Skip to main content
SpringerLink
Log in

Chirality and Symmetry Breaking in a Discrete Internal Space

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In previous papers the permutation group S 4 has been suggested as an ordering scheme for quarks and leptons, and the appearance of this finite symmetry group was taken as indication for the existence of a discrete inner symmetry space underlying elementary particle interactions. Here it is pointed out that a more suitable choice than the tetrahedral group S 4 is the pyritohedral group A 4×Z 2 because its vibrational spectrum exhibits exactly the mass multiplet structure of the 3 fermion generations. Furthermore it is noted that the same structure can also be obtained from a primordial symmetry breaking S 4→A 4. Since A 4 is a chiral group, while S 4 is achiral, an argument can be given why the chirality of the inner pyritohedral symmetry leads to parity violation of the weak interactions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. It seems then natural to assume that not only the internal symmetry is discrete but that physical space is a lattice, too. Although theories with a discrete inner symmetry over a continuous base manifold have been examined [10] they seem to me a bit artificial because they usually lead to domain walls and other discontinuities. Nevertheless, this point may be left open here, because for most arguments in this article it is not essential, whether physical space is discrete or continuous.

  2. The nontrivial groups of order 24 are D 8×Z 3 and Dic 12×Z 2, S 3×K, Dic 24, SL(2,3), Q 8×Z 3, A 4×Z 2, S 4 and a semidirect product of D 8 and Z 3 (in the notation of Ref. [13]).

  3. Note again that we are talking about Bloch waves in internal space while the Bloch waves in physical space reduce to ordinary free fermion fields in the limit of large distances (small spatial lattice constants).

  4. An alternative to spontaneous symmetry breaking is an explicit breaking of symmetries, which could be induced for example by a pseudoscalar chiral interaction among the lattice atoms. Such an interaction is given e.g. by the scalar triple product

    $$ H_{3}= f_3 \sum_{l,l',l'',s,s',s''} \vec{u} (l,s) \bigl[\vec{u}\bigl(l',s'\bigr) \times \vec{u} \bigl(l'',s''\bigr)\bigr] $$
    (29)

    of lattice vectors which is positive for even permutations and negative for odd ones. In the philosophy discussed in the main text, however, Eq. (29) is merely an effective interaction which could arise after the spontaneous symmetry breaking and may be used to describe chiral effects in the low energy regime.

  5. More generally, in SO(d 1,d 2) the spinor dimensions viewed over complex space coincide with the case of the (d 1+d 2)-dimensional Euclidean space.

References

  1. Mohapatra, R.N., Pati, J.C.: Phys. Rev. D 11, 566, 2558 (1975).

    Article  ADS  Google Scholar 

  2. Lampe, B.: Found. Phys. 39, 215 (2009)

    Article  ADS  MATH  Google Scholar 

  3. Lampe, B.: J. Phys. G 34, 1 (2007)

    Article  Google Scholar 

  4. Lampe, B.: arXiv:hep-ph/0610270 (2006). See also arXiv:hep-ph/9810417 (1998)

  5. Lampe, B.: Mod. Phys. Lett. A 23, 2835 (2008)

    Article  ADS  MATH  Google Scholar 

  6. Lampe, B.: arXiv:0805.3762v1 (2008)

  7. Rousseau, D.L., Bauman, R.P., Porto, S.P.S.: J. Raman Spectrosc. 10, 253 (1981)

    Article  ADS  Google Scholar 

  8. Kroumova, E., Aroyo, M.I., Perez Mato, J.M., Kirov, A., Capillas, C., Ivantchev, S., Wondratschek, H.: Phase Transit. 76, 155 (2003)

    Article  Google Scholar 

  9. Johnson, R.C.: Phys. Lett. 114B, 147 (1982)

    ADS  Google Scholar 

  10. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Nucl. Phys. B 241, 333 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Griffith, J.S.: Irreducible Tensor Methods for Molecular Symmetry Groups. Dover, New York (2006)

    Google Scholar 

  12. Besche, H.U., Eick, B., O’Brien, E.A.: Int. J. Algebra Comput. 12, 623 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Besche, H.U., Eick, B.: Electron. Res. Announc. Am. Math. Soc. 7, 1 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Herring, C., Kittel, C.: Phys. Rev. 81, 869 (1951)

    Article  ADS  MATH  Google Scholar 

  15. Dyson, F.J.: Phys. Rev. 102, 1217 (1956)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Alfe, D.: Comput. Phys. Commun. 180, 2622 (2009)

    Article  ADS  Google Scholar 

  17. Stokes, H.T., Boyer, L.L.: (2002). www.physics.byu.edu/~stokesh/isotropy.htm

  18. Lampe, B.: Cent. Eur. J. Phys. 2, 193 (2010)

    Google Scholar 

  19. Hasenfratz, A., Hasenfratz, P., Jansen, K., Kuti, J., Shen, Y.: Nucl. Phys. B 365, 79 (1991)

    Article  ADS  Google Scholar 

  20. Dimopoulos, S., Susskind, L.: Nucl. Phys. B 155, 237 (1979)

    Article  ADS  Google Scholar 

  21. Uehara, S., Yanagida, T.: Phys. Lett. B 165, 94 (1985)

    Article  ADS  Google Scholar 

  22. Amati, D., Barbieri, R., Davis, A.C., Veneziano, G.: Phys. Lett. B 102, 408 (1981)

    Article  ADS  Google Scholar 

  23. Smit, J.: Nucl. Phys. B 175, 307 (1980)

    Article  ADS  Google Scholar 

  24. Maraner, P., Pachos, J.K.: arXiv:0807.0826v2 (2009)

  25. Berruto, F., Grignani, G., Semenoff, G.W., Sodano, P.: Phys. Rev. D 57, 5070 (1998)

    Article  ADS  Google Scholar 

  26. Pickett, G.T., Gross, M., Okuyama, H.: Phys. Rev. Lett. 85, 200 (2000)

    Article  Google Scholar 

  27. Harris, A.B., Kamien, R.D., Lubensky, T.C.: arXiv:cond-mat/9901174v2 (1999)

  28. Cordeiro, C.E., de Mello, E.V., Continentino, M.A.: Z. Phys. B 85, 307 (1991)

    Article  ADS  Google Scholar 

  29. Cochran, W.: Phys. Rev. Lett. 3, 412 (1959)

    Article  ADS  Google Scholar 

  30. Salje, E.K.H., Wruck, B., Thomas, H.: Z. Phys., B Condens. Matter 82, 399 (1991)

    Article  ADS  Google Scholar 

  31. Sollich, P., Heine, V., Dove, M.T.: J. Phys. 6, 3171 (1994)

    Google Scholar 

  32. Massey, W.S.: Am. Math. Mon. 10, 697 (1990)

    MathSciNet  Google Scholar 

  33. Kumar, V.H.S., Suresh, P.K.: arXiv:gr-qc/0605016 (2006)

  34. Cacciatori, S.L., Scotti, A.: arXiv:hep-th/0503106 (2005)

  35. Dove, M.T., Cool, T., Palmer, D.C., Putnis, A., Salje, E.K.H., Winkler, B.: Am. Mineral. 78, 486 (1993)

    Google Scholar 

  36. Dixon, G.M.: Division Algebras. Kluwer Academic, Dordrecht (2009)

    Google Scholar 

  37. Conway, J., Smith, D.: On Octonions and Quaternions. AK Peters, Wellesley (2003)

    MATH  Google Scholar 

  38. Kantor, I.L., Solodovnikov, A.S.: Hypercomplex Numbers—An Elementary Introduction to Algebras. Springer, Berlin (1989)

    MATH  Google Scholar 

  39. Schafer, R.D.: An Introduction to Nonassociative Algebras. Academic Press, New York (1966)

    MATH  Google Scholar 

  40. Ross, G.G.: Grand Unified Theories. Oxford University Press, London (1984)

    Google Scholar 

  41. Ford, K., Wheeler, J.A.: Geons, Black Holes and Quantum Foam. Norton, New York (1995). ISBN 0-393-04642-7

    Google Scholar 

  42. Dzyaloshinsky, I.: J. Phys. Chem. Solids 4, 241 (1958)

    Article  ADS  Google Scholar 

  43. Moriya, T.: Phys. Rev. 120, 91 (1960)

    Article  ADS  Google Scholar 

  44. Zhang, J.J., Yuan, J.H., Zeng, Q.J., Zhang, J.P.: Commun. Theor. Phys. 56, 173 (2011)

    Article  ADS  Google Scholar 

  45. Adler, R.J., Santiago, D.I.: On gravity and the uncertainty principle. Mod. Phys. Lett. A 14, 1371 (1999). doi:10.1142/S0217732399001462

    Article  ADS  Google Scholar 

  46. Wen, X.-G.: arXiv:cond-mat/0107071v2 [cond-mat.str-el] (2001)

  47. Ambjorn, J., Jurkiewicz, J., Loll, R.: arXiv:hep-th/0002050 (2001)

  48. Yamamoto, H.: Phys. Rev. D 30, 1727 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  49. Weinberg, S., Witten, E.: Phys. Lett. B 96, 59 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  50. Holstein, T., Primakoff, H.: Phys. Rev. 58, 1098 (1940)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hans-Maurer-Str. 9, 84435, Lengdorf, Germany

    Bodo Lampe

Authors
  1. Bodo Lampe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bodo Lampe.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lampe, B. Chirality and Symmetry Breaking in a Discrete Internal Space. Int J Theor Phys 51, 3073–3100 (2012). https://doi.org/10.1007/s10773-012-1190-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-012-1190-y

Keywords

Search

Navigation