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25 pages, 473 KiB

Open AccessArticle

Octonion Internal Space Algebra for the Standard Model

by Ivan Todorov

Universe 2023, 9(5), 222; https://doi.org/10.3390/universe9050222 - 6 May 2023

Cited by 7 | Viewed by 2074

This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. [...] Read more.

O = C \oplus C^{3}

, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space

S

C ℓ_{10}

Majorana spinors is generated by the

C ℓ_{6} (\subset C ℓ_{10})

volume form,

ω_{6} = γ_{1} \dots γ_{6}

, and is left invariant by the Pati–Salam subgroup of

S p i n (10)

G_{PS} = S p i n (4) \times S p i n (6) / Z_{2}

. While the

S p i n (10)

invariant volume form

ω_{10} = γ_{1} \dots γ_{10}

C ℓ_{10}

is known to split

S

on a complex basis into left and right chiral (semi)spinors,

P = \frac{1}{2} (1 - i ω_{6})

is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of

G_{PS}

that preserves the sterile neutrino (which is identified with the Fock vacuum). The

Z_{2}

-graded internal space algebra

A

is then included in the projected tensor product

A \subset P C ℓ_{10} P = C ℓ_{4} \otimes C ℓ_{6}^{0}

. The Higgs field appears as the scalar term of a superconnection, an element of the odd part

C ℓ_{4}^{1}

of the first factor. The fact that the projection of

C ℓ_{10}

only involves the even part

C ℓ_{6}^{0}

of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio

\frac{m_{H}}{m_{W}}

of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle. Full article

(This article belongs to the Section Mathematical Physics)

29 pages, 460 KiB

Open AccessArticle

Clifford Odd and Even Objects in Even and Odd Dimensional Spaces Describing Internal Spaces of Fermion and Boson Fields

by Norma Susana Mankoč Borštnik

Symmetry 2023, 15(4), 818; https://doi.org/10.3390/sym15040818 - 28 Mar 2023

Cited by 4 | Viewed by 1666

Abstract

In a long series of works, it has been demonstrated that the spin-charge-family theory, assuming a simple starting action in even dimensional spaces with

d \geq (13 + 1)

, with massless fermions interacting with gravity only, offers the explanation for [...] Read more.

In a long series of works, it has been demonstrated that the spin-charge-family theory, assuming a simple starting action in even dimensional spaces with

d \geq (13 + 1)

, with massless fermions interacting with gravity only, offers the explanation for all assumed properties of the second quantized fermion and boson fields in the standard model, as well as offering predictions and explanations for several of the observed phenomena. The description of the internal spaces of the fermion and boson fields by the Clifford odd and even objects, respectively, justifies the choice of the simple starting action of the spin-charge-family theory. The main topic of the present article is the analysis of the properties of the internal spaces of the fermion and boson fields in odd dimensional spaces,

d = (2 n + 1)

, which can again be described by the Clifford odd and even objects, respectively. It turns out that the properties of fermion and boson fields differ essentially from their properties in even dimensional spaces, resembling the ghosts needed when looking for final solutions with Feynman diagrams. Full article

(This article belongs to the Special Issue Symmetry beyond the Standard Models of Cosmology, High Energy Physics and Quantum Field Theory)

14 pages, 326 KiB

Open AccessArticle

Relations between Clifford Algebra and Dirac Matrices in the Presence of Families

by Dragan Lukman, Mickael Komendyak and Norma Susana Mankoč Borštnik

Particles 2020, 3(3), 518-531; https://doi.org/10.3390/particles3030035 - 29 Jun 2020

Cited by 3 | Viewed by 2629

Abstract

The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of

γ^{a}

’s. Arranged into irreducible representations of “eigenvectors” of the Cartan subalgebra of the Lorentz algebra [...] Read more.

The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of

γ^{a}

’s. Arranged into irreducible representations of “eigenvectors” of the Cartan subalgebra of the Lorentz algebra

S^{a b}

(= \frac{i}{2} γ^{a} γ^{b} |_{a \neq b})

these objects form

2^{\frac{d}{2} - 1}

families with

2^{\frac{d}{2} - 1}

family members each. Family members of each family offer the description of all the observed quarks and leptons and antiquarks and antileptons, appearing in families. Families are reachable by

{\tilde{S}}^{a b}

= \frac{1}{2} {\tilde{γ}}^{a} {\tilde{γ}}^{b} |_{a \neq b}

. Creation operators, carrying the family member and family quantum numbers form the basis vectors. The action of the operators

γ^{a}

’s,

S^{a b}

{\tilde{γ}}^{a}

’s and

{\tilde{S}}^{a b}

, applying on the basis vectors, manifests as matrices. In this paper the basis vectors in

d = (3 + 1)

Clifford space are discussed, chosen in a way that the matrix representations of

γ^{a}

and of

S^{a b}

coincide for each family quantum number, determined by

{\tilde{S}}^{a b}

, with the Dirac matrices. The appearance of charges in Clifford space is discussed by embedding

d = (3 + 1)

space into

d = (5 + 1)

-dimensional space. The achievements and predictions of the spin-charge-family theory is also shortly presented. Full article

(This article belongs to the Special Issue Beyond the Standard Models in Particle Physics and Cosmology)

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