Advertisement

Foundations of Physics

, Volume 40, Issue 11, pp 1681–1699 | Cite as

Spin Path Integrals and Generations

  • Carl BrannenEmail author
Article

Abstract

The spin of a free electron is stable but its position is not. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have shown that the Feynman position path integral can be mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). Since the more common use of MUBs is in finite dimensional Hilbert spaces, this raises the question “what happens when spin path integrals are computed over products of MUBs?” Such an assumption makes spin no longer stable. We show that the usual spin-1/2 is obtained in the long-time limit in three orthogonal solutions that we associate with the three elementary particle generations. We give applications to the masses of the elementary leptons.

Keywords

Feynman path integral Mutually unbiased bases Quantum information theory Spin Elementary particles Generations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Zee, A.: Quantum Field Theory in a Nutshell. Princeton University Press, Princeton (2003) zbMATHGoogle Scholar
  2. 2.
    Svetlichny, G.: Feynman’s integral is about mutually unbiased bases. In: Proc. 7th Intl. Conf. Symm. Nonlin. Phys., p. 032, June 2008. arXiv:0708.3079 [quant-ph]
  3. 3.
    Tolar, J., Chadzitaskos, G.: Feynman’s path integral and mutually unbiased bases. J. Phys. A 24, 245306 (2009). arXiv:0904.0886 [quant-ph] CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Messiah, A.: Quantum Mechanics. Dover Publications, New York (1999) Google Scholar
  5. 5.
    Roe, B.P.: Particle Physics at the New Millenium. Springer, Berlin (1996) zbMATHGoogle Scholar
  6. 6.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics (Non-relativistic Theory). Butterworth-Heinemann, Stoneham-London (2002) Google Scholar
  7. 7.
    Dechoum, K., França, H.M., Malta, C.P.: Classical aspects of the Pauli-Schrödinger equation. Phys. Lett. A 248, 93–102 (1998). http://dx.doi.org/10.1016/S0375-9601(98)00682-3 zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    França, H.M.: The Stern-Gerlach phenomenon according to classical electrodynamics. Found. Phys. 39, 1177–1190 (2009). http://www.springerlink.com/content/9777268758078027/ zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Foster, B.Z., Jacobson, T.: Propagating spinors on a tetrahedral spacetime lattice (2003). hep-th/0310166v2
  10. 10.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965) zbMATHGoogle Scholar
  11. 11.
    Berry, M.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–47 (1984). http://rspa.royalsocietypublishing.org/content/392/1802/45.abstract zbMATHCrossRefADSGoogle Scholar
  12. 12.
    Pancharatnam, S.: Generalized theory of interference, and its applications. Part I. Coherent pencils. Proc. Indian. Acad. Sci. 44, 247–262 (1956) MathSciNetGoogle Scholar
  13. 13.
    Bhandari, R.: On geometric phase from pure projections. J. Mod. Opt. 45(10), 2187–2195 (1998). physics/9810045 ADSGoogle Scholar
  14. 14.
    Regge, T.: Introduction to complex orbital momenta. Il Nuovo Cimento 14, 951–976 (1959). http://www.springerlink.com/content/l787r567r01743x4/ zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rosen, G.: Heuristic development of a Dirac-Goldhaber model for lepton and quark structure. Mod. Phys. Lett. A 22, 283–288 (2007). http://www.worldscinet.com/mpla/22/2204/S0217732307022621.html CrossRefADSGoogle Scholar
  16. 16.
    Koide, Y.: Fermion-boson two body model of quarks and leptons and Cabibbo mixing. Lett. Nuovo Cimento 34, 201 (1982). http://www.springerlink.com/content/41u50161nj02n875/?p=892b40a1ed424866be9db40fb80e2c5c&pi=2 CrossRefGoogle Scholar
  17. 17.
    Koide, Y.: A fermion-boson composite model of quarks and leptons. Phys Lett. B 120, 161–165 (1983). http://linkinghub.elsevier.com/retrieve/pii/0370269383906445 CrossRefADSGoogle Scholar
  18. 18.
    Brannen, C.: The lepton masses (2006). http://www.brannenworks.com/MASSES2.pdf
  19. 19.
    Martin, R.: Results from the neutral current detector phase of the Sudbury Neutrino Observatory (2009). arXiv:0905.4907 [hep-ex]
  20. 20.
    Diwan, M.V.: Recent results from the MINOS experiment (2009). arXiv:0904.3706 [hep-ex]
  21. 21.
    Weinberg, S.: Foundations. The Quantum Theory of Fields, vol. 1. Cambridge University Press, Cambridge (1995) Google Scholar
  22. 22.
    Motl, L.: An analytical computation of asymptotic Schwarzschild quasinormal frequencies. Adv. Theor. Math. Phys. 6, 1135–1162 (2003). gr-qc/0212096 MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.RedmondUSA

Personalised recommendations