Groupoid approach to noncommutative quantization of gravity

Journal of Mathematical Physics

Volumn 38, Issue 11, 1997, Pages 5840-5853

  Heller, Michael ^a,d   Sasin, Wiesław ^b   Lambert, Dominique ^c  

^a VATICAN OBSERVATORY   (Italy)

^b WARSAW UNIVERSITY OF TECHNOLOGY   (Poland)

^c UNIVERSITY OF NAMUR   (Belgium)

^d NONE   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031530666     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532186     Document Type: Article

References (50)

1. M. Heller and W. Sasin, "Noncommutative structure of singularities in general relativity," J. Math. Phys. 37, 5665-5671 (1996).
2. A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); A. Sitarz, "Gravity from non-commutative geometry," Class. Quantum Grav. 11, 2127-2134 (1994); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy models," J. Math. Phys. 37, 4549-4556 (1996); A. Connes, "Gravity coupled with matter and the foundation of non commutative geometry," Commun. Math. Phys. 182, 155-176 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996.
3. A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); A. Sitarz, "Gravity from non-commutative geometry," Class. Quantum Grav. 11, 2127-2134 (1994); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy models," J. Math. Phys. 37, 4549-4556 (1996); A. Connes, "Gravity coupled with matter and the foundation of non commutative geometry," Commun. Math. Phys. 182, 155-176 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996.
4. A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); A. Sitarz, "Gravity from non-commutative geometry," Class. Quantum Grav. 11, 2127-2134 (1994); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy models," J. Math. Phys. 37, 4549-4556 (1996); A. Connes, "Gravity coupled with matter and the foundation of non commutative geometry," Commun. Math. Phys. 182, 155-176 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996.
5. A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); A. Sitarz, "Gravity from non-commutative geometry," Class. Quantum Grav. 11, 2127-2134 (1994); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy models," J. Math. Phys. 37, 4549-4556 (1996); A. Connes, "Gravity coupled with matter and the foundation of non commutative geometry," Commun. Math. Phys. 182, 155-176 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996.
6. A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); A. Sitarz, "Gravity from non-commutative geometry," Class. Quantum Grav. 11, 2127-2134 (1994); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy models," J. Math. Phys. 37, 4549-4556 (1996); A. Connes, "Gravity coupled with matter and the foundation of non commutative geometry," Commun. Math. Phys. 182, 155-176 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996.
7. M. Heller and W. Sasin, "Structured spaces and their application to relativistic physics," J. Math. Phys. 36, 3644-3662 (1995).
8. A. Connes, Noncommutative Geometry (Academic, New York, 1994).
9. B. G. Schmidt, "A new definition of singular points in general relativity," Gen. Relativ. Gravit, 1, 269-280 (1971).
10. B. Bosshard, "On the b-boundary of the closed Friedmann-model," Commun. Math. Phys. 46, 263-268 (1976).
11. R. A. Johnson, "The bundle boundary in some special cases," J. Math. Phys. 18, 898-902 (1977).
12. A. Connes, "Sur la theorie non commutative de l'integration," in Algèbres d'Opérateurs, Lecture Notes in Mathematics, No 725, edited by P. de la Harpe (Springer, Berlin, 1979), pp. 19-143.
13. J. Madore, An Introduction to Noncommutative Differential Geometry and Its Physical Applications (Cambridge University Press, Cambridge, 1995).
14. M. Dubois-Violette, "Dèrivations et calcul différentiel non commutatif," C. R. Acad. Sci. Paris 307, Sér. I, 403-408 (1988).
15. M. Dubois-Violette, "Some aspects of noncommutative geometry," preprint, 1995.
16. M. Dubois-Violette and P. W. Michor, "Dérivations et calcul différentiel non commutatif II," C. R. Acad. Sci. Paris 319, Sér. I, 927-931 (1944).
17. R. Coquereaux, "Connections, metrics, symmetries and scalar fields in commutative and non-commutative geometry," Class. Quantum Grav. 9, S41-S53 (1992).
18. M. Heller, "Commutative and non-commutative Einstein algebras," Acta Cosmol. Cracow 21, 111-130 (1995).
19. W. Sasin and M. Heller, "Non-commutative differential geometry," Acta Cosmol. Cracow 21, 235-245 (1995).
20. W. Sasin and Y. Eledrisi, "On a generalization of the Levi-Civita connections of a degenerate metric," Acta Cosmol. Cracow 21, 247-250 (1995).
21. For a more general definition of curvature in the noncommutative geometry see M. Spera, "A symplectic approach to Yang Mills theory for non commutative tori," Can. J. Math. 44, 368-387 (1992).
22. A. A. Kirillov, Elements of Representation Theory (Nauka, Moscow, 1978) (in Russian).
23. J. P. Serre, Représentations Linéaires des Groupes Finis (Hermann, Paris, 1967).
24. A similar formula appears in J. Madore, "Modification of Kaluza-Klein theory," Phys. Rev. D 41, 3709-3719 (1990); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy models," J. Math. Phys. 37, 4549-4556 (1996).
25. A similar formula appears in J. Madore, "Modification of Kaluza-Klein theory," Phys. Rev. D 41, 3709-3719 (1990); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy models," J. Math. Phys. 37, 4549-4556 (1996).
26. The Einstein-Hilbert action for various approaches to noncommutative general relativity was studied in A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy Models," J. Math. Phys. 37. 4549-4556 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996. See also other works quoted in Ref. 2.
27. The Einstein-Hilbert action for various approaches to noncommutative general relativity was studied in A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy Models," J. Math. Phys. 37. 4549-4556 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996. See also other works quoted in Ref. 2.
28. The Einstein-Hilbert action for various approaches to noncommutative general relativity was studied in A. H. Chamseddine, G. Felder, and J. Fröhlich, "Gravity in non-commutative geometry," Commun. Math. Phys. 155, 205-217 (1993); P. M. Hajac, "The Einstein action for algebras of matrix valued functions - Toy Models," J. Math. Phys. 37. 4549-4556 (1996); A. H. Chamseddine and A. Connes, "A universal action formula," preprint, 1996. See also other works quoted in Ref. 2.
29. M. Heller, "Einstein algebras and general relativity," Int. J. Theor. Phys. 31, 277-288 (1992); M. Heller and W. Sasin, "Sheaves of Einstein algebras," ibid. 34, 387-398 (1995).
30. M. Heller, "Einstein algebras and general relativity," Int. J. Theor. Phys. 31, 277-288 (1992); M. Heller and W. Sasin, "Sheaves of Einstein algebras," ibid. 34, 387-398 (1995).
31. M. Heller and W. Sasin, "The closed Friedman world model with the initial and final singularities as a non-commutative space," Banach Center Pub. 41, 153-162 (1997).
32. P. Jordan, J. von Neumann, and E. P. Wigner, "On an algebraic generalization of the quantum mechanical formalism," Ann. Math. 35, 29 (1934); I. E. Segal, "Postulates for general quantum mechanics," ibid. 48, 930-948 (1947); I. E. Segal, "Irreducible representations of operator algebras" Bull. Am. Math. Soc. 53, 73-88 (1947); R. Haag and D. Kastler, "An algebraic approach to quantum field theory," J. Math. Phys. 5, 848-861 (1964).
33. P. Jordan, J. von Neumann, and E. P. Wigner, "On an algebraic generalization of the quantum mechanical formalism," Ann. Math. 35, 29 (1934); I. E. Segal, "Postulates for general quantum mechanics," ibid. 48, 930-948 (1947); I. E. Segal, "Irreducible representations of operator algebras" Bull. Am. Math. Soc. 53, 73-88 (1947); R. Haag and D. Kastler, "An algebraic approach to quantum field theory," J. Math. Phys. 5, 848-861 (1964).
34. P. Jordan, J. von Neumann, and E. P. Wigner, "On an algebraic generalization of the quantum mechanical formalism," Ann. Math. 35, 29 (1934); I. E. Segal, "Postulates for general quantum mechanics," ibid. 48, 930-948 (1947); I. E. Segal, "Irreducible representations of operator algebras" Bull. Am. Math. Soc. 53, 73-88 (1947); R. Haag and D. Kastler, "An algebraic approach to quantum field theory," J. Math. Phys. 5, 848-861 (1964).
35. P. Jordan, J. von Neumann, and E. P. Wigner, "On an algebraic generalization of the quantum mechanical formalism," Ann. Math. 35, 29 (1934); I. E. Segal, "Postulates for general quantum mechanics," ibid. 48, 930-948 (1947); I. E. Segal, "Irreducible representations of operator algebras" Bull. Am. Math. Soc. 53, 73-88 (1947); R. Haag and D. Kastler, "An algebraic approach to quantum field theory," J. Math. Phys. 5, 848-861 (1964).
36. See, for instance, W. Thirring, Lehrbuch der Mathematischen Physik (Springer, Wien, 1979) Vol. 3; G. Emch, Mathematical and Conceptual Foundations of 20th Century Physics (North-Holland, Amsterdam, 1984).
37. See, for instance, W. Thirring, Lehrbuch der Mathematischen Physik (Springer, Wien, 1979) Vol. 3; G. Emch, Mathematical and Conceptual Foundations of 20th Century Physics (North-Holland, Amsterdam, 1984).
38. G. J. Murphy, C*-Algebras and Operator Theory (Academic, Boston, 1990).
39. Some notion of time in the noncommutative context was derived in A. Connes and C. Rovelli, "Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories," Class. Quantum Grav. 11, 2899-2917 (1994). See also C. Rovelli, "Quantum mechanics without time: A model," Phys. Rev. 42, 2638-2646 (1990); C. Rovelli, "Time in quantum gravity: An hypothesis," Phys. Rev. D 43, 442-456 (1991). The "emergence of time" in our model will be discussed in a forthcoming paper.
40. Some notion of time in the noncommutative context was derived in A. Connes and C. Rovelli, "Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories," Class. Quantum Grav. 11, 2899-2917 (1994). See also C. Rovelli, "Quantum mechanics without time: A model," Phys. Rev. 42, 2638-2646 (1990); C. Rovelli, "Time in quantum gravity: An hypothesis," Phys. Rev. D 43, 442-456 (1991). The "emergence of time" in our model will be discussed in a forthcoming paper.
41. Some notion of time in the noncommutative context was derived in A. Connes and C. Rovelli, "Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories," Class. Quantum Grav. 11, 2899-2917 (1994). See also C. Rovelli, "Quantum mechanics without time: A model," Phys. Rev. 42, 2638-2646 (1990); C. Rovelli, "Time in quantum gravity: An hypothesis," Phys. Rev. D 43, 442-456 (1991). The "emergence of time" in our model will be discussed in a forthcoming paper.
42. A. Connes, "Gravity coupled with matter and the foundation of non commutative geometry," Commun. Math. Phys. 182, 155-176 (1996).
43. A. Connes, "Noncommutative geometry and reality," J. Math. Phys. 36, 6194-6231 (1995).
44. A. Connes and J. Lott, "Particle models and noncommutative geometry," Nucl. Phys. B. Proc. Suppl. 18, 29-47 (1990).
45. The notion of a quantum groupoid was studied in J. H. Lu, "Hopf algebroids and quantum groupoids," Int. J. Math. 7, 47-70 (1996).
46. A. O. Barut and R. Ra̧czka, Theory of Group Representations and Applications (Polish Scientific, Warsaw, 1980). pp. 172 and 421-323.
47. G. Landi and G. Marmo, "Einstein algebras and the Kaluza-Klein monopole," Phys. Lett. B 210, 68-72 (1988); J. Madore, "Modification of Kaluza-Klein Theory," Phys. Rev. D 41, 3709-3719 (1990): J. Madore, An Introduction to Noncommutative Differential Geometry and Its Physical Applications (Cambridge University Press, Cambridge, 1995), pp. 180-187.
48. G. Landi and G. Marmo, "Einstein algebras and the Kaluza-Klein monopole," Phys. Lett. B 210, 68-72 (1988); J. Madore, "Modification of Kaluza-Klein Theory," Phys. Rev. D 41, 3709-3719 (1990): J. Madore, An Introduction to Noncommutative Differential Geometry and Its Physical Applications (Cambridge University Press, Cambridge, 1995), pp. 180-187.
49. G. Landi and G. Marmo, "Einstein algebras and the Kaluza-Klein monopole," Phys. Lett. B 210, 68-72 (1988); J. Madore, "Modification of Kaluza-Klein Theory," Phys. Rev. D 41, 3709-3719 (1990): J. Madore, An Introduction to Noncommutative Differential Geometry and Its Physical Applications (Cambridge University Press, Cambridge, 1995), pp. 180-187.
50. The following work could be regarded as a first step in the last direction: M. Heller and W. Sasin, "Superstructured spaces," Acta Cosmol. Cracow 21, 61-70 (1995).