Skip to main content
SpringerLink
Log in

Isotopic Liftings of Clifford Algebras and Applications in Elementary Particle Mass Matrices

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

Abstract

Isotopic liftings of algebraic structures are investigated in the context of Clifford algebras, where it is defined a new product involving an arbitrary, but fixed, element of the Clifford algebra. This element acts as the unit with respect to the introduced product, and is called isounit. We construct isotopies in both associative and non-associative arbitrary algebras, and examples of these constructions are exhibited using Clifford algebras, which although associative, can generate the octonionic, non-associative, algebra. The whole formalism is developed in a Clifford algebraic arena, giving also the necessary pre-requisites to introduce isotopies of the exterior algebra. The flavor hadronic symmetry of the six u,d,s,c,b,t quarks is shown to be exact, when the generators of the isotopic Lie algebra \(\mathfrak {su}(6)\) are constructed, and the unit of the isotopic Clifford algebra is shown to be a function of the six quark masses. The limits constraining the parameters, that are entries of the representation of the isounit in the isotopic group SU(6), are based on the most recent limits imposed on quark masses.

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

References

  1. Hestenes, D., Sobczik, G.: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht (1984)

    MATH  Google Scholar 

  2. Kadeisvili, J.V.: An introduction to the Lie–Santilli isotopic theory. Math. Methods Appl. Sci. 19, 1349–1395 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Santilli, R.M.: Isonumber and genonumbers of dimension 1, 2, 4, 8, their isoduals and pseudoduals, and hidden numbers of dimension 3, 5, 6, 7. Algebras Groups Geom. 10, 273–321 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Santilli, R.M.: Nonlocal integral isotopies of differential calculus, geometries and mechanics. Rend. Circ. Mat. Palermo Suppl. 42, 7–82 (1996)

    MathSciNet  Google Scholar 

  5. Santilli, R.M.: Elements of Hadronic Mechanics, Vol. I: Mathematical Foundations. Ukrainian Academy of Sciences, Kiev (1993)

    Google Scholar 

  6. Santilli, R.M.: Elements of Hadronic Mechanics, Vol. II: Theoretical Foundations. Ukrainian Academy of Sciences, Kiev (1993)

    Google Scholar 

  7. Santilli, R.M.: Elements of Hadronic Mechanics, Vol. III: Recent Advances, Experimental Verifications and Industrial Applications. Ukrainian Academy of Sciences, Kiev (in press). Preliminary version available in pdf format at http://www.i-b-r.org/Hadronic-Mechanics.htm

  8. Kadeisvili, J.V.: Santilli’s Isotopies of Contemporary Algebras, Geometries and Relativities, 2nd edn. Ukraine Academy of Sciences, Kiev (1997)

    MATH  Google Scholar 

  9. Santilli, R.M.: Lie-admissible invariant representation of irreversibility for matter and antimatter at the classical and operator levels. Nuovo Cimento B 121, 443–498 (2006)

    MathSciNet  Google Scholar 

  10. Santilli, R.M.: Isotopic breaking of gauge theories. Phys. Rev. D 20, 555–570 (1979)

    Article  ADS  Google Scholar 

  11. Kadeisvili, J.V., Kamiya, N., Santilli, R.M.: A characterization of isofields and their isoduals. Hadronic J. 16, 169–187 (1993)

    MATH  MathSciNet  Google Scholar 

  12. Klimyk, S.U., Santilli, R.M.: Standard isorepresentations of isotopic Q-operator deformations of Lie algebras. Algebras, Groups Geom. 10, 323–332 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Santilli, R.M.: Isotopic lifting of quark theories with exact confinement and convergent perturbative expansions. Commun. Theor. Phys. 4, 1–23 (1995)

    Google Scholar 

  14. Alvarez-Gaumé, L., et al. (eds.): Review of particle physics. Phys. Lett. B 592, 1–1109 (2006)

    Google Scholar 

  15. Lounesto, P.: Octonions and triality. Adv. Appl. Clifford Algebras 11(2), 191–213 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Baylis, W.: The paravector model of spacetime. In: Clifford (Geometric) Algebras with Applications in Physics, Mathematics and Engineering. Birkhäuser, Berlin (1995)

    Google Scholar 

  17. Santilli, R.M.: Isotopic lifting of SU(2) symmetry with application to nuclear physics. JINR (Joint Institute for Nuclear Research) Rapid Comm. 6, 24–38 (1993)

    Google Scholar 

  18. Santilli, R.M.: Isotopic lifting of quark theories. Int. J. Phys. 1, 1–26 (1995)

    Google Scholar 

  19. Santilli, R.M.: Isorepresentations of the Lie-isotopic SU(2) algebra with applications to nuclear physics and to local realism. Acta Appl. Math. 50, 177–190 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Animalu, A.O.E.: A gauge-invariant relativistic theory of the Rutherford–Santilli neutron. Hadronic J. 27(5), 99–624 (2004)

    Google Scholar 

  21. Cederwall, M., Preitschopf, C.R.: S 7 and its Kač–Moody algebra. Commun. Math. Phys. 167, 373 (1995), hep-th/9309030

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. da Rocha, R., Vaz Jr. J.: Clifford algebra-parametrized octonions and generalizations. J. Algebra 301, 459–473 (2006), math-ph/0603053

    Article  MATH  MathSciNet  Google Scholar 

  23. Cederwall, M.: Introduction to division algebras, sphere algebras and twistors. Talk presented at the Theoretical Physics Network Meeting at NORDITA, Copenhagen, Sept. 1993, hep-th/9310115

  24. da Rocha, R., Vaz Jr. J.: Twistors, Generalizations and Exceptional Structures, PoS(WC2004) (2004) 022 math-ph/0412037

  25. Sthanumoorthy, M., Misra, K.C. (eds.): Kač–Moody Lie Algebras and Related Topics: Ramanujan International Symposium on Kač–Moody Lie Algebras and Applications, AMS Colloquium, Providence (2004)

    Google Scholar 

  26. Bengtsson, I., Cederwall, M.: Particles, twistors and division algebras. Nucl. Phys. B 302, 81–103 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  27. Lounesto, P.: Counterexamples in Clifford algebras with CLICAL. In: Ablamowicz, R. et al. (eds.) Clifford Algebras with Numeric and Symbolic Computations, pp. 3–30. Birkhäuser, Boston (1996)

    Google Scholar 

  28. Lounesto, P.: Counterexamples in Clifford algebras. Adv. Appl. Cliff. Algebras 6, 69–104 (1996)

    MATH  MathSciNet  Google Scholar 

  29. Dixon, G.M.: Division Algebras: Octonions, Quaternions, Complex Numbers, and the Algebraic Design of Physics. Kluwer Academic, Dordrecht (1994)

    MATH  Google Scholar 

  30. Dixon, G.M.: Octonion XY-product. hep-th/9503053

  31. Dixon, G.M.: Octonion X-product orbits. hep-th 9410202

  32. Dixon, G.M.: Octonion X-product and octonion lattices. hep-th 9411063

  33. Keller, J., Rodríguez-Romo, S.: Multivectorial representation of Lie groups. Int. J. Theor. Phys. 30(2), 185–196 (1991)

    Article  MATH  Google Scholar 

  34. Chisholm, J., Farewell, R.: Unified spin gauge theories of the four fundamental forces. In: Micali, A. et al. (eds.) Clifford Algebras and Their Applications in Math. Physics. Kluwer Academic, Dordrecht (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09210-170, Santo André, SP, Brazil

    R. da Rocha

  2. Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Unicamp, 13083-970, Campinas, SP, Brazil

    R. da Rocha

  3. Departamento de Matemática Aplicada, IMECC, Unicamp, CP 6065, 13083-859, Campinas, SP, Brazil

    J. Vaz Jr.

Authors
  1. R. da Rocha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Vaz Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. da Rocha.

Rights and permissions

Reprints and permissions

About this article

Cite this article

da Rocha, R., Vaz, J. Isotopic Liftings of Clifford Algebras and Applications in Elementary Particle Mass Matrices. Int J Theor Phys 46, 2464–2487 (2007). https://doi.org/10.1007/s10773-007-9362-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-007-9362-x

Keywords

Search

Navigation