1997/10/02 by Jerzy Lewandowski, JERZY LEWANDOWSKI, Donald Marolf +1 · 75 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Commutator #Diffeomorphism #First class constraint #Hamiltonian (control theory) #Hamiltonian constraint #Homotopy and Cohomology in Algebraic Topology #Hypersurface #Infinitesimal #Loop quantum gravity #Noncommutative and Quantum Gravity Theories #Poisson bracket #Quantum #gr-qc #hep-th
paper · pdf · doi:10.1142/s0218271898000231
published in International Journal of Modern Physics D 07(02), 299-330 (World Scientific) · 30 pages RevTex, 2 figures included
arxiv created 1997/10/02 · openalex publication_date 1998/04/01 · arxiv updated 2015/06/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
This work introduces a new space [Formula: see text] 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 [Formula: see text] into itself, and so are actual operators in this space. Their commutator can be computed on [Formula: see text] 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 nontrivial action on [Formula: see text], the commutator of quantum constraints vanishes identically for a large class of proposals.