Representations of the Weyl algebra in quantum geometry

Communications in Mathematical Physics

Volumn 285, Issue 1, 2009, Pages 67-140

  Fleischhack, Christian ^a,b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b UNIVERSITY OF HAMBURG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 55949105267     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-008-0593-3     Document Type: Article

References (40)

1. A. Ashtekar 1987 New Hamiltonian formulation of general relativity Phys. Rev. D 36 1587 1602
2. Ashtekar A. Isham C.J. 1992 Representations of the holonomy algebras of gravity and nonabelian gauge theories Class. Quant. Grav. 9 1433 1468
3. A. Ashtekar J. Lewandowski 1995 Differential geometry on the space of connections via graphs and projective limits J. Geom. Phys. 17 191 230
4. A. Ashtekar J. Lewandowski 1995 Projective techniques and functional integration for gauge theories J. Math. Phys. 36 2170 2191
5. A. Ashtekar J. Lewandowski 1997 Quantum theory of geometry I: Area operators Class. Quant. Grav. 14 A55 A82
6. Ashtekar, A., Lewandowski, J.: Representation theory of analytic holonomy C*algebras. In: Baez, J.C. (ed.) Knots and Quantum Gravity (Riverside, CA, 1993), Oxford Lecture Series in Mathematics and its Applications 1, pp. 21-61, Oxford University Press, Oxford (1994)
7. J.C. Baez S. Sawin 1998 Diffeomorphism-invariant spin network states J. Funct. Anal. 158 253 266
8. J.C. Baez S. Sawin 1997 Functional integration on spaces of connections J. Funct. Anal. 150 1 26
9. Bahr, B., Thiemann, T.: Automorphisms in loop quantum gravity. http://arxiv.org/list/0711.0373[gr-qc] (2007)
10. E. Bierstone P.D. Milman 1988 Semianalytic and subanalytic sets Publ. Math. IHES 67 5 42
11. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 2 (Equilibrium States, Models in Quantum Statistical Mechanics). Springer-Verlag, New York (1996)
12. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1 (C*- and W*-Algebras, Symmetry Groups, Decomposition of States). Springer-Verlag, New York (1987)
13. Fleischhack, Ch.: Construction of generalized connections. http://arxiv.org/list/math-ph/0601005 (2006)
14. Ch. Fleischhack 2003 Hyphs and the Ashtekar-Lewandowski Measure J. Geom. Phys. 45 231 251
15. Ch. Fleischhack 2006 Irreducibility of the Weyl Algebra in Loop Quantum Gravity Phys. Rev. Lett. 97 061302
16. Fleischhack, Ch.: Mathematische und physikalische Aspekte verallgemeinerter Eichfeldtheorien im Ashtekarprogramm (Dissertation). Universität Leipzig (2001)
17. Ch. Fleischhack 2004 Proof of a Conjecture by Lewandowski and Thiemann Commun. Math. Phys. 249 331 352
18. Ch. Fleischhack 2003 Regular connections among generalized connections J. Geom. Phys. 47 469 483
19. Ch. Fleischhack 2000 Stratification of the Generalized Gauge Orbit Space Commun. Math. Phys. 214 607 649
20. P.X. Gallagher 1965 Zeros of group characters Math. Zeitschr. 87 363 364
21. R.M. Hardt 1975 Stratification of Real Analytic Mappings and Images Invent. Math. 28 193 208
22. W. Huebsch M. Morse 1962 Diffeomorphisms of manifolds Rend. Circ. Mat. Palermo 11 291 318
23. W. Kamiński J. Lewandowski M. Bobieński 2006 Background independent quantizations: the scalar field I Class. Quant. Grav. 23 2761 2270
24. W. Kamiński J. Lewandowski A. Okołów 2006 Background independent quantizations: the scalar field II Class. Quant. Grav. 23 5547 5586
25. J. Lewandowski A. Okołów 2005 Automorphism covariant representations of the holonomy-flux*-algebra Class. Quant. Grav. 22 657 680
26. J. Lewandowski A. Okołów H. Sahlmann T. Thiemann 2006 Uniqueness of diffeomorphism invariant states on holonomy-flux algebras Commun. Math. Phys. 267 703 733
27. J. Lewandowski T. Thiemann 1999 Diffeomorphism invariant quantum field theories of connections in terms of webs Class. Quant. Grav. 16 2299 2322
28. Łojasiewicz, S.: On semi-analytic and subanalytic geometry. In: Panoramas of mathematics (Warszawa, 1992/1994), Banach Center Publications 34, pp. 89-104, Polish Academy of Sciences, Warszawa (1995)
29. S. Łojasiewicz 1964 Triangulation of semi-analytic sets Ann. Scuola Norm. Sup. Pisa 18 449 474
30. A. Okołów J. Lewandowski 2003 Diffeomorphism covariant representations of the holonomy-flux*-algebra Class. Quant. Grav. 20 3543 3568
31. Pezzana, M.: Strong analytic triangulation for manifolds (Review of an article by Massimo Ferrarotti (Boll. Un. Mat. Ital. A 17, 79-84 (1980))).
32. Math. Rev. 81e, 32012 (1981)
33. Sahlmann, H.: Some results concering the representation theory of the algebra underlying loop quantum gravity. http://arxiv.org/list/gr-qc/0207111 (2002)
34. 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)
35. H. Sahlmann T. Thiemann 2006 Irreducibility of the Ashtekar-Isham- Lewandowski Representation Class. Quant. Grav. 23 4453 4472
36. 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)
37. Takesaki, M.: Theory of Operator Algebras I (Encyclopaedia of Mathematical Sciences 124). Springer-Verlag, Berlin (2002)
38. 1-complexes Annals of Math. 41 809 824
39. Whitney H.: Geometric Integration Theory. Princeton University Press, Princeton, NJ (1957)
40. Zeidler, E. (Hrsg.): Taschenbuch der Mathematik, Bd. 1. Teubner-Verlag, Leipzig, Stuttgart (1996)