On a Microscopic Representation of Space-Time III

  • Rolf DahmEmail author
Part of the following topical collections:
  1. Homage to Prof. W.A. Rodrigues Jr


Using the Dirac (Clifford) algebra \(\gamma ^{\mu }\) as initial stage of our discussion, we summarize previous work with respect to the isomorphic 15 dimensional Lie algebra su*(4) as complex embedding of sl(2,\(\mathbb {H}\)), the relation to the compact group SU(4) as well as subgroups and group chains. The main subject, however, is to relate these technical procedures to the geometrical (and physical) background which we see in projective and especially in line geometry of \(\mathbb {R}^{3}\). This line geometrical description, however, leads to applications and identifications of line Complexe and the discussion of technicalities versus identifications of classical line geometrical concepts, Dirac’s ‘square root of \(p^{2}\)’, the discussion of dynamics and the association of physical concepts like electromagnetism and relativity. We outline a generalizable framework and concept, and we close with a short summary and outlook.


Relativity Unification Quantum field theory Dirac theory Clifford algebra Geometry Projective geometry Line geometry Line Complex Complex geometry Congruences 

Mathematics Subject Classification

Primary 83E99 Secondary 14N99 



  1. 1.
    Bjorken, J.D., Drell, S.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1965)zbMATHGoogle Scholar
  2. 2.
    Blaschke, W.: Vorlesungen über Differentialgeometrie III (Die Grundlehren der Mathematischen Wissenschaft XXIX). Springer, Berlin (1928)Google Scholar
  3. 3.
    Clebsch, A.: Zum Gedächtniss an Julius Plücker. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Vol. 16 (1871)Google Scholar
  4. 4.
    Clebsch, A.: Ueber die Plückerschen Complexe. Math. Ann. 2, 1 (1869)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dahm, R.: On A Microscopic Representation of Space-Time II. In: Proceedings of ICCA 9, Weimar, 2011; to be published (2011)Google Scholar
  6. 6.
    Dahm, R.: On a microscopic representation of space-time VIII -On Relativity. Yad. Fis. Phys. Atom. Nuclei (2018) (accepted for publication)Google Scholar
  7. 7.
    Dahm, R.: Projective Geometry, Conformal and Lie Symmetries and Their Breakdown to a su(2). QTS 7, Prague (2011)Google Scholar
  8. 8.
    Dahm, R.: On a microscopic representation of space-time. Yad. Fis. 75, 1244 (2012)Google Scholar
  9. 9.
    Dahm, R.: On a microscopic representation of space-time. Phys. Atom. Nuclei 75, 1173 (2012)ADSCrossRefGoogle Scholar
  10. 10.
    Dahm, R.: Some Remarks on Rank-3 Lie Algebras in Physics. GOL X, Tallinn (2013)Google Scholar
  11. 11.
    Dahm, R.: On a microscopic representation of space-time VII-On Spin. J. Phys. Conf. Ser. 965, 012012 (2018)CrossRefGoogle Scholar
  12. 12.
    Ehlers, J., Pirani, F., Schild, A.: The geometry of free fall and light propagation. In: O’Raifeartaigh, L. (ed.) Studies in Relativity, pp. 63–84. Clarendon Press, Oxford (1972)Google Scholar
  13. 13.
    Enriques, F.: Vorlesungen über Projektive Geometrie. B. G. Teubner, Leipzig (1903)zbMATHGoogle Scholar
  14. 14.
    Gilmore, R.: Lie Groups, Lie Algebras and Some of Their Applications. Wiley, New York (1974)CrossRefGoogle Scholar
  15. 15.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, San Diego (1978)zbMATHGoogle Scholar
  16. 16.
    Jackson, J.D.: Klassische Elektrodynamik (1975, German Translation), 2nd edn. de Gruyter, Berlin (1983)Google Scholar
  17. 17.
    Kähler, E.: Innerer und Äusserer Differentialkalkül. Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin. Akademie-Verlag, Berlin (1960)Google Scholar
  18. 18.
    Klein, F.: Ueber die Transformation der Allgemeinen Gleichung des Zweiten Grades Zwischen Linien–Koordinaten auf eine Kanonische Form. Inauguraldissertation. Georgi, Bonn (1868)Google Scholar
  19. 19.
    Klein, F.: Ueber Liniengeometrie und metrische Geometrie. Math. Ann. 5, 257 (1872)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Klein, F.: Ueber gewisse in der Liniengeometrie auftretende Differentialgleichungen. Math. Ann. 5, 278 (1872)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Klein, F.: Zur Interpretation der complexen Elemente in der Geometrie. Göttinger Nachrichten 1872, 373 (1872)zbMATHGoogle Scholar
  22. 22.
    Klein, F.: Vorlesungen über nicht-euklidische Geometrie (Die Grundlehren der Mathematischen Wissenschaft XXVI). Springer, Berlin (1928)Google Scholar
  23. 23.
    Lurié, D.: Particles and Fields. Interscience Publishers, New York (1968)Google Scholar
  24. 24.
    Percacci, R.: Geometry of Nonlinear Field Theories. World Scientific, Singapore (1986)CrossRefGoogle Scholar
  25. 25.
    Plücker, J.: Neue Geometrie des Raumes. Clebsch, A., Klein, F. (eds). B. G. Teubner, Leipzig (1868/1869)Google Scholar
  26. 26.
    Plücker, J.: Discussion de la forme générale des ondes lumineuses. J. Reine Angew. Math. 19, 1 (1838)MathSciNetGoogle Scholar
  27. 27.
    Plücker, J.: On a new geometry of space. Proc. R. Soc. Lond. 14, 58 (1865)Google Scholar
  28. 28.
    Plücker, J.: Fundamental views regarding mechanics. Philos. Trans. R. Soc. Lond. 156, 361 (1866)ADSCrossRefGoogle Scholar
  29. 29.
    Reye, T.: Die Geometrie der Lage. Rümpler, Hannover (1866/1868)Google Scholar
  30. 30.
    Reye, T.: Synthetische Geometrie der Kugeln und linearen Kugelsysteme. Teubner, Leipzig (1879)zbMATHGoogle Scholar
  31. 31.
    Smilga, W.: Reverse Engineering Approach to Quantum Electrodynamics. arXiv:1004.0820 [physics.gen-ph] (2011)
  32. 32.
    Study, E.: Geometrie der Dynamen. Teubner, Leipzig (1903)zbMATHGoogle Scholar
  33. 33.
    von Staudt, G.K.C.: Beiträge zur Geometrie der Lage. Bauer und Raspe, Nürnberg (1856)Google Scholar
  34. 34.
    Weyl, H.: Raum-Zeit-Materie, 1970, 6th edn. Springer, Berlin (1918)zbMATHGoogle Scholar
  35. 35.
    Wiegand, R.: Ein streitbarer Gelehrter im 19. Jahrhundert. University Bonn, Pressestelle. Pressemitteilungen 21.5.2008 (2008)
  36. 36.
    Zindler, K.: Liniengeometrie mit Anwendungen. (Sammlung Schubert XXXIV/LI). G. J. Göschensche Verlagshandlung, Leipzig (1902/1906)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.beratung für ISMainzGermany

Personalised recommendations