How Clifford algebra helps understand second quantized quarks and leptons and corresponding vector and scalar boson fields, {\it opening a new step beyond the standard model}

This article presents the description of the internal spaces of fermion and boson fields in $d$-dimensional spaces, with the odd and even"basis vectors"which are the superposition of odd and even products of operators $\gamma^a$. While the Clifford odd"basis vectors"manifest properties of fermion fields, appearing in families, the Clifford even"basis vectors"demonstrate properties of the corresponding gauge fields. In $d\ge (13+1)$ the corresponding creation operators manifest in $d=(3+1)$ the properties of all the observed quarks and leptons, with the families included, and of their gauge boson fields, with the scalar fields included, making several predictions. The properties of the creation and annihilation operators for fermion and boson fields are illustrated on the case $d=(5+1)$, when $SO(5,1)$ demonstrates the symmetry of $SU(3)\times U(1)$.


Introduction
The standard model (corrected with the right-handed neutrinos) has been experimentally confirmed without raising any severe doubts so far on its assumptions, which, however, remain unexplained.
The standard model assumptions have several explanations in the literature, mostly with several new, not explained assumptions. The most popular are the grand unifying theories ( [1,2,3,4,5] and many others).
In a long series of works ( [6,7,8,9],and the references there in) the author has found, together with the collaborators ( [10,7,11,12,13,14] and the references therein), the phenomenological success with the model named the spin-charge-family theory with the properties: a. The internal space of fermions are described by the "basis vectors" which are superposition of odd products of anti-commuting objects (operators) 1 γ a (in the sense {γ a , γ b } + = 2η ab ), Sect. 2.1, vectors" differ essentially from their properties in even dimensional spaces, resembling the ghosts needed to make the contributions of the Feynman diagrams finite [18].
The theory seems very promising to offer a new insight into the second quantization of fermion and boson fields and to show the next step beyond the standard model.
The more work is put into the theory, the more phenomena the theory can explain. Other references used a different approach by trying to make the next step with Clifford algebra to the second quantized fermion, which might also be a boson field [39,40].
Let us present a simple starting action of the spin-charge-family theory ( [14] and the references therein) for massless fermions and anti-fermions which interact with massless gravitational fields only; with vielbeins (the gauge fields of momenta) and the two kinds of spin connection fields, the gauge fields of the two kinds of the Lorentz transformations in the internal space of fermions, of S ab andS ab , in d = 2(2n + 1)-dimensional space The vielbeins, f a α , and the two kinds of the spin connection fields, ω abα (the gauge fields of S ab ) andω abα (the gauge fields ofS ab ), manifest in d = (3 + 1) as the known vector gauge fields and the scalar gauge fields taking care of masses of quarks and leptons and antiquarks and antileptons and of the weak boson fields [11,8,9,12] 4 .
The action, Eq. (1), assumes two kinds of the spin connection gauge fields, due to two kinds of the operators: γ a andγ a . Let be pointed out that the description of the internal space of bosons with the Clifford even "basis vectors" offers as well two kinds of the Clifford even "basis vectors", as presented in d.ii..
In Sect. 2 the Grassmann and the Clifford algebras are explained, Subsect.2.1, and creation and annihilation operators described as tensor products of the "basis vectors" offering an explanation of the internal spaces of fermion (by the Clifford odd algebra) and boson (by the Clifford even algebra) fields and the basis in ordinary space.
In Subsect. 2.3, the properties of the Clifford odd and even "basis vectors" are demonstrated in the toy model in d = (5 + 1).
In Subsect. 2.4, the properties of the creation and annihilation operators for the second quantized fermion and boson fields in even dimensional spaces are described.
Sect. 3 presents what the reader could learn new from this article. In App. B, the answers of the spin-charge-family theory to some of the open questions of the standard model are discussed.
App. A, suggested by the referee, illustrates on the simplest case d = (3 + 1) (and d = (1 + 1); which offers only one "family" of fermions, d = (3 + 1) has two families) the properties of the Clifford odd and Clifford even"basis vectors" describing the internal spaces of fermion and boson fields, explaining in a pedagogical way in details their construction, manifestation of anti-commutativity (in the fermion case) and commutativity (in the boson case) of the tensor product of the "basis vectors" and the basis in ordinary space-time.
The referee suggested also several footnotes.
Ref. [18] in Subsect. 2.2.2. Although anti-commuting, the Clifford odd "basis vectors" manifest properties of the Clifford even "basis vectors" in even dimensional spaces. And the Clifford even "basis vectors", although commuting, manifest properties of the Clifford odd "basis vectors" in even dimensional spaces.
While the Grassmann algebra offers the description of the "anti-commuting integer spin second quantized fields" and of the "commuting integer spin second quantized fields" [13,14], the Clifford algebras which are superposition of odd products of either γ a 's orγ a 's offer the description of the second quantized half integer spin fermion fields, which from the point of the subgroups of the SO(d − 1, 1) group manifest spins and charges of fermions and antifermions in the fundamental representations of the group and subgroups, Table 4. The superposition of even products of either γ a 's orγ a 's offer the description of the commuting second quantized boson fields with integer spins (as we can see in [16,15] and shall see in this contribution) which from the point of the subgroups of the SO(d − 1, 1) group manifest spins and charges in the adjoint representations of the group and subgroups. The following postulate, which determines how doesγ a operate on γ a , reduces the two Clifford subalgebras, γ a andγ a , to one, to the one described by γ a [10,6,9,12] with (−) B = −1, if B is (a function of) odd products of γ a 's, otherwise (−) B = 1 [10], the vacuum state |ψ oc > is defined in Eq. (15) of Subsect. 2.2.
After the postulate of Eq. (7) it follows: a. The Clifford subalgebra described byγ a 's looses its meaning for the description of the internal space of quantum fields. b. The "basis vectors" which are superposition of odd or even products of γ a 's obey the postulates for the second quantized fields for fermions or bosons, respectively, Sect.2.2. c. It can be proven that the relations presented in Eq. (6) remain valid also after the postulate of Eq. (7). The proof is presented in Ref. ([14], App. I, Statement 3a). d. Each irreducible representation of the Clifford odd "basis vectors" described by γ a 's are equipped by the quantum numbers of the Cartan subalgebra members ofS ab , chosen in Eq. (8), as follows S 03 , S 12 , S 56 , · · · , S d−1 d , S 03 ,S 12 ,S 56 , · · · ,S d−1 d , After the postulate of Eq. (7) no vector space ofγ a 's needs to be taken into account for the description of the internal space of either fermions or bosons, in agreement with the observed properties of fermions and bosons. Also the Grassmann algebra is reduced to only one of the Clifford subalgebras. The operatorγ a will from now on be used to describe the properties of fermion "basis vectors", determining byS ab = i 4 (γ aγb −γ bγa ) the "family" quantum numbers of the irreducible representations of the Lorentz group in internal space of fermions, S ab , and the properties of bosons "basis vectors" determined by S ab = S ab +S ab . We shall see that while the fermion "basis vectors" appear in "families", the boson "basis vectors" have no "families" and manifest properties of the gauge fields of the corresponding fermion fields. In App. A the case of d = (3 + 1) is discussed. γ a 's equip each irreducible representation of the Lorentz group (with the infinitesimal generators S ab = i 4 {γ a , γ b } − ) when applying on the Clifford odd "basis vectors" (which are superposition of odd products of γ a ′ s ) with the family quantum numbers (determined byS ab = i 4 {γ a ,γ b } − ). Correspondingly the Clifford odd "basis vectors" (they are the superposition of odd products of γ a 's) form 2 d 2 −1 families, with the quantum number f , each family has 2 d 2 −1 members, m. They offer the description of the second quantized fermion fields.
The Clifford even "basis vectors" (they are the superposition of even products of γ a 's) have no families, as we shall see in what follows, but they do carry both quantum numbers, f and m, offering the description of the second quantized boson fields as the gauge fields of the second quantized fermion fields. The generators of the Lorentz transformations in the internal space of the Clifford even "basis vectors" are S ab = S ab +S ab .
Properties of the Clifford odd and the Clifford even "basis vectors" are discussed in the following subsection.

"Basis vectors" of fermions and bosons in even and odd dimensional spaces
This subsection is a short overview of similar sections of several articles of the author, like [17,15,18,13]. After the reduction of the two Clifford subalgebras to only one, Eq. (7), we only need to define "basis vectors" for the case that the internal space of second quantized fields is described by superposition of odd or even products γ a 's 5 .
Let us use the technique which makes "basis vectors" products of nilpotents and projectors [6,10] which are eigenvectors of the (chosen) Cartan subalgebra members, Eq. (8), of the Lorentz algebra in the space of γ a 's, either in the case of the Clifford odd or in the case of the Clifford even products of γ a 's. There are in even-dimensional spaces d 2 members of the Cartan subalgebra, Eq. (8). In odd-dimensional spaces there are d−1 2 members of the Cartan subalgebra. One finds in even dimensional spaces for any of the d 2 Cartan subalgebra member, S ab applying on a nilpotent ab (k) or on projector the relations with k 2 = η aa η bb 6 , demonstrating that the eigenvalues of S ab on nilpotents and projectors expressed with γ a differ from the eigenvalues ofS ab on nilpotents and projectors expressed with γ a , so thatS ab can be used to equip each irreducible representation of S ab with the "family" quantum number. 7 We define in even d the "basis vectors" as algebraic, * A , products of nilpotents and projectors so that each product is an eigenvector of all d 2 Cartan subalgebra members, Eq.(8). Fermion "basis vectors" are (algebraic, * A ,) products of odd number of nilpotents; each of them is the eigenvector of one of the Cartan subalgebra members, and the rest of the projectors; again is each projector the eigenvector of one of the Cartan subalgebra members. The boson "basis vectors" are (algebraic, * A ) products of an even number of nilpotents and the rest of the projectors. (In App. A, the reader can find concrete examples.) 5 In Ref. [14], the reader can find in Subsects. (3.2.1 and 3.2.2) definitions for the "basis vectors" for the Grassmann and the two Clifford subalgebras, which are products of nilpotents and projectors chosen to be the eigenvectors of the corresponding Cartan subalgebra members of the Lorentz algebras presented in Eq. (8). 6 Let us prove one of the relations in Eq. (10): S ab For k 2 = η aa η bb the first relation follows. 7 The reader can find the proof of Eq. (10) also in Ref. [14], App. (I).
It follows that the Clifford odd "basis vectors", which are the superposition of odd products of γ a , must include an odd number of nilpotents, at least one, while the superposition of an even products of γ a , that is Clifford even "basis vectors", must include an even number of nilpotents or only projectors.
We shall see that the Clifford odd "basis vectors" have properties appropriate to describe the internal space of the second quantized fermion fields while the Clifford even "basis vectors" have properties appropriate to describe the internal space of the second quantized boson fields.
Taking into account Eq. (6) one finds More relations are presented in App. C. The relations in Eq. (11) demonstrate that the properties of "basis vectors" which include an odd number of nilpotents, differ essentially from the "basis vectors", which include an even number of nilpotents.
One namely recognizes: i. Since the Hermitian conjugated partner of a nilpotent ab (k) † is η aa ab (−k) and since neither S ab nor S ab nor both can transform odd products of nilpotents to belong to one of the 2 ii. Since odd products of γ a anti-commute with another group of odd products of γ a , the Clifford odd "basis vectors" anti-commute, manifesting in a tensor product, * T , with the basis in ordinary space (together with the corresponding Hermitian conjugated partners) properties of the anti-commutation relations postulated by Dirac for the second quantized fermion fields 8 . The creation and annihilation operators, which include the internal space of fermions and bosons described by "basis vectors", the anti-commutativity or commutativity of which determine properties of the "basis vectors", fulfil the postulates of the second quantized fermion and boson fields. Basis of ordinary space commute as presented in Eq. (31). App. (A) discuses the creation and annihilation operators. The Clifford even "basis vectors" correspondingly fulfil, in a tensor product, * T , with the basis in ordinary space, the commutation relations for the second quantized boson fields. iii. The Clifford odd "basis vectors" have all the eigenvalues of the Cartan subalgebra members equal to either ± 1 2 or to ± i 2 . The Clifford even "basis vectors" have all the eigenvalues of the Cartan subalgebra members S ab = S ab +S ab equal to either ±1 and zero or to ±i and zero.
In odd-dimensional spaces the "basis vectors" can not be products of only nilpotents and projections. As we shall see in Subsect. 2.2.2, half of "basis vectors" can be chosen as products of nilpotents and projectors, the rest can be obtained from the first half by the application of S 0d on the first half. 8 So far, we multiply nilpotents and projectors, or products of nilpotents and projectors forming "basis vectors", among themselves. With the tensor product, * T , we include the basis in ordinary space.
We shall demonstrate, shortly overviewing [18], that the second half of the "basis vectors" have unusual properties: The Clifford odd "basis vectors have properties of the Clifford even "basis vectors", the Clifford even "basis vectors" have properties of the Clifford odd "basis vectors".

Clifford odd and even "basis vectors" in even d
Let us define Clifford odd and even "basis vectors" as products of nilpotents and projectors in evendimensional spaces.

a. Clifford odd "basis vectors"
This part overviews several papers with the same topic ( [14,18] and references therein). The Clifford odd "basis vectors" must be products of an odd number of nilpotents, and the rest, up to d 2 , of projectors, each nilpotent and each projector must be the "eigenstate" of one of the members of the Cartan subalgebra, Eq. (8), correspondingly are the "basis vectors" eigenstates of all the members of the Lorentz algebra: S ab 's determine 2 Let us start in d = 2(2n + 1) with the "basis vector"b 1 † 1 which is the product of only nilpotents, all the rest members belonging to the f = 1 family follow by the application of S 01 , S 03 , . . . , S 0d , S 15 , . . . , S 1d , S 5d . . . , S d−2 d . They are presented on the left-hand side. Their Hermitian conjugated partners are presented on the right-hand side. The algebraic product mark * A among nilpotents and projectors is skipped.
In d = 4n the choice of the starting "basis vector" with maximal number of nilpotents must have one projector The Hermitian conjugated partners of the Clifford odd "basis vectors"b m † 1 , presented in Eq. It is not difficult to see that all the "basis vectors" within any family, as well as the "basis vectors" among families, are orthogonal; that is, their algebraic product is zero. The same is true within their Hermitian conjugated partners. Both can be proved by the algebraic multiplication using Eqs. (11,47).
When we choose the vacuum state equal to for one of members m, which can be anyone of the odd irreducible representations f it follows that the Clifford odd "basis vectors" obey the relationŝ while the normalization < ψ oc |b m † f * Ab m † f * A |ψ oc >= 1 is used and the anti-commutation relation mean f . If we write the creation and annihilation operators for fermions as the tensor, * T , products of "basis vectors" and the basis in ordinary space, the creation and annihilation operators fulfil Dirac's anticommutation postulates since the "basis vectors" transfer their anti-commutativity to creation and annihilation operators; the ordinary basis namely commute as presented in Eqs. (31,32). Describing the internal space of fermions with the Clifford odd "basis vectors", makes creation operators fulfilling the Dirac postulates for the second quantized fermion fields: No postulates are needed. The creation and annihilation operators for fermions and bosons are discussed in App. A, in the part with the title "Creation and annihilation operators".
It turns out, therefore, that not only the Clifford odd "basis vectors" offer the description of the internal space of fermions, they explain the second quantization postulates for fermions as well. Table 1, presented in Subsect. 2.3, illustrates the properties of the Clifford odd "basis vectors" on the case of d = (5 + 1).

b. Clifford even "basis vectors"
This part proves that the Clifford even "basis vectors" are in even-dimensional spaces offering the description of the internal spaces of boson fields -the gauge fields of the corresponding Clifford odd "basis vectors": It is a new recognition, offering a new understanding of the second quantized fermion and boson fields [15].
The Clifford even "basis vectors" must be products of an even number of nilpotents and the rest, up to d 2 , of projectors; each nilpotent and each projector is chosen to be the "eigenstate" of one of the members of the Cartan subalgebra of the Lorentz algebra, S ab = S ab +S ab , Eq. (8). Correspondingly the "basis vectors" are the eigenstates of all the members of the Cartan subalgebra of the Lorentz algebra.
The Clifford even "basis vectors" appear in two groups, each group has 2  Let us call the Clifford even "basis vectors" iÂ m † f , where i = (I, II) denotes the two groups of Clifford even "basis vectors", while m and f determine membership of "basis vectors" in any of the two groups, I or II.
There are 2  Table 1, presented in Subsect. 2.3, illustrates properties of the Clifford odd and Clifford even "basis vectors" on the case of d = (5 + 1). Looking at this case it is easy to evaluate properties of either even or odd "basis vectors". We shall discuss in this subsection the general case by carefully inspecting properties of both kinds of "basis vectors".
The Clifford even "basis vectors" belonging to two different groups are orthogonal due to the fact that they differ in the sign of one nilpotent or one projector, or the algebraic product of a member of one group with a member of another group gives zero according to the first two lines of Eq. (47): The members of each of these two groups have the property For a chosen (m, f, f ') there is only one m ′ (out of 2 d 2 −1 ) which gives nonzero contribution. Two "basis vectors", iÂ m † f and iÂ m ′ † f ′ , the algebraic product, * A , of which gives non zero contribution, "scatter" into the third one iÂ m † f ' , for i = (I, II).
It remains to evaluate the algebraic application, * A , of the Clifford even "basis vectors" I,IIÂ m † f on the Clifford odd "basis vectors"b m ′ † f ' . One finds, taking into account Eq. (47), For each IÂ m † f there are among 2 Taking into account Eq.
Eqs. (20,21) demonstrates that IÂ m † f , applying onb m ′ † f ' , transforms the Clifford odd "basis vector" into another Clifford odd "basis vector" of the same family, transferring to the Clifford odd "basis vector" integer spins, or gives zero.
For "scattering" the Clifford even "basis vectors" IIÂ m † f on the Clifford odd "basis vectors"b m ′ † f ' it follows while we getb For eachb m † f there are among 2 (+) ) When the fermion field with the Cartan subalgebra family members quantum numbers (S 03 , S 12 , S 56 . . .
, keeping family members quantum numbers unchanged.
Eqs. (22,23) demonstrate that IIÂ m ′ † f ′ , "absorbed" byb m † f , transforms the Clifford odd "basis vector" into the Clifford odd "basis vector" of the same family member and of another family, or gives zero.
The Clifford even "basis vectors" offer the description of the internal space of the gauge fields of the corresponding fermion fields.
While the Clifford odd "basis vectors",b m † f , offer the description of the internal space of the second quantized anti-commuting fermion fields, appearing in families, the Clifford even "basis vectors", I,IIÂ m † f , offer the description of the internal space of the second quantized commuting boson fields, having no families and appearing in two groups. One of the two groups, IÂ m † f , transferring their integer quantum numbers to the Clifford odd "basis vectors",b m † f , changes the family members quantum numbers leaving the family quantum numbers unchanged. The second group, transferring their integer quantum numbers to the Clifford odd "basis vector", changes the family quantum numbers leaving the family members quantum numbers unchanged.
Both groups of Clifford even "basis vectors" manifest as the gauge fields of the corresponding fermion fields: One concerning the family members quantum numbers, the other concerning the family quantum numbers.
We shall discus properties of the Clifford even and odd "basis vectors" for d = (5 + 1)-dimensional internal spaces in Subsect. 2.3 in more details.

Clifford odd and even "basis vectors" in d odd
Let us shortly overview properties of the fermion and boson "basis vectors" in odd dimensional spaces, as presented in Ref. [18], Subsect. 2.2.
In even dimensional spaces the Clifford odd "basis vectors" fulfil the postulates for the second quantized fermion fields, Eq. (16), and the Clifford even "basis vectors" have the properties of the internal spaces of their corresponding gauge fields, Eqs. (19,20,23). In odd dimensional spaces, the Clifford odd and even "basis vectors" have unusual properties resembling properties of the internal spaces of the Faddeev-Popov ghosts, as we described in [18].
The rest of the "basis vectors" in odd dimensional spaces, 2 2n 2 −1 × 2 2n 2 −1 , follow if S 0 2n+1 is applied on these half of the "basis vectors". Since S 0 2n+1 are Clifford even operators, they do not change the oddness or evenness of the "basis vectors".
For the Clifford odd "basis vectors", the 2 d−1 2 −1 members appearing in 2 d−1 2 −1 families and representing the part which is the same as in even, d = 2n, dimensional space are present on the left-hand side of Eq. (24), the part obtained by applying S 0 2n+1 on the one of the left-hand side is presented on the right hand side. Below the "basis vectors" and their Hermitian conjugated partners are presented.
The application of S 0d orS 0d on the left-hand side of the "basis vectors" (and the Hermitian conjugated partners of both) generate the whole set of 2 × 2 d−2 members of the Clifford odd "basis vectors" and their Hermitian conjugated partners in d = (2n + 1)-dimensional space appearing on the left-hand side and the right-hand sides of Eq. (24).
2 −1 +k ′ on the right-hand side of Eq. (24) obtain properties of the two groups (they are orthogonal to each other; the algebraic products, * A , of a member from one group, and any member of another group give zero) with the Hermitian conjugated partners within the same group; they have properties of the Clifford even "basis vectors" from the point of view of the Hermiticity property: The operators γ a are up to a constant the self-adjoint operators, while S 0d transform one nilpotent into a projector.
S ab do not change the Clifford oddness ofb m † f , andb m f ;b m † f remain to be Clifford odd objects, however, with the properties of boson fields.
The right hand side of Eq. (24), although anti-commuting, is resembling the properties of the Clifford even "basis vectors" on the left hand side of Eq. (25), while the right-hand side of Eq. (25), although commuting, resembles the properties of the Clifford odd "basis vectors", from the left hand side of Eq. (24): γ a are up to a constant the self adjoint operators, while S 0d transform one nilpotent into a projector (or one projector into a nilpotent). However, S ab do not change Clifford evenness of I A m † f , i = (I, II).
Clifford even It can clearly be seen that the left-hand side of the Clifford odd "basis vectors" and the right-hand side of the Clifford even "basis vectors", although the former are the Clifford odd objects and the latter are Clifford even objects, have similar properties [18].

Example demonstrating properties of Clifford odd and even "basis
vectors" for d = (5 + 1) Subsect. 2.3 demonstrates the properties of the Clifford odd and even "basis vectors" in the special case when d = (5 + 1) to clear up the relations of the Clifford odd and even "basis vectors" to fermion and boson fields, respectively. Table 1 presents the 64 (= 2 d=6 ) "eigenvectors" of the Cartan subalgebra members of the Lorentz algebra, S ab and S ab , Eq. (8).
The Clifford odd "basis vectors" -they appear in 4 (= 2 d=6 2 −1 ) families, each family has 4 members -are products of an odd number of nilpotents, either of three or one. They appear in the group named odd Ib m † f . Their Hermitian conjugated partners appear in the second group named odd IIb m f . Within each of these two groups the members are mutually orthogonal (which can be checked by using Eq. (47) The "basis vectors" and their Hermitian conjugated partners are normalized as [−]) is normalized to one: < ψ oc |ψ oc >= 1.
The more extended overview of the properties of the Clifford odd "basis vectors" and their Hermitian conjugated partners for the case d = (5 + 1) can be found in Ref. [14].
The Clifford even "basis vectors" are products of an even number of nilpotents -of either two or none in this case. They are presented in Table 1 18). An overview of the Clifford even "basis vectors" and their Hermitian conjugated partners for the case d = (5 + 1) can be found in Ref. [15]. Table 1: 2 d = 64 "eigenvectors" of the Cartan subalgebra of the Clifford odd and even algebrasthe superposition of odd and even products of γ a 's -in d = (5 + 1)-dimensional space are presented, divided into four groups. The first group, odd I, is chosen to represent "basis vectors", namedb m † f , appearing in 2 d 2 −1 = 4 "families" (f = 1, 2, 3, 4), each "family" with 2 d 2 −1 = 4 "family" members (m = 1, 2, 3, 4). The second group, odd II, contains Hermitian conjugated partners of the first group for each family separately,b m f = (b m † f ) † . Either odd I or odd II are products of an odd number of nilpotents (one or three) and projectors (two or none). The "family" quantum numbers ofb m † f , that is the eigenvalues of (S 03 ,S 12 ,S 56 ), are for the first odd I group appearing above each "family", the quantum numbers of the family members (S 03 , S 12 , S 56 ) are written in the last three columns. For the Hermitian conjugated partners of odd I, presented in the group odd II, the quantum numbers (S 03 , S 12 , S 56 ) are presented above each group of the Hermitian conjugated partners, the last three columns tell eigenvalues of (S 03 ,S 12 ,S 56 ). The two groups with the even number of γ a 's, even I and even II, each group has their Hermitian conjugated partners within its group, have the quantum numbers f , that is the eigenvalues of (S 03 ,S 12 ,S 56 ), written above column of four members, the quantum numbers of the members, (S 03 , S 12 , S 56 ), are written in the last three columns. To find the quantum numbers of (S 03 , S 12 , S 56 ) one has to take into account that S ab = S ab +S ab .
Let us check what does the algebraic application, * A , of IÂ m=1 † f =4 , for example, presented in Table 1 in the first line of the fourth column of even I, do on the Clifford odd "basis vector"b m=2 † f =2 , presented in odd I as the second member of the second column. (This can easily be evaluated by taking into account Eq. (47) for any m.) The sign → means that the relation is valid up to the constant. The Hermitian conjugated partner of ). This means that Clifford even "basis vector" changes the family members quantum numbers of the Clifford odd "basis vector", leaving the family quantum numbers unchanged.
One can find that the algebraic application, * A , of IÂ 1 † 3 (≡

56
[+]) onb 1 † 1 leads to the same family member of the same family f = 1, namely tob 1 † 1 . Calculating the eigenvalues of the Cartan subalgebra members, Eq. (8), before and after the algebraic multiplication, * A , assures us that IÂ m † 3 carry the integer eigenvalues of the Cartan subalgebra members, namely of S ab = S ab +S ab , since they transfer to the Clifford odd "basis vector" integer eigenvalues of the Cartan subalgebra members, changing the Clifford odd "basis vector" into another Clifford odd "basis vector" of the same family.
We, therefore, confirm that the algebraic application of  2, 3, 4). All IIÂm f giving non zero contributions, keep the family member quantum numbers of the Clifford odd "basis vectors" unchanged, changing the family quantum number. All the rest give zero contribution.
The Cartan subalgebra has in d = (5 + 1)-dimensional space 3 members. To illustrate that the Clifford even "basis vectors" have the properties of the gauge fields of the corresponding Clifford odd , respectively, and one singlet denoted by the square. (τ 3 = 0, τ 8 = 0, τ ′ = − 1 2 ). The triplet and the singlet appear in four families, with the family quantum numbers presented in the last three columns of Table 2. .
"basis vectors" let us study properties of the SU(3) ×U(1) subgroups of the Clifford odd and Clifford even "basis vectors". We need the relations between S ab and (τ 3 , τ 8 , τ ' ) The corresponding relations for (τ 3 ,τ 8 ,τ ′ ) can be read from Eq. (29), if replacing S ab byS ab . The corresponding relations for superposition of the Cartan subalgebra elements (τ ′ , τ 3 , τ 8 ) for S ab = S ab +S ab follow if in Eq. (29) S ab is replaced by S ab .
The corresponding table for the Clifford even "basis vectors" IIÂm f are not presented. IIÂm f carry, namely, the same quantum numbers (τ 3 , τ 8 , τ ' ) as IÂm f . There are only products of nilpotents and projectors which distinguish among IÂm f and IIÂm f , causing differences in properties with respect to the Clifford odd "basis vectors"; IIÂm In the case that the group SO(5, 1) -manifesting as SU(3) × U(1) and representing the colour group with quantum numbers (τ 3 , τ 8 ) and the "fermion" group with the quantum number τ , -is  f mb [+i] embedded into SO(13, 1) the triplet would represent quarks (and antiquarks), and the singlet leptons (and antileptons). The corresponding gauge fields, presented in Table 3 and Fig. 2, if belonging to the sextet, would transform the triplet of quarks among themselves, changing the colour and leaving the "fermion" quantum number equal to 1 6 . Table 3 presents the Clifford even "basis vectors" IÂ m † f for d = (5 + 1) with the properties: i. They are products of an even number of nilpotents, iv. They have properties of the boson gauge fields. When the Clifford even "basis vectors", IÂ m † f , apply on the Clifford odd "basis vectors" (offering the description of the fermion fields) they transform the Clifford odd "basis vectors" into another Clifford odd "basis vectors" of the same family, transferring to the Clifford odd "basis vectors" the integer spins with respect to the SO(d − 1, 1) group, while with respect to subgroups of the SO(d−1, 1) group they transfer appropriate superposition of the eigenvalues (manifesting the properties of the adjoint representations of the corresponding subgroups.) If, for example, IÂ 1 † 3 applies on a singletb 1 † 1 keeps the internal space ofb 1 † 1 unchanged (it can change only momentum), while if IÂ 2 † 3 applies onb 1 † 1 transforms it to a member of a triplet, tob 2 † 1 .

Second quantized fermion and boson fields with internal spaces described by Clifford "basis vectors" in even dimensional spaces
We learned in the previous Subsects. (2.2, 2.3) that in even dimensional spaces (d = 2(2n+1) or d = 4n) the Clifford odd and the Clifford even "basis vectors", which are the superposition of the Clifford odd and the Clifford even products of γ a 's, respectively, offer the description of the internal spaces of fermion and boson fields. The Clifford odd algebra offers 2 . Nilpotents and projectors are (chosen to be) eigenvectors of the Cartan subalgebra members of the Lorentz algebra in the internal space of S ab for the Clifford odd "basis vectors" and of S ab (= S ab +S ab ) for the Clifford even "basis vectors".
To define the creation operators, for fermions or bosons, besides the "basis vectors" defining the internal space of fermions and bosons, the basis in ordinary space in momentum or coordinate representation is needed. Here Ref. [14], Subsect. 3.3 and App. J is overviewed.
Let us introduce the momentum part of the single-particle states. (The extended version is presented in Ref. [14] in Subsect. 3.3 and App. J.) with the normalization < 0 p | 0 p >= 1. While the quantized operatorsˆ p andˆ x commute {p i ,p j } − = 0 and {x k ,x l } − = 0, it follows for {p i ,x j } − = iη ij . One correspondingly finds . The internal space of either fermion or boson fields has the finite number of "basis vectors", 2 The creation operators for either fermions or bosons must be tensor products, * T , of both contributions, the "basis vectors" describing the internal space of fermions or bosons and the basis in ordinary momentum or coordinate space.
The creation operators for a free massless fermion of the energy p 0 = | p|, belonging to a family f and to a superposition of family members m applying on the vacuum state |ψ oc > * T |0 p > can be written as ( [14], Subsect.3.3.2, and the references therein) where the vacuum state for fermions |ψ oc > * T |0 p > includes both spaces, the internal part, Eq.(15), and the momentum part, Eq. (31) (in a tensor product for a starting single particle state with zero momentum, from which one obtains the other single fermion states of the same "basis vector" by the operatorb † p which pushes the momentum by an amount p 10 ). 10 The creation operators and their Hermitian conjugated annihilation operators in the coordinate representation can be read in [14] and the references therein Eqs. (55,57,64) and the references therein).
The creation operators and annihilation operators for fermion fields fulfil the anti-commutation relations for the second quantized fermion fields 11 12 .
The creation operatorsb s † f ( p) and their Hermitian conjugated partners annihilation operatorsb s f ( p), creating and annihilating the single fermion states, respectively, fulfil when applying the vacuum state, |ψ oc > * T |0 p >, the anti-commutation relations for the second quantized fermions, postulated by Dirac (Ref. [14], Subsect. 3.3.1, Sect. 5). 13 To write the creation operators for boson fields, we must take into account that boson gauge fields have the space index α, describing the α component of the boson field in the ordinary space 14 . We, therefore, add the space index α as follows.
We treat free massless bosons of momentum p and energy p 0 = | p| and of particular "basis vectors" iÂ m † f 's which are eigenvectors of all the Cartan subalgebra members 15 , i C m f α carry the space index α of the boson field. Creation operators operate on the vacuum state |ψ ocev > * T |0 p > with the internal 11 Let us evaluate: 12 Two fermion states (formed from two creation operators applying on the vacuum state) with the orthogonal basis part in ordinary space (with two different momenta in ordinary space in the case of free fields) "do not meet"; correspondingly, each can carry the same "basis vector". They must differ in the internal basis if they have the identical ordinary part of the basis. (Otherwise, the tensor product, * TH , of such two fermion states is zero.) Illustration: Let us treat an atom with many electrons. Each electron has a spin of either 1/2 or −1/2. Their orthogonal basis in ordinary space allows them to have the internal spin ±1/2 (leading to total angular momentum either ±1/2 or larger due to the angular momentum in ordinary space). As mentioned in the introduction section in a.iii. the Hilbert space of the second quantized fermion states is represented by the tensor products, * TH , of all possible members of creation operators from zero to infinity applying on the simple vacuum state. For any of these members the scalar product is obtained by multiplying from the left hand side by their Hermitian conjugated partner. 13 The anti-commutation relations of Eq. (34) are valid also if we replace the vacuum state, |ψ oc > |0 p >, by the Hilbert space of the Clifford fermions generated by the tensor products multiplication, * TH , of any number of the Clifford odd fermion states of all possible internal quantum numbers and all possible momenta (that is, of any number ofb s † f ( p) of any (s, f, p)), Ref. ([14], Sect. 5.).
14 In the spin-charge-family theory the Higgs's scalars origin in the boson gauge fields with the vector index (7,8), Ref. ([14], Sect. 7.4.1, and the references therein). 15 In the general case, the energy eigenstates of bosons are in a superposition of iÂ m † f , for either i = I or i = II. One example, which uses the superposition of the Cartan subalgebra eigenstates manifesting the SU (3) × U (1) subgroups of the group SO(5, 1), is presented in Fig. 2. space part just a constant, |ψ ocev >= | 1 >, and for a starting single boson state with zero momentum from which one obtains the other single boson states with the same "basis vector" by the operatorsb † p which push the momentum by an amount p, making also i C m f α depending on p. For the creation operators for boson fields in a coordinate representation one finds using Eqs. (31,32) iÂm To understand what new the Clifford algebra description of the internal space of fermion and boson fields, Eqs. (35,36,33), bring to our understanding of the second quantized fermion and boson fields and what new can we learn from this offer, we need to relate ab c ab ω abα and mf II C m f α . The gravity fields, the vielbeins and the two kinds of spin connection fields, f a α , ω abα ,ω abα , respectively, are in the spin-charge-family theory (unifying spins, charges and families of fermions and offering not only the explanation for all the assumptions of the standard model but also for the increasing number of phenomena observed so far) the only boson fields in d = (13 + 1), observed in d = (3 + 1) besides as gravity also as all the other boson fields with the Higgs's scalars included [11].
We, therefore, need to relate: Let be repeated that IÂ m † f are chosen to be the eigenvectors of the Cartan subalgebra members, Eq. (8). Correspondingly we can relate a particular IÂ m † f I C m f α with such a superposition of ω abα 's, which is the eigenvector with the same values of the Cartan subalgebra members as there is a particular IÂ m † f C mf α . We can do this in two ways: i. Using the first relation in Eq. (37). On the left hand side of this relation S ab 's apply onb m † f part ofb m † f ( p). On the right hand side IÂ m † f apply as well on the same "basis vector"b m † f . ii. Using the second relation, in which S cd apply on the left hand side on ω abα 's, on each ω abα separately; c ab mf are constants to be determined from the second relation, where on the right-hand side of this relation S cd (= S cd +S cd ) apply on the "basis vector" IÂ m † f of the corresponding gauge field 16 .
We must treat equivalently also IIÂ m † f II C m f α andω abα .
Let us conclude this section by pointing out that either the Clifford odd "basis vectors",b m † f , or the Clifford even "basis vectors", iÂ m † f , i = (I, II), have each in any even d, 2

Conclusions
In the spin-charge-family theory [6,8,11,9,22,12,14] the Clifford odd algebra describes the internal space of fermion fields. The Clifford odd "basis vectors" -the superposition of odd products of γ a 's -in a tensor product with the basis in ordinary space form the creation and annihilation operators, in which the anti-commutativity of the "basis vectors" is transferred to the creation and annihilation operators for fermions, explaining the second quantization postulates for fermion fields.
The Clifford odd "basis vectors" have all the properties of fermions: Half integer spins concerning the Cartan subalgebra members of the Lorentz algebra in the internal space of fermions in even dimensional spaces (d = 2(2n + 1) or d = 4n), as discussed in Subsects. (2.2, 2.4) (and in App A in a pedagogical way). With respect to the subgroups of the SO(d − 1, 1) group the Clifford odd "basis vectors" appear in the fundamental representations, as illustrated in Subsects. 2.3.
In this article, it is demonstrated that Clifford even algebra is offering the description of the internal space of boson fields. The Clifford even "basis vectors" -the superposition of even products of γ a 'sin a tensor product with the basis in ordinary space form the creation and annihilation operators which manifest the commuting properties of the second quantized boson fields, offering the explanation for the second quantization postulates for boson fields [16,15]. The Clifford even "basis vectors" have all the properties of boson fields: Integer spins for the Cartan subalgebra members of the Lorentz algebra in the internal space of bosons, as discussed in Subsects. 2.2.
With respect to the subgroups of the SO(d − 1, 1) group the Clifford even "basis vectors" manifest the adjoint representations, as illustrated in Subsect. 2.3.
The operators in each of the two Clifford subalgebras appear in even-dimensional spaces in two groups of 2 There are as well the Clifford even operators (the even products of either γ a 's in one subalgebra or of γ a 's in another subalgebra) which again appear in two groups of 2  Table 1.
The Grassmann algebra operators are expressible with the operators of the two Clifford subalgebras and opposite, Eq. (5). The two Clifford sub-algebras are independent of each other, Eq. (6), forming two independent spaces.
Either the Grassmann algebra [12] or the two Clifford subalgebras can be used to describe the internal space of anti-commuting objects, if the superposition of odd products of operators (θ a 's or γ a 's, orγ a 's) are used to describe the internal space of these objects. The commuting objects must be a superposition of even products of operators (θ a 's or γ a 's orγ a 's).
No integer spin anti-commuting objects have been observed so far, and to describe the internal space of the so far observed fermions only one of the two Clifford odd subalgebras are needed.
The problem can be solved by reducing the two Clifford subalgebras to only one, the one (chosen to be) determined by γ a 's. The decision thatγ a 's apply on γ a as follows: {γ a B = (−) B i Bγ a } |ψ oc >, Eq. (7), (with (−) B = −1, if B is a function of an odd products of γ a 's, otherwise (−) B = 1) enables that 2 d 2 −1 irreducible representations of S ab = i 2 {γ a , γ b } − (each with the 2 d 2 −1 members) obtain the family quantum numbers determined byS ab = i 2 {γ a ,γ b } − . The decision to use in the spin-charge-family theory in d = 2(2n+1), n ≥ 3 (d ≥ (13+1) indeed), the superposition of the odd products of the Clifford algebra elements γ a 's to describe the internal space of fermions which interact with gravity only (with the vielbeins, the gauge fields of momenta, and the two kinds of the spin connection fields, the gauge fields of S ab andS ab , respectively), Eq. (1), offers not only the explanation for all the assumed properties of fermions and bosons in the standard model, with the appearance of the families of quarks and leptons and antiquarks and antileptons ( [14] and the references therein) and of the corresponding vector gauge fields and the Higgs's scalars included [11], but also for the appearance of the dark matter [35] in the universe, for the explanation of the matter/antimatter asymmetry in the universe [8], and for several other observed phenomena, making several predictions [7,33,34,36].
The recognition that the use of the superposition of the even products of the Clifford algebra elements γ a 's to describe the internal space of boson fields, what appears to manifest all the properties of the observed boson fields, as demonstrated in this article, makes clear that the Clifford algebra offers not only the explanation for the postulates of the second quantized anti-commuting fermion fields but also for the postulates of the second quantized boson fields.
This recognition, however, offers the possibility to relate where the relations among IÂ m † f I C m f α and IIÂ m † f II C m f α with respect to ω abα andω abα , not discussed directly in this article, need additional study and explanation.
Although the properties of the Clifford odd and even "basis vectors" and correspondingly of the creation and annihilation operators for fermion and boson fields are, hopefully, demonstrated in this article, yet the proposed way of the second quantization of fields, the fermion and the boson ones needs further study to find out what new can the description of the internal space of fermions and bosons bring into the understanding of the second quantized fields.
This study showing up that the Clifford algebra can be used to describe the internal spaces of fermion and boson fields equivalently, offering correspondingly the explanation for the second quantization postulates for fermion and boson fields is opening a new insight into the quantum field theory, since studies of the interaction of fermion fields with boson fields and of boson fields with boson fields so far looks very promising.
The study of properties of the second quantized boson fields, the internal space of which is described by Clifford even algebra has just started and needs further consideration. members in any of two "families" of the group ofb m † account the above concrete evaluations, the relations of Eq. (16) for our particular casê one sees that I A 2 † Recognizing that internal spaces of fermion fields and their corresponding boson gauge fields are describable in even dimensional spaces by the Clifford odd and even "basis vectors", respectively, it becomes evidently that when including the basis in ordinary space, we must take into account that boson gauge fields have the space index α, which describes the α component of the boson fields in ordinary space.
We multiply, therefore, as presented in Eq. (35), the Clifford even "basis vectors" with the coefficient i C m f α carrying the space index α so that the creation operators iÂ m † f α ( p) =b † p * T i C m f α iÂ m † f , i = (I, II) carry the space index α 22 . The self-adjoint "basis vectors", like ( iÂ 1 † 1α , iÂ 2 † 2α , i = (I, II)), do not change quantum numbers of the Clifford odd "basis vectors", since they have internal quantum numbers equal to zero.
In higher dimensional space, like in d = (5 + 1), IÂ 1 † 3 , presented in Table 3, could represent the internal space of a photon field, which transfers to, for example, a fermion and anti-fermion pair with the internal space described by (b 1 † 1 ,b 3 † 1 ), presented in Table 2, the momentum in ordinary space. The subgroup structure of SU(3) gauge fields can be recognized in Fig. 2.
Properties of the gauge fields iÂ m † f α need further studies. In even dimensional spaces, the Clifford odd and even "basis vectors", describing internal spaces of fermion and boson fields, offer the explanation for the second quantized postulates for fermion and boson fields [17]. There are many suggestions in the literature for unifying charges in larger groups, adding additional groups for describing families [1,2,3,4,5], or by going to higher dimensional spaces of the Kaluza-Kline like theories [24,25,26,27,28,29,31,30], what also the spin-charge-family is.
Let me present some open questions of the standard model and briefly tell the answers offered by the spincharge family theory.
A. Where do fermions -quarks and leptons and antiquarks and antileptons -and their families originate? The answer offered by the spin-charge-family theory: In d = (13+1) one irreducible representation of SO(13, 1) analysed with respect to subgroups SO(7, 1) (containing subgroups of SO(3, 1) × SU (2) × SU (2)) and SO(6) (containing subgroups of SU (3) × U (1)) offers the Clifford odd "basis vectors", describing the internal spaces of quarks and leptons and antiquarks and antileptons, Table 4, as assumed by the standard model. The Clifford odd "basis vectors" appear in families. B. Why are charges of quarks so different from charges of leptons, and why have left-handed family members so different charges from the right-handed ones? The answer offered by the spin-charge-family theory: The SO(7, 1) part of the "basis vectors" is identical for quarks and leptons and identical for antiquarks and antileptons, Table 4, they distinguish only in the SU (3), the colour or anticolour part, and in the fermion or antifermion U (1) quantum numbers. All families have the same content of SO(7, 1), SU (3) and U (1) with respect to S ab . They distinguish only in the family quantum number, determined byS ab . The difference between left-handed and right-handed members appears due to the difference in one quantum numbers of the two SU (2) groups, as seen in Table 4. C. Why do family members -quarks and leptons -manifest such different masses if they all start as massless, as (elegantly) assumed by the standard model?
The answer offered by the spin-charge-family theory: Masses of quarks and leptons are in this theory determined by the spin connection fields ω stσ , the gauge fields of S ab 23 , and byω stσ , the gauge fields ofS ab , which are the same for quarks and leptons 24 . Triplets and singlets are scalar gauge fields with the space index σ = (7,8). They have, with respect to the space index, the quantum numbers of the Higgs scalars, Ref. ([14], Table 8, Eq. (110,111)). D. What is the origin of boson fields, of vector fields which are the gauge fields of fermions, and the Higgs' scalars and the Yukawa couplings? Have all boson fields, with gravity and scalar fields included a common origin? The answer offered by the spin-charge-family theory: In a simple starting action, Eq. (1), boson fields origin in gravity -in vielbeins and two kinds of spin connection fields, ω abα andω abα , in d = (13 + 1) -and manifest in d = (3 + 1) as vector gauge fields, α = (0, 1, 2, 3), or scalar gauge fields, α ≥ 5 [11], ( [14], Sect. 6 and references therein). Boson gauge fields are massless as there are fermion fields. The breaks of the starting symmetry makes some gauge fields massive. This article describes the internal space of boson fields by the Clifford even basis vectors, manifesting as the boson gauge fields of the corresponding fermion fields described by the Clifford odd "basis vectors". The description of the boson fields with the Clifford even "basis vectors" confirms the existence of two kinds of spin connection fields as we see in Sects. 2.2 and2.3, but also open a door to a new understanding of gravity. According to the starting action, Eq. (1), all gauge fields start in d ≥ (13 + 1) as gravity. E. How are scalar fields connected with the origin of families? How many scalar fields determine properties of the so far (and others possibly be) observed fermions and of weak bosons? The answer offered by the spin-charge-family theory: The interaction between quarks and leptons and the scalar gauge fields, which at the electroweak brake obtain constant values, causes that quarks and leptons and the weak bosons become massive. There are three singlets, they distinguish among quarks and leptons, and two triplets, they do not distinguish among quarks and leptons, which give masses to the lower four families 25 . F. Where does the dark matter originate? The answer offered by the spin-charge-family theory: The theory predicts two groups of four families at low energy. The stable of the upper four groups are candidates to form the dark matter [35]. G. Where does the "ordinary" matter-antimatter asymmetry originate? The answer offered by the spin-charge-family theory: The theory predicts scalars triplets and antitriplets with the space index α = (9, 10, 11, 12, 13, 14) [8].
H. How can we understand the second quantized fermion and boson fields? The answer offered by the spin-charge-family theory: The main contribution of this article, Sect. 2, is the description of the internal spaces of fermion and boson fields with the superposition of odd (for fermions) and even (for bosons) products of γ a . The corresponding creation and annihilation operators, which are tensor, * T , products of (finite number) "basis vectors" and (infinite) basis in ordinary space inherit anti-commutativity or commutativity from the corresponding "basis vectors", explaining the postulates for the second quantized fermion and boson fields. I. What is the dimension of space? (3 + 1)?, ((d − 1) + 1)?, ∞? The answer offered by the spin-charge-family theory: We observe (3 + 1)-dimensional space. In order that one irreducible representation (one family) of the Clifford odd "basis vectors", analysed with respect to subgroups SO(3, 1)× SO(4) ×SU (3) ×U (1) of the group SO(13, 1) includes all quarks and leptons and antiquarks and antileptons, the space must have d ≥ (13 + 1). (Since the only "elegantly" acceptable numbers are 0 and ∞, the space-time could be ∞.) The SO(10) theory [2], for example, unifies the charges of fermions and bosons separately. Analysing SO(10) with respect to the corresponding subgroups, the charges of fermions appear in fundamental representations and bosons in adjoint representations 26 .
There are additional open questions answers of which the spin-charge-family the theory offers. D One family representation of Clifford odd "basis vectors" in d = (13 + 1) This appendix, is following App. D of Ref. [18], with a short comment on the corresponding gauge vector and scalar fields and fermion and boson representations in d = (14 + 1)-dimensional space included. In even dimensional space d = (13 + 1) ( [15], App. A), one irreducible representation of the Clifford odd "basis vectors", analysed from the point of view of the subgroups SO(3, 1) × SO(4) (included in SO(7, 1)) and SO(7, 1)×SO(6) (included in SO(13, 1), while SO(6) breaks into SU(3)×U(1)), contains the Clifford odd "basis vectors" describing internal spaces of quarks and leptons and antiquarks, and antileptons with the quantum numbers assumed by the standard model before the electroweak break. Since SO(4) contains two SU(2) groups, Y = τ 23 + τ 4 , one irreducible representation includes the right-handed neutrinos and the left-handed antineutrinos, which are not in the standard model scheme.
The Clifford even "basis vectors", analysed to the same subgroups, offer the description of the internal spaces of the corresponding vector and scalar fields, appearing in the standard model before the electroweak break [16,15]; as explained in Subsect. 2.2.1.
For an overview of the properties of the vector and scalar gauge fields in the spin-charge-family theory, the reader is invited to see Refs. ( [14,11] and the references therein). The vector gauge fields, expressed as the superposition of spin connections and vielbeins, carrying the space index m = (0, 1, 2, 3), manifest properties of the observed boson fields. The scalar gauge fields, causing the electroweak break, carry the space index s = (7,8) and determine the symmetry of mass matrices of quarks and leptons. [23, ?, ?]).
In this Table 4  , for the hyper charge Y = τ 23 + τ 4 , and electromagnetic charge Q = Y + τ 13 , one reproduces all the quantum numbers of quarks, leptons, and antiquarks, and antileptons. One notices that the SO(7, 1) part is the same for quarks and leptons and the same for antiquarks and antileptons. Quarks distinguish from leptons only in the colour and "fermion" quantum numbers and antiquarks distinguish from antileptons only in the anti-colour and "anti-fermion" quantum numbers.
Let me point out that in addition to the electroweak break of the standard model the break at ≥ 10 16 GeV is needed ( [14], and references therein). The condensate of the two right-handed neutrinos causes this break (Ref. [14], Table 6); it interacts with all the scalar and vector gauge fields, except the weak, U(1), SU(3) and the gravitational field in d = (3 + 1), leaving these gauge fields massless up to the electroweak break, when the scalar fields, leaving massless only the electromagnetic, colour and gravitational fields, cause masses of fermions and weak bosons.
The theory predicts two groups of four families: To the lower group of four families, the three so far observed contribute. The theory predicts the symmetry of both groups to be SU(2) × SU(2) × U(1), Ref. ([14], Sect. 7.3), which enable to calculate mixing matrices of quarks and leptons for the accurately enough measured 3 × 3 sub-matrix of the 4 × 4 unitary matrix. No sterile neutrinos are needed, and no symmetry of the mass matrices must be guessed [36].
In the literature, one finds a lot of papers trying to reproduce mass matrices and measured mixing matrices for quarks and leptons [41,42,43,44,45,47,46].
The stable of the upper four families predicted by the spin-charge-family theory is a candidate for the dark matter, as discussed in Refs. [35,14]. In the literature, there are several works suggesting candidates for the dark matter and also for matter/antimatter asymmetry [49,48].