Loop constraints: A habitat and their algebra

TL;DR

The paper investigates whether loop-quantized Euclidean Hamiltonian constraints reproduce the classical constraint algebra after regulator removal. It introduces a vertex-smooth habitat for states, enabling unregulated constraint operators to act and for their commutators to be defined. The authors show that, on this habitat, RST-like unregulated constraints have vanishing commutators for broad classes of proposals, while symmetrizations can be anomalous on subspaces, unless projections are used to enforce a preferred structure. Together, these results illuminate fundamental limitations of regulator-removal loop constructions in capturing the full gravity constraint algebra and guide future work (including the Lorentzian case and GLMP).

Abstract

This work introduces a new space $\T'_*$ of `vertex-smooth' states for use in the loop approach to quantum gravity. Such states provide a natural domain for Euclidean Hamiltonian constraint operators of the type introduced by Thiemann (and using certain ideas of Rovelli and Smolin). In particular, such operators map $\T'_*$ into itself, and so are actual operators in this space. Their commutator can be computed on $\T'_*$ and compared with the classical hypersurface deformation algebra. Although the classical Poisson bracket of Hamiltonian constraints yields an inverse metric times an infinitesimal diffeomorphism generator, and despite the fact that the diffeomorphism generator has a well-defined non-trivial action on $\T'_*$, the commutator of quantum constraints vanishes identically for a large class of proposals.