what you meant. I thought the passive/active wording.unusual so.I didnt really understood your point $\endgroup$ – lalala Jul 15 '18 at 16:46
  • $\begingroup$ @lalala but in LQG volume is also not invariant under the spatial diffeomorphism. $\endgroup$ – o.nemoul Jul 15 '18 at 17:51
  • $\begingroup$ @SolenodonParadoxus if you mean that the fixed region is in the coordinates chart and not in the real manifold, i will agree with you. But why do we consider the region inside the coordinates system? $\endgroup$ – o.nemoul Jul 15 '18 at 17:54

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