Further results on geometric operators in quantum gravity

Abstract

We investigate some properties of geometric operators in canonical quantum gravity in the connection approach `a la Ashtekar, which are associated with volume, area and length of spatial regions. We motivate the construction of analogous discretized lattice quantities, compute various quantum commutators of the type [area,volume], [area,length] and [volume, length], and find 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 with 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.