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Hanusch, Maximilian: Invariant connections and symmetry reduction in loop quantum gravity. 2014
Inhalt
Introduction
Quantum Gravity
Loop Quantum Gravity
Challenges
Mathematical Context
Invariant and Generalized Connections
Symmetry Reduction
Projective Structures and Normalized Radon Measures
Aims and Organization
Preliminaries
Notations
Principal Fibre Bundles
Parallel Transports
Automorphisms and Invariant Connections
Projective Structures and Radon Measures
Special Mathematical Background
Loop Quantum Gravity Case
Bounded Functions and Maps on Spectra
Maps on Spectra
Bounded Functions
The Bohr Compactification
Basic Definitions
Abelian Group Structures and Quasi-Characters
Spectral Extensions of Group Actions
Group Actions on Spectra
Invariant and Generalized Connections
Homomorphisms of Paths and Invariance
Homomorphisms of Paths
Invariant Homomorphisms
Summary
Modification of Invariant Homomorphisms
Analytic and Lie Algebra Generated Curves
Basic Properties of Analytic Curves
Basic Properties of Lie Algebra Generated Curves
Modifications along Lie Algebra Generated Curves
Inclusion Relations
Modifications along Free Segments
Summary
Measures on Quantum-Reduced Configuration Spaces
The Reduced Ashtekar-Lewandowski Measure
Lie Algebra Generated Configuration Spaces
Tychonoff Products
Normalized Radon Measures
Independence from the Choices
Summary
Homogeneous Isotropic LQC
Setting
Group Structures, Actions and some Measure Theoretical Aspects
Quantization vs. Reduction
Motivation of the Construction
Projective Structures on R
Radon Measures on R
Summary
A Characterization of Invariant Connections
-Coverings
Characterization of Invariant Connections
Reduction of Invariant Connections
Reconstruction of Invariant Connections
Particular Cases and Applications
-Coverings and the Induced Action
(Almost) Fibre Transitivity
Trivial Bundles – Applications to LQG
Summary
Conclusions and Outlook
Appendix to Preliminaries
Projective Structures and Radon Measures
Appendix to Special Mathematical Background
Homogeneous Isotropic Connections
Abelian Group Structures and Quasi-Characters
Appendix to Spectral Extensions of Group Actions
Invariant Generalized Connections
Appendix to Modification of Invariant Homomorphisms
Analytic and Lie Algebra Generated Curves
Inclusion Relations
Appendix to Lie Algebra Generated Configuration Spaces
Lemma and Definition 6.7
Lemma 6.11
Completeness and Independence of the Family in Example 6.14
Appendix to Homogeneous Isotropic LQC
Proof of the Lemma in Subsection 7.4
Appendix to A Characterization of Invariant Connections
The Proof of the Case in Subsection 8.3.3
A Result used in the End of Section 8.3
Spherically Symmetric Connections
Symbols
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