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Some Quantum Symmetries and Their Breaking II

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Abstract

We consider symmetry breaking in the context of vector bundle theory, which arises quite naturally not only when attempting to “gauge” symmetry groups, but also as a means of localizing those global symmetry breaking effects known as spontaneous. We review such spontaneous symmetry breaking first for a simplified version of the Goldstone scenario for the case of global symmetries, and then in a localized form which is applied to a derivation of some of the phenomena associated with superconduction in both its forms, type I and type II. We then extend these procedures to effect the Higgs mechanism of electroweak theory, and finally we describe an extension to the flavor symmetries of the lightest quarks, including a brief discussion of CP-violation in the neutral kaon system. A largely self-contained primer of vector bundle theory is provided in Sect. 4, which supplies most of the results required thereafter.

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References

  1. Abelev, B.I., et al. (STAR Collaboration): Azimuthal charged-particle correlations and possible local strong parity violation. Phys. Rev. Lett. 103, 251601 (2009). doi:10.1103/PhysRevLett.103.251601

    Article  ADS  Google Scholar 

  2. Abrikosov, A.: Nobel Lecture, available on the website nobelprize.org (2003)

  3. Atiyah, M.F.: K-Theory. Benjamin, New York (1967)

    Google Scholar 

  4. Avis, S.J., Isham, C.J.: Quantum field theory and fibre bundles in a general spacetime. In: Lévy, M., Deser, S. (eds.) Recent Developments in Gravitation, Cargése 1978. Plenum, New York (1978)

    Google Scholar 

  5. Beringer, J., et al. (Particle Data Group): Review of particle physics. Phys. Rev. D 86, 010001 (2012)

    Article  ADS  Google Scholar 

  6. Burgess, C.P.: Goldstone and pseudo-Goldstone bosons in nuclear, particle and condensed-matter physics. In: Effective Theories in Matter, Nuclear Physics Summer School and Symposium, Seoul National University, Korea (1999). arXiv:hep-th/9808176v3

  7. Cartier, P.: Quantum mechanical commutation relations and theta functions. In: Borel, A., Mostow, G.D. (eds.) Proceedings of Symposia in Pure Mathematics Volume IX: Algebraic Groups and Discontinuous Subgroups. American Mathematical Society, Providence (1966)

    Google Scholar 

  8. Derdzinski, A.: Geometry of the Standard Model of Elementary Particles. Springer, Berlin (1992)

    MATH  Google Scholar 

  9. Geroch, R.P.: Spinor structure of space-times in General Relativity. I. J. Math. Phys. 9, 1739 (1968)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Geroch, R.P.: Spinor structure of space-times in General Relativity. II. J. Math. Phys. 11, 343 (1970)

    Article  ADS  MATH  Google Scholar 

  11. Grothendieck, A.: A General Theory of Fibre Spaces with Structure Sheaf. University of Kansas, Lawrence (1955) (second edn. 1958)

    Google Scholar 

  12. Hirzebruch, F.: Topological Methods in Algebraic Geometry. Springer, Berlin (1965)

    Google Scholar 

  13. Husemoller, D.: Fibre Bundles. McGraw–Hill, New York (1966). Second edition, Springer, Berlin (1975)

    MATH  Google Scholar 

  14. Jost, J.: Nonlinear Methods in Riemannian and Kählerian Geometry. Birkhäuser, Boston (1988)

    Book  MATH  Google Scholar 

  15. Karoubi, M.: K-Theory. An Introduction. Springer, Berlin (1978)

    MATH  Google Scholar 

  16. Kharzeev, D.E., McLerran, L.D., Warringa, H.J.: The effects of topological charge change in heavy ion collisions: “Event by event P and CP violation” (2007). arXiv:0711.0950v1 [hep-ph]

  17. Mallios, A.: Geometry of Vector Sheaves I, II. Kluwer Academic, Dordrecht (1998)

    Book  Google Scholar 

  18. Mallios, A.: Modern Differential Geometry in Gauge Theories I, II. Birkhäuser, Boston (2006)

    Google Scholar 

  19. Nestruev, J.: Smooth Manifolds and Observables. Springer, Berlin (2003)

    MATH  Google Scholar 

  20. Selesnick, S.A.: Rank one projective modules over certain Fourier algebras. In: Fourman, M.P., Mulvey, C.J., Scott, D.S. (eds.) Applications of Sheaves. Lecture Notes in Mathematics, vol. 753. Springer, Berlin (1979)

    Chapter  Google Scholar 

  21. Selesnick, S.A.: Second quantization, projective modules, and local gauge invariance. Int. J. Theor. Phys. 22(1), 29 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. Selesnick, S.A.: CP violation in the Cabibbo-rotated nonlinear σ-model. Nuovo Cimento A 101, 249 (1989)

    Article  ADS  Google Scholar 

  23. Selesnick, S.A.: Dirac’s equation on the quantum net. J. Math. Phys. 35(8), 3936 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Selesnick, S.A.: Quanta, Logic and Spacetime, 2nd edn. World Scientific, Singapore (2003)

    Book  MATH  Google Scholar 

  25. Selesnick, S.A.: Foundation for quantum computing II. Int. J. Theor. Phys. (2006). doi:10.1007/s10773-006-9254-5

    Google Scholar 

  26. Selesnick, S.A.: Some quantum symmetries and their breaking I. Int. J. Theor. Phys. 51(3), 871 (2012). doi:10.1007/s10773-011-0964-y

    Article  MathSciNet  MATH  Google Scholar 

  27. Tinkham, M.: Introduction to Superconductivity, 2nd edn. Dover, Mineola (1996)

    Google Scholar 

  28. Vaisman, I.: Cohomology and Differential Forms. Marcel Dekker, New York (1973)

    MATH  Google Scholar 

  29. Warner, F.: Foundations of Differentiable Manifolds and Lie Groups. Springer, Berlin (2010)

    Google Scholar 

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Acknowledgements

Thanks are owed to Cliff Burgess, Dmitri Kharzeev and Ivan Selesnick for help in the preparation of this paper.

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  1. Physics and Astronomy, University of Missouri - St. Louis, St. Louis, MO, 63121, USA

    S. A. Selesnick

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  1. S. A. Selesnick
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Selesnick, S.A. Some Quantum Symmetries and Their Breaking II. Int J Theor Phys 52, 1088–1121 (2013). https://doi.org/10.1007/s10773-012-1423-0

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