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A Dodecalogue of Basic Didactics from Applications of Abstract Differential Geometry to Quantum Gravity

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Abstract

We summarize the twelve most important in our view novel concepts that have arisen, based on results that have been obtained, from various applications of Abstract Differential Geometry (ADG) to Quantum Gravity (QG). The present document may be used as a concise, yet informal, discursive and peripatetic conceptual guide-cum-terminological glossary to the voluminous technical research literature on the subject. In a bonus section at the end, we dwell on the significance of introducing new conceptual terminology in future QG research by means of ‘poetic language’.

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References

  1. Auyang, S.Y.: How is Quantum Field Theory Possible? Oxford University Press, New York (1995)

    Google Scholar 

  2. Clark, P.: Philosophy of Science Today. Clarendon, Oxford (2003)

    Google Scholar 

  3. Dirac, P.A.M.: Quantized singularities in the electromagnetic field. Proc. Roy. Soc. London A 133, 60 (1931)

    Article  MATH  ADS  Google Scholar 

  4. Einstein, A.: The Meaning of Relativity, 5th edn. Princeton University Press, Princeton (1956)

    Google Scholar 

  5. Einstein, A.: Math. Intell. 12(2), 31 (1990)

    MathSciNet  Google Scholar 

  6. Faddeev, L.D.: Modern mathematical physics: what it should be. In: Fokas, A., Grigoryan, A., Kibble, T., Zegarlinski, B. (eds.) Mathematical Physics 2000, Imperial College Press, London (2000)

    Google Scholar 

  7. Feynman, R.P.: The Character of Physical Law. Penguin Books, London (1992)

    Google Scholar 

  8. Feynman, R.P.: Feynman lectures on gravitation. In: Morinigo, F.B., Wagner, W.G., Hatfield, B. (eds.) Notes, Penguin Books, London (1999)

    Google Scholar 

  9. Finkelstein, D.R.: Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg. Springer, Berlin (1996)

    MATH  Google Scholar 

  10. Isham, C.J.: Canonical groups and the quantization of geometry and topology, In: Ashtekar, A., Stachel, J. (eds.) Conceptual Problems of Quantum Gravity, Birkhäuser, Basel (1991)

    Google Scholar 

  11. Kandinsky, W.: Watercolors, Drawings, Writings (Cassour, J., added author). Primo Tempo (1961)

  12. Kazantzakis, N.: Salvatores Dei: Spiritual Exercises. Simon & Schuster, New York (1960), translated by Kimon Friar, a Touchstone Book

    Google Scholar 

  13. Leibniz, G.W.: Discourse on Metaphysics and the Monadology. Great Books in Philosophy Series. Prometheus Books, Amherst (1992), translated by Montgomery, G.R.

    Google Scholar 

  14. Mallios, A.: Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry, vols. 1–2. Kluwer Academic, Dordrecht (1998). There is also a Russian translation of this 2-volume book by MIR Publishers, Moscow (vol. 1, 2000 and vol. 2, 2001)

    Google Scholar 

  15. Mallios, A.: On an axiomatic treatment of differential geometry via vector sheaves. Appl. Math. Jpn. (Int. Plaza) 48, 93 (1998) (invited paper)

    MATH  MathSciNet  Google Scholar 

  16. Mallios, A.: On an axiomatic approach to geometric prequantization: a classification scheme à la Kostant–Souriau–Kirillov. J. Math. Sci. (N.Y.) 95, 2648 (1999) (invited paper)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mallios, A.: Abstract differential geometry, general relativity and singularities. In: Abe, J.M., Tanaka, S. (eds.) Unsolved Problems in Mathematics for the 21st Century: A Tribute to Kiyoshi Iséki’s 80th Birthday, p. 77. IOS, Amsterdam (2001) (invited paper)

    Google Scholar 

  18. Mallios, A.: Modern Differential Geometry in Gauge Theories, vol. I: Maxwell Fields. Birkhäuser, Boston (2005)

    MATH  Google Scholar 

  19. Mallios, A.: Modern Differential Geometry in Gauge Theories, vol. II: Yang–Mills Fields. Birkhäuser, Boston (2006)

    Google Scholar 

  20. Mallios, A.: K-theory of topological algebras and second quantization. Acta Universitatis Ouluensis–Scientiae Rerum Naturalium, A 408, 145 (2004), math-ph/0207035

    MathSciNet  Google Scholar 

  21. Mallios, A.: Abstract differential geometry, singularities and physical applications. In: Strantzalos, P., Fragoulopoulou, M. (eds.) Topological Algebras with Applications to Differential Geometry and Mathematical Physics. Proceedings of a Fest-Colloquium in Honour of Professor Anastasios Mallios, 16–18 September 1999. University of Athens Publications, Athens (2002)

  22. Mallios, A.: Quantum gravity and “singularities”. Note Math. 25, 57 (2006) (invited paper), physics/0405111

    MATH  MathSciNet  Google Scholar 

  23. Mallios, A.: Geometry and physics today. In: Raptis, I. (ed.) Special Proceedings issue for Glafka–2004: Iconoclastic Approaches to Quantum Gravity, Int. J. Theor. Phys. (2006) (invited paper), gr-qc/0506088

  24. Mallios, A.: Remarks on “singularities”. gr-qc/0202028 (2002)

  25. Mallios, A., Raptis, I.: Finitary spacetime sheaves of quantum causal sets: curving quantum causality. Int. J. Theor. Phys. 40, 1885 (2001), gr-qc/0102097

    Article  MATH  MathSciNet  Google Scholar 

  26. Mallios, A., Raptis, I.: Finitary Čech–de Rham cohomology. Int. J. Theor. Phys. 41, 1857 (2002), gr-qc/0110033

    Article  MATH  MathSciNet  Google Scholar 

  27. Mallios, A., Raptis, I.: Finitary, causal and quantal vacuum Einstein gravity. Int. J. Theor. Phys. 42, 1479 (2003), gr-qc/0209048

    Article  MATH  MathSciNet  Google Scholar 

  28. Mallios, A., Raptis, I.: \(\mathcal {C}^{\infty}\) -smooth singularities exposed: chimeras of the differential spacetime manifold. Research monograph (2005, in preparation), gr-qc/0411121

  29. Mallios, A., Rosinger, E.E.: Abstract differential geometry, differential algebras of generalized functions and de Rham cohomology. Acta Appl. Math. 55, 231 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mallios, A., Rosinger, E.E.: Space-time foam dense singularities and de Rham cohomology. Acta Appl. Math. 67, 59 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mallios, A., Rosinger, E.E.: Dense singularities and de Rham cohomology. In: Strantzalos, P., Fragoulopoulou, M. (eds.) Topological Algebras with Applications to Differential Geometry and Mathematical Physics, Proceedings of a Fest-Colloquium in Honour of Professor Anastasios Mallios, 16–18 September1999. Department of Mathematics, University of Athens Publications, Athens (2002)

  32. Manin, Y.I.: Mathematics and Physics. Birkhäuser, Boston (1980)

    Google Scholar 

  33. Midgley, M.: Science as Salvation: A Modern Myth and its Meaning. Routledge, London (2004)

    Google Scholar 

  34. Nehamas, A.: Nietzsche: Life as Literature. Harvard University Press, Harvard (1985)

    Google Scholar 

  35. Plotnitsky, A.: Complementarity: anti-epistemology after Bohr and Derrida. Duke University Press, Dusham (1994)

    Google Scholar 

  36. Raptis, I.: Algebraic quantization of causal sets. Int. J. Theor. Phys. 39, 1233 (2000), gr-qc/9906103

    Article  MATH  MathSciNet  Google Scholar 

  37. Raptis, I.: Finitary spacetime sheaves. Int. J. Theor. Phys. 39, 1703 (2000), gr-qc/0102108

    Article  MATH  MathSciNet  Google Scholar 

  38. Raptis, I.: Finitary-algebraic ‘resolution’ of the inner Schwarzschild singularity. Int. J. Theor. Phys. 45, 85 (2006), gr-qc/0408045

    MATH  MathSciNet  Google Scholar 

  39. Raptis, I.: Finitary topos for locally finite, causal and quantal vacuum Einstein gravity. Int. J. Theor. Phys. (2007, to appear), gr-qc/0507100

  40. Raptis, I.: Categorical quantum gravity. Int. J. Theor. Phys. 45, 1383 (2006), gr-qc/0509089

    Article  MATH  MathSciNet  Google Scholar 

  41. Raptis, I.: ‘Third’ quantization of vacuum Einstein gravity and free yang–mills theories. Int. J. Theor. Phys. (2007, to appear), gr-qc/0606021

  42. Raptis, I., Zapatrin, R.R.: Quantization of discretized spacetimes and the correspondence principle. Int. J. Theor. Phys. 39, 1 (2000), gr-qc/9904079

    Article  MATH  MathSciNet  Google Scholar 

  43. Raptis, I., Zapatrin, R.R.: Algebraic description of spacetime foam. Class. Quantum Gravity 20, 4187 (2001), gr-qc/0102048

    Article  ADS  MathSciNet  Google Scholar 

  44. Schopenhauer, A.: Essays and Aphorisms. Penguin, London (1970)

    Google Scholar 

  45. Sklar, L.: In: Sklar, L. (ed.) The Philosophy of Science: A Collection of Essays, vols. 1–5. Garland, New York (2000)

    Google Scholar 

  46. Stachel, J.J.: The other Einstein: Einstein contra field theory. In: Beller, M., Cohen, R.S., Renn, J. (eds.) Einstein in Context, Cambridge University Press, Cambridge (1993)

    Google Scholar 

  47. Stevens, W.: Adagia (included in Opus Posthumous). Vintage Books, New York (1990)

    Google Scholar 

  48. ’t Hooft, G.: Obstacles on the way towards the quantization of space, time and matter, ITP-University of Utrecht, pre-print SPIN-2000/20 (2001)

  49. Tolstoy, I.: The Knowledge and the Power: Reflections on the History of Science. Canongate, Edinburgh (1990)

    Google Scholar 

  50. Wheatley, M.J., Kellner-Rogers, M.: A Simpler Way, 1st edn. Berrett–Koehler, San Francisco (1996)

    Google Scholar 

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Authors and Affiliations

  1. Department of Education and Professional Studies, Franklin-Wilkins Building, Waterloo Road, London, SE1 9NN, UK

    Ioannis Raptis

  2. Theoretical Physics Group, The Blackett Laboratory, Imperial College, Prince Consort Road, South Kensington, London, SW7 2BZ, UK

    Ioannis Raptis

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  1. Ioannis Raptis
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Raptis, I. A Dodecalogue of Basic Didactics from Applications of Abstract Differential Geometry to Quantum Gravity. Int J Theor Phys 46, 3009–3021 (2007). https://doi.org/10.1007/s10773-007-9416-0

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