i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 84 — #100 i
i
i
i
i
i
BLED WORKSHOPS
IN PHYSICS
VOL. 21, NO. 1
Proceedings to the 23rd [Virtual]
Workshop, Volume 1
What Comes Beyond . . . (p. 84)
Bled, Slovenia, July 4–12, 2020
7 How Far has so Far the Spin-Charge-Family
Theory Succeeded To explain the Standard Model
Assumptions, the Matter-Antimatter Asymmetry, the
Appearance of the Dark Matter, the Second Quantized
Fermion Fields. . . , Making Several Predictions
N.S. Mankoč Borštnik
University of Ljubljana, Slovenia
Abstract. The assumptions of the standard model, which 50 years ago offered an elegant
new step towards understanding basic fermion and boson fields, are still waiting for an
explanation. The spin-charge-family theory is promising not only in explaining the standard
model postulates but also in explaining the cosmological observations, like there are the
appearance of the dark matter, of the matter-antimatter asymmetry, making several predictions.
This theory assumes that the internal degrees of freedom of fermions (spins, handedness
and all the charges) are described by the Clifford algebra objects in d ≥ (13+1)-dimensional
space. Fermions interact with only the gravity (the vielbeins and the two kinds of the spin
connection fields, which manifest in d = (3 + 1) as all the vector gauge fields as well as the
scalar fields - the higgs and the Yukawa couplings). The theory describes the internal space
of fermions with the Clifford objects which are products of odd numbers of γa objects,
what offers the explanation for quantum numbers of quarks and leptons and anti-quarks
and ani-leptons, with family included. In this talk I overview shortly the achievements
of the spin-charge-family theory so far and in particular the explanation of the second
quantization procedure offered by the description of the internal space of fermions with
the anticommuting Clifford algebra objects of the odd character. The theory needs still to
answer many open questions that it could be accepted as the next step beyond the standard
model.
Povzetek. Privzetki Standardnega Modela, ki je pred 50 leti ponudil eleganten opis osnovnih
fermionskh in bozonskih polj, so še vedno nepojasnjeni. Teorija spinov-nabojev-družin ponuja,
poleg razlage privzetkov Standardnega Modela, tudi razlago nekaterih kozmoloških opažanj,
kot je pojav temne snovi, asimetrije snovi in antisnovi, ponudi pa tudi več napovedi. Teorija
privzame, da so notranje prostostne stopnje fermionov (spin, ročnost in vsi naboji) opisane
z objekti Cliffordove algebre v prostoru z razsežnostjo d ≥ (13 + 1). Fermioni interagirajo
samo z gravitacijskim poljem (s tetradami in dvema vrstama spinskih povezav), ki se
v prostoru d = (3 + 1) predstavi kot običajna gravitacija, kot vsa poznana vektorska
umeritvena polja ter kot skalarna umeritvena polja, ki pojacnijo pojav Higgsovega skalarja
in Yukawinih sklopitev. Notranje prostostne stopnje fermionov opisuje avtorica teorije s
Cliffordovo algebro, ki ponudi razumevanje privzetkov za lastnosti kvarkov in leptonov
in njihovih družin, v Standardnem Modelu. V predavanju avtorica na kratko predstavi
dosedanje dosežke Teorije spinov-nabojev-družin, napovedi teorije ter tudi odprta vprašanja.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 85 — #101 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 85
Poudarek predavanja je na ponudbi drugačne poti do druge kvantizacije fermionov kot je
splošno privzeta Diracova. Opis notranjega prostora fermionov z objekti, ki antikomutirajo,
pojasni antikomutacjske lastnosti fermionov v drugi kvantizaciji. Predstavi tudi odprta
vprašanja, ki jih je potrebno rešiti, da bo teorija lahko sprejeta kot nov korak k razumevanju
vesolja in osnovnih gradnikov vesolja.
Keywords: Beyond the standard model, Gravity as the only gauge fields, Kaluza-
Klein-like theories, Higher dimensional spaces, Dark matter, Matter/antimatter
asymmetry, Four families of quarks and leptons, Second quantization of fermion
fields in Clifford and in Grassmann space, Explanation of the Dirac postulates
7.1 Introduction
Let us start with the motivation for the spin-charge-family theory.
The standard model offered an elegant new step towards understanding ele-
mentary fermion and boson fields by postulating (the inspiration came from the
experiments):
a. The existence of massless fermion family members with the spins and charges
in the fundamental representation of the groups, a.i. the quarks as colour triplets
and colouress leptons, a.ii the left handed members as the weak doublets, a.ii.
the right handed weak chargeless members, a.iii. the left handed quarks differ-
ing from the right handed leptons in the hyper charge, a.iv. all the right handed
members differing among themselves in hyper charges, a.v. antifermions carry
the corresponding anticharges of fermions and opposite handedness, a.vi. the
number of massless families, determined by experiments (there is no right handed
neutrino postulated, since it would carry none of the so far observed charges, and
correspondingly there is also no left handed antineutrino allowed).
b. The existence of massless vector gauge fields to the observed charges of quarks
and leptons, carrying charges in the adjoint representations of the corresponding
charged groups.
c. The existence of the massive weak doublet scalar higgs, c.i. carrying the weak
charge ±1
2
and the hypercharge ±1
2
(as it would be in the fundamental represen-
tation of the two groups), c.ii. gaining at some step of the expanding universe
the nonzero vacuum expectation value, c.iii. breaking the weak and the hyper
charge and correspondingly breaking the mass protection, c.iv. taking care of the
properties of fermions and of the weak bosons masses, c.v. as well as of the Yukawa
couplings.
d. The presentation of fermions and bosons as second quantized fields.
e. The gravitational field in d = (3+ 1) as independent gauge field.
The standard model assumptions have been confirmed without raising any
doubts so far, but also by offering no explanations for the assumptions. The last
among the fields postulated by the standard model, the scalar higgs, was detected
in June 2012, the gravitational waves were detected in February 2016.
The standard model has in the literature several explanations, mostly with
many new not explained assumptions. The most popular seem to be the grand
unifying theories [14–30]. At least SO(10) offers the explanation for the potulates
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 86 — #102 i
i
i
i
i
i
86 N.S. Mankoč Borštnik
from a.i. to a.iv, partly to b. — but does not explain the assumptions a.v. up to
a.vi., c. and d., and does not connect gravity with gauge vector and scalar fields.
What questions should one ask to be able to find a trustworthy next step
beyond the standard models of elementary particle physics and cosmology, which
would offer understanding of not yet understood phenomena?
i. Where do fermions, quarks and leptons, originate and why do they differ from
the boson fields in spins, charges and statistics?
ii. How can one describe the internal degrees of fermions to explain the Dirac’s
postulates of the second quantization?
iii. Why are charges of quarks and leptons so different, why have the left handed
family members so different charges from the right handed ones and why does
the handedness relate charges to anticharges?
iv. Where do families of quarks and leptons originate and how many families do
exist?
v. Why do family members – quarks and leptons — manifest so different masses if
they all start as massless?
vi. How is the origin of the scalar field (the Higgs’s scalar) and the Yukawa cou-
plings connected with the origin of families and how many scalar fields determine
properties of the so far (and others possibly be) observed fermions and masses of
weak bosons? (The Yukawa couplings certainly speak for the existence of several
scalar fields with the properties of Higgs’s scalar.) Why is the Higgs’s scalar, or
are all scalar fields of similar properties as the higgs, if there are several, doublets
with respect to the weak and the hyper charge? Do possibly exist also scalar fields
with the colour charges in the fundamental representation and where, if they are,
do they manifest?
vii. Where do the so far observed (and others possibly non observed) vector gauge
fields originate? Do they have anything in common with the scalar fields and the
gravitational fields?
viii. Where does the dark matter originate?
ix. Where does the ”ordinary” matter-antimatter asymmetry originate?
x. Where does the dark energy originate and why is it so small?
xi. What is the dimension of space? (3+ 1)?, ((d− 1) + 1)?,∞?
And many others.
My working hypotheses is that a trustworthy next step must offer answers to
several open questions, the more answers to the above open questions the step
covers the greater the possibilities of the theory being the right next step.
I am proposing the spin-charge-family theory [1–10], offering so far the answers
from i. to ix. of the above questions; The more work is invested in this theory the
more answers to the above open questions the theory offers.
Let me make in what follows a short introduction into the spin-charge-family
theory to show briefly up the way the theory is offering the answers to the above
mentioned open questions. A more detailed presentation of the theory and its
achievements are presented in Sect. 7.2.
The spin-charge-family theory is a kind of the Kaluza-Klein like theories [8,
31–38] due to the assumption that in d ≥ 5 — in the spin-charge-family theory
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 87 — #103 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 87
d ≥ (13+ 1) — fermions interact with the gravity only 1, treating consequently all
the vector gauge fields, the scalar gauge fields, and the gravity in an equivalent
way, offering answers to the above questions vi. and vii..
In the spin-charge-family theory the fermion internal space is described by the
”basis vectors”, which are the superposition of the odd products of the Clifford
algebra objects. There are two kinds of the Clifford algebra objects [1, 2, 12, 45, 46].
In d = (13 + 1)-dimensional space the odd Clifford algebra objects of one kind
offer in d = (3 + 1) the description of the spins and all the charges of fermions
and antifermions, since both — fermions and antifermions — appear in the same
irreducible representation of one of the two Lorentz groups in the internal space of
fermions, what consequently explains the connection among the spins, handedness
and charges of fermions, answering the questions i. and iii..
The other kind takes care of the family quantum numbers of fermions, distin-
guishing among different irreducible representations [3, 4, 7, 9], and offering a part
answer to iv..
The creation operators, creating the single particle states, are tensor products
of the superposition of the finite number of the Clifford odd ”basis vectors” of
the internal space and of the infinite basis in the momentum space. The ”basis
vectors” of the internal space transfer their oddness to the creation operators and
correspondingly guarantees the oddness of the single fermion states, since the
vacuum state has an even Clifford character.
The Hilbert space of fermions is formed from all possible tensor products
of any number of single fermion creation operators, operating on the vacuum
state [12].
The spin-charge-family theory offers correspondingly answers to the questions
from i. to iv., explaining the common origin of spins and charges of fermions and
antifermions, of all the quantum numbers of quarks and leptons and antiquarks
and antileptons postulated by the standard model, as well as of the origin of families.
The theory explains as well the Dirac postulates of the second quantization of the
ferrmion fields.
Fermions interact with the vielbeins and the two kinds of the spin connection
fields, the gauge fields of the momenta and of the two kinds of the generators of
the Lorentz transformations, determined by the two kinds of the Clifford algebra
objects [3–10, 12].
The spin connection fields of one kind manifest in d = (3+ 1) as the vector
gauge fields of the charges of fermions, as the gravitational fields and also as the
scalar gauge fields [5], to which also the scalar fields which are the gauge field of
the second kind of the spin connection fields contribute. These offer answers to the
questions vi. and vii., while explaining the common origin of the gravity, the vector
gauge fields of the charges and the scalar gauge fields. The scalar gauge fields of
1 Correspondingly the spin-charge-family theory shares with the Kaluza-Klein like theories
their weak points, at least: a. Not yet solved the quantization problem of the gravitational
field. b. The spontaneous break of the starting symmetry, which would at low energies
manifest the observed almost massless fermions [32]. Concerning this second point we
proved on the toy model of d = (5+ 1) that the break of symmetry can lead to (almost)
massless fermions [68–70].
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 88 — #104 i
i
i
i
i
i
88 N.S. Mankoč Borštnik
both origins — of both generators of the Lorentz transformations in internal space
of fermions — determine the scalar higgs and the Yukawa couplings, which all are
in the standard model postulated.
The two kinds of the Clifford algebra objects require the existence of the two
groups of four families of quarks and leptons and antiquarks and antileptons. The
two groups distinguish from each other with respect to the family quantum num-
bers and correspondingly with respect to the interaction with the different two
groups of the scalar gauge fields, which determine masses of these two groups of
families after the break of the weak and hyper charge symmetries. Consequently:
a. To the observed three families of quarks and leptons and antiquarks and antilep-
tons there must exist the fourth family [3, 9, 49, 51, 53, 54]. b. The second group of
the four families offers the explanation for the existence of the dark matter [52, 61].
The quantum numbers of the weak charge and the hyper charge of the scalar
fields, taking care of the masses of the two groups of four families, depend on the
space index of the scalar fields. The scalar fields with the space indexes 7 and 8 do
carry the weak and the hyper charge as assumed by the standard model, explaining
the origin of scalar higgs and Yukawa couplings [3, 9, 49, 51, 53, 54], what adds the
explanation to the question vi..
There appear in the spin-charge-family the scalar fields with the space indexes
9− 14, which are the colour triplets [4,61]. They cause the transitions of antiquarks
and antileptons into quarks and back. In the expanding universe under the non
equilibrium conditions they offer the explanation of today’s dominance of ordinary
matter in the observed part of the universe.
It remains to tell how does in the spin-charge-family appear the spontaneous
breaking of the starting symmetry in d = (13 + 1), first with the appearance of
the condensate of two right handed neutrinos [3, 4, 9], and then when scalar fields
with space index (7, 8) obtain nonzero vacuum expectation values.
The detailed, although still short, presentation of the spin-charge-family theory
is presented in Sects. 7.2and 7.2.1.
7.2 Short presentation of the spin-charge-family theory
The spin-charge-family theory assumes a simple starting action for fermions, cou-
pled to only gravitational field in d ≥ (13 + 1)-dimensional space through the
vielbeins fαa, the gauge fields of momenta, and the two kinds of the spin connec-
tion fields,ωabα and ω̃abα, the gauge fields of the two kinds of the generators of
the Lorentz transformations of the Clifford algebras, and with the internal space
of fermions described by the anticommuting ”basis vectors” of one of the two
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 89 — #105 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 89
Clifford algebras
A =
∫
ddx E
1
2
(ψ̄ γap0aψ) + h.c.+∫
ddx E (αR+ α̃ R̃) ,
p0a = f
α
ap0α +
1
2E
{pα, Ef
α
a}− ,
p0α = pα −
1
2
Sabωabα −
1
2
S̃abω̃abα ,
R =
1
2
{fα[afβb] (ωabα,β −ωcaαω
c
bβ)}+ h.c. ,
R̃ =
1
2
{fα[afβb] (ω̃abα,β − ω̃caα ω̃
c
bβ)}+ h.c. . (7.1)
Here 2 fα[afβb] = fαafβb − fαbfβa.
As written in the introduction, the tensor products of the superposition of
the finite number of anticommuting ”basis vectors” and of the infinite basis in the
momentum space offer the description of the fermion creation and annihilation
anticommuting operators. The creation and annihilation operators explain the
Dirac postulates of the second quantized fermions, Sect. (7.2.1,7.2.1, 7.2.1).
The single fermion states manifest in d = (3 + 1) space the spins and all
the charges of the observed quarks and leptons and antiquarks and antileptons,
Table 7.3, as well as families, Table 7.4, predicting the fourth family [49–51, 53, 54,
57, 58] to the observed three families and offering the explanation for the observed
dark matter [52, 61].
The spin connection gauge fields manifest in d = (3 + 1) as the ordinary
gravity, the known vector gauge fields, the scalar gauge fields [5] with the prop-
erties of higgs explaining the higgses and the Yukawa couplings, predicting new
vector and scalar fields, which offer explanation for the dark matter [52] and for
matter/antimatter asymmetry [4].
To be in agreement with the observations in d = (3+ 1) the manifoldM(13+1)
must break first intoM(7+1) ×M(6) (which manifests as SO(7, 1) ×SU(3) ×U(1)),
affecting both internal degrees of freedom - the one represented by γa and the one
represented by γ̃a [3].
There is a scalar condensate (Table 7.5) of two right handed neutrinos with
the family quantum numbers of the group of four families (the one which does
not include the observed three families), Table 7.4, which bring masses of the scale
∝ 1016 GeV or higher to all the vector and scalar gauge fields, which interact with
the condensate [4].
2 fαa are inverted vielbeins to eaα with the properties eaαfαb = δab, eaαfβa = δβα, E =
det(eaα). Latin indices a, b, ..,m, n, .., s, t, .. denote a tangent space (a flat index), while
Greek indices α, β, .., µ, ν, ..σ, τ, .. denote an Einstein index (a curved index). Letters from
the beginning of both the alphabets indicate a general index (a, b, c, .. and α, β, γ, .. ),
from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m,n, .. and
µ, ν, ..), indexes from the bottom of the alphabets indicate the compactified dimensions
(s, t, .. and σ, τ, ..). We assume the signature ηab = diag{1,−1,−1, · · · ,−1}.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 90 — #106 i
i
i
i
i
i
90 N.S. Mankoč Borštnik
Since the left handed spinors couple differently (with respect to M(7+1)) to
scalar fields than the right handed ones, the break can leave massless and mass
protected 2((7+1)/2−1) families [68]. The rest of families get heavy masses 3.
There is additional breaking of symmetry: The manifoldM(7+1) breaks fur-
ther to M(3+1)× SU(2) × SU(2) included in M(4). These electroweak break is
caused by the scalar fields with the space index (7, 8). They carry due to the space
index the weak charge and hyper charge [3, 4].
I shall shortly present the influence of the breaks with the condensate and with
the scalar fields (the electroweak break) when presenting properties of fermions
and vector and scalar gauge fields in d = (3+ 1).
7.2.1 Properties of fermion fields in the spin-charge-family theory
Let us start with the properties of the fermion fields in the spin-charge-family theory.
Fermion fields, which are the superposition of tensor products of the anticom-
muting ”basis vectors” describing fermions internal degrees of freedom and of
commuting basis in the momentum (coordinate) space, manifest the anticommut-
ing properties already on the single fermion level [13], demonstrating that the first
quantized fermions are the approximation to the second quantized fields.
There are two kinds of the anticommuting objects [1–3,9,12] — the Grassmann
coordinates and correspondingly the Grassmann operators, θa and ∂
∂θa
, and the
Clifford coordinates/operators, γa and γ̃a, expressible with one another. Either
the Grassmann or the two Clifford algebras offer in d-dimensional space 2 · 2d
operators (the Grassmann algebra has 2d − 1 products of θa’s and 2d − 1 products
of ∂
∂θa
’s and the identity, the two Clifford algebras have each 2d − 1 products of
γa’ and 2d − 1 products of γ̃a’s and the identity) with the properties [12, 13]
{θa, θb}+ = 0 , {
∂
∂θa
,
∂
∂θb
}+ = 0 , {θa,
∂
∂θb
}+ = δab ,
(θa)† = ηaa
∂
∂θa
, (
∂
∂θa
)† = ηaaθa ,
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ , {γ
a, γ̃b}+ = 0 ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a ,
(a, b) = (0, 1, 2, 3, 5, · · · , d) . (7.2)
The identity is the self adjoint member. The signature ηab = diag{1,−1,−1, · · · ,−1}
is assumed.
The two algebras are expressible with one another
3 A toy model [68, 69] was studied in d = (5 + 1) with the same action as in Eq. (7.1).
The break from d = (5 + 1) to d = (3 + 1)× an almost S2 was studied. For a particular
choice of vielbeins and for a class of spin connection fields the manifoldM(5+1) breaks
into M(3+1) times an almost S2, while 2((3+1)/2−1) families remain massless and mass
protected. Equivalent assumption, although not yet proved how does it really work, is
made in the d = (13 + 1) case. This study is in progress quite some time.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 91 — #107 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 91
γa = (θa +
∂
∂θa
) , γ̃a = i (θa −
∂
∂θa
) ,
θa =
1
2
(γa − iγ̃a) ,
∂
∂θa
=
1
2
(γa + iγ̃a) . (7.3)
Let me add the generators of the Lorentz transformations in both algebras
Sab = i (θa
∂
∂θb
− θb
∂
∂θa
) , (Sab)† = ηaaηbbSab ,
Sab =
i
4
(γaγb − γbγa) , S̃ab =
i
4
(γ̃aγ̃b − γ̃bγ̃a) ,
Sab = Sab + S̃ab , {Sab, S̃ab}− = 0 ,
{Sab, γc}− = i(η
bcγa − ηacγb) , {Sab, γ̃c}− = 0 ,
{S̃ab, γ̃c}− = i(η
bcγ̃a − ηacγ̃b) , {S̃ab, γc}− = 0 , (7.4)
The Grassmann algebra offers the description of the integer spin fermions,
with the charges in the adjoint representations. Both Clifford algebras offer the
description of the half integer spin fermions with charges in the fundamental rep-
resentations. Both algebras, the Grassmann algebra and the two Clifford algebras,
can be separated into odd and even parts with odd and even products of algebra
elements.
While in the Grassmann algebra the Hermitian conjugated partners of prod-
ucts of θa’s are the corresponding products of ∂
∂θa
’s, Eq. (7.2), and opposite, in the
Clifford algebras the Hermitian conjugated partners are less transparent, due to
the relations γa† = ηaaγa and γ̃a† = ηaaγ̃a, Eq. (7.2).
In order to resolve the problem of the Hermitian conjugated partners in the
Clifford case and also to be able to make predictions of the theory to be compared
with the experimental results, let us arrange products of θa’s as well as products
of either γa’s or γ̃a’s into irreducible representations with respect to the Lorentz
group with the generators [2] presented in Eq. (7.4) and to arrange the members of
each irreducible representation to be eigenstates of the Cartan subalgebra
S03,S12,S56, · · · ,Sd−1 d ,
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d . (7.5)
The easiest way to achieve this is to find the eigenstates of each member of the
Cartan subalgebras separately.
The observed fermions have the half integer spin and charges in the funda-
mental representations, and there are no fermions observed yet with the integer
spins and charges in the adjoint representations. The spin-charge-family theory must
correspondingly use the Clifford algebras. However, there are also no experimen-
tal evidences that there is any need for two independent representations offered
by the two kinds of the Clifford algebra objects, γa’s and γ̃a’s.
Let us therefore start the discussion about the description of the internal space
of fermions by taking into account the two Clifford algebras and let us leave the
discussion on the Grassmann algebra for later, Ref. [13].
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 92 — #108 i
i
i
i
i
i
92 N.S. Mankoč Borštnik
We can make a choice for the members of the irreducible representations
of the two Lorentz groups to be the ”eigenvectors” of the corresponding Cartan
subalgebras of Eq. (7.5), taking into account Eq. (7.2). If Sab and S̃ab represents
each one of the (d
2
for even d) members of the Cartan subalgebra elements, we
easily check that
Sab
ab
(k) =
k
2
ab
(k) ,
ab
(k):=
1
2
(γa +
ηaa
ik
γb) , (
ab
(k))2 = 0 ,
ab
(k)
†
= ηaa
ab
(−k) ,
Sab
ab
[k] =
k
2
ab
[k] ,
ab
[k]:=
1
2
(1+
i
k
γaγb) , (
ab
[k])2 =
ab
[k] ,
ab
[k]
†
=
ab
[k] ,
S̃ab
ab
˜(k) =
k
2
ab
˜(k) ,
ab
˜(k):=
1
2
(γa +
ηaa
ik
γb) , (
ab
˜(k))2 = 0 ,
ab
˜(k)
†
= ηaa
ab
˜(−k) ,
S̃ab
ab
˜[k] =
k
2
ab
˜[k] ,
ab
˜[k]:=
1
2
(1+
i
k
γaγb) , (
ab
˜[k])2 =
ab
˜[k] ,
ab
˜[k]
†
=
ab
˜[k] . (7.6)
The notation
ab
(k),
ab
[k],
ab
˜(k) and
ab
˜[k] is introduced to simplify the discussions. The
Clifford ”vectors” — nilpotents (
ab
(k)
ab
(k)= 0,
ab
˜(k)
ab
˜(k)= 0) and projectors (
ab
[k]
ab
[k]=
ab
[k]
,
ab
˜[k]
ab
˜[k]=
ab
˜[k] — of both algebras are normalized up to a phase [2, 12, 13].
Both have half integer spins. The ”eigenvalues” of the operator S03 for the
”eigenvectors” 1
2
(γ0 ∓ γ3), for example, are equal to ± i
2
, respectively, for the
”vectors” 1
2
(1± γ0γ3) are ± i
2
, respectively, while all the rest ”eigenvectors” have
”eigenvalues” ± 1
2
. One finds equivalently for the ”eigenvectors” of the operator
S̃03: for 1
2
(γ̃0 ∓ γ̃3) the ”eigenvalues” ± i
2
, respectively, and for the ”eigenvectors”
1
2
(1± γ̃0γ̃3) the ”eigenvalues” k = ± i
2
, respectively, while all the rest ”eigenvec-
tors” have k = ± 1
2
.
It is useful to know some additional relations among nilpotents and projectors,
taken from Ref. [3]
ab
(k)
ab
[k] = 0 ,
ab
[k]
ab
(k)=
ab
(k) ,
ab
(k)
ab
[−k] =
ab
(k) ,
ab
[k]
ab
(−k)= 0 ,
ab
(k)
ab
(−k) = ηaa
ab
[k] ,
ab
[k]
ab
[−k]= 0 . (7.7)
The same relations are valid also if one replaces
ab
(k) with
ab
˜(k) and
ab
[k] with
ab
˜[k].
The ”basis vectors” are products of d
2
eigenvectors of all the Cartan subalgebra
members. For the description of the internal space of fermions only those ”basis
vectors” which are products of an odd number of nilpotents, the rest can be
projectors, are acceptable, since they anticommute algebraically, what we expect for
the single fermion states of the second quantized fields.
To make clear what the anticommutation of the basis vectors mean, let us
start with the first ”basic vector”, denoting it as b̂m=1†f=1 , with f defining different
irreducible representations and m a member in the representation f. Then its
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 93 — #109 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 93
Hermitian conjugated partner is b̂mf = (b̂
m†
f )
†. Let us make a choice of the starting
”basic vector” for the Clifford algebra of the kind γa’s with an odd products of the
nilpotents
b̂m=1†f=1 : =
0 3
(+i)
1 2
[+]
5 6
[+]
7 8
(+)
9 10
(+)
11 12
[−]
13 14
[−] · · ·
d−3d−2
[−]
d−1d
[−] ,
(b̂m=1†f=1 )
† = b̂m=1f=1 =
d−1d
[−]
d−3d−2
[−] · · ·
13 14
[−]
11 12
[−]
9 10
(−)
7 8
(−)
56
[+]
12
[+]
03
(−i) , (7.8)
the rest products in · · ·
d−3d−2
[−]
d−1d
[−] are assumed to be all projectors with k = −1,
[−]. All the rest members of this irreducible representation are reachable by Sab.
Let us see how do Sab’s transform the ”basis vectors”.
Sac
ab
(k)
cd
(k) = −
i
2
ηaaηcc
ab
[−k]
cd
[−k] ,
Sac
ab
[k]
cd
[k] =
i
2
ab
(−k)
cd
(−k) ,
Sac
ab
(k)
cd
[k] = −
i
2
ηaa
ab
[−k]
cd
(−k) ,
Sac
ab
[k]
cd
(k) =
i
2
ηcc
ab
(−k)
cd
[−k] ,
(7.9)
We learn from Eq. (7.50) that S01 transforms b̂m=1†f=1 into, let us call it b̂
m=2†
f=1 ,
b̂m=2†f=1 =
0 3
[−i]
1 2
(+)
5 6
[+]
7 8
(+)
9 10
(+)
11 12
[−]
13 14
[−] · · ·
d−3d−2
[−]
d−1d
[−] .
Application of all possible Sdg generates 2
d
2
−1 members of each Clifford
odd irreducible representation. To each irreducible representation the Hermitian
conjugated irreducible representation belongs.
The Hermitian conjugated partner of the starting ”basic vector” of an odd
product of nilpotents obviously belong to another irreducible representation, since
it is not reachable by Sab. Each Scd namely transforms a pair of projectors into
a pair of nilpotents, a pair of nilpotents into a pair of projectors, and a pair of a
nilpotent and a projector into a pair of a projector and a nilpotens, changing in
each member of a pair its k into −k. The Hermitian conjugation transforms in b̂m†f
an odd number of nilpotents, each carrying its own k, into the same number of
nilpotents, each carrying then −k 4.
From Eq. (7.50) we learn that the starting member b̂m=1†f=2 of the next irre-
ducible representation can be obtained from b̂m=1†f=1 by replacing, for example,
0 3
(+i)
1 2
[+] in b̂m=1†f=1 with
0 3
[+i]
1 2
(+). This new ”basis vector” does not belong to either
the starting irreducible representation, or to the Hermitian conjugated partners
of the starting irreducible representation, due to the way how it is creating: S01
transforms
0 3
(+i)
1 2
[+] into
0 3
[−i]
1 2
(−), the Hermitian conjugation transforms
0 3
(−i)
1 2
[+].
4 The ”basis vectors” with an even number of nilpotents have in even dimensional spaces
the property that there is one member of each representation which is self adjoint, the
one which is the product of only projectors.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 94 — #110 i
i
i
i
i
i
94 N.S. Mankoč Borštnik
Exchanging all possible pairs in the starting ”basis vector” by keeping the
same k’s while transforming a pair of nilpotents into a pair of projectors, a pair of
projectors into a pair of nilpotents and a pair of a nilpotent and a projector into a
pair of the projector and the nilpotent, we generate 2
d
2
−1 irreducible representa-
tions with 2
d
2
−1 members each.
The Hemitian conjugation then generates 2
d
2
−1· 2d2−1 partners to the 2d2−1
members of each of the 2
d
2
−1 irreducible representations.
One can find that the algebraic product of b̂mf ∗Ab̂m†f is the same for all
m of a particular irreducible representation f (since b̂mf (2 S
ab)†∗A(2 Sab)b̂m†f =
b̂mf ∗Ab̂m†f , due to the relation (−2iSab)†(−2iSab) = 1).
Each irreducible representation contributes different algebraic product b̂mf ∗Ab̂m†f .
For the representation of Eq. (7.8) the product b̂m=1f=1 ∗ab̂m=1†f=1 is equal to
|ψoc > |f=1 =
0 3
[−i]
1 2
[+]
5 6
[+]
7 8
[−]
9 10
[−]
11 12
[−]
13 14
[−] · · ·
d−3d−2
[−]
d−1d
[−] .
This can be checked by using Eq. (7.7).
Defining the vacuum state |ψoc > for the vector space determined by γa’s as
a sum of all different products of
∑2d2−1
f=1 b̂
m
f ∗Ab̂m†f , ∀ m, and for d = 2n+ 1, one
obtains
|ψoc > =
03
[−i]
12
[−]
56
[−] · · ·
d−1 d
[−] +
03
[+i]
12
[+]
56
[−] · · ·
d−1 d
[−]
+
03
[+i]
12
[−]
56
[+] · · ·
d−1 d
[−] + · · · |1 > ,
for d = 2(2n+ 1) . (7.10)
Let me add that the application of any member of the Cartan subalgebras on the
vacuum state, Sab|ψoc >= 0, S̃ab|ψoc >= 0, ∀ Sab and S̃ab belonging to Cartan
subalgeras of Eq. (7.5).
One finds that all the members of all the irreducible representations fulfill
together with their Hermitian conjugated partners the relations
b̂mf ∗A |ψoc > = 0 · |ψoc > ,
b̂m†f ∗A |ψoc > = |ψ
m
f > ,
{b̂mf , b̂
m ′
f ′ }∗A+|ψoc > = 0 |ψoc > ,
{b̂mf , b̂
m ′†
f }∗A+|ψoc > = δ
mm ′ |ψoc > ,
{b̂m†f , b̂
m ′†
f ′ }∗A+|ψoc > = 0 · |ψoc > , (7.11)
for each f. ∗A represents the algebraic multiplication of b̂m†f ’s and b̂m
′
f ′ ’s among
themselves and with the vacuum state |ψoc > of Eq.(7.10).
The relations of Eq. (7.11) almost manifest the anticommutation relations for
the second quantized fermion fields postulated by Dirac [67]. It is pointed out
almost, since the relation
{b̂mf , b̂
m ′†
f ′ }∗A+|ψoc > = δ
mm ′δff
′
|ψoc > (7.12)
is not fulfilled. There are, namely, besides b̂mf , 2
d
2
−1 − 1members of the Hermitian
conjugated partners belonging each to a different irreducible representation, which
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 95 — #111 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 95
give a nonzero contribution — not an identity as b̂mf does — when multiplying
b̂m†f from the left hand side. b̂
m
f ′ ∗A b̂
m†
f 6= 0
for 2
d
2
−1 − 1 different f ′ 6= f, while b̂mf ∗A b̂m†f = 1.
Let me illustrate this on the example of b̂m=1†f=1 of Eq. (7.8). Besides (b̂
m=1†
f=1 )
† =
b̂m=1f=1 =
d−1d
[−]
d−3d−2
[−] · · ·
13 14
[−]
11 12
[−]
9 10
(−)
7 8
(−)
56
[+]
12
[+]
03
(−i) also
d−1d
[−]
d−3d−2
[−] · · ·
13 14
[−]
11 12
[−]
9 10
(−)
7 8
(−)
56
[+]
12
(−)
03
[+i] ,
d−1d
[−]
d−3d−2
[−] · · ·
13 14
[−]
11 12
[−]
9 10
(−)
7 8
(−)
56
(−)
12
[+]
03
[+i] ,
etc (7.13)
applied on b̂m=1†f=1 , give a nonzero contributions.
But index f determine different irreducible representations and we can not
expect that the algebraic anticommutation relations will be fulfilled also among
different irreducible representations. Different irreducible representations should
be treated in tensor products.
All the discussions about the Clifford algebra with γa’s, appearing after
Eq. (7.7), can be as well repeated also for the Clifford algebra with γ̃a’s.
The Dirac’s postulates for the second quantized fermion fields include the
infinite basis in momentum space, while we treated so far the finite dimensional
internal space of fermions. Before extending the vector basis space by making the
tensor product of internal space and the momentum space let us recognize that the
observed quarks and leptons and antiquarks and antileptons do not at all suggest
that there might be two different internal spaces, which could be described by
two kinds of the Clifford algebra objects. Let us therefore first reduce the Clifford
space by the postulate, which leave only γa’s as the algebra describing the internal
degrees of freedom of fermions, while γ̃a’s are used to give quantum numbers to
different irreducible representations.
Reduction of the Clifford space It is needed to give to each irreducible repre-
sentation of the Lorentz transformations in the internal space of fermions the
quantum number, which will distinguish among the 2
d
2
−1 different irreducible
representations. If we keep the Clifford algebra with γa’s to describe the internal
space of fermions, then γ̃a’s, or rather S̃ab’s, can be used to determine ”family”
quantum number of each irreducible representation of the Lorentz algebra in the
Clifford space of γa’s.
We want that all the relations among γa’s and γ̃a’s, presented in Eq. (7.2),
remain unchanged, while the eigenvalues of the Cartan subalgebra of S̃ab are
expected to be changed.
The postulate [2, 7, 9, 10, 12, 46]
γ̃aB = (−)B i Bγa , (7.14)
with (−)B = −1, if B is a function of an odd product of γa’s, otherwise (−)B =
1 [46], does just that 5
5 Eq. (7.14) requires that γ̃a(a0 + abγb + abcγbγc + · · · ) = (ia0γa + (−i)abγbγa +
iabcγ
bγcγa + · · · ), what means that the relation γ̃a a0 = ia0γa is only one of the
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 96 — #112 i
i
i
i
i
i
96 N.S. Mankoč Borštnik
It is not difficult to check that the relations in Eq. (7.2), concerning γ̃a’s are
still valid: {γa, γb}+ = 2ηab = {γ̃a, γ̃b}+ , {γa, γ̃b}+ = 0 , (γa)† = ηaa γa , (γ̃a)† =
ηaa γ̃a.
After this postulate the vector space of γ̃a’s is ”frozen out”. And also the
Grassmann algebra space is now reduced to θa = γa and ∂
∂θa
= 0 6. No vector
space of γ̃a’s exists any longer, what is in agreement with the observed properties
of fermions. While the anticommutation relations among γa’s and γ̃a’s remain the
same as in Eq. (7.2), it follows for the eigenvalues of S̃ab
Sab
ab
(k)=
k
2
ab
(k) , S̃ab
ab
(k)=
k
2
ab
(k) ,
Sab
ab
[k]=
k
2
ab
[k] , S̃ab
ab
[k]= −
k
2
ab
[k] , (7.15)
showing that the eigenvalues of Sab on the nilpotents and projectors of γa’s differ
from the eigenvalues of S̃ab on the nilpotents and projectors of γa’s. The members
of the Cartan subalgebra of S̃ab, Eq. (7.5), can now be used to give to the irreducible
representations of Sab the ”family” quantum numbers.
Let me mention that if one arranges the space of odd products of γa’s with
respect to Sab(= Sab + S̃ab), these new ”basis vector” will form multiplets with
integer spins and charges in adjoint representations. Like the ”basis vectors” ex-
pressed by Grassmann algebra do in Ref. [13], Table I, but this time with θa’s
replaced by γa’s.
relations included into Eq. (7.14). Another relation, for example, is γ̃aγa = (−i)γaγa =
−iηaa. One correspondingly finds {γ̃a, γ̃b}+ = 2ηab = γ̃aγ̃b + γ̃bγ̃a = γ̃aiγb + γ̃biγa =
iγb(−i)γa + iγa(−i)γb = 2ηab. {γ̃a, γb}+ = 0 = γ̃aγb + γbγ̃a = γb(−i)γa + γbiγa = 0.
{γ̃a, γa}+ = 0 = γ̃
aγa + γaγ̃a = γa(−i)γa + γaiγa = 0.
6 Let me show how does the Grassmann space loose the Hermitian conjugated partners
to θa’s, while θa’s become equal to γa’s. My statement that Eq. (7.14) requires θa = γa
and ∂
∂θa
= 0 can be proved as follows. There are only two requirements which have
to be analyzed in details, γ̃a(α) = iαγa, α is any constant and γ̃aγa = −iγaγa. Both
relations apply on |ψoc >: In the Grassmann case the vacuum state is identity | 1 >, while
in the Clifford algebra the vacuum state is the sum of even products of γa’s as seen in
Eq. (7.10), which applies on identity. Let us express γa’s, γ̃a’s and |ψoc > in terms of θa’s
and ∂
∂θa
as written in Eq. (7.3). Eq. (7.3) requires that γa = (θa + ∂
∂θa
), γ̃a = i(θa − ∂
∂θa
).
Let us put these expressions into Eq. (7.14) and let |ψoc > be expressed in terms of θa’s.
Taking into account that θa’s applying on identity gives θa’s back while ∂
∂θa
applying on
identity gives zero, it follows that |ψoc >= a0 + aabθaθb + · · · , the rest of expansion is
irrelevant for the proof. The constant α can be skipped, since constants appear in |ψoc >=
a0+aabθ
aθb+ · · · anyhow. The first relation [γ̃a = iγa]|ψoc >, expressed with θa’s and
∂
∂θa
, reads: i(θa − ∂
∂θa
)(a0 + aabθ
aθb + · · · ) = i(θa + ∂
∂θa
)(a0 + aabθ
aθb + · · · ). From
this we find iθaa0 = ia0θa and i(− ∂∂θa )aabθ
aθb = i ∂
∂θa
abθ
aθb, requiring that ∂
∂θa
= 0
(as an operator Hermitian conjugated to θa for ∀ a). These relation requires that the
derivatives should not exist any longer, if the relation should hold. Then it follows from
γa = (θa + ∂
∂θa
) that θa = γa, which means that the Grassmann space has no meaning
any longer, the only remaining space is the space of the Clifford products of odd number
of γa’s, on which γa’s and γ̃a’s operate: [γ̃a = iγa]|ψoc > and [γ̃aγb = −iγbγa]|ψoc >.
This complicates the proof .
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 97 — #113 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 97
It is useful to notice that γa transform
ab
(k) into
ab
[−k], never to
ab
[k], while γ̃a
transform
ab
(k) into
ab
[k], never to
ab
[−k]
γa
ab
(k)= ηaa
ab
[−k], γb
ab
(k)= −ik
ab
[−k],
γa
ab
[k]=
ab
(−k), γb
ab
[k]= −ikηaa
ab
(−k) ,
γ̃a
ab
(k)= −iηaa
ab
[k], γ̃b
ab
(k)= −k
ab
[k],
γ̃a
ab
[k]= i
ab
(k), γ̃b
ab
[k]= −kηaa
ab
(k) . (7.16)
Some additional applications of γ̃a’s and S̃ab’s on nilpotents and projectors
expressed by the γa’s can be found in App. 7.4.
Each irreducible representation has now the ”family” quantum number, de-
termined by S̃ab of the Cartan subalgebra of Eq. (7.5). Now we can replace the
fourth equation in Eq. (7.11) — {b̂mf , b̂
m ′†
f }∗A+|ψoc >= δ
mm ′ |ψoc > — with the
relation in Eq. (7.12) — {b̂mf , b̂
m ′†
f ′ }∗A+|ψoc >= δ
mm ′δff ′ |ψoc >.
Each family contributes in even dimensional spaces one summand of d
2
projectors to the vacuum state |ψoc > of fermions.
Correspondingly the ”basic vectors” and their Hermitian conjugated partners
fulfill algebraically the anticommutation relations of Dirac’s second quantized
fermions: Different irreducible representations carry different ”family” quantum
numbers and to each ”family” quantum number only one Hermitian conjugated
partner with the same ”family” quantum number belongs. Also each summand of
the vacuum state, Eq. (7.10), belongs to a particular ”family”.
One can easily check that each ”basic vector” b̂m†f , applied algebraically on
|ψoc >, gives nonzero contribution on the summand in the odd number of γa’s,
determined by b̂mf b̂
m†
f , which is the same for all m of particular f, representing
therefore the corresponding state |ψfm >, while on all other summands b̂
m†
f gives
zero, b̂mf applying on |ψoc > gives zero for all f and allm.
To define creation and annihilation operators, which determine on the vacuum
state the single fermion states, we ought to make the tensor products of the 2
d
2
−1
× 2d2−1 ”basis vectors”, describing the internal space of fermions and of infinite
basis of momenta.
The oddness of the products of the odd number of γa’s guarantees the anticommuting
properties of all the objects which include an odd number of γa’s.
The creation and annihilation operators, derived as tensor products of the
”basis vectors” and the basis in momenum space, will fulfill the Dirac’s postulates
of the second quantized fermions without postulating them, as Dirac did. They
follow by themselves from the fact that the creation and annihilation operators are
superposition of odd products of γa’s.
Second quantized fermion fields Since the nonrelativistic quantum theory is an
approximation of the relativistic second quantized field theory — as the relativistic
classical physics is an approximation of the quantum physics, and as the nonrela-
tivistic classical physics, which we use the most of time, is the approximation of
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 98 — #114 i
i
i
i
i
i
98 N.S. Mankoč Borštnik
the relativistic classical physics — let us try to recognize what properties should
the single particle states have to form the Hilbert space of second quantized fields.
In the references [10, 12, 13] the properties of the single fermion states, the
tensor products among which form the Hilbert space, are discussed in details.
In this talk I am presenting this topic from the point of view of the spin-charge-
family. This theory offers, as written in the introduction, the explanation for the
appearance of the spin (and handedness in the case of massless fermions), of all
the charges, as well as of the families fermions. The number of families depends
on the way how does the symmetry of the space breaks from d = (13 + 1) to
d = (3+ 1).
In Table 7.3 one irreducible representation of SO(13 + 1) of one family (be-
longing to the one of the two groups of four families which includes the so far
observed three families) is presented. The first ”basis vector” describes the inter-
nal degrees of freedom of the right handed quark ûc1†R , of the first family with
(S̃03, S̃12, S̃56, S̃78) equal to (1
2
,−1
2
,−1
2
, 1
2
), presented in Table 7.4 as ûc1†R1 . The ”ba-
sis vector” b̂m=1†f=1 , Eq. 7.8, represents for d = (13 + 1) just this û
c1†
R1 quark, and
b̂m=1f=1 is its Hermitian conjugated partner.
The ”basis vector” b̂m=1†f=2 represents for d = (13+1) the right handed u-quark
with all the properties of ûc1†R1 except for the family quantum numbers, which
are now equal to (−1
2
, 1
2
,−1
2
, 1
2
). One can read in Table 7.3 that the spin of this
right handed quark ûc1†R is +
1
2
, the weak SU(2) charge is zero, the colour charge
is (1
2
, 1
2
√
3
). It carries the additional SU(2) charge equal to 1
2
and the ”fermion”
quantum number — τ4 charge — equal to 1
6
.
When solving the equations of motion for free massless fermions, which
follow from the action, presented in Eq. (7.1), under the assumption, that at low en-
ergies the momentum of this right handed quark is pa = (p0, p1, p2, p3, 0, · · · , 0),
the solution s = 1 is the superposition
ûsf=1†R (~p) = β(û
c1†
R↑ + p
1 + ip2
|p0|+ |p3|
ûc1†R↓ ) , (7.17)
with |p0| = |~p|, with ↑, ↓ denoting spin ±1
2
, respectively, and with β∗β = |p
0|+|p3|
2|p0|
normalizing the state.
There are steps from the d = (13 + 1) dimensional space to the step where
momentum in higher dimensions do not contribute to dynamics in d = (3 + 1),
while the break of symmetry makes the internal degrees of freedom (spins and
families) to manifest as the spin and charges as presented in Table 7.3 and families
as presented at Table 7.4. One finds the detailed presentations in Ref.( [3–5, 9, 49,
52, 70] and the references therein).
Let us here represent the general solutions of equations of motion for free
massless fermions with the internal space of fermions described by the ”basis
vectors” b̂m†f , fulfilling the relations of Eq. (7.11), for each family f separately, and
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 99 — #115 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 99
also with respect to different families, b̂mf ∗A b̂
m ′†
f = δ
mm ′δff ′ ,
b̂sf†(~p)|p0=|~p|
def
=
∑
m
csfm (~p, |p
0| = |~p|) b̂m†f ,
b̂
sf†
tot(~p,~x)
def
= (b̂sf†(~p) e−i(p
0x0−~p·~x))||p0|=|~p| ,∑
m
(csf∗m(~p) · cs
′f ′
m(~p))||p0|=|~p| = δ
ss ′δff ′ , (7.18)
s represents different orthonormalized solutions of the equations of motion,
csfm(~p, |p
0| = |~p|) are coefficients, depending on momentum |~p| with |p0| = |~p|.
For the case of the right handed u-quarks of Eq. (7.17) the two nonzero coefficients
are β and β p
1+ip2
|p0|+|p3|
.
Creation operators of an odd Clifford character b̂
sf†
tot(~p) create the single
particle states, < x|ψsf(p̃,p0) > |p0=|p̃|, manifesting the oddness of the creation
operators
< x|ψsf(p̃,p0) > |p0=|p̃| =
∫
dp0δ(p0 − |~p|) b̂sf†(~p) e−ipax
a ∗A |ψoc >
= (b̂sf†(~p) · e−i(p0x0−ε~p·~x))|p0=|~p| ∗A |ψoc > , (7.19)
with the property∫
dd−1x
(
√
2π)d−1
< ψs
′f ′( ~p ′, p ′0 = | ~p ′|)|x >< x||ψsf(~p, p0 = |~p|) >=∫
dd−1x
(
√
2π)d−1
eip
′
ax
a
|p ′0=| ~p ′| e
−ipax
a
|p0|=|~p|
· < ψoc| (b̂s
′f ′( ~p ′) b̂sf†(~p)) ∗A |ψoc >= δss
′δff
′
δ(~p− ~p ′) . (7.20)
One further finds the single particle fermion states in the coordinate representation
|ψsf(x̃, x0) >=
∫+∞
−∞
dd−1p
(
√
2π)d−1
(b̂sf†(p̃) e−i(p
0x0−εp̃·x̃)|p0=|p̃| ∗A |ψoc >=∑
m
b̂m†f |ψoc>
∫+∞
−∞
dd−1p
(
√
2π)d−1
(csfm(~p) e
−i(p0x0−ε~p·~x))|p0=|~p| =∑
m
b̂m†f |ψoc > c
sf
m(−i
∂
∂xa
, |p0| = |(−i
∂
∂xa
|) δ(~x) , (7.21)
where it is taken into account that b̂sf†(~p)|p0=|~p| |ψoc >=
∑
m c
sf
m (~p, |p
0| =
|~p|) b̂m†f , Eq. (7.18), and that
∫
dd−1x
(
√
2π)d−1
eip
′
ax
a
e−ipax
a
= δ(~p − ~p ′). ε = ±1, de-
pending on handedness and spin of solutions.
Taking into account the above derivations, leading to∫
dp0δ(p0 − |~p|)ei(p
0x0−p0x0) = 1
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 100 — #116 i
i
i
i
i
i
100 N.S. Mankoč Borštnik
and < ψoc| b̂sf(~p, p0)∗A b̂
s ′f ′†(~p, p0) |ψoc >= δ
ss ′δff
′
, one finds
< ψsf(~x, x0)|ψs
′f ′(~x ′, x0) >=
=
∫+∞
−∞
dd−1p
(
√
2π)d−1
∫+∞
−∞ δ(p
0 − |~p|) < ψsf(~x, x0)|~p >< ~p|ψs
′f ′ (~x ′, x0) >
=
∫+∞
−∞
dd−1p
(
√
2π)d−1
e−i~p·~x ei~p·
~x ′
∫
dp0 δ(p0 − |~p|)
< ψoc|b̂sf (~p, p0)∗A b̂
s ′f ′† (~p, p0) ∗A |ψoc >=
= δss
′
δff ′ δ(~x− ~x
′) . (7.22)
The scalar product < ψsf(~x, x0) |ψs
′f ′(~x ′, x0) > has obviously the desired proper-
ties of the second quantized states.
The new creation operators b̂
sf†
tot(~p,~x), which are generated on the tensor
products of both spaces, internal and momentum, fulfill obviously the below
anticommutation relations when applied on |ψoc >
{b̂
sf
tot(~p,~x) , b̂
sf†
tot(~p
′,~x)}+ ∗T |ψoc > = δ
ss ′ δff ′ δ(~p− ~p ′) |ψoc > ,
{b̂
sf
tot(~p,~x) , b̂
s ′f ′
tot (
~p ′,~x)}+ ∗T |ψoc > = 0 · |ψoc > ,
{b̂
sf†
tot(~p,~x) , b̂
s ′f ′†
tot (~p
′,~x)}+ ∗T |ψoc > = 0 · |ψoc > ,
b̂
sf†
tot(~p,~x) ∗T |ψoc> = |ψ
sf(~p, ,~x) > ,
b̂
sf
tot(~p, ,~x) ∗T |ψoc > = 0 · |ψoc > ,
|p0| = |~p| . (7.23)
It is not difficult to show that b̂
sf
tot(~p,~x) and b̂
sf†
tot(~p,~x) manifest the same anticom-
mutation relations also on tensor products of an arbitrary chosen products of sets
of single fermion states [13].
Hilbert space of fermion fields The tensor products of any number of any sets
of the single fermion creation operators b̂
sf†
tot(~p,~x) (fulfilling together with their
Hermitian conjugated partners annihilation operators b̂
sf
tot(~p,~x) the anticommuta-
tion relations of Eq. (7.23)) form the Hilbert space of the second quantized fermion
fields. The number of the sets is infinite. The internal space, defined by b̂mf , con-
tributes in d-dimensional space for each chosen momentum ~p (and for a parameter
~x) the finite number, 22
d
2
−1·2
d
2
−1
, of such sets, the total Hilbert space has, due to
the infinite basis in the momentum (or coordinate) space, the infinite number of
sets
NH =
∞∏
~p
22
d−2
. (7.24)
The number operator is defined as
Nsf~p = b̂
sf†
tot(~p,~x) ∗T b̂
sf
tot(~p,~x) ,
Nsf~p |ψoc > = 0 · |ψoc > . (7.25)
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 101 — #117 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 101
The vacuum state contains no fermions.
The Clifford odd objects b̂
sf†
tot(~p,~x) demonstrate their oddness also with re-
spect to the whole Hilbert spaceH, that is with respect to any tensor product of
members of any sets of creation operators b̂
sf†
tot(~p
′,~x)). Correspondingly the anti-
commutation relations follow also for the application of b̂
sf†
tot(~p,~x) and b̂
sf
tot(~p,~x)
onH
{b̂
sf
tot(~p,~x) , b̂
s ′f ′†
tot (~p
′,~x)}∗T+H = δss
′
δff ′ δ(~p− ~p
′) H ,
{b̂
sf†
tot(~p,~x) , b̂
s ′f ′†
tot (~p
′,~x)}∗T+ H = 0 · H ,
{b̂
sf†
tot(~p,~x), b̂
s ′f ′†
tot (~p
′,~x)}∗T+ H = 0 · H . (7.26)
I presented in this talk the derivation of the creation and annihilation operator
of the second quantized fermion fields, which obey the Dirac’s postulates for
the second quantized fermion fields without postulating them, just by analyzing
properties of creation and annihilation operators obtained as tensor products of the
”basis vectors” of an odd Clifford algebra and of the basis in either momentum or
coordinate space. In Ref. [10–13] the relation between the creation and annihilation
operators, postulated by Dirac and the ones presented in this talk are discussed.
Properties of fermions in d = (3+ 1) This section follows quite a lot Refs. [3, 4].
With respect to the last years I have not succeeded to improve much the part
presented in this subsection. I have been working on the symmetries of the spin-
charge-family theory and in particular on how can the theory, using the Clifford
algebra to describe all the internal properties of fermions — spins, charges and
families — help to explain the assumptions of the second quantized fermion fields.
I shall therefore review the other achievements of the theory very briefly.
In Eq. (7.1) the starting action is presented for fermion and boson fields in
d = (13 + 1). In order that predictions of the spin-charge-family theory are in
agreement with the observed properties of quarks and leptons and antiquarks and
antileptons, of the vector gauge fields and of the scalar gauge fields (manfesting as
the higgs and Yukawa couplings), the manifoldM(13+1) ought to break first into
M(7+1) ×M(6) (which manifests as SO(7, 1) ×SU(3) ×U(1)), affecting fermions,
vector gauge fields and scalar gauge fields.
This first break is caused by the scalar condensate of two right handed neutri-
nos, presented in Table 7.5, Sect. 7.5 which interact with all the scalar gauge fields
(with the gauge fields with the space index (5, 6, 7, · · · , 14), as well as with those
vector gauge fields (with the gauge fields with the space index (0, 1, 2, 3), which
couple to the condensate. The only vector gauge fields which do not interact with
the condensate and remain consequently massless are the weak charge, colour
charge and hyper charge vector gauge fields.
Since the left handed fermions couple differently to scalar fields than the
right handed ones, the break can leave massless and mass protected 2((7+1)/2−1)
families [68]. The rest of families get heavy masses 7.
7 A toy model [68, 69] was studied in d = (5 + 1) with the same action as in Eq. (7.1).
The break from d = (5 + 1) to d = (3 + 1)× an almost S2 was studied. For a particular
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 102 — #118 i
i
i
i
i
i
102 N.S. Mankoč Borštnik
The fermion families are arranged into twice two groups of massless four
families, with respect to family quantum numbers as presented in Table 7.4 in
Sect. 7.5, each group manifesting SU(2)⊂SO(3,1) × SU(2)⊂SO(4) symmetry, one
group manifesting SU(2)L × SU(2)L symmetry, the other SU(2)R × SU(2)R sym-
metry.
The nonzero vacuum expectation values of the scalar fields with the space
index (7, 8), which carry the weak and hyper charges, break the mass protection
and make family massive [7, 9].
The breaks of the staring symmetry make the spins in higher dimensions to
manifest as charges in d = (3+ 1).
The superposition of the Lorentz members of the Clifford algebra, manifesting
in d = (3+ 1) the spins, Eq. (7.52), charges, Eqs. (7.53, 7.54) and families, Eqs (7.55,
7.56). are presented in Sect. 7.5.
Let me rewrite the fermion part of the action, Eq. (7.1), by taking into account
the degrees of freedom the action manifests in d = (3+ 1) in the way that we can
clearly see that the action does manifest in the low energy regime by the standard
model required properties of fermions, of vector gauge fields and of scalar gauge
fields [1–3, 7, 9, 51–53, 71, 72].
Lf = ψ̄γm(pm −
∑
A,i
gAiτAiAAim )ψ+
{
∑
s=7,8
ψ̄γsp0s ψ}+
{
∑
t=5,6,9,...,14
ψ̄γtp0t ψ} , (7.27)
where p0s = ps − 12S
s ′s"ωs ′s"s −
1
2
S̃abω̃abs, p0t = pt − 12S
t ′t"ωt ′t"t −
1
2
S̃abω̃abt,
with m ∈ (0, 1, 2, 3), s ∈ (7, 8), (s ′, s") ∈ (5, 6, 7, 8), (a, b) (appearing in S̃ab)
run within either (0, 1, 2, 3) or (5, 6, 7, 8), t runs ∈ (5, . . . , 14), (t ′, t") run either
∈ (5, 6, 7, 8) or ∈ (9, 10, . . . , 14). The spinor function ψ represents all family mem-
bers of all the 2
7+1
2
−1 = 8 families.
a. The first line of Eq. (7.27) determines in d = (3 + 1) the kinematics
and dynamics of fermion fields, coupled to the vector gauge fields [3, 5, 9]. The
vector gauge fields are the superposition of the spin connection fields ωstm,
m = (0, 1, 2, 3), (s, t) = (5, 6, · · · , 13, 14), the gauge fields of Sst. They are shortly
presented in Sect. 7.34.
The operators τAi (τAi =
∑
a,b c
Ai
ab S
ab, Sab are the generators of the
Lorentz transformations in the Clifford space of γa’s) are presented in Eqs. (7.53,
7.54) of Sect. 7.5. They represent the colour charge, ~τ3, the weak charge, ~τ1, and
the hyper charge, Y = τ4 + τ23, τ4 is the fermion charge, originating in SO(6) ⊂
SO(13, 1), τ23 belongs together with ~τ1 of SU(2)weak to SO(4) group (⊂ SO(13+
1)).
choice of vielbeins and for a class of spin connection fields the manifoldM(5+1) breaks
into M(3+1) times an almost S2, while 2((3+1)/2−1) families remain massless and mass
protected. Equivalent assumption, although not yet proved how does it really work, is
made in the d = (13 + 1) case. This study is in progress.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 103 — #119 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 103
One fermion irreducible representation of the Lorentz group contains, as seen in
Table 7.3, quarks and leptons and antiquarks and antileptons, belonging to the first
family in Table 7.4. One can notice that the SO(7, 1) subgroup content of the
SO(13, 1) group is the same for the quarks and leptons and the same for the
antiquarks and antileptons. Quarks distinguish from leptons, and antiquarks from
antileptons, only in the SO(6) ⊂ SO(13, 1) part, that is in the colour (τ33, τ38) part
and in the fermion quantum number τ4. The quarks distinguish from antiquarks,
and leptons from antileptons, in the handedness, in the colour part and in the τ4
part, explaining the relation between handedness and charges of fermions and
antifermions 8.
The vector gauge fields, which interact with the condensate, presented in
Table 7.5, become massive. The vector gauge fields not interacting with the condensate —
the weak, colour and hyper charged vector gauge fields — remain massless, in agreement
with by the standard model assumed gauge fields before the electroweak break of
the mass protection,
After the electroweak break, caused by the scalar fields, the only conserved
charges are the colour and the electromagnetic charge Q = τ13 + Y, Y = τ4 + τ23.
b. The second line of Eq. (7.27) is the mass term, responsible in d = (3+1) for
the masses of fermions. The interaction of fermions with the superposition of the
spin connection fields with the space index s = (7, 8), which gain nonzero vacuum
expectation values, cause the electroweak break, bringing masses to fermions
and antifermions and to the weak vector gauge fields. They are superposition
of either ωs ′t ′s or ω̃abs. These scalar fields explain the appearance of the higgs and
Yukawa couplings of the standard model. Their properties are shortly presented in
Subsect. 7.2.2.
These scalar gauge fields split into two groups of four families, one group
manifesting the symmetry — S̃U(2)
(S̃O(3,1),L)
×S̃U(2)
(S̃O(4),L)
×U(1) — and the
other the symmetry — S̃U(2)
(S̃O(3,1),R)
×S̃U(2)
(S̃O(4),R)
×U(1), Eq. (7.37). The
scalar gauge fields, manifesting SU(2)L,R × SU(2)L,R, are the superposition of the
gauge fields ω̃abs, s = (7, 8), (a, b) = either (0, 1, 2, 3) or (5, 6, 7, 8), manifesting
as twice two triplets interacting each with one of the two groups of four families,
presented in Table 7.4. The three U(1) singlet scalar gauge fields are superposition
ofωs ′t ′s, s = (7, 8), (s ′, t ′) = (5, 6, 7, 8, 9, · · · , 14), with the sum of Ss
′t ′ arranged
into superposition of τ13, τ23 and τ4. The three triplets interact with both groups
of quarks and leptons and antiquarks and antileptons.
Each of the two groups have well defined symmetry of mass matrices, what limits
the number of free parameters.
To one of the groups of four families the observed quarks and leptons be-
long [51, 54, 57, 58].
We predict the mixing matrices for quarks, taking as the input the masses
of the fourth family, since the elements for the 3 × 3 submatrix of the 4 × 4
8 Ref. [8] points out that the connection between handedness and charges for fermions and
antifermions, both appearing in the same irreducible representation, explains the triangle
anomalies in the standard model with no need to connect ”by hand” the handedness and
charges of fermions and antifermions.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 104 — #120 i
i
i
i
i
i
104 N.S. Mankoč Borštnik
mixing matrix are (far) not accurately enough measured, that we could predict
masses of the fourth family quarks [8, 51, 54]. The newer are the experimental
data the better is the agreement of the measured mixing matrix elements with our
predictions [54, 58] at least so far.
The stable of the upper four families offers the explanation for the dark mat-
terappearance and it is so far in agreement with experimental evidences of the
dark matter [52, 61].
I discuss predictions of the spin-charge-family theory for the properties of the
lower four families and of the dark matter in Sect. 7.3.
c. The third line of Eq. (7.27) represents the scalar fields, which cause
transitions from antileptons and antiquarks into quarks and leptons and back,
offering the explanation for the matter/antimatter asymmetry in the expanding
universe at non equilibrium conditions [4]. They are colour triplets with respect
to the space index equal to (9, 10, 11, 12, 13, 14), while they carry the quantum
numbers with respect to the superposition of Sab in adjoint representations, as
can be seen in Table 7.2 and in Fig. 7.1 of Subsect. 7.2.2. I discuss properties of
these scalar fields, offered by the spin-charge-family theory, in Sect. 7.3.
7.2.2 Properties of vector and scalar gauge fields in spin-charge-family theory
In the starting action, Eq. (7.1), the second line represents the action for gauge
fields in d = (13+ 1)-dimensional space, with the index gf denoting gauge fields,
vector or scalar,
Agf =
∫
ddx E (αR+ α̃ R̃) ,
R =
1
2
{fα[afβb] (ωabα,β −ωcaαω
c
bβ)}+ h.c. ,
R̃ =
1
2
{fα[afβb] (ω̃abα,β − ω̃caα ω̃
c
bβ)}+ h.c. , (7.28)
which in the spin-charge-family theory manifests after the break of the starting
symmetry in d = (3 + 1) as the action for all observed vector and scalar gauge
fields. Here fβa and eaα are vielbeins and inverted vielbeins respectively
eaαf
β
a = δ
β
α , e
a
αf
α
b = δ
a
b , (7.29)
E = det(eaα).
Varying the action of Eq. (7.28) with respect to the spin connection fields, the
expression for the spin connection fieldsωeab follows
ωab
e =
1
2E
{eeα ∂β(Ef
α
[af
β
b]) − eaα ∂β(Ef
α
[bf
βe])
− ebα∂β(Ef
α[efβa])}
+
1
4
{Ψ̄(γe Sab − γ[aSb]
e)Ψ}
−
1
d− 2
{δea[
1
E
edα∂β(Ef
α
[df
β
b]) + Ψ̄γdS
d
b Ψ]
− δeb[
1
E
edα∂β(Ef
α
[df
β
a]) + Ψ̄γdS
d
a Ψ]} . (7.30)
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 105 — #121 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 105
If replacing Sab in Eq. (7.30) with S̃ab, the expression for the spin connection fields
ω̃ab
e follows.
In Ref. [5] it is proven that in spaces with the desired symmetry the vielbein
can be expressed with the gauge fields, if only one of the two spin connection
fields are present
fσm =
∑
A
~τAσ ~AAm , (7.31)
with
AAim =
∑
st
cAistω
st
m ,
τAiσ =
∑
st
cAist (esτ f
σ
t − etτ f
σ
s)x
τ ,
τAi =
∑
st
cAist S
st . (7.32)
If fermions are not present them spin connections of both kinds are uniquely
determined by vielbeins, as can be noticed from Eq. (7.30). If fermions are present,
carrying both — family members and family quantum numbers — then vielbeins
and both kinds of spin connections are influenced by the presence of fermions,
which carry different family and family members quantum numbers.
The scalar (gauge) fields, carrying the space index s = (5, 6, . . . , d), offer in
the spin-charge-family for s = (7, 8) the explanation for the origin of the Higgs’s
scalar and the Yukawa couplings of the standard model, while scalars with the space
index s = (9, 10, . . . , 14) offer the explanation for the proton decay, as well as for
the matter/antimatter asymmetry in the universe.
We use the notation
τAi =
∑
a,b
cAiab S
ab ,
{τAi, τBj}− = iδ
ABfAijkτAk ,
AAia =
∑
s,t
cAistω
st
a , (7.33)
a = m = (0, 1, 2, 3) for vector gauge fields and a = s = (5, 6, . . . , 14) for scalar
aguge fields.
The explicit expressions for cAiab, and correspondingly for τAi, and AAia , are
written in Sect. 7.5.
Vector gauge fields in d = (3 + 1) In the spin-charge-family theory there are
besides the gravity, the colour and the weak SU(2)I vector gauge fields, also the
second SU(2)II and the U(1)τ4 vector gauge fields. The U(1)τ4 vector gauge field
is the vector gauge field of τ4(= −1
3
(S9 10 + S11 12 + S13 14)) - the fermion charge.
The hyper charge vector gauge field of the standard model is the superposition of
the third component of the second SU(2)II vector gauge fields and the U(1)τ4
vector gauge field (AYm = cos θ2Aτ
4
m + sin θ2A23m , θ2 is the angle of the break
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 106 — #122 i
i
i
i
i
i
106 N.S. Mankoč Borštnik
of the SU(2)II × U(1)τ4 symmetry to U(1)Y at the scale ≥ 1016 or higher, [9]
and references therein). After the appearance of the condensate, presented in
Table 7.5, there are namely only the gravity, the colour, the weak SU(2)I and
the U(1)Y hyper charge vector gauge fields, which remain massless. The two
components of the second SU(2)II vector gauge fields and the superposition
AY
′
m = − sin θ2Aτ
4
m +cos θ2A23m , which is the gauge field of Y ′(= − tan
2 θ2τ
4+τ23)
gain high masses due to the interaction with the condensate. All the vector gauge
fields are expressible with the spin connection fieldsωstm,
AAim =
∑
s,t
cAist ω
st
m . (7.34)
Let me present expressions for the two SU(2) vector gauge fields, SU(2)I and
SU(2)II
~A1m = ~A1m = (ω58m −ω67m,ω57m +ω68m,ω56m −ω78m) ,
~A2m = ~A2m = (ω58m +ω67m,ω57m −ω68m,ω56m +ω78m) . (7.35)
The reader can similarly construct all the other vector gauge fields from the
coefficients for the corresponding charges, or find the expressions in Refs. [4, 7, 9]
and references therein.
The electroweak break, caused by the non zero expectation values of the
scalar gauge fields, carrying the space index s = (7, 8), makes the weak and the
hyper charge massive. The only vector gauge fields which remains massless are
the electromagnetic and the colour vector gauge fields — the observed two.
Scalar gauge fields in d = (3+1) There are in the spin-charge-family theory scalar
fields taking care of the masses of quarks and leptons: They have the space index
s = (7, 8) and carry with respect to the space index the weak charge τ13 = ±1
2
and the hyper charge Y = ∓1
2
. With respect to τAi =
∑
ab c
Ai
abS
ab and τ̃Ai =∑
ab c
Ai
abS̃
ab they carry charges and family charges in adjoint representations,
Table 7.1, Eq. (7.39).
There are scalar fields transforming antileptons and antiquarks into quarks
and leptons and back. They carry space index s = (9, 10, . . . , 14), They are with
respect to the space index colour triplets, while they carry charges τAi and τ̃Ai in
adjoint representations.
The infinitesimal generators Sab, which apply on the spin connectionsωbde
(= fαe ωbdα) and ω̃b̃d̃e (= f
α
e ω̃b̃d̃α), on either the space index e or any of the
indices (b, d, b̃, d̃), as follows
SabAd...e...g = i (ηaeAd...b...g − ηbeAd...a...g) , (7.36)
(see Section IV. and Appendix B in Ref. [9]).
Scalar gauge fields determining scalar higgs and Yukawa couplings
Let me introduce a common notation AAis for all the scalar gauge fields with
s = (7, 8), independently of whether they originate in ωabs — in this case Ai
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 107 — #123 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 107
= (Q,Q ′, Y ′) - or in ω̃ãb̃s — in this case all the family quantum numbers of all
eight families contribute. All these gauge fields contribute to the masses of the
quarks and leptons and the antiquarks and antileptons after gaining nonzero
vacuum expectation values.
AAis represents (A
Q
s , A
Q ′
s , A
Y ′
s ,
~̃A1̃s ,
~̃A
ÑL̃
s ,
~̃A2̃s ,
~̃A
ÑR̃
s ) ,
τAi represents (Q, Q ′, Y ′, ~̃τ1, ~̃NL, ~̃τ2, ~̃NR) . (7.37)
Here τAi represent all the operators, which apply on the fermions. These scalars,
the gauge scalar fields of the generators τAi and τ̃Ai, are expressible in terms of
the spin connection fields (Ref. [9], Eqs. (10, 22, A8, A9)).
Let me demonstrate [9] that all the scalar fields with the space index (7, 8)
carry with respect to this space index the weak and the hyper charge (∓1
2
, ±1
2
),
respectively. This means that all these scalars have properties as required for the
Higgs in the standard model.
We need to know the application of the operators τ13 (= 1
2
(S56 − S78), Y
(= τ4+ τ23) andQ (= τ13+ Y), Eq (7.53, 7.54, 7.58), with Sab defined in Eq. (7.36),
on the scalar fields with the space index s = (7, 8).
To compare the properties of the scalar fields with those of the Higgs’s scalar
of the standard model let the scalar fields be eigenstates of τ13 = 1
2
(S56 − S78).
I rewrite for this purpose the second line of Eq. (7.27) as follows, ignoring the
momentum ps, s = (5, 6, . . . , d), since it is expected that solutions with nonzero
momenta in higher dimensions do not contribute to the masses of fermion fields at
low energies in d = (3+1). We pay correspondingly no attention to the momentum
ps , s ∈ (5, . . . , 8), when having in mind the lowest energy solutions, manifesting
at low energies.)∑
s=(7,8),A,i
ψ̄ γs (−τAiAAis )ψ =
−ψ̄ {
78
(+) τAi (AAi7 − iA
Ai
8 )+
78
(−) (τAi (AAi7 + iA
Ai
8 ) }ψ ,
78
(±)= 1
2
(γ7 ± i γ8 ) , AAi78
(±)
:= (AAi7 ∓ iAAi8 ) , (7.38)
with the summation over A and i performed, since AAis represent the scalar fields
(AQs , A
Q ′
s , AY
′
s ) determined byωs ′,s ′′,s and those determined by (ω̃a,b,s Ã4̃s ,
~̃A1̃s ,
~̃A2̃s ,
~̃AÑRs and
~̃AÑLs ).
The application of the operators τ13, Y (Y = 1
2
(S56+S78)− 1
3
(S9 10+S11 12+
S13 14)) andQ on the scalar fields (AAi7 ∓iAAi8 ) with respect to the space index s =
(7, 8), by taking into account Eq. (7.36) to make the application of the generators
Sab on the space indexes, gives
τ13 (AAi7 ∓ iAAi8 ) = ±
1
2
(AAi7 ∓ iAAi8 ) ,
Y (AAi7 ∓ iAAi8 ) = ∓
1
2
(AAi7 ∓ iAAi8 ) ,
Q (AAi7 ∓ iAAi8 ) = 0 . (7.39)
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 108 — #124 i
i
i
i
i
i
108 N.S. Mankoč Borštnik
Since τ4, Y, τ13 and τ1+, τ1− give zero if applied on (AQs , A
Q ′
s and AY
′
s ) with
respect to the quantum numbers (Q,Q ′, Y ′), and since Y and τ13 commute with
the family quantum numbers, one sees that the scalar fields AAis ( =(A
Q
s , AYs , AY
′
s ,
Ã4̃s , Ã
Q̃
s , ~̃A1̃s ,
~̃A2̃s ,
~̃AÑRs ,
~̃AÑLs )), rewritten as AAi78
(±)
= (AAi7 ∓ iAAi8 ) , are eigenstates
of τ13 and Y, having the quantum numbers of the standard model Higgs’ scalar.
These superposition of AAi78
(±)
are presented in Table 7.1 as two doublets with
respect to the weak charge τ13, with the eigenvalue of τ23 (the second SU(2)II
charge) equal to either −1
2
or +1
2
, respectively. The operators τ1± = τ11 ± iτ12
name superposition τ13 τ23 spin τ4 Q
AAi78
(−)
AAi7 + iA
Ai
8 +
1
2
− 1
2
0 0 0
AAi56
(−)
AAi5 + iA
Ai
6 −
1
2
− 1
2
0 0 -1
AAi78
(+)
AAi7 − iA
Ai
8 −
1
2
+ 1
2
0 0 0
AAi56
(+)
AAi5 − iA
Ai
6 +
1
2
+ 1
2
0 0 +1
Table 7.1. The two scalar weak doublets, one with τ23 = − 1
2
and the other with τ23 = + 1
2
,
both with the ”fermion” quantum number τ4 = 0, are presented. In this table all the scalar
fields carry besides the quantum numbers determined by the space index also the quantum
numbers A and i from Eq. (7.37). The table is taken from Ref. [9].
τ
1±
=
1
2
[(S58 − S67) ∓ i (S57 + S68)] , (7.40)
transform one member of a doublet from Table 7.1 into another member of the
same doublet, keeping τ23 (= 1
2
(S56 + S78)) unchanged, clarifying the above
statement.
It is not difficult to show that the scalar fields AAi78
(±)
are triplets as the gauge
fields of the family quantum numbers ( ~̃NR, ~̃NL, ~̃τ2, ~̃τ1; Eqs. (7.55, 7.56, 7.36)) or
singlets as the gauge fields of Q = τ13 + Y, Q ′ = − tan2 ϑ1Y +τ13 and Y ′ =
− tan2 ϑ2τ4 + τ23.
Let us do this for ÃNLi78
(±)
and for AQ78
(±)
, taking into account Eq. (7.52) (where we
replace Sab by Sab) and Eq. (7.36), and recognizing that ÃNL±78
(±)
= ÃNL178
(±)
∓ i ÃNL278
(±)
.
Ã
ÑL±
78
(±)
= {(ω̃
23
78
(±)
+ i ω̃
01
78
(±)
) ∓ i (ω̃
31
78
(±)
+ i ω̃
02
78
(±)
)} ,
ÃÑL378
(±)
= (ω̃
12
78
(±)
+ i ω̃
03
78
(±)
) ,
AQ78
(±)
= ω
56
78
(±)
− (ω
9 10
78
(±)
+ω
11 12
78
(±)
+ω
13 14
78
(±)
) .
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 109 — #125 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 109
One finds
Ñ3L Ã
ÑL±
78
(±)
= ± ÃÑL±78
(±)
, Ñ3L Ã
ÑL3
78
(±)
= 0 ,
QAQ78
(±)
= 0 . (7.41)
with Q = S56 + τ4 = S56 − 1
3
(S9 10 + S11 12 + S13 14), and with τ4 defined in
Eq. (7.54), if replacing Sab by Sab from Eq. (7.36). Similarly one finds properties
with respect to the Ai quantum numbers for all the scalar fields AAi78
(±)
.
After the appearance of the condensate (Table 7.5), which breaks the SU(2)II
symmetry and brings masses to all the scalar fields, the weak ~τ1 and the hyper
charge Y remain the conserved charges.
At the electroweak scale the scalar gauge fields with the space index (7, 8),
with the Lagrange density
Lsg = E
∑
A,i
{(pmA
Ai
s )
† (pmAAis ) − (−λ
Ai + (m ′Ai)
2))AAi†s A
Ai
s
+
∑
B,j
ΛAiBjAAi†s A
Ai
s A
Bj†
s A
Bj
s } , (7.42)
gain nonzero vacuum expectation values and cause the electroweak break 9. The
above Lagrange density needs to be studied. At this stage is just postulated.
The two groups of four families became massive. The mass matrices mani-
fest either S̃U(2)
S̃O(3,1)L
× S̃U(2)
S̃U(4)L
×U(1) symmetry, this is the case for the
lower four families of the eight families, presented in Table 7.4, or S̃U(2)
S̃O(3,1)R
×
S̃U(2)
S̃U(4)R
× U(1) symmetry, this is the case for the higher four families, pre-
sented in Table 7.4. The same three U(1) singlet fields contribute to the masses of
both groups, the two SU(2) triplet fields are for each of the two groups different,
although manifesting the same symmetries.
The mass matrix of family member — quarks and leptons — are 4×4matrices.
The observed three families of quarks and leptons form the 3×3 submatrices of the
4× 4matrices. The symmetry of the mass matrices, manifesting in all orders [57],
limits the number of free parameters.
All the scalars, the two triplets and the three singlets, are doublets with respect
to the weak charge, contributing to the weak and the hyper charge of the fermions
so that they transform the right handed members into the left handed onces.
Mα =

−a1 − a e d b
e −a2 − a b d
d b a2 − a e
b d e a1 − a

α
, (7.43)
with α representing family members — quarks and leptons of left and right
handedness [49–51, 53, 54, 58].
9 The expression for the Lagrange density of Eq. (7.42) is only estimated, more or less
guessed, I have no estimate yet for the constants.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 110 — #126 i
i
i
i
i
i
110 N.S. Mankoč Borštnik
The mass matrices of the upper four families have the same symmetry as
the mass matrices of the lower four families, but the scalar fields determining
the masses of the upper four families have different properties (nonzero vacuum
expectation values, masses and coupling constants) than those of the lower four,
giving to quarks and leptons of the upper four families much higher masses in
comparison with the lower four families of quarks and leptons, what offers the
explanation for the appearance of the dark matter, studied at Refs. [52, 61].
Scalar fields transforming antiquarks and antileptons into quarks and lep-
tons
I follow in this part to a great deal similar part in Ref. [3].
To the matter-antimatter asymmetry the terms contribute, which cause tran-
sitions from antileptons into quarks and from antiquarks into quarks and back.
These are terms included into the third line of Eq. (7.27). Let me rewrite this part
of the fermion action
Lf ′ = ψ† γ0 γt
{
∑
t=(9,10,...14)
[
pt − (
1
2
Ss
′s"ωs ′s"t +
1
2
St
′t ′′ ωt ′t"t +
1
2
S̃ab ω̃abt )
]}
ψ,
as follows
Lf" = ψ† γ0(−) {
∑
+,−
∑
(t t ′)
tt ′
(±©) ·
[ τ2+A2+
tt ′
(±©)
+ τ2−A2−
tt ′
(±©)
+ τ23A23
tt ′
(±©)
+ τ1+A1+
tt ′
(±©)
+ τ1−A1−
tt ′
(±©)
+ τ13A13
tt ′
(±©)
+ τ̃2+ Ã2+
tt ′
(±©)
+ τ̃2− Ã2−
tt ′
(±©)
+ τ̃23 Ã23
tt ′
(±©)
+ τ̃1+ Ã1+
tt ′
(±©)
+ τ̃1− Ã1−
tt ′
(±©)
+ τ̃13 Ã13
tt ′
(±©)
+ Ñ+R Ã
NR+
tt ′
(±©)
+ Ñ−R Ã
NR−
tt ′
(±©)
+ Ñ3R Ã
NR3
tt ′
(±©)
+ Ñ+L Ã
NL+
tt ′
(±©)
+ Ñ−L Ã
NL−
tt ′
(±©)
+ Ñ3L Ã
NL3
tt ′
(±©)
+
∑
i
τ3iA3i
tt ′
(±©)
+ τ4A4
tt ′
(±©)
+
∑
i
τ̃3i Ã3i
tt ′
(±©)
+ τ̃4 Ã4
tt ′
(±©)
] }ψ , (7.44)
where (t, t ′) run in pairs over [(9, 10), (11, 12), (13, 14)] and the summation must
go over + and − of tt ′
(±©)
.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 111 — #127 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 111
In Eq. (7.44) the relations below are used
∑
t,s ′,s ′′
γt
1
2
Ss
′s"ωs ′s"t =
∑
+,−
∑
(t t ′)
tt ′
(±©) 1
2
Ss
′s"ω
s"s"
tt ′
(±©)
,
ω
s"s"
tt ′
(±©)
: = ω
s"s"
tt ′
(±)
= (ωs ′s"t ∓ iωs ′s"t ′) ,
tt ′
(±©): = = 1
2
(γt ± γt ′) ,
tt ′
(±©) 1
2
Ss
′s"ω
s"s"
tt ′
(±©)
=
tt ′
(±©) { τ2+A2+
tt ′
(±©)
+ τ2−A2−
tt ′
(±©)
+ τ23A23
tt ′
(±©)
+ τ1+A1+
tt ′
(±©)
+ τ1−A1−
tt ′
(±©)
+ τ13A13
tt ′
(±©)
} ,
A
2±
tt ′
(±©)
= (ω
58
tt ′
(±©)
+ω
67
tt ′
(±©)
) ∓ i(ω
57
tt ′
(±©)
−ω
68
tt ′
(±©)
) ,
A23
tt ′
(±©)
= (ω
56
tt ′
(±©)
+ω
78
tt ′
(±©)
) ,
A
1±
tt ′
(±©)
= (ω
58
tt ′
(±©)
−ω
67
tt ′
(±©)
) ∓ i(ω
57
tt ′
(±©)
+ω
68
tt ′
(±©)
) ,
A13
tt ′
(±©)
= (ω
56
tt ′
(±©)
−ω
78
tt ′
(±©)
) ,
(t t ′) ∈ ((9 10), (11 12), (13 14)) . (7.45)
The rest of expressions in Eq. (7.45) are obtained in a similar way. They are pre-
sented in Eq. (7.62).
The scalar fields with the scalar index s = (9, 10, · · · , 14), presented in Ta-
ble 7.2, carry one of the triplet colour charges and the ”fermion” charge equal
to twice the quark ”fermion” charge, or the antitriplet colour charges and the
”antifermion” charge. They carry in addition the quantum numbers of the adjoint
representations originating in Sab or in S̃ab 10.
If the antiquark ūc̄2L , from the line 43 presented in Table 7.3, with the ”fermion”
charge τ4 = −1
6
, the weak charge τ13 = 0, the second SU(2)II charge τ23 = −12 ,
the colour charge (τ33, τ38) = (1
2
,− 1
2
√
3
), the hyper charge Y(= τ4 + τ23 =) −2
3
and the electromagnetic charge Q(= Y + τ13 =) −2
3
submits the A29 10
(⊕)
scalar field,
it transforms into uc3R from the line 17 of Table 7.3, carrying the quantum numbers
τ4 = 1
6
, τ13 = 0, τ23 = 1
2
, (τ33, τ38) = (0,− 1√
3
), Y = 2
3
and Q = 2
3
. These two
quarks, dc1R and u
c3
R can bind together with u
c2
R from the 9
th line of the same table
(at low enough energy, after the electroweak transition, and if they belong to a
superposition with the left handed partners to the first family) -into the colour
chargeless baryon - a proton. This transition is presented in Figure 7.1.
The opposite transition at low energies would make the proton decay.
10 Although carrying the colour charge in one of the triplet or antitriplet states, these fields
can not be interpreted as superpartners of the quarks since they do not have quantum
numbers as required by, let say, the N = 1 supersymmetry. The hyper charges and the
electromagnetic charges are namely not those required by the supersymmetric partners
to the family members.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 112 — #128 i
i
i
i
i
i
112 N.S. Mankoč Borštnik
field prop. τ4 τ13 τ23 (τ33, τ38) Y Q τ̃4 τ̃13 τ̃23 Ñ3L Ñ
3
R
A
1±
9 10
(±©)
scalar ∓© 1
3
± 1 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
+ ∓ 1 0 0 0 0 0
A139 10
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
A
1±
11 12
(±©)
scalar ∓© 1
3
∓ 1 0 (∓© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
+ ∓ 1 0 0 0 0 0
A1311 12
(±©)
scalar ∓© 1
3
0 0 (∓© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
A
1±
13 14
(±©)
scalar ∓© 1
3
∓ 1 0 (0, ∓© 1√
3
) ∓© 1
3
∓© 1
3
+ ∓ 1 0 0 0 0 0
A1313 14
(±©)
scalar ∓© 1
3
0 0 (0, ∓© 1√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
A
2±
9 10
(±©)
scalar ∓© 1
3
0 ± 1 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
+ ∓ 1 ∓© 1
3
+ ∓ 1 0 0 0 0 0
A239 10
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
· · ·
Ã
1±
910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 ± 1 0 0 0
Ã13910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
· · ·
Ã
2±
910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 ± 1 0 0
Ã23910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
· · ·
Ã
NL±
910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 ± 1 0
Ã
NL3
910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
· · ·
Ã
NR±
910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 ± 1
Ã
NR3
910
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
· · ·
A3i9 10
(±©)
scalar ∓© 1
3
0 0 (± 1+ ±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
· · ·
A49 10
(±©)
scalar ∓© 1
3
0 0 (±© 1
2
, ±© 1
2
√
3
) ∓© 1
3
∓© 1
3
0 0 0 0 0
· · ·
~A3m vector 0 0 0 octet 0 0 0 0 0 0 0
A4m vector 0 0 0 0 0 0 0 0 0 0 0
Table 7.2. Quantum numbers of the scalar gauge fields carrying the space index t =
(9, 10, · · · , 14), appearing in Eq. (7.27), are presented. The space degrees of freedom con-
tribute one of the triplets values to the colour charge of all these scalar fields. These scalars
are with respect to the two SU(2) charges, (τ13 and ~τ2), and the two S̃U(2) charges, (~̃τ1 and
~̃τ2), triplets (that is in the adjoint representations of the corresponding groups), and they
all carry twice the ”fermion” number (τ4) of the quarks. The quantum numbers of the two
vector gauge fields, the colour and the U(1)II ones, are added.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 113 — #129 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 113
uc2R
τ4= 1
6
,τ13=0,τ23= 1
2
(τ33,τ38)=(− 1
2
, 1
2
√
3
)
Y= 2
3
,Q= 2
3
uc2R
ūc̄2L
τ4=− 1
6
,τ13=0,τ23=− 1
2
(τ33,τ38)=( 1
2
,− 1
2
√
3
)
Y=− 2
3
,Q=− 2
3
uc3R
τ4= 1
6
,τ13=0,τ23= 1
2
(τ33,τ38)=(0,− 1√
3
)
Y= 2
3
,Q= 2
3
ē+L
τ4= 1
2
,τ13=0,τ23= 1
2
(τ33,τ38)=(0,0)
Y=1,Q=1
dc1R
τ4= 1
6
,τ13=0,τ23=− 1
2
(τ33,τ38)=( 1
2
, 1
2
√
3
)
Y=− 1
3
,Q=− 1
3
•
A29 10
(+)
,
τ4=2×(− 1
6
),τ13=0,τ23=−1
(τ33,τ38)=( 1
2
, 1
2
√
3
)
Y=− 4
3
,Q=− 4
3
•
Fig. 7.1. The birth of a ”right handed proton” out of an positron ē +L , antiquark ū
c̄2
L and
quark (spectator) uc2R . The family quantum number can be any.
7.3 Achievements and conclusions
It remains to point out the achievements of the spin-charge-family theory so far and
tell the open problems.
Achievements:
a. The simple starting action, Eq. (7.1), with the Clifford algebra used to describe
the internal space of fermions, which in d ≥ (13 + 1) interact with the vielbeins
and the two kinds of spin connection fields, offers a.i. that one irreducible
representation of the Lorentz algebra in internal space manifests in d = (3+ 1) all
the fermions and antifermions with the spins and charges of the standard model,
a.ii. that eight irreducible representations define in d = (3+ 1) (after the reduction
of the Clifford algebra from two kinds to only one kind) two times four families,
a.iii. that the two kinds of the spin connection fields manifest in d = (3+ 1) all the
vector gauge fields of the standard model, a.iv. that the scalar fields with respect
to d = (3+ 1), carrying the weak and the hyper charge ±1
2
and ∓1
2
, respectively,
forming two groups of scalar fields manifesting each the SU(2) × SU(2) × U(1)
symmetry, offer the explanation for the Higgs’s scalar and Yukawa couplings
of the standard model giving masses to two groups of four families — the lower
four families predicting the fourth family of quarks and leptons to the observed
three, the stable of the upper four families offering explanation for the dark matter,
a.v. that both groups of four families together spread masses from almost zero
to ≥ 1016 GeV, a.vi. that the scalar gauge fields manifesting as colour triplets
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 114 — #130 i
i
i
i
i
i
114 N.S. Mankoč Borštnik
and antitriplets offer the explanation for the matter/antimatter asymmetry of the
ordinary matter.
b. The decision to describe the internal space of fermions with the Clifford
odd algebra enables to define the creation operators as tensor products of finite
number of ”basis vectors” of internal space and infinite basis in ordinary space
applying on the vacuum state, which fulfill together the their Hermitian conjugated
annihilation operators the anticommutation relations postulated by Dirac for the
second quantized fields. The single fermion states have therefore by themselves
the anticommuting character. Tensor products of any number and any kind of
the single fermion creation operators define the second quantized fermion fields
forming the whole Hilbert space.
Predictions:
The spin-charge-family theory offers several explanations as discuss in Sects. 7.1
and 7.2 and also several predictions.
A. Prediction of the fourth family to the observed three families, Subsect. 7.2.1.
Taking into account the experimental data for masses of the observed families of
quarks and the corresponding mixing matrix we fit 6 parameters of the two quark
mass matrices, presented in Eq. (7.43), to twice 3measured massess of quarks and
to 6measured parameters of the mixing matrix.
Althrough any accurate 3×3 submatrix of the 4×4 unitary matrix determines
the 4 × 4 matrix completely, neither the quark nor the lepton mixing matrix is
measured accurately enough that it would be possible to determine three complex
phases of the 4 × 4 quark mixing matrix and the mixing matrix elements of the
fourth family quarks to the other three family members. We therefore assume that
mass matrices are symmetric and real, while making a choice for the masses of the
fourth family.
Results are presented for two choices ofmu4 = md4 , Ref. [54], [arxiv:1412.5866]:
1.mu4 = 700 GeV,md4 = 700 GeV.....new1
2.mu4 = 1 200 GeV,md4 = 1200 GeV.....new2
|V(ud)| =

expn 0.97425± 0.00022 0.2253± 0.0008 0.00413± 0.00049
new1 0.97423(4) 0.22539(7) 0.00299 0.00776(1)
new2 0.97423[5] 0.22538[42] 0.00299 0.00793[466]
expn 0.225± 0.008 0.986± 0.016 0.0411± 0.0013
new1 0.22534(3) 0.97335 0.04245(6) 0.00349(60)
new2 0.22531[5] 0.97336[5] 0.04248 0.00002[216]
expn 0.0084± 0.0006 0.0400± 0.0027 1.021± 0.032
new1 0.00667(6) 0.04203(4) 0.99909 0.00038
new2 0.00667 0.04206[5] 0.99909 0.00024[21]
new1 0.00677(60) 0.00517(26) 0.00020 0.99996
new2 0.00773 0.00178 0.00022 0.99997[9]

.
(7.46)
One can see that the above results for the mixing matrices of the lower three
families are in agreement with what Ref. [55] requires, namely that
Vu1d4 > Vu1d3 , Vu2d4 < Vu1d4 , and Vu3d4 < Vu1d4 .
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 115 — #131 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 115
Since we have not yet fit the mass matrix of Eq. (7.43) to the newest experi-
mental data [56], which appear after our Bled 2020, the evaluation for our 4× 4
quark mixing matrix with the new data and correspondingly a new prediction is
not yet offered.
Let me repeat the discussion of Ref. [58] that the existence of the fourth family
to the observed three is still not in disagreement with the latest experimental data
although some phenomenologists say different.
B. The spin-charge-family theory predicts in the low energy regime (up to 1016
GeV or higher) the existence of two decoupled groups of four families, which at
the electroweak break become massive [52]. The stable family of the upper group
of four families (with almost zero Yukawa couplings to the lower group of four
families) is the candidate for the dark matter, Subsect. 7.2.1.
I review here briefly the estimations done in Ref. [52]. We used the simple
hydrogen-like model to evaluate properties of the fifth family heavy baryons,
taking into account that for masses of the order of a few TeV or larger the force
among the constituents of the fifth family baryons is determined mostly by one
gluon exchange. The fifth family neutron is estimated as the most stable nucleon.
The ”nuclear interaction” among these baryons is found to have very interesting
properties. We studied scattering amplitudes among fifth family neutrons and
with the ordinary matter.
We followed the behaviour of the fifth family quarks and antiquarks in the
plasma of the expanding universe, through the freezing out procedure, solving the
Boltzmann equations, through the colour phase transition, while forming neutrons,
up to the present dark matter, taking into account the cosmological evidences, the
direct experimental evidences and all others known properties of the dark matter.
The cosmological evolution suggested the limits for the masses of the fifth
family quarks
10 TeV < mq5 c
2 < a few · 102 TeV (7.47)
and for the scattering cross sections
10−8 fm2 < σc5 < 10
−6 fm2 , (7.48)
while the measured density of the dark matter does not put much limitation on
the properties of heavy enough clusters.
The direct measurements limit the fifth family quark mass to
several 10TeV < mq5c
2 < 105 TeV . (7.49)
We also find that our fifth family baryons of the mass of several 10 TeV/c2 have
for a factor more than 100 times too small scattering amplitude with the ordinary
matter to cause a measurable heat flux on the Earth’s surface.
C. The spin-charge-family theory predicts several scalar fields with the weak and
the hyper charge of the Higg’s scalar (±1
2
, ∓1
2
) — two triplets and three singlets
— offering the explanation for the existence of the Higgs’s scalar and Yukawa
couplings, Subsect. 7.2.2.
The additional two triplets and the same three singlets determine properties
of the upper four families of quarks and leptons, Subsect. 7.2.2.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 116 — #132 i
i
i
i
i
i
116 N.S. Mankoč Borštnik
D. The spin-charge-family theory predicts several scalar fields which are colour
triplets or antitriplets, offering the explanation for the matter/antimatter asymme-
try in the (nonequilibrium) expanding universe as well as the proton decay [4],
Subsect. 7.2.2.
E. The mass matrices of the two fourth family groups are close to democratic
one, causing spreading of the fermion masses from 10−8 MeV to 1016 GeV or even
higher.
I conclude by saying that there are still a lot of open problems to be solved.
Some of them are common to the other theories, like the Kaluza-Klein-like theories,
the others require to extract as much as possible from the offer of the theory. We
need collaborators, since the more work is put into the spin-charge-family theory
the more explanations for the observed phenomena follow.
7.4 APPENDIX: Useful relations
From Eq. (7.16) it follows
Sac
ab
(k)
cd
(k) = −
i
2
ηaaηcc
ab
[−k]
cd
[−k] , S̃ac
ab
(k)
cd
(k) =
i
2
ηaaηcc
ab
[k]
cd
[k] ,
Sac
ab
[k]
cd
[k] =
i
2
ab
(−k)
cd
(−k) , S̃ac
ab
[k]
cd
[k] = −
i
2
ab
(k)
cd
(k) ,
Sac
ab
(k)
cd
[k] = −
i
2
ηaa
ab
[−k]
cd
(−k) , S̃ac
ab
(k)
cd
[k] = −
i
2
ηaa
ab
[k]
cd
(k) ,
Sac
ab
[k]
cd
(k) =
i
2
ηcc
ab
(−k)
cd
[−k] , S̃ac
ab
[k]
cd
(k) =
i
2
ηcc
ab
(k)
cd
[k] . (7.50)
By using Eq. (7.14) one finds the relations
ab
˜(k)
ab
(k) = 0 ,
ab
˜(−k)
ab
(k)= −i ηaa
ab
[k] ,
ab
˜(k)
ab
[k] = i
ab
(k) ,
ab
˜(k)
ab
[−k]= 0 ,
ab
˜[k]
ab
(k) =
ab
(k) ,
ab
˜[−k]
ab
(k)= 0 ,
ab
˜[k]
ab
[k] = 0 ,
ab
˜[−k]
ab
[k]=
ab
[k] . (7.51)
7.5 APPENDIX: One irreducible representation of the internal
space and families described by the Clifford algebra γa
Below the subgroups of the starting groups SO(13, 1) and ˜SO(13, 1) are presented,
manifesting in d = (3 + 1) the spins, charges and families of fermions in the
spin-charge-family theory. Table 7.3, representing one SO(13, 1) irreducible repre-
sentation of fermions — quarks and leptons and antiquarks and antileptons —
uses these expressions.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 117 — #133 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 117
a.i. The generators of the two SU(2) (⊂ SO(3, 1) ⊂ SO(7, 1) ⊂ SO(13, 1)) groups,
describing spins of fermions
~N±(= ~N(L,R)) : =
1
2
(S23 ± iS01, S31 ± iS02, S12 ± iS03) , (7.52)
are presented.
a.ii. The generators of the two SU(2) (SU(2) ⊂ SO(4) ⊂ SO(7, 1) ⊂ SO(13, 1))
groups, describing the two kinds of weak charges of fermions
~τ1 : =
1
2
(S58 − S67, S57 + S68, S56 − S78) ,
~τ2 : =
1
2
(S58 + S67, S57 − S68, S56 + S78) , (7.53)
are presented.
a.iii. The SU(3) and U(1) subgroups of SO(6) ⊂ SO(13, 1), describing the
colour charge and the ”fermion” charge of fermions
~τ3 :=
1
2
{S9 12 − S10 11 , S9 11 + S10 12, S9 10 − S11 12,
S9 14 − S10 13, S9 13 + S10 14 , S11 14 − S12 13 ,
S11 13 + S12 14,
1√
3
(S9 10 + S11 12 − 2S13 14)} ,
τ4 := −
1
3
(S9 10 + S11 12 + S13 14) , (7.54)
are presented.
b.i. The two S̃U(2) subgroups of ˜SO(3, 1) (⊂ S̃O(7, 1) ⊂ S̃O(13, 1)), describing
families of fermions
~̃NL,R : =
1
2
(S̃23 ± iS̃01, S̃31 ± iS̃02, S̃12 ± iS̃03) , (7.55)
are presented.
b.ii. The two S̃U(2) subgroups of S̃O(4) (⊂ S̃O(7, 1) ⊂ S̃O(13, 1)), describing
families of fermions
~̃τ1 : =
1
2
(S̃58 − S̃67, S̃57 + S̃68, S̃56 − S̃78) ,
~̃τ2 : =
1
2
(S̃58 + S̃67, S̃57 − S̃68, S̃56 + S̃78) , (7.56)
are presented.
b.iii. The group Ũ(1), the subgroup of S̃O(6) (⊂ S̃O(13, 1)), describing family
quantum numbers of fermions
τ̃4 := −
1
3
(S̃9 10 + S̃11 12 + S̃13 14) , (7.57)
are presented.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 118 — #134 i
i
i
i
i
i
118 N.S. Mankoč Borštnik
c. Relations among the hyper, weak and the second SU(2) charges
Y := τ4 + τ23 , Y ′ := −τ4 tan2 ϑ2 + τ23 , Q := τ13 + Y , Q ′ := −Y tan2 ϑ1 + τ13 ,
Ỹ := τ̃4 + τ̃23 , Ỹ ′ := −τ̃4 tan2 ϑ2 + τ̃23 , Q̃ := Ỹ + τ̃13 , Q̃ ′ = −Ỹ tan2 ϑ1 + τ̃13 ,
(7.58)
are presented.
Below are some of the above expressions written in terms of nilpotents and
projectors
N±+ = N
1
+ ± iN2+ = −
03
(∓i)
12
(±) , N±− = N1− ± iN2− =
03
(±i)
12
(±) ,
ѱ+ = −
03
˜(∓i)
12
˜(±) , ѱ− =
03
˜(±i)
12
˜(±) ,
τ1± = (∓)
56
(±)
78
(∓) , τ2∓ = (∓)
56
(∓)
78
(∓) ,
τ̃1± = (∓)
56
˜(±)
78
˜(∓) , τ̃2∓ = (∓)
56
˜(∓)
78
˜(∓) . (7.59)
i ab̂†
i
Γ(3+1) S12 τ13 τ23 τ33 τ38 τ4 Y Q
(Anti)octet, Γ(7+1) = (−1) 1 , Γ(6) = (1) − 1
of (anti)quarks and (anti)leptons
1 ûc1†
R
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] 1 1
2
0 1
2
1
2
1
2
√
3
1
6
2
3
2
3
2 ûc1†
R
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] 1 − 1
2
0 1
2
1
2
1
2
√
3
1
6
2
3
2
3
3 d̂c1†
R
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] 1 1
2
0 − 1
2
1
2
1
2
√
3
1
6
− 1
3
− 1
3
4 d̂c1†
R
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] 1 − 1
2
0 − 1
2
1
2
1
2
√
3
1
6
− 1
3
− 1
3
5 d̂c1†
L
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] -1 1
2
− 1
2
0 1
2
1
2
√
3
1
6
1
6
− 1
3
6 d̂c1†
L
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] -1 − 1
2
− 1
2
0 1
2
1
2
√
3
1
6
1
6
− 1
3
7 ûc1†
L
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] -1 1
2
1
2
0 1
2
1
2
√
3
1
6
1
6
2
3
8 ûc1†
L
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] -1 − 1
2
1
2
0 1
2
1
2
√
3
1
6
1
6
2
3
9 ûc2†
R
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] 1 1
2
0 1
2
− 1
2
1
2
√
3
1
6
2
3
2
3
10 ûc2†
R
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] 1 − 1
2
0 1
2
− 1
2
1
2
√
3
1
6
2
3
2
3
11 d̂c2†
R
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] 1 1
2
0 − 1
2
− 1
2
1
2
√
3
1
6
− 1
3
− 1
3
12 d̂c2†
R
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] 1 − 1
2
0 − 1
2
− 1
2
1
2
√
3
1
6
− 1
3
− 1
3
13 d̂c2†
L
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] -1 1
2
− 1
2
0 − 1
2
1
2
√
3
1
6
1
6
− 1
3
14 d̂c2†
L
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] -1 − 1
2
− 1
2
0 − 1
2
1
2
√
3
1
6
1
6
− 1
3
15 ûc2†
L
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] -1 1
2
1
2
0 − 1
2
1
2
√
3
1
6
1
6
2
3
16 ûc2†
L
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] -1 − 1
2
1
2
0 − 1
2
1
2
√
3
1
6
1
6
2
3
17 ûc3†
R
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) 1 1
2
0 1
2
0 − 1√
3
1
6
2
3
2
3
18 ûc3†
R
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) 1 − 1
2
0 1
2
0 − 1√
3
1
6
2
3
2
3
19 d̂c3†
R
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) 1 1
2
0 − 1
2
0 − 1√
3
1
6
− 1
3
− 1
3
20 d̂c3†
R
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) 1 − 1
2
0 − 1
2
0 − 1√
3
1
6
− 1
3
− 1
3
Continued on next page
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 119 — #135 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 119
i ab̂†
i
Γ(3+1) S12 τ13 τ23 τ33 τ38 τ4 Y Q
(Anti)octet, Γ(7+1) = (−1) 1 , Γ(6) = (1) − 1
of (anti)quarks and (anti)leptons
21 d̂c3†
L
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) -1 1
2
− 1
2
0 0 − 1√
3
1
6
1
6
− 1
3
22 d̂c3†
L
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) -1 − 1
2
− 1
2
0 0 − 1√
3
1
6
1
6
− 1
3
23 ûc3†
L
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) -1 1
2
1
2
0 0 − 1√
3
1
6
1
6
2
3
24 ûc3†
L
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) -1 − 1
2
1
2
0 0 − 1√
3
1
6
1
6
2
3
25 ν̂†
R
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) 1 1
2
0 1
2
0 0 − 1
2
0 0
26 ν̂†
R
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) 1 − 1
2
0 1
2
0 0 − 1
2
0 0
27 ê†
R
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) 1 1
2
0 − 1
2
0 0 − 1
2
−1 −1
28 ê†
R
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) 1 − 1
2
0 − 1
2
0 0 − 1
2
−1 −1
29 ê†
L
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) -1 1
2
− 1
2
0 0 0 − 1
2
− 1
2
−1
30 ê†
L
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) -1 − 1
2
− 1
2
0 0 0 − 1
2
− 1
2
−1
31 ν̂†
L
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) -1 1
2
1
2
0 0 0 − 1
2
− 1
2
0
32 ν̂†
L
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) -1 − 1
2
1
2
0 0 0 − 1
2
− 1
2
0
33 ^̄dc̄1†
L
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) -1 1
2
0 1
2
− 1
2
− 1
2
√
3
− 1
6
1
3
1
3
34 ^̄dc̄1†
L
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) -1 − 1
2
0 1
2
− 1
2
− 1
2
√
3
− 1
6
1
3
1
3
35 ūc̄1†
L
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) -1 1
2
0 − 1
2
− 1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
36 ūc̄1†
L
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) -1 − 1
2
0 − 1
2
− 1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
37 ^̄dc̄1†
R
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) 1 1
2
1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
38 ^̄dc̄1†
R
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) 1 − 1
2
1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
39 ^̄uc̄1†
R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) 1 1
2
− 1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
40 ^̄uc̄1†
R
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) 1 − 1
2
− 1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
41 ^̄dc̄2†
L
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) -1 1
2
0 1
2
1
2
− 1
2
√
3
− 1
6
1
3
1
3
42 ^̄dc̄2†
L
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) -1 − 1
2
0 1
2
1
2
− 1
2
√
3
− 1
6
1
3
1
3
43 ^̄uc̄2†
L
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) -1 1
2
0 − 1
2
1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
44 ^̄uc̄2†
L
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) -1 − 1
2
0 − 1
2
1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
45 ^̄dc̄2†
R
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) 1 1
2
1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
46 ^̄dc̄2†
R
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) 1 − 1
2
1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
47 ^̄uc̄2†
R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) 1 1
2
− 1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
48 ^̄uc̄2†
R
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) 1 − 1
2
− 1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
49 ^̄dc̄3†
L
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] -1 1
2
0 1
2
0 1√
3
− 1
6
1
3
1
3
50 ^̄dc̄3†
L
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] -1 − 1
2
0 1
2
0 1√
3
− 1
6
1
3
1
3
51 ^̄uc̄3†
L
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] -1 1
2
0 − 1
2
0 1√
3
− 1
6
− 2
3
− 2
3
52 ^̄uc̄3†
L
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] -1 − 1
2
0 − 1
2
0 1√
3
− 1
6
− 2
3
− 2
3
53 ^̄dc̄3†
R
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] 1 1
2
1
2
0 0 1√
3
− 1
6
− 1
6
1
3
54 ^̄dc̄3†
R
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] 1 − 1
2
1
2
0 0 1√
3
− 1
6
− 1
6
1
3
55 ^̄uc̄3†
R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] 1 1
2
− 1
2
0 0 1√
3
− 1
6
− 1
6
− 2
3
56 ^̄uc̄3†
R
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] 1 − 1
2
− 1
2
0 0 1√
3
− 1
6
− 1
6
− 2
3
57 ^̄e†
L
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] -1 1
2
0 1
2
0 0 1
2
1 1
58 ^̄e†
L
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] -1 − 1
2
0 1
2
0 0 1
2
1 1
59 ^̄ν†
L
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] -1 1
2
0 − 1
2
0 0 1
2
0 0
60 ^̄ν†
L
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] -1 − 1
2
0 − 1
2
0 0 1
2
0 0
61 ^̄ν†
R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] 1 1
2
− 1
2
0 0 0 1
2
1
2
0
62 ^̄ν†
R
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] 1 − 1
2
− 1
2
0 0 0 1
2
1
2
0
63 ^̄e†
R
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] 1 1
2
1
2
0 0 0 1
2
1
2
1
64 ^̄e†
R
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] 1 − 1
2
1
2
0 0 0 1
2
1
2
1
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 120 — #136 i
i
i
i
i
i
120 N.S. Mankoč Borštnik
Table 7.3. The left handed (Γ(13,1) = −1), multiplet of creation operators of fermions — the members of one fundamental representation of
the SO(13, 1) group, manifesting the subgroup SO(7, 1) of the colour charged quarks and antiquarks and the colourless leptons and antileptons —
is presented in the massless basis as products of nilpotents and projectors. The multiplet contains the left handed (Γ(3+1) = −1 weak (SU(2)I)
charged (τ13 = ± 1
2
, (~τ1 = 1
2
(S58 − S67, S57 + S68, S56 − S78)) and SU(2)II chargeless (τ
23 = 0, ~τ2 = 1
2
(S58 +
S67, S57 − S68, S56 + S78)) quarks and leptons and the right handed (Γ(3+1) = 1), weak (SU(2)I) chargeless and SU(2)II charged
(τ23 = ± 1
2
) quarks and leptons, both with the spin S12 up and down (± 1
2
, respectively). The creation operators of quarks distinguish from those
of leptons only in the SU(3) × U(1) part: Quarks are triplets of three colours ( (τ33, τ38) = [( 1
2
, 1
2
√
3
), (− 1
2
, 1
2
√
3
), (0,− 1√
3
)],
(~τ3 = 1
2
(S9 12 − S10 11, S9 11 + S10 12, S9 10 − S11 12, S9 14 − S10 13, S9 13 + S10 14, S11 14 − S12 13,
S11 13+S12 14, 1√
3
(S9 10+S11 12−2S13 14)), carrying the ”fermion charge” (τ4 = 1
6
, = − 1
3
(S9 10+S11 12+S13 14).
The colourless leptons carry the ”fermion charge” (τ4 = − 1
2
). In the same multiplet of creation operators the left handed weak (SU(2)I ) chargeless
and SU(2)II charged antiquarks and antileptons and the right handed weak (SU(2)I ) charged and SU(2)II chargeless antiquarks and antileptons
are included. Antiquarks distinguish from antileptons again only in the SU(3) × U(1) part: Anti-quarks are antitriplets, carrying the ”fermion charge”
(τ4 = − 1
6
). The anti-colourless antileptons carry the ”fermion” charge (τ4 = 1
2
). Y = (τ23 + τ4) is the hyper charge, the electromagnetic charge
isQ = (τ13 + Y). The creation operators of opposite charges (antifermion creation operators) are reachable from the particle ones besides bySab also by
the application of the discrete symmetry operator CN PN , presented in Refs. [65, 66]. The reader can find this Weyl representation also in Refs. [4, 9, 71, 72] and
in the references therein.
Table 7.3 represents in the spin-charge-family theory the creation operators for
observed quarks and leptons and antiquarks and antileptons for a particular family,
Table (7.4). Hermitian conjugation of the creation operators of Table 7.3 generates
the corresponding annihilation operators, fulfilling together with the creation
operators anticommutation relations for fermions of Eq. (7.23).
The condensate of two right handed neutrinos with the family quantum
numbers of the upper four families, causing the break of the starting symmetry
SO(13, 1) into SO(7, 1)× SU(3)×U(1), is presented in Table 7.5.
7.6 APPENDIX: Expressions for scalar fields in term ofωs ′s ′′s
and ω̃abs
The scalar fields, responsible for masses of the family members and of the heavy
bosons [6, 7] after gaining nonzero vacuum expectation values and triggering the
electroweak break, are presented in the second line of Eq. (7.27). These scalar fields
are included in the covariant derivatives as −1
2
Ss
′s"ωs ′s"s −
1
2
S̃abω̃abs, s ∈ (7, 8),
(a, b), ∈ (0, . . . , 3), (5, . . . , 8).
One can express the scalar fields carrying the quantum numbers of the sub-
groups of the family groups, expressed in terms of ω̃abs (they contribute to mass
matrices of quarks and leptons and to masses of the heavy bosons), if taking into
account Eqs. (7.55, 7.56, 7.58),∑
a,b
−
1
2
S̃ab ω̃abs = −(~̃τ
1̃ ~̃A1̃s +
~̃NL̃
~̃A
ÑL̃
s + ~̃τ
2̃ ~̃A2̃s +
~̃NR̃
~̃A
ÑR̃
s ) ,
~̃A1̃s = (ω̃58s − ω̃67s, ω̃57s + ω̃68s, ω̃56s − ω̃78s) ,
~̃A
ÑL̃
s = (ω̃23s + i ω̃01s, ω̃31s + i ω̃02s, ω̃12s + i ω̃03s) ,
~̃A2̃s = (ω̃58s + ω̃67s, ω̃57s − ω̃68s, ω̃56s + ω̃78s) ,
~̃A
ÑR̃
s = (ω̃23s − i ω̃01s, ω̃31s − i ω̃02s, ω̃12s − i ω̃03s) ,
(s ∈ (7, 8)) . (7.60)
Scalars, expressed in terms ofωabc (contributing as well to the mass matrices of
quarks and leptons and to masses of the heavy bosons) follow, if using Eqs. (7.53,
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 121 — #137 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 121
τ̃
1
3
τ̃
2
3
Ñ
3 L
Ñ
3 R
τ̃
4
I
û
c
1
†
R
1
0
3
(+
i)
1
2
[+
]
|
5
6
[+
]
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
1
0
3
(+
i)
1
2
[+
]
|
5
6
[+
]
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
−
1 2
0
−
1 2
0
−
1 2
I
û
c
1
†
R
2
0
3
[+
i]
1
2
(+
)
|
5
6
[+
]
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
2
0
3
[+
i]
1
2
(+
)
|
5
6
[+
]
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
−
1 2
0
1 2
0
−
1 2
I
û
c
1
†
R
3
0
3
(+
i)
1
2
[+
]
|
5
6
(+
)
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
3
0
3
(+
i)
1
2
[+
]
|
5
6
(+
)
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
1 2
0
−
1 2
0
−
1 2
I
û
c
1
†
R
4
0
3
[+
i]
1
2
(+
)
|
5
6
(+
)
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
4
0
3
[+
i]
1
2
(+
)
|
5
6
(+
)
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
1 2
0
1 2
0
−
1 2
II
û
c
1
†
R
5
0
3
[+
i]
1
2
[+
]
|
5
6
[+
]
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
5
0
3
[+
i]
1
2
[+
]
|
5
6
[+
]
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
0
−
1 2
0
−
1 2
−
1 2
II
û
c
1
†
R
6
0
3
(+
i)
1
2
(+
)
|
5
6
[+
]
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
6
0
3
(+
i)
1
2
(+
)
|
5
6
[+
]
7
8
[+
]
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
0
−
1 2
0
1 2
−
1 2
II
û
c
1
†
R
7
0
3
[+
i]
1
2
[+
]
|
5
6
(+
)
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
7
0
3
[+
i]
1
2
[+
]
|
5
6
(+
)
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
0
1 2
0
−
1 2
−
1 2
II
û
c
1
†
R
8
0
3
(+
i)
1
2
(+
)
|
5
6
(+
)
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
(−
)
1
3
1
4
(−
)
ν̂
† R
8
0
3
(+
i)
1
2
(+
)
|
5
6
(+
)
7
8
(+
)
||
9
1
0
(+
)
1
1
1
2
[+
]
1
3
1
4
[+
]
0
1 2
0
1 2
−
1 3
Ta
bl
e
7.
4.
E
ig
ht
fa
m
ili
es
of
cr
ea
ti
on
op
er
at
or
s
of
û
c
1
†
R
—
th
e
ri
gh
th
an
d
ed
u
-q
u
ar
k
w
it
h
sp
in
1 2
an
d
th
e
co
lo
u
r
ch
ar
ge
(τ
3
3
=
1
/
2
,τ
3
8
=
1
/
(2
√
3
))
,
ap
pe
ar
in
g
in
th
e
fi
rs
tl
in
e
of
Ta
bl
e
7.
3
—
an
d
of
th
e
co
lo
u
rl
es
s
ri
gh
th
an
d
ed
ne
u
tr
in
o
ν̂
† R
—
of
sp
in
1 2
,a
pp
ea
ri
ng
in
th
e
2
5
t
h
lin
e
of
Ta
bl
e
7.
3
—
ar
e
pr
es
en
te
d
in
th
e
le
ft
an
d
in
th
e
ri
gh
tc
ol
um
n,
re
sp
ec
ti
ve
ly
.T
ab
le
is
ta
ke
n
fr
om
[9
].
Fa
m
ili
es
be
lo
ng
to
tw
o
gr
ou
ps
of
fo
ur
fa
m
ili
es
,o
ne
(I
)i
s
a
do
ub
le
t
w
it
h
re
sp
ec
tt
o
(~̃ N
L
an
d
~̃ τ
(1
)
)a
nd
a
si
ng
le
tw
it
h
re
sp
ec
tt
o
(~̃ N
R
an
d
~̃ τ
(2
)
),
th
e
ot
he
r
(I
I)
is
a
si
ng
le
tw
it
h
re
sp
ec
tt
o
(~̃ N
L
an
d
~̃ τ
(1
)
)a
nd
a
d
ou
bl
et
w
it
h
w
ith
re
sp
ec
tt
o
(~̃ N
R
an
d
~̃ τ
(2
)
),
Eq
.(
7.
52
).
A
ll
th
e
fa
m
ili
es
fo
llo
w
fr
om
th
e
st
ar
tin
g
on
e
by
th
e
ap
pl
ic
at
io
n
of
th
e
op
er
at
or
s
( Ñ
± R
,L
,τ̃
(2
,1
)±
),
Eq
.(
7.
59
).
Th
e
ge
ne
ra
to
rs
(N
± R
,L
,τ
(2
,1
)±
)(
Eq
.(
7.
59
))
tr
an
sf
or
m
û
† 1R
to
al
lt
he
m
em
be
rs
of
on
e
fa
m
ily
of
th
e
sa
m
e
co
lo
ur
.T
he
sa
m
e
ge
ne
ra
to
rs
tr
an
sf
or
m
eq
ui
va
le
nt
ly
th
e
ri
gh
th
an
de
d
ne
ut
ri
no
ν̂
† 1R
to
al
lt
he
co
lo
ur
le
ss
m
em
be
rs
of
th
e
sa
m
e
fa
m
ily
.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 122 — #138 i
i
i
i
i
i
122 N.S. Mankoč Borštnik
state S03 S12 τ13 τ23 τ4 Y Q τ̃13 τ̃23 τ̃4 Ỹ Q̃ Ñ3L Ñ
3
R
(|νVIII1R >1 |ν
VIII
2R >2) 0 0 0 1 −1 0 0 0 1 −1 0 0 0 1
(|νVIII1R >1 |e
VIII
2R >2) 0 0 0 0 −1 −1 −1 0 1 −1 0 0 0 1
(|eVIII1R >1 |e
VIII
2R >2) 0 0 0 −1 −1 −2 −2 0 1 −1 0 0 0 1
Table 7.5. The condensate of the two right handed neutrinos νR, with the quantum numbers
of the VIIIth family, coupled to spin zero and belonging to a triplet with respect to the
generators τ2i, is presented, together with its two partners. The condensate carries ~τ1 = 0,
τ23 = 1, τ4 = −1 and Q = 0 = Y. The triplet carries τ̃4 = −1, τ̃23 = 1 and Ñ3R = 1,
Ñ3L = 0, Ỹ = 0, Q̃ = 0. The family quantum numbers of quarks and leptons are presented in
Table 7.4.
7.54, 7.58) ∑
s ′,s ′′
−
1
2
Ss
′s"ωs ′s"s = −(g
23 τ23A23s + g
13 τ13A13s + g
4 τ4A4s) ,
g13 τ13A13s + g
23 τ23A23s + g
4 τ4A4s = g
QQAQs + g
Q ′ Q ′AQ
′
s + g
Y ′ Y ′AY
′
s ,
A4s = −(ω9 10 s +ω11 12 s +ω13 14 s) ,
A13s = (ω56s −ω78s) , A
23
s = (ω56s +ω78s) ,
AQs = sin ϑ1A
13
s + cos ϑ1A
Y
s ,
AQ
′
s = cos ϑ1A
13
s − sin ϑ1A
Y
s ,
AY
′
s = cos ϑ2A
23
s − sin ϑ2A
4
s ,
(s ∈ (7, 8)) . (7.61)
Scalar fields from Eq. (7.60) interact with quarks and leptons and antiquarks and
antileptons through the family quantum numbers, while those from Eq. (7.61)
interact through the family members quantum numbers. In Eq. (7.61) the coupling
constants are explicitly written in order to see the analogy with the gauge fields of
the standard model.
Expressions for the vector gauge fields in terms of the spin connection fields
and the vielbeins, which correspond to the colour charge (~A3m), the SU(2)II charge
(~A2m), the weak SU(2)I charge (~A1m) and the U(1) charge originating in SO(6)
(~A4m), can be found by taking into account Eqs. (7.53, 7.54). Equivalently one finds
the vector gauge fields in the ”tilde” sector, or one just uses the expressions from
Eqs. (7.61, 7.60), if replacing the scalar index swith the vector indexm.
The expression for
∑
tab γ
t 1
2
S̃ab ω̃abt, used in Eq. (7.45) (S̃ab are the in-
finitesimal generators of either S̃O(3, 1) or S̃O(4), while ω̃abt belong to the cor-
responding gauge fields with t = (9, . . . , 14)), and obtained by using Eqs. (7.55 -
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 123 — #139 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 123
7.59), are
∑
abt
γt
1
2
S̃ab ω̃abt =
∑
+−tt ′ab
tt ′
(±©) 1
2
S̃ab ω̃
ab
tt ′
(±©)
=
∑
+−tt ′
tt ′
(±©) { τ̃2+ Ã2+
tt ′
(±©)
+ τ̃2− Ã2−
tt ′
(±©)
+ τ̃23 Ã23
tt ′
(±©)
+
τ̃1+ Ã1+
tt ′
(±©)
+ τ̃1− Ã1−
tt ′
(±©)
+ τ̃13 Ã13
tt ′
(±©)
+
Ñ+R Ã
NR+
tt ′
(±©)
+ Ñ−R Ã
NR−
tt ′
(±©)
+ Ñ3R Ã
NR3
tt ′
(±©)
+
Ñ+L Ã
NL+
tt ′
(±©)
+ Ñ−L Ã
NL−
tt ′
(±©)
+ Ñ3L Ã
NL3
tt ′
(±©)
} ,
Ã
NR±
tt ′
(±©)
= (ω̃
23
tt ′
(±©)
− i ω̃
01
tt ′
(±©)
) ∓ i(ω̃
31
tt ′
(±©)
− i ω̃
02
tt ′
(±©)
) ,
ÃNR3
tt ′
(±©)
= (ω̃
12
tt ′
(±©)
− i ω̃
03
tt ′
(±©)
) ,
Ã
NL±
tt ′
(±©)
= (ω̃
23
tt ′
(±©)
+ i ω̃
01
tt ′
(±©)
) ∓ i(ω̃
31
tt ′
(±©)
+ i ω̃
02
tt ′
(±©)
) ,
ÃNR3
tt ′
(±©)
= (ω̃
12
tt ′
(±©)
+ i ω̃
03
tt ′
(±©)
) .
(7.62)
The term
∑
tt ′t ′′ γ
t 1
2
St
′t"ωt ′t"t in Eq. (7.27) can be rewritten with respect to
the generators St
′t" and the corresponding gauge fieldsωs ′s"t as one colour octet
scalar field and one U(1)II singlet scalar field (Eq. 7.54)
γt
1
2
St"t
′"ωt"t ′"t =
∑
+,−
∑
(t t ′)
tt ′
(±©) { ~τ3 · ~A3
tt ′
(±©)
+ τ4 ·A4
tt ′
(±©)
} ,
(t t ′) ∈ ((9 10), 11 12), 13 14)) . (7.63)
Acknowledgements
The author thanks Department of Physics, FMF, University of Ljubljana, Society
of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the
research on the spin-charge-family theory, and Matjaž Breskvar of Beyond Semicon-
ductor for donations, in particular for sponsoring the annual workshops entitled
”What comes beyond the standard models” at Bled.
References
1. N. Mankoč Borštnik, ”Spin connection as a superpartner of a vielbein”, Phys. Lett. B
292 (1992) 25-29.
2. N. Mankoč Borštnik, ”Spinor and vector representations in four dimensional Grass-
mann space”, J. of Math. Phys. 34 (1993) 3731-3745.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 124 — #140 i
i
i
i
i
i
124 N.S. Mankoč Borštnik
3. N.S. Mankoč Borštnik, ”Spin-charge-family theory is offering next step in understand-
ing elementary particles and fields and correspondingly universe”, Proceedings to
the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences,
Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Rela-
tivistic Dynamics of Particles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981,
arXiv:1607.01618v2].
4. N.S. Mankoč Borštnik, ”Matter-antimatter asymmetry in the spin-charge-family theory”,
Phys. Rev. D 91 (2015) 065004 [arXiv:1409.7791].
5. N.S. Mankoč Borštnik, D. Lukman, ”Vector and scalar gauge fields with respect to
d = (3 + 1) in Kaluza-Klein theories and in the spin-charge-family theory”, Eur. Phys. J. C
77 (2017) 231.
6. N.S. Mankoč Borštnik, ”The spin-charge-family theory explains why the scalar Higgs
carries the weak charge ± 1
2
and the hyper charge ∓ 1
2
”, Proceedings to the 17th Work-
shop ”What comes beyond the standard models”, Bled, 20-28 of July, 2014, Ed. N.S.
Mankoč Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December
2014, p.163-82 [ arXiv:1502.06786v1] [arXiv:1409.4981].
7. N.S. Mankoč Borštnik N S, ”The spin-charge-family theory is explaining the origin
of families, of the Higgs and the Yukawa couplings”, J. of Modern Phys. 4 (2013) 823
[arXiv:1312.1542].
8. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”The spin-charge-family theory offers under-
standing of the triangle anomalies cancellation in the standard model”, Fortschritte der
Physik, Progress of Physics (2017) 1700046.
9. N.S. Mankoč Borštnik, ”The explanation for the origin of the Higgs scalar and for the
Yukawa couplings by the spin-charge-family theory”, J.of Mod. Physics 6 (2015) 2244-2274,
http://dx.org./10.4236/jmp.2015.615230 [arXiv:1409.4981].
10. N.S. Mankoč Borštnik and H.B. Nielsen, ”Why nature made a choice of Clifford
and not Grassmann coordinates”, Proceedings to the 20th Workshop ”What comes
beyond the standard models”, Bled, 9-17 of July, 2017, Ed. N.S. Mankoč Borštnik,
H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana, December 2017, p. 89-120
[arXiv:1802.05554v1v2].
11. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”New way of second quantized theory of
fermions with either Clifford or Grassmann coordinates and spin-charge-family the-
ory ” [arXiv:1802.05554v4,arXiv:1902.10628],
12. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”Understanding the second quantization of
fermions in Clifford and in Grassmann space” New way of second quantization of fermions
— Part I and Part II, Proceedings to the 22nd Workshop ”What comes beyond the standard
models”, 6 - 14 of July, 2019, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, DMFA
Založništvo, Ljubljana, December 2019, [arXiv:1802.05554v4, arXiv:1902.10628].
13. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”Understanding the second quantization of
fermions in Clifford and in Grassmann space” New way of second quantization of fermions
— Part I and Part II, in this proceedings [arXiv:2007.03517, arXiv:2007.03516].
14. H. Georgi, in Particles and Fields (edited by C. E. Carlson), A.I.P., 1975; Google Scholar.
15. H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193.
16. J. Pati and A. Salam, Phys.Rev. D 8 (1973) 1240.
17. H. Georgy and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.
18. Y. M. Cho, J. Math. Phys. 16 (1975) 2029.
19. Y. M. Cho, P. G. O.Freund, Phys. Rev. D 12 (1975) 1711.
20. A. Zee, Proceedings of the first Kyoto summer institute on grand unified theories and related
topics, Kyoto, Japan, June-July 1981, Ed. by M. Konuma, T. Kaskawa, World Scientific
Singapore.
21. A. Salam, J. Strathdee, Ann. Phys. (N.Y.) 141 (1982) 316.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 125 — #141 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 125
22. S. Randjbar-Daemi, A. Salam, J. Strathdee, Nucl. Phys. B 242 (1984) 447.
23. W. Mecklenburg, Fortschr. Phys. 32 (1984) 207.
24. Z. Horvath, L. Palla, E. Crammer, J. Scherk, Nucl. Phys. B 127 (1977) 57.
25. T. Asaka, W. Buchmuller, Phys. Lett. B 523 (2001) 199.
26. G. Chapline, R. Slansky, Nucl. Phys. B 209 (1982) 461.
27. R. Jackiw and K. Johnson, Phys. Rev. D 8 (1973) 2386.
28. I. Antoniadis, Phys. Lett. B 246 (1990) 377.
29. P. Ramond, Field Theory, A Modern Primer, Frontier in Physics, Addison-Wesley Pub.,
ISBN 0-201-54611-6.
30. P. Horawa, E. Witten, Nucl. Phys. B 460 (1966) 506.
31. T. Kaluza, ”On the unification problem in Physics”, Sitzungsber. d. Berl. Acad. (1918) 204,
O. Klein, ”Quantum theory and five-dimensional relativity”, Zeit. Phys. 37(1926) 895.
32. E. Witten, “Search for realistic Kaluza-Klein theory”,Nucl. Phys. B 186 (1981) 412.
33. M. Duff, B. Nilsson, C. Pope, Phys. Rep. C 130 (1984)1, M. Duff, B. Nilsson, C. Pope, N.
Warner, Phys. Lett. B 149 (1984) 60.
34. T. Appelquist, H. C. Cheng, B. A. Dobrescu, Phys. Rev. D 64 (2001) 035002.
35. M. Saposhnikov, P. TinyakovP 2001 Phys. Lett. B 515 (2001) 442 [arXiv:hep-
th/0102161v2].
36. C. Wetterich,Nucl. Phys. B 253 (1985) 366.
37. The authors of the works presented in An introduction to Kaluza-Klein theories, Ed. by H.
C. Lee, World Scientific, Singapore 1983.
38. M. Blagojević, Gravitation and gauge symmetries, IoP Publishing, Bristol 2002.
39. L. Alvarez-Gaumé, ”An Introduction to Anomalies”, Erice School Math. Phys.
1985:0093.
40. A. Bilal, ”Lectures on anomalies” [arXiv:0802.0634].
41. L. Alvarez-Gaumé, J.M. Gracia-Bondı́a, C.M. Martin, ”Anomaly Cancellation and the
Gauge Group of the Standard Model in NCG” [hep-th/9506115].
42. B. Belfatto, R. Beradze, Z. Berezhiani, ”The CKM unitarity problem: A trace of new
physics at the TeV scale?” [arXiv:1906.02714v1].
43. Z. Berezhiani, private communication with N.S. Mankoč Borštnik.
44. D. Lukman, N.S. Mankoč Borštnik, ”Representations in Grassmann space and fermion
degrees of freedom”, [arXiv:1805.06318].
45. N.S. Mankoč Borštnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep-
th/0111257].
46. N.S. Mankoč Borštnik, H.B.F. Nielsen, “How to generate families of spinors”, J. of Math.
Phys. 44 4817 (2003) [arXiv:hep-th/0303224].
47. N.S. Mankoč Borštnik and H.B. Nielsen, Dirac-Kähler approach connected to quantum
mechanics in Grassmann space,Phys. Rev. D 62, 044010 (2000) arXiv:[hep-th/9911032].
48. N.S. Mankoč Borštnik and H.B. Nielsen, ”Second quantization of spinors and Clifford
algebra objects”, Proceedings to the 8th Workshop ”What Comes Beyond the Standard
Models”, Bled, July 19 - 29, 2005, Ed. by Norma Mankoč Borštnik, Holger Bech Nielsen,
Colin Froggatt, Dragan Lukman, DMFA Založništvo, Ljubljana December 2005, p.63-71,
hep-ph/0512061.
49. M. Breskvar, D. Lukman, N. S. Mankoč Borštnik, ”On the Origin of Families of Fermions
and Their Mass Matrices — Approximate Analyses of Properties of Four Families Within
Approach Unifying Spins and Charges”, Proceedings to the 9th Workshop ”What Comes
Beyond the Standard Models”, Bled, Sept. 16 - 26, 2006, Ed. by Norma Mankoč Borštnik,
Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Založništvo, Ljubljana
December 2006, p.25-50, hep-ph/0612250.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 126 — #142 i
i
i
i
i
i
126 N.S. Mankoč Borštnik
50. G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoč Borštnik, ”Families of Quarks and
Leptons and Their Mass Matrices”, Proceedings to the 10th international workshop
”What Comes Beyond the Standard Model”, 17 -27 of July, 2007, Ed. Norma Mankoč
Borštnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Založništvo,
Ljubljana December 2007, p.53-70, hep-ph/0711.4681.
51. G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoč Borštnik, ”Predictions for four families
by the Approach unifying spins and charges” New J. of Phys. 10 (2008) 093002, hep-
ph/0606159, hep/ph-07082846.
52. G. Bregar, N.S. Mankoč Borštnik, ”Does dark matter consist of baryons of new stable
family quarks?”, Phys. Rev. D 80, 083534 (2009), 1-16
53. G. Bregar, N.S. Mankoč Borštnik, ”Can we predict the fourth family masses for quarks
and leptons?”, Proceedings (arxiv:1403.4441) to the 16 th Workshop ”What comes
beyond the standard models”, Bled, 14-21 of July, 2013, Ed. N.S. Mankoč Borštnik,
H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December 2013, p. 31-51,
http://arxiv.org/abs/1212.4055.
54. G. Bregar, N.S. Mankoč Borštnik, ”The new experimental data for the quarks mixing
matrix are in better agreement with the spin-charge-family theory predictions”, Proceed-
ings to the 17thWorkshop ”What comes beyond the standard models”, Bled, 20-28 of
July, 2014, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo,
Ljubljana December 2014, p.20-45 [ arXiv:1502.06786v1] [arxiv:1412.5866].
55. B. Belfatto, R. Beradze, Z. Berezhiani, ”The CKM unitarity problem: A trace of new
physics at the TeV scale?”, [arXiv:1906.02714].
56. Review of Particle, Particle Data Group , P A Zyla, R M Barnett, J Beringer, O Dahl, D
A Dwyer, D E Groom, C -J Lin, K S Lugovsky, E Pianori ...., Author Notes, Progress
of Theoretical and Experimental Physics, Volume 2020, Issue 8, August 2020, 083C01,
https://doi.org/10.1093/ptep/ptaa104, 14 August 2020.
57. A. Hernandez-Galeana and N.S. Mankoč Borštnik, ””The symmetry of 4×4mass matri-
ces predicted by the spin-charge-family theory — SU(2)× SU(2)×U(1) — remains in all
loop corrections”, Proceedings to the 21st Workshop ”What comes beyond the standard
models”, 23 of June - 1 of July, 2017, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman,
DMFA Založništvo, Ljubljana, December 2018 [arXiv:1902.02691, arXiv:1902.10628].
58. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”Do the present experiments exclude the exis-
tence of the fourth family members?”, Proceedings to the 19thWorkshop ”What comes
beyond the standard models”, Bled, 11-19 of July, 2016, Ed. N.S. Mankoč Borštnik,
H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December 2016, p.128-146
[arXiv:1703.09699].
59. A. Ali in discussions and in private communication at the Singapore Conference on
New Physics at the Large Hadron Collider, 29 February - 4 March 2016.
60. M. Neubert, in duscussions at the Singapore Conference on New Physics at the Large
Hadron Collider, 29 February - 4 March 2016.
61. N.S. Mankoč Borštnik, M. Rosina, ”Are superheavy stable quark clusters viable candi-
dates for the dark matter?”, International Journal of Modern Physics D (IJMPD) 24 (No.
13) (2015) 1545003.
62. D. Hestenes, G. Sobcyk, ”Clifford algebra to geometric calculus”, Reidel 1984.
63. P. Lounesto, P. Clifford algebras and spinors, Cambridge Univ. Press.2001.
64. M. Pavšič, ”Quantized fields á la Clifford and unification” [arXiv:1707.05695].
65. N.S. Mankoč Borštnik and H.B.F. Nielsen, ”Discrete symmetries in the Kaluza-Klein
theories”, JHEP 04:165, 2014 [arXiv:1212.2362].
66. T.Troha, D. Lukman, N.S. Mankoč Borštnik, ”Massless and massive representations in
the spinor technique”, .Int. J Mod. Phys. A 29, 1450124 (2014).
67. P.A.M. Dirac Proc. Roy. Soc. (London), A 117 (1928) 610.
i
i
“proc20Vol1” — 2020/12/6 — 22:10 — page 127 — #143 i
i
i
i
i
i
7 How Far has so Far the Spin-Charge-Family Theory. . . 127
68. D. Lukman, N.S. Mankoč Borštnik and H.B. Nielsen, ”An effective two dimensionality
cases bring a new hope to the Kaluza-Klein-like theories”, New J. Phys. 13:103027, 2011.
69. D. Lukman and N.S. Mankoč Borštnik, ”Spinor states on a curved infinite disc with non-
zero spin-connection fields”, J. Phys. A: Math. Theor. 45:465401, 2012 [arxiv:1205.1714,
arxiv:1312.541, arXiv:hep-ph/0412208 p.64-84].
70. D. Lukman, N.S. Mankoč Borštnik and H.B. Nielsen, ”Families of spinors in d = (1+ 5)
with a zweibein and two kinds of spin connection fields on an almost S2”, Proceedings
to the 15th Workshop ”What comes beyond the standard models”, Bled, 9-19 of July,
2012, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana
December 2012, 157-166, [arXiv:1302.4305].
71. A.Borštnik Bračič, N. Mankoč Borštnik,“The approach Unifying Spins and Charges
and Its Predictions“, Proceedings to the Euroconference on Symmetries Beyond the
Standard Model”, Portorož, July 12 - 17, 2003, Ed. by Norma Mankoč Borštnik, Holger
Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Založništvo, Ljubljana December
2003, p. 31-57, [arXiv:hep-ph/0401043, arXiv:hep-ph/0401055].
72. A. Borštnik Bračič, N. S. Mankoč Borštnik, ”On the origin of families of fermions and
their mass matrices”, hep-ph/0512062, Phys Rev. D 74 073013-28 (2006).
73. N.S. Mankoč Borštnik, H.B. Nielsen, “Particular boundary condition ensures that
a fermion in d=1+5, compactified on a finite disk, manifests in d=1+3 as massless
spinor with a charge 1/2, mass protected and chirally coupled to the gauge field”,
hep-th/0612126, arxiv:0710.1956, Phys. Lett. B 663, Issue 3, 22 May 2008, Pages 265-269.
74. H.A. Bethe, ”Intermediate quantum mechanics”, W.A. Benjamin, 1964 (New York,
Amsterdam).
75. C. Itzykson, J.B. Zuber, ”Quantum field theory”, McGraw-Hill, 1980 (New York).
76. N.S. Mankoč Borštnik, H. B. Nielsen, ”Fermions with no fundamental charges call for
extra dimensions”, Phys. Lett. B 644 (2007) 198-202 [arXiv:hep-th/0608006].