Abstract
We review uniqueness results for the kinematical part of loop quantum gravity. After sketching the general loop formalism, the holonomy-flux and the Weyl algebras are introduced. In both cases, then, diffeomorphism invariant representations are described.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abhay Ashtekar: New Variables for classical and quantum gravity. Phys. Rev. Lett. 57 (1986) 2244–2247.
Abhay Ashtekar and Chris J. Isham: Representations of the holonomy algebras of gravity and nonabelian gauge theories. Class. Quant. Grav. 9 (1992) 1433–1468. e-print: hep-th/9202053.
Abhay Ashtekar and Jerzy Lewandowski: Projective techniques and functional integration for gauge theories. J. Math. Phys. 36 (1995) 2170–2191. e-print: gr-qc/9411046.
Abhay Ashtekar and Jerzy Lewandowski: Representation theory of analytic holonomy C* algebras. In: Knots and Quantum Gravity (Riverside, CA, 1993), edited by John C. Baez, pp. 21–61, Oxford Lecture Series in Mathematics and its Applications 1 (Oxford University Press, Oxford, 1994). e-print: gr-qc/9311010.
Abhay Ashtekar: Gravity and the Quantum. New J. Phys. 7 (2005) 189. e-print: gr-qc/0410054.
Abhay Ashtekar and Jerzy Levandowski: Background Independent Quantum Gravity: A Status Report. Class. Quant. Grav. 21 (2004) R53. e-print:gr-qc/0404018.
John C. Baez: Spin network states in gauge theory. Adv. Math. 117 (1996) 253–272. e-print: gr-qc/9411007.
Edward Bierstone and Pierre D. Milman: Semianalytic and subanalytic sets. Publ. Math. IHES 67 (1988) 5–42.
Martin Bojowald: Absence of Singularity in Loop Quantum Cosmology. Phys. Rev. Lett. 86 (2001) 5227–5230. e-print: gr-qc/0102069.
Ola Bratteli and Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics, vol. 2 (Equilibrium States, Models in Quantum Statistical Mechanics). Springer-Verlag, New York, 1996.
Christian Fleischhack: Construction of Generalized Connections. MPI-MIS preprint 105/2005. e-print: math-ph/0601005.
Christian Fleischhack: Irreducibility of the Weyl Algebra in Loop Quantum Gravity. Preprint.
Christian Fleischhack: Quantization Restrictions for Diffeomorphism Invariant Gauge Theories. In: Complex Analysis, Operator Theory and Applications to Mathematical Physics (Wien, 2005), edited by Friedrich Haslinger, Emil Straube, and Harald Upmeier. (to appear)
Christian Fleischhack: Representations of the Weyl Algebra in Quantum Geometry. e-print: math-ph/0407006.
Patrick X. Gallagher: Zeros of group characters. Math. Zeitschr. 87 (1965) 363–364.
R. M. Hardt: Stratification of Real Analytic Mappings and Images. Invent. Math. 28 (1975) 193–208.
Jerzy Lewandowski, Andrzej Okołów, Hanno Sahlmann, and Thomas Thiemann: Uniqueness of diffeomorphism invariant states on holonomy-flux algebras. e-print: gr-qc/0504147.
Stanisław Łojasiewicz: Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa 18 (1964) 449–474.
Jonathan Rosenberg: A selective history of the Stone-von Neumann theorem. Contemp. Math. 365 (2004) 331–353.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Fleischhack, C. (2006). Kinematical Uniqueness of Loop Quantum Gravity. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds) Quantum Gravity. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7978-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7978-0_10
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7977-3
Online ISBN: 978-3-7643-7978-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)