Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (original) (raw)

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.

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References

  1. Ashtekar A., (1991) Lectures on non-perturbative canonical gravity. Notes prepared in collaboration with R. S. Tate. Singapore, World Scientific
    Google Scholar
  2. Łojasiewicz S. (1964) Triangulation of semi-analytic sets. Ann. Scuola. Norm. Sup. Pisa 18, 449–474
    MathSciNet Google Scholar
  3. Bierstone E., Milman P.D. (1988) Semianalytic and Subanalytic sets. Publ. Maths. IHES 67, 5–42
    MATH MathSciNet Google Scholar
  4. Rovelli C. (1998) Loop quantum gravity. Living Rev. Rel. 1: 1
    MathSciNet Google Scholar
  5. Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge: Cambridge University Press, in press; a prelimary version is available a http://aixiv.org/list/ gr-qc/0110034, 2001
  6. Ashtekar A., Lewandowski J. (2004) Background independent quantum gravity: A status report. Class. Quant. Grav. 21, R53
    Article MATH ADS MathSciNet Google Scholar
  7. Rovelli, C.: Quantum Gravity. Cambridge: Cambridge University Press, in press, 2004
  8. Rovelli C. (1991) Ashtekar formulation of general relativity and loop space non-perturbative quantum gravity: a report. Class. Quant. Grav. 8, 1613–1675
    Article MATH ADS MathSciNet Google Scholar
  9. Ashtekar A., Lewandowski J. (1997) Quantum theory of geometry I: Area operators. Class. Quant. Grav. 14, A55–A82
    Article MATH ADS MathSciNet Google Scholar
  10. Ashtekar A., Corichi A., Zapata J.A. (1998) Quantum theory of geometry III: Non-commutativity of Riemannian structures. Class. Quant. Grav. 15, 2955–2972
    Article MATH ADS MathSciNet Google Scholar
  11. Fairbairn W. Rovelli C. (2004) Separable Hilbert space in loop quantum gravity, J. Math. Phys. 45, 2802–2814
    Article ADS MathSciNet Google Scholar
  12. Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. In: Operator Theory: Advances and Applications. Vol. 37, Basel: Birkhäuser, 1990
  13. Sahlmann, H.: Some comments on the representation theory of the algebra underlying loop quantum gravity. http://arxiv.org/list/ gr-qc/0207111, 2002
  14. Sahlmann, H.: When do measures on the space of connections support the triad operators of loop quantum gravity? http://arxiv.org/list/ gr-qc/0207112, 2002
  15. Okołow A., Lewandowski J. (2003) Diffeomorphism covariant representations of the holonomy – flux *-algebra. Class. Quant. Grav. 20, 3543–3568
    Article ADS Google Scholar
  16. Sahlmann, H., Thiemann, T.: On the superselection theory of the weyl algebra for diffeomorphism invariant quantum gauge theories. http://arxiv.org/list/ gr-qc/0302090, 2003
  17. Zapata J.A. (1998) Combinatorial space from loop quantum gravity. Gen. Rel. Grav. 30: 1229
    Article MATH ADS MathSciNet Google Scholar
  18. Velhinho J.M. (2004) On the structure of the space of generalized connections. Int. J. Geom. Meth. Mod. Phys. 1, 311–334
    Article MATH MathSciNet Google Scholar
  19. Ashtekar A., Lewandowski J. (1995) Differential Geometry on the Space of Connections via Graphs and Projective Limits. J. Geom. Phys. 17, 191–230
    Article MATH ADS MathSciNet Google Scholar
  20. Okołow A., Lewandowski J. (2004) Automorphism covariant representations of the holonomy-flux *-algebra. Class. Quant. Grav. 22, 657
    Article ADS Google Scholar
  21. Ashtekar A., Isham C.J. (1992) Representation of the holonomy algebras of gravity and non-Abelian gauge theories. Class. Quant. Grav. 9, 1433–1467
    Article MATH ADS MathSciNet Google Scholar
  22. Ashtekar A., Lewandowski J., (1994) Representation theory of analytic holonomy algebras. In: Baez J.C. (eds) Knots and Quantum Gravity. Oxford, Oxford University Press
    Google Scholar
  23. Baez J.C. (1994) Generalized measures in gauge theory. Lett. Math. Phys. 31, 213–223
    Article MATH ADS MathSciNet Google Scholar
  24. Marolf D., Mourão J. (1995) On the support of the Ashtekar–Lewandowski measure, Commun. Math. Phys. 170, 583–606
    Article MATH ADS Google Scholar
  25. Ashtekar A., Lewandowski J. (1995) Projective techniques and functional integration. J. Math. Phys. 36, 2170–2191
    Article MATH ADS MathSciNet Google Scholar
  26. Ashtekar A., Lewandowski J., Marolf D., Mourão J., Thiemann T. (1995) Quantization of diffeomorphism invariant theories of connections with local degrees of freedom. J. Math. Phys. 36, 6456–6493
    Article MATH ADS MathSciNet Google Scholar
  27. Lewandowski J., Marolf D. (1998) Loop constraints: A habitat and their algebra. Int. J. Mod. Phys. D7, 299–330
    Article MATH ADS MathSciNet Google Scholar
  28. Fleischhack, C., Personal communication
  29. Fleischhack, C.: Representations of the Weyl algebra in quantum geometry. http://arxiv.org/list/ math-ph/0407006, 2004
  30. Reed M., Simon B. (1980) Methods of Modern Mathematical Physics Vol.1: Functional Analysis. New York, Academic Press
    Google Scholar

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Authors and Affiliations

  1. Physics Department, Center for Gravitational Physics and Geometry, 104 Davey, Penn State, University Park, PA, 16802, USA
    Jerzy Lewandowski & Hanno Sahlmann
  2. Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoża 69, 00-681, Warszawa, Poland
    Jerzy Lewandowski & Andrzej Okołów
  3. Albert Einstein Institut, MPI f. Gravitationsphysik, Am Mühlenberg 1, 14476, Golm, Germany
    Thomas Thiemann
  4. Perimeter Institute for Theoretical Physics and University of Waterloo, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada
    Thomas Thiemann
  5. Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, 70803-4001, USA
    Andrzej Okołów

Authors

  1. Jerzy Lewandowski
  2. Andrzej Okołów
  3. Hanno Sahlmann
  4. Thomas Thiemann

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Correspondence toThomas Thiemann.

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Communicated by Y. Kawahigashi

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Lewandowski, J., Okołów, A., Sahlmann, H. et al. Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras.Commun. Math. Phys. 267, 703–733 (2006). https://doi.org/10.1007/s00220-006-0100-7

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