#### marcus

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This is very exciting. I have wondered about this merely as a spectator, because e.g. AFAIK the spinfoam formalism has not confirmed that about the geometric operators. What they say is that discrete spectrum HAS NOT BEEN PROVEN yet for the geometric operators, so it could go either way. Also I think there are versions of LQG which do not reproduce the discreteness result. This is sure to cause turmoil havoc and release a lot of creativity!

http://arxiv.org/abs/0708.1721

Bianca Dittrich, Thomas Thiemann

12 pages

(Submitted on 13 Aug 2007)

"One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli's partial and complete observables provides a possible mechanism for turning a non gauge invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that 'fundamental discreteness at Planck scale in LQG' is an empty statement. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators."

http://arxiv.org/abs/0708.1721

**Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?**Bianca Dittrich, Thomas Thiemann

12 pages

(Submitted on 13 Aug 2007)

"One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli's partial and complete observables provides a possible mechanism for turning a non gauge invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that 'fundamental discreteness at Planck scale in LQG' is an empty statement. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators."

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