Further results on geometric operators in quantum gravity

TL;DR

This paper explores the properties of geometric operators in canonical quantum gravity, revealing that their quantum commutators are generally non-zero, which suggests fundamental features of the quantum geometry in loop quantum gravity.

Contribution

It introduces discretized lattice analogs of geometric operators and analyzes their quantum commutators, highlighting potential universal anomalies in their algebra within loop quantum gravity.

Findings

Quantum commutators of geometric operators are generally non-vanishing.

Some lattice computations match continuum results for fixed spin-network states.

Anomalous commutators may be a general feature of the continuum loop representation.

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.