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Abstract

We investigate some properties of geometric operators in canonical quantum gravity in the connection approach à la Ashtekar, which are associated with the volume, area and length of spatial regions. We give the motivations for the construction of analogous discretized lattice quantities, compute various quantum commutators of the type [area, volume], [area, length] and [volume, length], and find that they are generally non-vanishing. Although our calculations are performed mostly within a lattice-regularized approach, some are - for special, fixed spin-network configurations - identical to corresponding continuum computations. Comparison with the structure of the discretized theory leads us to conclude that anomalous commutators may be a general feature of operators constructed along similar lines within a continuum loop representation of quantum general relativity; the validity of the lattice approach remains unaffected.