Relations between Clifford Algebra and Dirac Matrices in the Presence of Families
Abstract
:1. Introduction
- i.
- the existence of massless family members with the charges in fundamental representations of the groups—the colour triplet quarks and colourless leptons the left handed members as the weak charged doublets, the right handed weak chargeless members, the left handed quarks distinguishing in the hyper charge from the left handed leptons, each right handed member having a different hyper charge, the existence of the corresponding anti fermions—the existence of massless families to each family member,
- ii.
- the existence of massless vector gauge fields to the observed charges of the family members, carrying charges in the adjoint representations of the charge groups,
- iii.
- the existence of the massive scalar fields with the nonzero vacuum expectation value carrying the weak charge and the hyper charge (like if it would be in the fundamental representation of the weak group), determining the masses of quarks and leptons and of the weak bosons, the existence of the Yukawa couplings determining the properties of fermions.
Mathematical Background
2. Properties of Vectors in Clifford Space
2.1. Action
2.2. Creation and Annihilation Operators in Space Embedded in Space
2.3. , , and Matrices in
3. Conclusions
- i.
- That the appearance of two kinds of the Clifford algebra objects, ’s and ’s, offers the explanation for the appearance of families of quarks and leptons, without postulating the family groups.
- ii.
- The appearance of charges in part if . To reproduce all the properties of fermions manifesting as quarks and leptons in the dimension of space time must be .
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Short Overview of Achievements of Spin-Charge-Family Theory
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Lukman, D.; Komendyak, M.; Mankoč Borštnik, N.S. Relations between Clifford Algebra and Dirac Matrices in the Presence of Families. Particles 2020, 3, 518-531. https://doi.org/10.3390/particles3030035
Lukman D, Komendyak M, Mankoč Borštnik NS. Relations between Clifford Algebra and Dirac Matrices in the Presence of Families. Particles. 2020; 3(3):518-531. https://doi.org/10.3390/particles3030035
Chicago/Turabian StyleLukman, Dragan, Mickael Komendyak, and Norma Susana Mankoč Borštnik. 2020. "Relations between Clifford Algebra and Dirac Matrices in the Presence of Families" Particles 3, no. 3: 518-531. https://doi.org/10.3390/particles3030035