Short question: is the concept of length meaningful in LQG?
Hurkyl in my opinion the most developed and applied branch of LQG
is LQC and Bojowald just this month published a very brief summary of the current status of his field.
one could take this as representative of LQG
here you will see area operators/eigenvalues
and volume operators/eigenvalues
you will not see a length operator, or a length eigenvalue
over and over again, in LQG analysis, I have seen length arise as the square root of area, and it could also emerge as the volume of something divided by its cross-section area.
And so I have gotten the impression that length is less of a basic or primitive measurement and more derived.
I have the impression that Yes the concept of length IS meaningful in LQG and, although not in bojowald's brief paper, one encounters lengths quite often! Although it may be less basic to LQG than either area or volume and more of a convenient derived concept.
Do you have any ideas about this? I could be interesting to pursue it some.
(also I am just someone on the sidelines giving my impression, an LQG expert might contradict me and say length was equally basic with area and volume)
Perhaps it is worth note a related question in quantum mechanics, where it seems that the symplectic area (in phase space) is more important that the position (in configuration space).
Actually, Thomas looked at the length operator in LQG in his paper gr-qc/9606092. It is done with his usual technique based on the volume operator.
But this length operator is not particularly enlightening... the area operator is the "fundamental" geometrical operator in LQG because we are using a gauge theory formalism with a connection one-form. The natural object for a one-form is to be integrated along a line. Then as the metric (the triad) is conjugated to the connection, the natural object to consider is its integrated form over the dual of a line i.e a (d-1) manifold if you are working in a (d,1) spacetime. In 2+1 gravity, it is a a line too, in 3+1, it is a surface. I guess that this fits pretty well with the holographic principle stuff...
Also, one must keep in mind that the geometrical operators are defined on the non-diffeomorphic invariant Hilbert space in LQG. If one works in the diffeomorphic invariant kinematical space, then it is much harder to understand what one means by the distance between two objects.. what do you think?..
Separate names with a comma.