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 S a b ( = i 2 γ a γ b | a ≠ b ) these objects form 2 d 2 − 1 families with 2 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 S ˜ a b = 1 2 γ ˜ a γ ˜ b | a ≠ 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 , γ ˜ a ’s and 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 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.
Highlights
IntroductionMotivation: More than 50 years ago the electroweak (and colour) standard model offered an elegant new step in understanding the origin of elementary fermion and boson fields by postulating 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
The fact that the odd Clifford algebra offers the description of quarks and leptons and antiquarks and antileptons as assumed by the standard model suggests that the standard model vector gauge fields—gravitational, colour, weak and hyper— have the common origin
We demonstrate in this paper the appearance of families, when internal degrees of freedom are described with the odd Clifford algebra in d = (3 + 1), and the appearance of charges and families, when internal degrees of freedom are described with the odd Clifford algebra in d = (5 + 1), we present the basis vectors of each irreducible representation of each family in both cases as well as the corresponding matrix representations of the operators of the Clifford algebra objects
Summary
Motivation: More than 50 years ago the electroweak (and colour) standard model offered an elegant new step in understanding the origin of elementary fermion and boson fields by postulating 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. The spin-charge-family theory [1,2,3,4,5,6], describing the internal degrees of freedom with the odd Clifford algebra in d ≥ (13 + 1), explains all the postulated properties of quarks and leptons and antiquarks and antileptons [[2,3,4] and the references therein]. The fact that the odd Clifford algebra offers the description of quarks and leptons and antiquarks and antileptons as assumed by the standard model suggests that the standard model vector gauge fields—gravitational, colour, weak and hyper— have the common origin. Statement 3: The vielbeins and the two kinds of the spin connection fields in d = (13 + 1) manifest in d = (3 + 1) all the known vector gauge fields postulated by the standard model as well a several scalar fields, those carrying the weak and hyper charge equal to We make the choice of the basis vectors in the way that the matrix elements coincide for each family with the Dirac ones up to a phase
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