Loop constraints: A habitat and their algebra

TL;DR

This paper introduces a new space of states in loop quantum gravity that allows for well-defined Hamiltonian operators and investigates their algebra, revealing that the quantum constraint commutator vanishes under many proposals.

Contribution

It defines a new vertex-smooth state space for loop quantum gravity and analyzes the algebra of Hamiltonian constraints within this framework.

Findings

Hamiltonian operators map the new space into itself.

The quantum constraint commutator vanishes for many proposals.

Comparison with classical hypersurface deformation algebra shows discrepancies.

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.