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Mathematical Physics, Analysis and Geometry

, Volume 15, Issue 4, pp 299–315 | Cite as

Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

  • Johannes Brunnemann
  • Christian FleischhackEmail author
Article

Abstract

Let \({\cal A}\) be the affine space of all connections in an SU(2) principal fibre bundle over ℝ3. The set of homogeneous isotropic connections forms a line l in \({\cal A}\). We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding \(l \hookrightarrow {\cal A}\) does not continuously extend to an embedding \(\overline{l} \hookrightarrow \overline{\cal A}\) of the respective compactifications. Here, the Bohr compactification \(\overline{l}\) corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and \(\overline{\cal A}\) to that of loop quantum gravity. Analogous results are given for the anisotropic case.

Keywords

Parallel transports Spaces of connections Almost periodicity Qualitative theory of ODEs Loop quantum gravity Cosmological models 

Mathematics Subject Classifications (2010)

34C27 53C05 83F05 

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References

  1. 1.
    Anosov, D.V., Arnold, V.I. (eds.): Dynamical Systems I (Encyclopaedia of Mathematical Sciences 1). Springer, Berlin (1994)Google Scholar
  2. 2.
    Ashtekar, A., Bojowald, M., Lewandowski, J.: Mathematical structure of loop quantum cosmology. Adv. Theor. Math. Phys. 9, 233–268 (2003). e-print: gr-qc/0304074 MathSciNetGoogle Scholar
  3. 3.
    Ashtekar, A., Lewandowski, J.: Background independent quantum gravity: a status report. Class. Quantum Gravity 21, R53–R152 (2004). e-print: gr-qc/0404018 MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Bohr, H.: Fastperiodische Funktionen. Verlag von Julius Springer, Berlin (1932)Google Scholar
  5. 5.
    Bojowald, M.: Absence of singularity in loop quantum cosmology. Phys. Rev. Lett. 86, 5227–5230 (2001). e-print: gr-qc/0102069 MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Engle, J.: Relating loop quantum cosmology to loop quantum gravity: symmetric sectors and embeddings. Class. Quantum Gravity 24, 5777–5802 (2007). e-print: gr-qc/0701132 (Previous title: On the physical interpretation of states in loop quantum cosmology)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Fleischhack, Ch.: Mathematische und physikalische Aspekte verallgemeinerter Eichfeldtheorien im Ashtekarprogramm (Dissertation). Universität Leipzig (2001)Google Scholar
  8. 8.
    Koslowski, T.A.: Holonomies of isotropic SU(2) connections on ℝ3 (in preparation)Google Scholar
  9. 9.
    Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1990)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany
  2. 2.Department MathematikUniversität HamburgHamburgGermany
  3. 3.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany

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