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The precise analog of the theta-quantization ambiguity of Yang-Mills theoryexists for the real SU(2) connection formulation of general relativity. As inthe former case theta labels representations of large gauge transformations,which are super-selection sectors in loop quantum gravity. We show that unlesstheta=0, the (kinematical) geometric operators such as area and volume are notwell defined on spin network states. More precisely the intersection of theirdomain with the dense set Cyl in the kinematical Hilbert space H of loopquantum gravity is empty. The absence of a well defined notion of area operatoracting on spin network states seems at first in conflict with the expectedfinite black hole entropy. However, we show that the black hole (isolated)horizon area--which in contrast to kinematical area is a (Dirac) physicalobservable--is indeed well defined, and quantized so that the black holeentropy is proportional to the area. The effect of theta is negligible in thesemiclassical limit where proportionality to area holds.

The theta parameter in loop quantum gravity: effects on quantum geometry and black hole entropy