2005/04/30 by Jerzy Lewandowski, Andrzej Okolow, Andrzej Okołów +2 · 16 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Black Holes and Theoretical Physics #Diffeomorphism #Hamiltonian (control theory) #Holonomy #Invariant (physics) #Loop quantum gravity #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Observable #Physics #Pure mathematics #Quantum #Quantum gravity #Quantum mechanics #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1007/s00220-006-0100-7
published as Commun.Math.Phys.267:703-733,2006 · 38 pages, one figure. v2: Minor changes, final version, as published in CMP
openalex publication_date 2006/08/21 · arxiv created 2006/09/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms. While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.