Skip to main content
SpringerLink
Log in

The Underlying Fiber Bundle Geometry of the CAM Gauge Model of the Standard Model of Particle Physics: \(\pmb {SU(3)}\)

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

The CAM (Composition-Algebra based Methodology) (Wolk in Int J Mod Phys A 35:2050037–2050075, 2020; Pap Phys 9:090002–09007, 2017; Phys Scr 94:025301–025307, 2019; Adv Appl Clifford Algebras 27(4):3225–3234, 2017; J Appl Math Phys 6:1537–1538, 2018; Phys Scr 94:105301–105306, 2019; Adv Appl Clifford Algebras 30:4–14, 2020) [46,47,48,49,50,51,52] gauge model’s fiber bundle framework—previously shown to induce the SU(2) and U(1) Lagrangians of the Standard Model—is extended to the SU(3) Lagrangian.

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. The Composition Algebra-based Methodology (CAM) as set forth in References [46,47,48,49,50,51,52] and shown to be a gauge theory [52]. Composition algebras are algebras \(\mathbb {A}\) such that for any two elements the algebraic norm of their product equals the product of their norms [12, 25, 36]: \(\left\| xy\right\| =\left\| x\right\| \left\| y\right\| \;\forall x,y\in \mathbb {A}\). These composition algebras exist in 1, 2, 4 and 8 dimensions, corresponding to \(\mathbb {K}=\left\{ \mathbb {R},\mathbb {C},\mathbb {H},\mathbb {O}\right\} \) and their split versions \(\mathbb {K}^{\prime }=\left\{ \mathbb {C}^{\prime },\mathbb {H}^{\prime },\mathbb {O}^{\prime }\right\} \) [12, 35]. The \(\mathbb {K}\) algebras are the only division algebras (composition algebras without zero divisors) [12, 36].

  2. Excepting use of, so far, \(S^{0}=\left\{ -1,1\right\} \), which is trivially isomorphic to the unit reals [9].

  3. Ref. [31, Preface].

  4. Ref. [31, p. 2].

  5. See Ref. [52, Sect. 8] for further discussion of this topic.

  6. See Ref. [52, Sects. 5.1–5.3 & 9.3.1]. The sphere \(S_{\mathbb {O}_{u}}^{7}\) has no metric defined on it—it is a smooth (differentio-topological) manifold containing the smooth division-algebraic operation (See also Ref. [52, Fn. (aaa)].

  7. Where we note that \(S_{\mathbb {O}_{u}}^{7}\rightarrow {\mathscr {S}}^{7}\) signifies that the unit 7-sphere with smooth octonionic operator structure \(S_{\mathbb {O}_{u}}^{7}\) now acts as an operator fiber \({\mathscr {S}}^{7}\), as with \(S_{\mathbb {H}_{u}}^{3}\rightarrow {\mathscr {S}}^{3}\) and \(S_{\mathbb {C}_{u}}^{1}\rightarrow {\mathscr {S}}^{1}\) for the SU(2) and U(1) interactions, respectively [52].

  8. See Ref. [52, Sect. 5.3] for further discussion of the wedge action. The algebraic-topological wedge operation \(\vee \) identifies distinct points on two or more manifolds to a single point. Thus for example \(S^{1}\vee S^{1}\) is equivalent to a figure-eight—being two circles touching at a point [21]. The \(\vee \)-operation in effect connects or couples manifolds together at a common point.

  9. See Ref. [52, Sects. 5.4.1& 5.5] for \(B_{1}\) and \(B_{3}\).

  10. These operators are defined in Eq. (5) below and specifically addressed in Reference [52] and throughout the CAM papers (cf. Ref. [46]).

  11. See Ref. [46, Eqs. (1)–(8)] for the \(\eta \partial \) derivation. See also Ref. [50, Sect. 2] and Ref. [47, Sect. 3.ii].

  12. Ref. [1, Chap. 3 & Sect. 3.3].

  13. Ref. [52, Sect. 7.3.2] and [49, Fn. (14) & Sect. 2.2].

  14. Refs. [50, Sect. 2.2.1], [48]. U(1) lacks the commutator and therefore its Lie algebra structure constants are identically zero.

  15. Ref. [52, Sect. 5].

  16. Ref. [52, Sect. 5.6].

  17. The requirement of complexification with \(\mathrm {i}\in \mathbb {C}\) was previously addressed [52].

  18. Ref. [52, Sects. 5.5 and 5.6].

  19. Ref. [52, Eq. (32)]; Ref. [50, Sect. 2.3].

  20. Ref. [50, Sect. 2.2.1, Eqs. (11)–(14)].

  21. In analogous manner as a vector space \(\mathrm {V}\) is complexified with \(\mathbb {C}\)-unit \(\mathrm {i}\): \(\mathrm {V}\overset{\otimes \mathrm {i}}{\rightarrow }\mathcal {\mathbb {V}}\) (See Ref. [34, Sect. 8.2.2)].

  22. Since \(\mathbb {R},\mathbb {C},\mathbb {H}\subset \mathbb {C}\ell _{1,7}\), we can write all division algebras in the form \(\mathbb {K}\oplus \ell \mathbb {K}\); e.g., \(\mathbb {H}=\mathbb {C}\oplus \ell \mathbb {C}\) [12, 35].

  23. See Ref. [52, Sects. 5.1 & 9.3.1].

  24. See Ref. [50, Eq. (2)] and references therein; Ref. [49, Eq. (9)] and references therein; Ref. [46, Eq. (6)] and references therein.

  25. Since a metric g on a manifold M is the structure that smoothly assigns to each point \(p\in M\) a metric \(g_{p}\) on \(T_{p}M\) [5, 13] by which dot and cross products manifest [5, 13, 40], the form of (15) requires operation within a manifold equipped with a metric.

  26. Ref. [18, Eq. (10.84)].

  27. Ref. [6, Eqs. (9.29)–(9.30)].

  28. See Ref. [46, Sect. III] for producing the linear portion of \(G_{\mu \nu }\).

  29. Local gauge symmetry is preserved if the fields have equal mass [41, 43].

  30. Ref. [46, Sect. ii, Eq. (9)].

  31. Ref. [18, pp. 235–236].

  32. Ref. [27, Sect. 4.5].

  33. Ref. [52, Sect. ii, Eq. (9)]. See also Ref. [49], where this structure constant constraint was shown to be the essence of why there can be no gauge-mediated proton decay via SU(5) or any other purported GUT group, and thus why such decay has never been observed.

  34. Ref. [50, Sects. 2.2.2 & 3].

  35. Ref. [50, Sect. 2.2.2].

  36. Unlike for the electric and strong interactions; e.g., the electron does not strongly interact and the neutrino does not electrically interact.

  37. Ref. [50, Sect. 2.2.2].

  38. Ref. [18, Eq. (10.36)].

  39. Ref. [41, Sect. 7.8.2].

  40. Ref. [40, Sect. 4.8.6].

  41. See Ref. [50, Sect. 2.2.2].

  42. See Ref. [41, p. 154].

  43. Ref. [52, Sects. 5.5–5.6].

  44. See Ref. [50, Sect. 2.2.2].

  45. According to the equivalence principle [7, 8, 37, 44, 45], locally flat space-time inertial frame \(\left( \mathbb {M},\eta \right) \) is recoverable in a sufficiently small neighborhood of any space-time point \(p\in (\mathbb {B},{\mathbf {g}})\) (See Ref. [7, Sect. 5.2.2 (p. 88) & Sect. 6.1.1 (p. 102)]; Ref. [37, Sects. 16 & 51)].

  46. See Ref. [52, Secs. 5.5–5.6] for analogous formalisms of SU(2) and U(1).

  47. Ref. [52, Eqs. (28)–(33)].

  48. See Ref. [18, Eq. (10.88)]; Ref. [6, Eq. (9.36)].

  49. Ref. [52, Sect. 9].

References

  1. Abłamowicz, R., Sobczyk, G.: Lectures on Clifford (Geometric) Algebras and Applications. Birkhäuser, New York (2004)

    Book  MATH  Google Scholar 

  2. Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72, 20–104 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adams, J.F., Atiyah, M.F.: K-theory and the Hopf invariant. Q. J. Math. 17, 31–44 (1966)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Atiyah, M.F.: K-Theory. W. A. Benjamin Inc, New York (1967)

    MATH  Google Scholar 

  5. Baez, J., Muniain, J.P.: Gauge Fields. Knots and Gravity. World Scientific Publishing, Hackensack (2013)

    MATH  Google Scholar 

  6. Bailin, D., Love, A.: Introduction to Gauge Field Theory, rev edn. Taylor & Francis Group, Philadelphia (1993)

    MATH  Google Scholar 

  7. Cheng, T.-P.: Relativity. Gravitation and Cosmology. Oxford University Press, Oxford (2016)

    Google Scholar 

  8. D’Inverno, R.: Introducing Einstein’s Relativity. Clarendon Press, Oxford (1992)

    MATH  Google Scholar 

  9. Dixon, G.M.: Division Algebras: Octionions, Quaternions. Complex Numbers and the Algebraic Design of Physics. Kluwer Academic Publishers, Norwell (1994)

    Book  Google Scholar 

  10. Dixon, G.M.: Division algebras: family replication. J. Math. Phys. 45, 3878–3882 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  12. Dray, T., Manogue, C.A.: The Geometry of the Octonions. World Scientific, New Jersey (2015)

    Book  MATH  Google Scholar 

  13. Fecko, M.: Differential Geometry and Lie groups for Physicists. Cambridge University Press, New York (2011)

    MATH  Google Scholar 

  14. Frankel, T.: The Geometry of Physics, an Introduction, 3rd edn. Cambridge University Press, New York (2012)

    MATH  Google Scholar 

  15. Furey, C.: Three generations, two unbroken gauge symmetries and one eight-dimensional algebra. Phys. Lett. B 785, 84–89 (2018)

    Article  ADS  MATH  Google Scholar 

  16. Furey, C.: \(SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\) as a symmetry of division algebraic operators. Eur. Phys. J. C 78, 375–384 (2018)

    Article  ADS  Google Scholar 

  17. Gogberashvili, M.: Standard model of particle physics from split octonions. Prog. Phys. 12, 1–5 (2016)

    Google Scholar 

  18. Griffiths, D.: Introduction to Elementary Particles, 2 rev edn. Wiley-VCH, Weinheim (2008)

    MATH  Google Scholar 

  19. Günaydin, M., Gürsey, F.: Quark structure and the octonions. Math. Phys. 14, 11–26 (1973)

    Article  MathSciNet  Google Scholar 

  20. Günaydin, M., Gürsey, F.: Quark statistics and octonions. Phys. Rev. D 9, 3387–3391 (1974)

    Article  ADS  Google Scholar 

  21. Hatcher, A.: Algebraic Topology. Cambridge University Press, New York (2001)

    MATH  Google Scholar 

  22. Hay, G.E.: Vector and Tensor Analysis. Dover Publications Inc, New York (1953)

    MATH  Google Scholar 

  23. Haywood, S.: Symmetries and Conservation Laws in Particle Physics. Imperial College Press, London (2011)

    MATH  Google Scholar 

  24. Kane, G.: String Theory and the Real World. Morgan & Claypool Publishers, San Rafeal (2017)

    Book  MATH  Google Scholar 

  25. Lee, J.M.: Introduction to Smooth Manifolds, 2nd edn. Springer, New York (2013)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Ma, Z.: Group Theory for Physicists. World Scientific, Singapore (2007)

    Book  MATH  Google Scholar 

  28. Maia, M.D.: Geometry of the Fundamental Interactions. Springer, New York (2011)

    Book  MATH  Google Scholar 

  29. Manogue, C.A., Dray, T.: Octonions, \(E6\), and particle physics. J. Phys. Conf. Ser. 254, 012005–012017 (2010)

    Article  Google Scholar 

  30. McInnes, B.: The Semi-Spin Groups in String Theory. arXiv:hep-th/9906059v1 [hep-th] (1999)

  31. Mielke, E.W.: Geometrodynamics of Gauge Fields: On the Geometry of Yang-Mills and Gravitational Gauge Theories, 2nd edn. Springer, Switzerland (2017)

    Book  MATH  Google Scholar 

  32. Milnor, J.W.: Topology from the Differential Viewpoint, rev edn. Princeton University Press, New Jersey (1965)

    MATH  Google Scholar 

  33. Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, New Jersey (1974)

    Book  MATH  Google Scholar 

  34. Nakahara, M.: Geometry, Topology and Physics, 2nd edn. IOP Publishing Ltd. (2003)

  35. Nash, C., Sen, S.: Topology and Geometry for Physicists. Dover Publications Inc., New York (2011)

    MATH  Google Scholar 

  36. Okubo, S.: Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  37. Pauli, W.: Theory of Relativity. Dover Publications Inc, New York (1958)

    MATH  Google Scholar 

  38. Penrose, R., Rindler, W.: Spinors and Spacetime. Cambridge University Press, New York (1984)

    Book  MATH  Google Scholar 

  39. Pushpa, K., Bisht, P.S., Tianjun, L., Negi, O.: Quaternion octonion reformulation of grand unified theories. Int. J. Theor. Phys. 51, 3228–3241 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Robinson, M.: Symmetry and the Standard Model. Springer, New York (2011)

    Book  MATH  Google Scholar 

  41. Schwichtenberg, J.: Physics from Symmetry. Springer, Switzerland (2015)

    Book  MATH  Google Scholar 

  42. Stoica, O.C.: Leptons, quarks and gauge from the complex Clifford algebra \(C\ell _{6}\). Adv. Appl. Clifford Algebras 18, 52–86 (2018)

    Article  MATH  Google Scholar 

  43. Quigg, C.: Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, 2nd edn. Princeton University Press, Princeton (2013)

    MATH  Google Scholar 

  44. Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)

    Book  MATH  Google Scholar 

  45. Weinberg, S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York (1972)

    Google Scholar 

  46. Wolk, B.: An Alternative derivation of the Dirac operator generating intrinsic Lagrangian local gauge invariance. Pap. Phys. 9, 090002–09007 (2017)

    Article  Google Scholar 

  47. Wolk, B.: On an intrinsically local gauge symmetric \(SU(3)\) field theory for quantum chromodynamics. Adv. Appl. Clifford Algebras 27(4), 3225–3234 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wolk, B.: Addendum. J. Appl. Math. Phys. 6, 1537–1538 (2018)

    Article  Google Scholar 

  49. Wolk, B.: The division algebraic constraint on gauge mediated proton decay. Phys. Scr. 94, 105301–105306 (2019)

    Article  ADS  Google Scholar 

  50. Wolk, B.: An alternative formalism for generating pre-Higgs \(SU(2)_{L}\otimes U(1)_{Y}\) electroweak unification that intrinsically accommodates \(SU(2)\) left-chiral asymmetry. Phys. Scr. 94, 025301–025307 (2019)

    Article  ADS  Google Scholar 

  51. Wolk, B.: The Riemann curvature tensor and Higgs scalar field within CAM theory. Adv. Appl. Clifford Algebras 30, 4–14 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wolk, B.: The underlying geometry of the CAM gauge model of the standard model of particle physics–\(SU(2)\otimes U(1)\). Int. J. Mod. Phys. A 35, 2050037–2050075 (2020)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tallahassee, FL, 32301, USA

    Brian Jonathan Wolk

Authors
  1. Brian Jonathan Wolk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brian Jonathan Wolk.

Additional information

Communicated by Roldão da Rocha.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wolk, B.J. The Underlying Fiber Bundle Geometry of the CAM Gauge Model of the Standard Model of Particle Physics: \(\pmb {SU(3)}\). Adv. Appl. Clifford Algebras 31, 26 (2021). https://doi.org/10.1007/s00006-021-01127-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-021-01127-6

Keywords

Mathematics Subject Classification

Search

Navigation