Automorphism covariant representations of the holonomy-flux lowast-algebra
Open accessAbstract
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 .
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Given a flux vector field X_{S,f}, the map Ad ̃φ maps the generalized flow into the flow of the flux vector field X_{φ(S), ̃φ∗f}.
“Given a flux vector field X S,f , the map Ad ˜ ϕ defined above maps the generalized flow (17) into a new flow Ad ˜ ϕ θ ( · ) . It is easy to check, that the new flow is the flow of the flux vector field X ϕ ( S ) , ˜ ϕ ∗ f [20], hence”
From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [20]
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- Authors
- Okołów, A., & Lewandowski, J.
- Affiliations
- University of Warsaw (PL)
- Pennsylvania State University (US)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2005-01-25
- Volume
- 22
- Issue
- 4
- Pages
- 657-679
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W3101097682
- Cited by
- 18
- References
- 20
- Citations by year
- 2013: 1 · 2012: 1
- FWCI
- 3.41
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- loop quantum gravity, principal bundles, cylindrical functions, flux, generalized connections, gauge transformations
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