×

The homological Kähler-de Rham differential mechanism. I: Application in general theory of relativity. (English) Zbl 1219.83037

Summary: The mechanism of differential geometric calculus is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalars defined together with the associated de Rham complex. In this communication, we demonstrate that the dynamical mechanism of physical fields can be formulated by purely algebraic means, in terms of the homological Kähler-de Rham differential schema, constructed by connection inducing functors and their associated curvatures, independently of any background substratum. In this context, we show explicitly that the application of this mechanism in General Relativity, instantiating the case of gravitational dynamics, is related with the absolute representability of the theory in the field of real numbers, a byproduct of which is the fixed background manifold construct of this theory. Furthermore, the background independence of the homological differential mechanism is of particular importance for the formulation of dynamics in quantum theory, where the adherence to a fixed manifold substratum is problematic due to singularities or other topological defects.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53B35 Local differential geometry of Hermitian and Kählerian structures
83C75 Space-time singularities, cosmic censorship, etc.
83C45 Quantization of the gravitational field
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, vol. 150 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1995. · Zbl 0819.13001
[2] S. I. Gelfand and Y. I. Manin, Methods of Homological Algebra, Springer, Berlin, Germany, 1996. · Zbl 0855.18001
[3] A. Mallios, Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry, Vols 1-2, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. · Zbl 0904.18001
[4] A. Ashtekar and J. Lewandowski, “Background independent quantum gravity: a status report,” Classical and Quantum Gravity, vol. 21, no. 15, pp. R53-R152, 2004. · Zbl 1077.83017 · doi:10.1088/0264-9381/21/15/R01
[5] C. J. Isham and J. Butterfield, “Some possible roles for topos theory in quantum theory and quantum gravity,” Foundations of Physics, vol. 30, no. 10, pp. 1707-1735, 2000. · doi:10.1023/A:1026406502316
[6] L. Crane, “Clock and category: is quantum gravity algebraic?” Journal of Mathematical Physics, vol. 36, no. 11, pp. 6180-6193, 1995. · Zbl 0848.57035 · doi:10.1063/1.531240
[7] F. Antonsen, “Logics and quantum gravity,” International Journal of Theoretical Physics, vol. 33, no. 10, pp. 1985-2017, 1994. · Zbl 0818.03031 · doi:10.1007/BF00675166
[8] S. A. Selesnick, Quanta, Logic and Spacetime, World Scientific, River Edge, NJ, USA, 2nd edition, 2004. · Zbl 1138.81323
[9] C. Isham, “Some reflections on the status of conventional quantum theory when applied to quantum gravity,” in The Future of the Theoretical Physics and Cosmology (Cambridge, 2002): Celebrating Stephen Hawking’s 60th Birthday, G. W. Gibbons, E. P. S. Shellard, and S. J. Rankin, Eds., pp. 384-408, Cambridge University Press, Cambridge, UK, 2003.
[10] C. Rovelli, Quantum Gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 2004. · Zbl 0962.83507 · doi:10.1017/CBO9780511755804
[11] R. Penrose, “The problem of spacetime singularities: implications for quantum gravity?” in The Future of the Theoretical Physics and Cosmology (Cambridge, 2002): Celebrating Stephen Hawking’s 60th Birthday, G. W. Gibbons, E. P. S. Shellard, and S. J. Rankin, Eds., pp. 51-73, Cambridge University Press, Cambridge, UK, 2003.
[12] L. Smolin, “The case for background independence,” in The Structural Foundations of Quantum Gravity, pp. 196-239, Oxford University Press, Oxford, UK, 2006. · Zbl 1116.83013
[13] R. D. Sorkin, “A specimen of theory construction from quantum gravity,” in The Creation of Ideas in Physics, J. Leplin, Ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
[14] J. Stachel, “Einstein and quantum mechanics,” in Conceptual Problems of Quantum Gravity, A. Ashtekar and J. Stachel, Eds., Birkhäuser, Boston, Mass, USA, 1991.
[15] M. Atiyah and I. MacDonald, Introduction to Commutative Algebra, Addison Wesley, Reading, Mass, USA, 1969. · Zbl 0175.03601
[16] S. MacLane, Categories for the Working Mathematician, vol. 5 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1971. · Zbl 0705.18001
[17] G. E. Bredon, Sheaf Theory, vol. 170 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1997. · Zbl 0874.55001
[18] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic, Universitext, Springer, New York, NY, USA, 1994.
[19] J. L. Bell, Toposes and Local Set Theories, vol. 14 of Oxford Logic Guides, The Clarendon Press, Oxford University Press, New York, NY, USA, 1988. · Zbl 0649.18004
[20] M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des Topos et Cohomologie Étale des Schémas. Tome 2, vol. 270 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1972. · Zbl 0234.00007
[21] E. Zafiris, “Quantum observables algebras and abstract differential geometry: the topos-theoretic dynamics of diagrams of commutative algebraic localizations,” International Journal of Theoretical Physics, vol. 46, no. 2, pp. 319-382, 2007. · Zbl 1113.81071 · doi:10.1007/s10773-006-9223-z
[22] A. Mallios and E. E. Rosinger, “Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology,” Acta Applicandae Mathematicae, vol. 55, no. 3, pp. 231-250, 1999. · Zbl 0929.18005 · doi:10.1023/A:1006106718337
[23] A. Mallios and E. E. Rosinger, “Space-time foam dense singularities and de Rham cohomology,” Acta Applicandae Mathematicae, vol. 67, no. 1, pp. 59-89, 2001. · Zbl 1005.46020 · doi:10.1023/A:1010663502915
[24] S. A. Selesnick, “Second quantization, projective modules, and local gauge invariance,” International Journal of Theoretical Physics, vol. 22, no. 1, pp. 29-53, 1983. · Zbl 0518.58024 · doi:10.1007/BF02086896
[25] S. A. Selesnick, “Correspondence principle for the quantum net,” International Journal of Theoretical Physics, vol. 30, no. 10, pp. 1273-1292, 1991. · doi:10.1007/BF01026175
[26] H. Matsumura, Commutative Algebra, W. A. Benjamin, New York, NY, USA, 1970. · Zbl 0211.06501
[27] A. Mallios, “Quantum gravity and “singularities”,” Note di Matematica, vol. 25, no. 2, pp. 57-76, 2005/06.
[28] A. Mallios, “Remarks on “singularities”,” in Progress in Mathematical Physics, F. Columbus, Ed., Nova Science Publishers, Hauppauge, NY, USA, 2003.
[29] E. Zafiris, “Generalized topological covering systems on quantum events’ structures,” Journal of Physics A, vol. 39, no. 6, pp. 1485-1505, 2006. · Zbl 1100.81002 · doi:10.1088/0305-4470/39/6/020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.