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Communications in Mathematical Physics

, Volume 267, Issue 3, pp 703–733 | Cite as

Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

  • Jerzy Lewandowski
  • Andrzej Okołów
  • Hanno Sahlmann
  • Thomas Thiemann
Article

Abstract

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.

Keywords

Cotangent Bundle Loop Quantum Gravity Generalize Connection Cylindrical Function Cyclic Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Jerzy Lewandowski
    • 1
    • 2
  • Andrzej Okołów
    • 2
    • 5
  • Hanno Sahlmann
    • 1
  • Thomas Thiemann
    • 3
    • 4
  1. 1.Physics DepartmentCenter for Gravitational Physics and GeometryUniversity ParkUSA
  2. 2.Instytut Fizyki TeoretycznejUniwersytet WarszawskiWarszawaPoland
  3. 3.Albert Einstein InstitutMPI f. GravitationsphysikGolmGermany
  4. 4.Perimeter Institute for Theoretical Physics and University of WaterlooWaterlooCanada
  5. 5.Department of Physics and AstronomyLouisiana State UniversityBaton RougeUSA

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