Loop constraints: A habitat and their algebra

Details Loop constraints: A habitat and their algebra Loop constraints: A habitat and their algebra

1997-10-02

arxiv.org/abs/gr-qc/9710016v1

Int.J.Mod.Phys. D7 (1998) 299-330

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

30 pages RevTex, 2 figures included

Scientific paper

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.

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