2004/05/31 by Andrzej Okolow, Andrzej Okołów, Jerzy Lewandowski · 6 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Black Holes and Theoretical Physics #Noncommutative and Quantum Gravity Theories #gr-qc
paper · pdf · doi:10.1088/0264-9381/22/4/002
published as Class.Quant.Grav. 22 (2005) 657-680 · 34 pages, 1 figure, LaTeX2e, minor clarifying remarks
openalex publication_date 2005/01/25 · arxiv created 2005/02/11 · arxiv updated 2009/12/01 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/30
We continue the analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of loop quantum gravity. We consider an arbitrary principal bundle of a compact connected structure group and, following Sahlmann's ideas (Sahlmann 2002 Preprint gr-qc/0207111), define a holonomy-flux ∗-algebra whose elements correspond to the elementary variables. There exists a natural action of automorphisms of the bundle on the algebra; this action generalizes the action of analytic diffeomorphisms and gauge transformations on the algebra considered in earlier works. We define the automorphism covariance of a ∗-representation of the algebra on a Hilbert space and prove that the only Hilbert space admitting such a representation is a direct sum of the spaces L 2 , given by a unique measure on the space of generalized connections. This result is a generalization of our previous work (Okołów and Lewandowski 2003 Class. Quantum Grav. 20 3543–67 ( Preprint gr-qc/0302059)) where we assumed that the principal bundle is trivial and its base manifold is .