Communications in Mathematical Physics

, Volume 298, Issue 2, pp 419–436 | Cite as

There is No “Theory of Everything” Inside E8

Article

Abstract

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any “Theory of Everything” obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin’s classic papers and are written for mathematicians.

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References

  1. 1.
    Lisi, A.G.: An exceptionally simple theory of everything. http://arxiv.org/abs/0711.0770 V1 [hep-th], 2007
  2. 2.
    Borel, A.: Linear Algebraic Groups. Vol. 126 of Graduate Texts in Mathematics, 2nd ed., New York: Springer-Verlag, 1991Google Scholar
  3. 3.
    Serre, J.-P.: Local Fields. Vol. 67 of Graduate Texts in Mathematics. New York-Berlin: Springer, 1979Google Scholar
  4. 4.
    Tits J.: Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque. J. Reine Angew. Math. 247, 196–220 (1971)MATHMathSciNetGoogle Scholar
  5. 5.
    Georgi H., Glashow S.: Unity of all elementary particle forces. Phys. Rev. Lett. 32, 438–441 (1974)CrossRefADSGoogle Scholar
  6. 6.
    Pati J., Salam A.: Lepton number as the fourth color. Phys. Rev. D10, 275–289 (1974)ADSGoogle Scholar
  7. 7.
    Distler, J.: Superconnections for dummies. Weblog entry, May 12, 2008, available at http://golem.ph.utexas.edu/~distler/blog/archives/001680.html
  8. 8.
    Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. Princeton Landmarks in Mathematics and Physics, Princeton, NJ: Princeton University Press, 2000Google Scholar
  9. 9.
    Weinberg, S.: The Quantum Theory of Fields. 3 volumes. Cambridge: Cambridge University Press, 1995-2000Google Scholar
  10. 10.
    ’t Hooft, G.: In: Recent Developments in Gauge Theories, Cargèse France, ’t Hooft, G., Itzykson, C., Jaffe, A., Lehmann, H., Mitter, P.K., Singer, I., Stora, R., eds., no. 59 in Proceedings of the Nato Advanced Study Series B, London: Plenum Press, 1980Google Scholar
  11. 11.
    Lisi, A.G.: First person: A. Garrett Lisi, Financial Times, (April 25, 2009), available at http://www.ft.com/cms/s/2/ebead98a-2d71-11de-9eba-00144feabdc0.html
  12. 12.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. vol. 34 of Graduate Studies in Mathematics, Providence, RI: Amer. Math. Soc., 2001Google Scholar
  13. 13.
    Serre J.-P.: On the values of the characters of compact Lie groups. Oberwolfach Reports 1(1), 666–667 (2004)Google Scholar
  14. 14.
    Elashvili, A., Kac, V., Vinberg, E.: On exceptional nilpotents in semisimple Lie algebras. http://arxiv.org/abs/0812.1571, V1 [math.GR], 2008
  15. 15.
    Dynkin, E.: Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc. Transl. (2) 6, 111–244, (1957) [Russian original: Mat. Sbornik N.S. 30(72), 349–462, (1952)]Google Scholar
  16. 16.
    Bourbaki, N.: Lie Groups and Lie Algebras: Chapters 7–9. Berlin: Springer-Verlag, 2005Google Scholar
  17. 17.
    Gross B., Nebe G.: Globally maximal arithmetic groups. J. Algebra 272(2), 625–642 (2004)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Humphreys, J.: Introduction to Lie Algebras and Representation Theory. Vol. 9 of Graduate Texts in Mathematics. Third printing, revised Berlin-Heidelberg-New York: Springer-Verlag, 1980Google Scholar
  19. 19.
    Collingwood D., McGovern W.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrant Reinhold, New York (1993)MATHGoogle Scholar
  20. 20.
    Onishchik, A., Vinberg, E.: Lie Groups and Lie Algebras III. Vol. 41 of Encyclopaedia Math. Sci. Berlin-Heidelberg-New York: Springer, 1994Google Scholar
  21. 21.
    Carter R.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley-Interscience, New York (1985)MATHGoogle Scholar
  22. 22.
    Bourbaki, N. Lie Groups and Lie Algebras: Chapters 4–6, Berlin: Springer-Verlag, 2002Google Scholar
  23. 23.
    Vavilov N.: Do it yourself structure constants for Lie algebras of type E . J. Math. Sci. (N.Y.) 120(4), 1513–1548 (2004)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Èlašvili A.: Centralizers of nilpotent elements in semisimple Lie algebras. Sakharth. SSR Mecn. Akad. Math. Inst. Šrom. 46, 109–132 (1975)Google Scholar
  25. 25.
    McKay, W., Patera, J.: Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras. Vol. 69 of Lecture Notes in Pure and Applied Mathematics. New York: Marcel Dekker Inc., 1981Google Scholar
  26. 26.
    Springer, T., Steinberg, R.: Conjugacy classes. In: Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Vol. 131 of Lecture Notes in Math., Berlin: Springer, 1970, pp. 167–266Google Scholar
  27. 27.
    Steinberg R.: Lectures on Chevalley Groups. Yale University, New Haven, CT (1968)Google Scholar
  28. 28.
    Tits, J.: Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen. Vol. 40 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1967Google Scholar
  29. 29.
    Garibaldi R.: Clifford algebras of hyperbolic involutions. Math. Zeit. 236, 321–349 (2001)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Platonov V., Rapinchuk A.: Algebraic Groups and Number Theory. Academic Press, Boston, MA (1994)MATHGoogle Scholar
  31. 31.
    Grisaru M.T., Pendleton H.N.: Soft spin 3/2 fermions require gravity and supersymmetry. Phys. Lett. B67, 323 (1977)ADSGoogle Scholar
  32. 32.
    Schon J., Weidner M.: Gauged N=4 supergravities. JHEP 05, 034 (2006)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Theory Group, Department of Physics, and Texas Cosmology CenterUniversity of TexasAustinUSA
  2. 2.Department of Mathematics & Computer Science, 400 Dowman Dr.Emory UniversityAtlantaUSA

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