Finitary Čech-de Rham cohomology

International Journal of Theoretical Physics

Volumn 41, Issue 10, 2002, Pages 1857-1902

  Mallios, Anastasios ^a   Raptis, Ioannis ^b  

^a UNIVERSITY OF ATHENS   (Greece)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Abstract differential geometry; Causal sets; Quantum gravity; Sheaf cohomology; Sheaf theory

Indexed keywords

EID: 0036432367     PISSN: 00207748     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021000806312     Document Type: Article

References (81)

1. Alexandrov, P. S. (1956). Combinatorial Topology, Vol. 1, Greylock, Rochester, New York.
2. Alexandrov, P. S. (1961). Elementary Concepts of Topology, Dover Publications, New York.
3. Bekenstein, J. D. (1973). Black holes and entropy. Physical Review D 7, 2333.
4. Böhm, A. (1979). Quantum Mechanics, Springer-Verlag, Berlin.
5. Bombelli, L., Lee, J., Meyer, D., and Sorkin, R. D. (1987). Space-time as a causal set. Physical Review Letters 59, 521.
6. Bott, R. and Tu, L. W. Differential Forms in Algebraic Topology, (1981). Springer, Berlin, Graduate Text in Mathematics, Vol. 82.
7. Bourbaki, N. (1967). Elements of Mathematics: General Topology 4th edn., Vol. 3 (Translated from French), Addison-Wesley, Reading, MA.
8. Breslav, R. B., and Zapatrin, R. R. (2000). Differential structure of Greechie logics. International Journal of Theoretical Physics 39, 1027. quant-ph/99903011.
9. Breslav, R. B., Parfionov, G. N., and Zapatrin, R. R. (1999). Topology measurement within the histories approach, Hadronic Journal 22, 225. quant-ph/9903011.
10. Butterfield, J. and Isham, C. J. (1998). A topos perspective on the Kochen-Specker the-orem: I. Quantum states as generalized valuations, International Journal of Theoretical Physics 37, 2669.
11. Butterfield, J. and Isham, C. J. (1999). A topos perspective on the Kochen Specker the-orem: II. Conceptual aspects and classical analogues, International Journal of Theoretical Physics 38, 827.
12. Butterfield, J. and Isham, C. J. (2000). Some possible roles for topos theory in quantum theory and quantum gravity, Foundations of Physics 30, 1707. grqc/9910005.
13. Butterfield, J., Hamilton, J., and Isham, C. J. (2000). A topos perspective on the Kochen Specker theorem: III. Von Neumann algebras as the base category, International Journal of Theoretical Physics 39, 2667. quant-ph/99911020.
14. Connes, A. (1994). Noncommutative Geometry, Academic Press, New York.
15. Crane, L. (1995). Clock and category: Is quantum gravity algebraic?, Journal of Mathematical Physics 36, 6180.
16. Dimakis, A. and Müller-Hoissen, F. (1999). Discrete Riemannian geometry, Journal of Mathematical Physics 40, 1518.
17. Dimakis, A., Müller-Hoissen, F., and Vanderseypen, F. (1995). Discrete differential manifolds and dynamics of networks, Journal of Mathematical Physics 36, 3771.
18. Dubuc, E. J. (1979). Sur le modèles de la géometrié différentielle synthétique, Cahiers de Topologie et Geometrie Difiérentielle 20, 231.
19. ∞ schemes, American Journal of Mathematics 103, 683.
20. Dugundji, J. (1966). Topology, Allyn and Bacon, Boston.
21. Eilenberg, S. and Steenrod, N. (1952). Foundation of Algebraic Topology, Princeton University Press, Princeton, NJ.
22. Einstein, A. (1924/1991). Über den Äther. Schweizerische Naturforschende Gesellachaft Verhanflungen 105, 85. English translation by Simon Saunders, On the ether. In The Philosophy of Vacuum, H. Brown and S. Saunders, eds., Clarendon Press, Oxford.
23. Einstein, A. (1924/1991). Über den Äther. Schweizerische Naturforschende Gesellachaft Verhanflungen 105, 85. English translation by Simon Saunders, On the ether. In The Philosophy of Vacuum, H. Brown and S. Saunders, eds., Clarendon Press, Oxford.
24. Einstein, A. (1936). Physics and reality, Journal of the Franklin Institute 221, 313.
25. Einstein, A. (1950). Out of my Later Years, Philosophical Library, New York.
26. Einstein, A. (1956). The Meaning of Relativity, Princeton University Press, Princeton, NJ. (3rd extended edition, 1990.)
27. Finkelstein, D. (1958). Past future asymmetry of the gravitational field of a point particle, Physical Review 110, 965.
28. Finkelstein, D. (1988). "Superconducting" causal nets, International Journal of Theoretical Physics 27, 473.
29. Finkelstein, D. R. (1996). Quantum Relativity. A Synthesis of the Idead of Einstein and Heisenberg, Springer-Verlag, Berlin.
30. Geroch, R. (1968a). What is a singularity in general relativity?, Annals of Physics 48, 526.
31. Geroch,R. (1968b). Local characterization of singularities in general relativity, Journal of Mathematical Physics 9, 450.
32. Geroch, R. (1972). Einstein algebras, Communications in Mathematical Physics 26, 271.
33. Hawking, S. W. (1975). Particle creation by black holes, Communications in Mathematical Physics 43, 199.
34. Hawking, S. W. (1976). Black holes and thermodynamics, Physical Review D 13, 191.
35. Isham, C. J. (1997). Topos theory and consistent histories: The internal logic of the set of all consistent sets, International Journal of Theoretical Physics 36, 785.
36. Kastler, D. (1986). Introduction to Alain Connes' non-commutative differential geometry. In Fields and Geometry 1986: Proceedings of the XXIInd Winter School and Workshop of Theoretical Physics, Karpacz, Poland, A. Jadczyk, ed., World Scientific, Singapore.
37. Lavendhomme, R. (1996). Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht.
38. Mallios, A. (1992). On an abstract form of Weil's integrality theorem, Note di Matematica 12, 167
39. Mallios, A. (1998a). Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry, Vols. 1-2, Kluwer, Dordrecht.
40. Mallios, A. (1988b). On an axiomatic treatment of differential geometry via vector sheaves. Applications, Mathematica Japonica (International Plaza), 48, 93. Invited paper.
41. Mallios, A. (1999). On an axiomatic approach to geometric prequantization: A classification scheme à la Kostan-Souriau-Kirillov, Journal of Mathematical Sciences (New York), 95, 2648. Invited paper.
42. Mallios, A. (2001). K-Theory-J of topological algebras and second quantization. Extended paper version of a homonymous talk delivered at the International Conference on Topological Algebras and Applications, Oulu, Finland.
43. Mallios, A. (in press). Abstract differential geometry, general relativity and singularities. In Unsolved Problems in Mathematics for the 21st Century. A Tribute to Kiyoshi Iseki's 80th Birthday, IOS Press, Amsterdam. Invited paper.
44. Mallios, A. (in preparation). Gauge theories from the point of view of abstract differential geometry, 2-volume work, continuation of Mallios (1998a).
45. Mallios, A. and Raptis, I. (2001). Finitary spacetime sheaves of quantum causal sets: Curving quantum causality, International Journal of Theoretical Physics 40, 1885. gr-qc/0102097.
46. ∞-smooth singularities: Chimeras of the spacetime manifold.
47. Mallios, A. and Raptis, I. (2002b). Finitary, causal, and quantal vacuum Einstein gravity. Preprint gr-gc/0209048.
48. Mallios, A. and Rosinger, E. E. (1999). Abstract differential geometry, differential algebras of generalized functions and de Rham cohomology, Acta Applicandae Mathematicae 55, 231.
49. Mallios, A. and Rosinger, E. E. (2001). Space-time foam dense singularities and de Rham cohomology, Acta Applicandae Mathematicae, 67, 59.
50. Manin, Yu. I., (1988). Gauge Theory and Complex Geometry, Springer-Verlag, Berlin.
51. O'Donnell, C. J. and Spiegel, E. (1997). Incidence Algebras, Marcel Dekker, New York. Monographs and Textbooks in Pure and Applied Mathematics.
52. Penrose, R. (1977). Space-time singularities. In Proceedings of the First Marcel Grossmann Meeting on General Relativity, North-Holland, Amsterdam.
53. Raptis, I. (2000a). Algebraic quantization of causal sets, International Journal of Theoretical Physics 39, 1233. gr-qc/9906103.
54. Raptis, I. (2000b). Finitary spacetime sheaves, International Journal of Theoretical Physics 39, 1703. gr-qc/0102108.
55. Raptis, I. (2001a). Non-commutative topology for curved quantum causality. Pre-print. gr-qc/0101082.
56. Raptis, I. (200lb). Presheaves, sheaves and their topoi in quantum gravity and quantum logic. Paper version of a talk titled "Reflections on a possible 'quantum topos' structure where curved quantum causality meets 'warped' quantum logic" given at the 5th biannual International Quantum Structures Association Conference, in Cesena, Italy, (March April 2001). Preprint: gr-gc/0110064.
57. Raptis, I. (submitted) Sheafifying consistent histories. Preprint quant-ph/0107037.
58. Raptis, I. and Zapatrin, R. R. (2000). Quantization of discretized spacetimes and the correspondence principle, International Journal of Theoretical Physics 39, 1. gr-qc/9904079.
59. Raptis, I. and Zapatrin, R. R. (2001). Algebraic description of spacetime foam, Classical and Quantum Gravity 20, 4187. gr-qc/0102048.
60. Regge, T. (1961). General relativity without coordinates, Nuovo Cimento 19, 558.
61. Rideout, D. P. and Sorldn, R. D. (2000). A classical sequential growth dynamics for causal sets, Physical Review D, 61, 024002. gr-qc/9904062.
62. Rota, G.-C. (1968). On the foundation of combinatorial theory, I. The theory of Möbius functions, Zeitschrift fur Wahrscheinlichkeitstheorie 2, 340.
63. Selesnick, S. A. (1983). Second quantization, projective modules, and local gauge invariance, International Journal of Theoretical Physics 22, 29.
64. Simms, D. J. and Woodhouse, N. M. J. (1976). Lectures on Geometric Quantization, Springer, Berlin. Lecture Notes in Physics Vol. 53.
65. Solian, A. (1977). Theory of Modules. An Introduction to the Theory of Module Categories (Translated from the Romanian by Mioara Buiculescu), Wiley, London.
66. Sorkin, R. D. (1990a). Does a discrete order underlie spacetime and its metric? In Proceedings of the Third Canadian Conference on General Relativity and Relativistic Astrophysics, F. Cooperstock, and B. Tupper, eds., World Scientific, Singapore.
67. Sorkin, R. D. (1990b). Spacetime and causal sets. In Proceedings of of the SILARG VII Conference, Cocoyoc, Mexico, preprint.
68. Sorkin, R. D. (1991). Finitary substitute for continuous topology, International Journal of Theoretical Physics 30, 923.
69. Sorkin, R. D. (1995). A specimen of theory construction from quantum gravity. In The Creation of Ideas in Physics, J. Leplin, ed., Kluwer, Dordrecht.
70. Sorkin, R. D. (1997). Forks in the road, on the way to quantum gravity. Talk given at the symposium on Directions in General Relativity, University of Maryland, College Park in May 1993 in the honour of Dieter Brill and Charles Misner. gr-qc/9706002.
71. Sorkin, R. D. (in preparation). The causal set as the deep structure of spacetime.
72. Souriau, J. M. (1977). Geometric quantization and general relativity. In Proceedings of the First Marcel Grossmann Meeting on General Relativity, North-Holland, Amsterdam.
73. Stachel, J. (1991). Einstein and quantum mechanics. In Conceptual Problems of Quantum Gravity, Ashtekar, A. and Stachel, J. eds., Birkhäuser, Boston.
74. Stanley, R. P. (1986). Enumerative Combinatorics Wadsworth and Brooks, Monterey, CA.
75. Strooker, J. R. (1978). Introduction to Categories, Homological Algebra and Sheaf Cohomology, Cambridge University Press, Cambridge, UK.
76. von Westenholz, C. (1981). Differential Forms in Mathematical Physics, North-Holland, Amsterdam.
77. Wheeler, J. A. (1990). Information, physics, quantum: The search for links. In Complexity, Entropy and the Physics of Infomation, W. H. Zurek, ed., Addison-Wesley, Reading, MA.
78. Woodhouse, N. M. J. (1997). Geometric Quantization, 2nd ed., Clarendon Press, Oxford.
79. Zapatrin, R. R. (1993). Pre-Regge calculus: Topology via logic, International Journal of Theoretical Physics 32, 779.
80. Zapatrin, R. R. (1998). Finitary algebraic superspace, International Journal of Theoretical Physics 37, 799.
81. Zapatrin, R. R. (to appear). Incidence algebras of simplicial complexes. Pure Mathematics and its Applications math. CO/0001065.