Inhalt

•  
   Overview
•  
   Contents
•  
   Preface
•  
   I Quantisation of gravity in the Loop Quantum Gravity approach
•  
  An overview about General Quantum Physics and Quantum Gravity
•  
   The Ashtekar formulation of Classical Gravity
•  
   Quantisation procedures of a classical system
•  
  Some algebras in Physics
•  
   Algebras in Quantum Mechanics
•  
   Algebras in Quantum Field Theory
•  
   Algebras for Lattice Gauge Theories and Statistical Mechanics
•  
  General mathematical concepts for the construction of C*-algebras
•  
   Pontryagin duality, quantum groups and cross-product algebras
•  
   O*- and C*-algebras generated by unbounded and bounded operators
•  
   A short summary about algebras in Loop Quantum Gravity and Cosmology
•  
   Algebras in Loop Quantum Cosmology
•  
   Algebras in Loop Quantum Gravity
•  
   Comparison of QM, QFT and LQG algebras
•  
  Quantum constraints, KMS-Theory and dynamics
•  
   The implementation of quantum constraints on algebras of Loop Quantum Gravity
•  
   The classical hypersurface deformation constraint algebra
•  
   The quantum spatial diffeomorphism constraints
•  
   The quantum Hamilton constraint
•  
   The classical Thiemann Master constraint, Dirac and complete observables
•  
   KMS-Theory in Generally Covariant Theories
•  
   A summary of physical algebras of quantum operators in LQG
•  
  The configuration and momentum space of loop quantum gravity
•  
   Path, gauge and Lie groupoids
•  
   Loop spaces, loop and holonomy group
•  
   Fundamental groupoids of path spaces
•  
   Finite path groupoids and graph systems
•  
   General Lie and gauge groupoids
•  
   Transformations in a Lie groupoid
•  
  Duality of connections and holonomies
•  
   Infinitesimal geometric objects for a gauge theory
•  
   Integrated infinitesimals, path connections, holonomy groupoids and holonomy maps in groupoids
•  
  Holonomy maps and transformations in groupoids and graph systems
•  
   Holonomy maps for Yang Mills theories
•  
   Holonomy maps for gravitational theories
•  
   Holonomy maps and transformations for a gauge theory
•  
   Holonomy maps for finite path groupoids, graph systems and transformations
•  
   Holonomy maps for path groupoids
•  
  Classical and quantum flux variables
•  
   The classical flux variables
•  
   The Lie algebra-valued quantum flux operators associated to surfaces and graphs
•  
   The group-valued quantum flux operators associated to surfaces and graphs
•  
   The group-valued quantum flux operators associated to surfaces and finite path groupoids
•  
   II Quantum algebras of Loop Quantum Gravity and Cosmology
•  
  The quantum algebras of Loop Quantum Cosmology
•  
   The algebra of almost periodic functions
•  
   Weyl algebras over pre-symplectic spaces and Weyl algebra of LQC
•  
  The smooth holonomy C*-algebra
•  
   The algebra of almost periodic functions on the loop group
•  
   The cylindrical function C*-algebra for path groupoids
•  
   The modified Wilson C*-algebra
•  
   The analytic holonomy C*-algebra and Weyl C*-algebra
•  
   Dynamical systems of actions of the flux group on the analytic holonomy C*-algebra
•  
   Dynamical systems of actions of the group of bisections on two C*-algebras
•  
   Weyl C*-algebras associated to surfaces and inductive limits of finite graph systems
•  
   Flux and graph-diffeomorphism group-invariant states of the Weyl C*-algebra for surfaces
•  
   The holonomy-flux von Neumann algebra and the Weyl C*-algebra for surfaces
•  
   The holonomy-flux cross-product C*-algebra
•  
   The flux and flux transformation group, n.c. and heat-kernel-holonomy C*-algebra
•  
   The holonomy-flux cross-product C*-algebra for surface sets
•  
   The holonomy-flux cross-product C*-algebra for a finite graph system and a surface set
•  
   The holonomy-flux cross-product C*-algebra for surfaces
•  
   The holonomy-flux-graph-diffeomorphism cross-product C*-algebra
•  
   The group and the transformation group C*-algebra in Loop Quantum Cosmology
•  
  Analytic holonomy and holonomy-flux cross-product *-algebras
•  
   Some analytic holonomy *-algebras
•  
   The holonomy-flux cross-product *-algebra
•  
   The construction of the holonomy-flux cross-product *-algebra
•  
   Heisenberg holonomy-flux cross-product *-algebras
•  
   Representations and states of the holonomy-flux cross-product *-algebra
•  
   Tensor products of the holonomy-flux cross-product *-algebra
•  
   The localised holonomy-flux cross-product *-algebra
•  
   The localised holonomy *-algebra
•  
   A representation of the general localised part of the localised holonomy-flux cross-product *-algebra
•  
   C*-dynamical systems, KMS-states and the localised holonomy-flux cross-product *-algebra
•  
   The modified quantum Hamilton constraint operator
•  
   The holonomy-flux Nelson transform C*-algebra
•  
  Holonomy groupoid and holonomy-flux groupoid C*-algebras for gauge theories
•  
   The construction of the holonomy groupoid C*-algebra for gauge theories
•  
   Cross-product C*-algebras for gauge theories
•  
   Covariant holonomy groupoid formulation of LQG
•  
   Conclusion and Outlook
•  
   III Comparison tables, Appendix, Symbols, Index and References
•  
   Comparison tables
•  
  Appendix
•  
  Some mathematical objects in Differential Geometry
•  
   Infinitesimal connections on principal bundles
•  
  Some mathematical objects in Operator Algebra Theory
•  
   Topological spaces and groups
•  
   Hopf *-algebras
•  
   Operator Theory in the O*-algebra framework
•  
   Operator Theory in the C*-algebra framework
•  
   Banach *-algebras
•  
   Symbols
•  
   Index
•  
   References