Non-Ultralocality and Causality in the Relational Framework of Canonical Quantum Gravity

Abstract

The relational framework of canonical quantum gravity with non-ultralocal constraints is explored. After demonstrating the absence of anomalies, a spatially discretized version of the relational framework is introduced. This allows the application of Lieb-Robinson bounds to on-shell monotonic gauge-flow when there is a continuous external "time" parameter. An explicit Lieb-Robinson bound is derived for the differential on-shell evolution of the operator norm of the commutator of discretized Dirac observables, demonstrating how a local light cone-like causal structure emerges. Ultra-local constraints do not permit such a structure to arise via Lieb-Robinson bounds. Gauge and (3+1)-diff invariance of the light-cone is discussed along with the issues of quantum fluctuations, the nature of the non-localities, the spatial continuum limit, and possible links to non-commutative geometry.