How far has so far Spin-Charge-Family theory succeeded to explain Standard Model assumptions, matter-antimatter asymmetry, appearance of Dark Matter, second quantized fermion fields, making several predictions
HHow far has so far the Spin-Charge-Family theory succeeded to explainthe Standard Model assumptions, the matter-antimatter asymmetry, theappearance of the Dark Matter, the second quantized fermion fields...., makingseveral predictions
N.S. Mankoˇc BorˇstnikUniversity of Ljubljana, Slovenia (Dated: printed:December 18, 2020)The assumptions of the standard model , which 50 years ago offered an elegant new steptowards 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 pos-tulates but also in explaining the cosmological observations, like there are the appearanceof the dark matter , of the matter-antimatter asymmetry , making several predictions. Thistheory assumes that the internal degrees of freedom of fermions (spins, handedness andall the charges) are described by the Clifford algebra objects in d ≥ (13 + 1)-dimensionalspace. Fermions interact with only the gravity (the vielbeins and the two kinds of the spinconnection fields, which manifest in d = (3 + 1) as all the vector gauge fields as well asthe scalar fields - the higgs and the Yukawa couplings). The theory describes the internalspace of fermions with the Clifford objects which are products of odd numbers of γ a objects,what offers the explanation for quantum numbers of quarks and leptons and anti-quarks andani-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 quantizationprocedure offered by the description of the internal space of fermions with the anticommutingClifford algebra objects of the odd character. The theory needs still to answer many openquestions that it could be accepted as the next step beyond the standard model . I. INTRODUCTION
Let us start with the motivation for the spin-charge-family theory.The standard model offered an elegant new step towards understanding elementary fermion andboson fields by postulating (the inspiration came from the experiments): a. The existence of massless fermion family members with the spins and charges in the fundamentalrepresentation of the groups, a.i. the quarks as colour triplets and colouress leptons, a.ii theleft handed members as the weak doublets, a.ii. the right handed weak chargeless members, a.iii. the left handed quarks differing from the right handed leptons in the hyper charge, a.iv. a r X i v : . [ phy s i c s . g e n - ph ] D ec all the right handed members differing among themselves in hyper charges, a.v. antifermionscarry the corresponding anticharges of fermions and opposite handedness, a.vi. the number ofmassless families, determined by experiments (there is no right handed neutrino postulated, sinceit would carry none of the so far observed charges, and correspondingly there is also no left handedantineutrino 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 ± and the hypercharge ± (as it would be in the fundamental representation 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 theYukawa 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, butalso 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 detectedin February 2016.The standard model has in the literature several explanations, mostly with many new not ex-plained assumptions. The most popular seem to be the grand unifying theories [14–30]. At least SO (10) offers the explanation for the potulates from a.i. to a.iv , partly to b. — but does notexplain the assumptions a.v. up to a.vi. , c. and d. , and does not connect gravity with gaugevector and scalar fields.What questions should one ask to be able to find a trustworthy next step beyond the standardmodels of elementary particle physics and cosmology, which would offer understanding of not yetunderstood phenomena? i. Where do fermions, quarks and leptons, originate and why do they differ from the boson fieldsin spins, charges and statistics? ii.
How can one describe the internal degrees of fermions to explain the Dirac’s postulates of thesecond quantization? iii.
Why are charges of quarks and leptons so different, why have the left handed family membersso different charges from the right handed ones and why does the handedness relate charges toanticharges? 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 startas massless? vi.
How is the origin of the scalar field (the Higgs’s scalar) and the Yukawa couplings connectedwith the origin of families and how many scalar fields determine properties of the so far (and otherspossibly be) observed fermions and masses of weak bosons? (The Yukawa couplings certainly speakfor the existence of several scalar fields with the properties of Higgs’s scalar.) Why is the Higgs’sscalar, or are all scalar fields of similar properties as the higgs, if there are several, doublets withrespect to the weak and the hyper charge? Do possibly exist also scalar fields with the colourcharges 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 openquestions, the more answers to the above open questions the step covers the greater the possibilitiesof 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 aboveopen questions the theory offers.Let me make in what follows a short introduction into the spin-charge-family theory to showbriefly up the way the theory is offering the answers to the above mentioned open questions. Amore detailed presentation of the theory and its achievements are presented in Sect. II.The spin-charge-family theory is a kind of the Kaluza-Klein like theories [8, 31–38] due to theassumption that in d ≥ spin-charge-family theory d ≥ (13 + 1) — fermions interactwith the gravity only [81], 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 twokinds of the Clifford algebra objects [1, 2, 12, 49, 50]. In d = (13 + 1)-dimensional space the oddClifford algebra objects of one kind offer in d = (3 + 1) the description of the spins and all thecharges of fermions and antifermions, since both — fermions and antifermions — appear in thesame 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, distinguishing amongdifferent 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 superposi-tion of the finite number of the Clifford odd ”basis vectors” of the internal space and of the infinitebasis in the momentum space. The ”basis vectors” of the internal space transfer their oddness tothe 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 ofsingle 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 quantumnumbers 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 secondquantization of the ferrmion fields.Fermions interact with the vielbeins and the two kinds of the spin connection fields, the gaugefields 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 ofthe charges of fermions, as the gravitational fields and also as the scalar gauge fields [5], to whichalso the scalar fields which are the gauge field of the second kind of the spin connection fieldscontribute. These offer answers to the questions vi. and vii. , while explaining the common originof the gravity, the vector gauge fields of the charges and the scalar gauge fields. The scalar gaugefields of both origins — of both generators of the Lorentz transformations in internal space offermions — determine the scalar higgs and the Yukawa couplings, which all are in the standardmodel postulated.The two kinds of the Clifford algebra objects require the existence of the two groups of fourfamilies of quarks and leptons and antiquarks and antileptons. The two groups distinguish fromeach other with respect to the family quantum numbers and correspondingly with respect to theinteraction with the different two groups of the scalar gauge fields, which determine masses of thesetwo 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 antileptons there mustexist the fourth family [3, 9, 53, 55, 57, 58]. b. The second group of the four families offers theexplanation for the existence of the d ark matter [56, 65].The quantum numbers of the weak charge and the hyper charge of the scalar fields, taking careof the masses of the two groups of four families, depend on the space index of the scalar fields. Thescalar fields with the space indexes 7 and 8 do carry the weak and the hyper charge as assumed bythe standard model , explaining the origin of scalar higgs and Yukawa couplings [3, 9, 53, 55, 57, 58],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 arethe colour triplets [4, 65]. They cause the transitions of antiquarks and antileptons into quarks andback. In the expanding universe under the non equilibrium conditions they offer the explanationof 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 thestarting symmetry in d = (13 + 1), first with the appearance of the condensate of two right handedneutrinos [3, 4, 9], and then when scalar fields with space index (7 ,
8) obtain nonzero vacuumexpectation values.The detailed, although still short, presentation of the spin-charge-family theory is presented inSects. IIand II A 2.
II. SHORT PRESENTATION OF THE
SPIN-CHARGE-FAMILY
THEORY
The spin-charge-family theory assumes a simple starting action for fermions, coupled to onlygravitational field in d ≥ (13 + 1)-dimensional space through the vielbeins f αa , the gauge fieldsof momenta, and the two kinds of the spin connection fields, ω abα and ˜ ω abα , the gauge fields ofthe two kinds of the generators of the Lorentz transformations of the Clifford algebras, and withthe internal space of fermions described by the anticommuting ”basis vectors” of one of the twoClifford algebras A = (cid:90) d d x E
12 ( ¯ ψ γ a p a ψ ) + h.c. + (cid:90) d d x E ( α R + ˜ α ˜ R ) ,p a = f αa p α + 12 E { p α , Ef αa } − ,p α = p α − S ab ω abα −
12 ˜ S ab ˜ ω abα ,R = 12 { f α [ a f βb ] ( ω abα,β − ω caα ω cbβ ) } + h.c. , ˜ R = 12 { f α [ a f βb ] (˜ ω abα,β − ˜ ω caα ˜ ω cbβ ) } + h.c. . (1)Here [82] f α [ a f βb ] = f αa f βb − f αb f βa .As written in the introduction, the tensor products of the superposition of the finite number ofanticommuting ”basis vectors” and of the infinite basis in the momentum space offer the descriptionof the fermion creation and annihilation anticommuting operators. The creation and annihilationoperators explain the Dirac postulates of the second quantized fermions, Sect. (II A,II A 2, II A 3).The single fermion states manifest in d = (3 + 1) space the spins and all the charges of theobserved quarks and leptons and antiquarks and antileptons, Table III, as well as families, Table IV,predicting the fourth family [53–55, 57, 58, 61, 62] to the observed three families and offering theexplanation for the observed dark matter [56, 65].The spin connection gauge fields manifest in d = (3 + 1) as the ordinary gravity, the knownvector gauge fields, the scalar gauge fields [5] with the properties of higgs explaining the higgsesand the Yukawa couplings, predicting new vector and scalar fields, which offer explanation for the dark matter [56] and for matter/antimatter asymmetry [4].To be in agreement with the observations in d = (3 + 1) the manifold M (13+1) must break firstinto M (7+1) × M (6) (which manifests as SO (7 , × SU (3) × U (1)), affecting both internal degreesof freedom - the one represented by γ a and the one represented by ˜ γ a [3].There is a scalar condensate (Table V) of two right handed neutrinos with the family quantumnumbers of the group of four families (the one which does not include the observed three families),Table IV, which bring masses of the scale ∝ GeV or higher to all the vector and scalar gaugefields, which interact with the condensate [4].Since the left handed spinors couple differently (with respect to M (7+1) ) to scalar fields thanthe right handed ones, the break can leave massless and mass protected 2 ((7+1) / − families [72].The rest of families get heavy masses [83].There is additional breaking of symmetry: The manifold M (7+1) breaks further to M (3+1) × SU (2) × SU (2) included in M (4) . These electroweak break is caused by the scalar fields with thespace index (7 , d = (3 + 1). A. 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 anticommuting ”basisvectors” describing fermions internal degrees of freedom and of commuting basis in the momentum(coordinate) space, manifest the anticommuting properties already on the single fermion level [13],demonstrating that the first quantized fermions are the approximation to the second quantizedfields.There are two kinds of the anticommuting objects [1–3, 9, 12] — the Grassmann coordinatesand correspondingly the Grassmann operators, θ a and ∂∂θ a , and the Clifford coordinates/operators, γ a and ˜ γ a , expressible with one another. Either the Grassmann or the two Clifford algebras offerin d -dimensional space 2 · d operators (the Grassmann algebra has 2 d − θ a ’s and2 d − ∂∂θ a ’s and the identity, the two Clifford algebras have each 2 d − γ a ’ and 2 d − γ a ’s and the identity) with the properties [12, 13] { θ a , θ b } + = 0 , { ∂∂θ a , ∂∂θ b } + = 0 , { θ a , ∂∂θ b } + = δ ab , ( θ a ) † = η aa ∂∂θ a , ( ∂∂θ a ) † = η aa θ a , { γ a , γ b } + = 2 η ab = { ˜ γ a , ˜ γ b } + , { γ a , ˜ γ b } + = 0 , ( γ a ) † = η aa γ a , (˜ γ a ) † = η aa ˜ γ a , ( a, b ) = (0 , , , , , · · · , d ) . (2)The identity is the self adjoint member. The signature η ab = diag { , − , − , · · · , − } is assumed.The two algebras are expressible with one another γ a = ( θ a + ∂∂θ a ) , ˜ γ a = i ( θ a − ∂∂θ a ) ,θ a = 12 ( γ a − i ˜ γ a ) , ∂∂θ a = 12 ( γ a + i ˜ γ a ) . (3)Let me add the generators of the Lorentz transformations in both algebras S ab = i ( θ a ∂∂θ b − θ b ∂∂θ a ) , ( S ab ) † = η aa η bb S ab ,S ab = i γ a γ b − γ b γ a ) , ˜ S ab = i γ a ˜ γ b − ˜ γ b ˜ γ a ) , S ab = S ab + ˜ S ab , { S ab , ˜ S ab } − = 0 , { S ab , γ c } − = i ( η bc γ a − η ac γ b ) , { S ab , ˜ γ c } − = 0 , { ˜ S ab , ˜ γ c } − = i ( η bc ˜ γ a − η ac ˜ γ b ) , { ˜ S ab , γ c } − = 0 , (4)The Grassmann algebra offers the description of the integer spin fermions, with the charges inthe adjoint representations. Both Clifford algebras offer the description of the half integer spinfermions with charges in the fundamental representations. Both algebras, the Grassmann algebraand the two Clifford algebras, can be separated into odd and even parts with odd and even productsof algebra elements.While in the Grassmann algebra the Hermitian conjugated partners of products of θ a ’s arethe corresponding products of ∂∂θ a ’s, Eq. (2), and opposite, in the Clifford algebras the Hermitianconjugated partners are less transparent, due to the relations γ a † = η aa γ a and ˜ γ a † = η aa ˜ γ a , Eq. (2).In order to resolve the problem of the Hermitian conjugated partners in the Clifford case andalso to be able to make predictions of the theory to be compared with the experimental results, letus arrange products of θ a ’s as well as products of either γ a ’s or ˜ γ a ’s into irreducible representationswith respect to the Lorentz group with the generators [2] presented in Eq. (4) and to arrange themembers of each irreducible representation to be eigenstates of the Cartan subalgebra S , S , S , · · · , S d − d ,S , S , S , · · · , S d − d , ˜ S , ˜ S , ˜ S , · · · , ˜ S d − d . (5)The easiest way to achieve this is to find the eigenstates of each member of the Cartan subalgebrasseparately.The observed fermions have the half integer spin and charges in the fundamental representa-tions, and there are no fermions observed yet with the integer spins and charges in the adjointrepresentations. The spin-charge-family theory must correspondingly use the Clifford algebras.However, there are also no experimental evidences that there is any need for two independentrepresentations offered by the two kinds of the Clifford algebra objects, γ a ’s and ˜ γ a ’s.Let us therefore start the discussion about the description of the internal space of fermions bytaking into account the two Clifford algebras and let us leave the discussion on the Grassmannalgebra for later, Ref. [13].We can make a choice for the members of the irreducible representations of the two Lorentzgroups to be the ”eigenvectors” of the corresponding Cartan subalgebras of Eq. (5), taking intoaccount Eq. (2). If S ab and ˜ S ab represents each one of the ( d for even d ) members of the Cartansubalgebra elements, we easily check that S ab ab ( k ) = k ab ( k ) , ab ( k ):= 12 ( γ a + η aa ik γ b ) , ( ab ( k )) = 0 , ab ( k ) † = η aa ab ( − k ) ,S ab ab [ k ] = k ab [ k ] , ab [ k ]:= 12 (1 + ik γ a γ b ) , ( ab [ k ]) = ab [ k ] , ab [ k ] † = ab [ k ] , ˜ S ab ab ˜( k ) = k ab ˜( k ) , ab ˜( k ):= 12 ( γ a + η aa ik γ b ) , ( ab ˜( k )) = 0 , ab ˜( k ) † = η aa ab ˜( − k ) , ˜ S ab ab ˜[ k ] = k ab ˜[ k ] , ab ˜[ k ]:= 12 (1 + ik γ a γ b ) , ( ab ˜[ k ]) = ab ˜[ k ] , ab ˜[ k ] † = ab ˜[ k ] . (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 arenormalized up to a phase [2, 12, 13].Both have half integer spins. The ”eigenvalues” of the operator S for the ”eigenvectors” ( γ ∓ γ ), for example, are equal to ± i , respectively, for the ”vectors” (1 ± γ γ ) are ± i ,respectively, while all the rest ”eigenvectors” have ”eigenvalues” ± . One finds equivalently forthe ”eigenvectors” of the operator ˜ S : for ( ˜ γ ∓ ˜ γ ) the ”eigenvalues” ± i , respectively, andfor the ”eigenvectors” (1 ± ˜ γ ˜ γ ) the ”eigenvalues” k = ± i , respectively, while all the rest”eigenvectors” have k = ± .It is useful to know some additional relations among nilpotents and projectors, taken fromRef. [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 ) = η aa ab [ k ] , ab [ k ] ab [ − k ]= 0 . (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 eigenvectors of all the Cartan subalgebra members. Forthe description of the internal space of fermions only those ”basis vectors” which are products of0an odd number of nilpotents, the rest can be projectors, are acceptable, since they anticommutealgebraically , 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”basis vector”, denoting it as ˆ b m =1 † f =1 , with f defining different irreducible representations and m amember in the representation f . Then its Hermitian conjugated partner is ˆ b mf = (ˆ b m † f ) † . Let usmake a choice of the starting ”basis vector” for the Clifford algebra of the kind γ a ’s with an oddproducts of the nilpotents ˆ b m =1 † f =1 : = (+ i ) [+] [+] (+) (+)
11 12 [ − ]
13 14 [ − ] · · · d − d − [ − ] d − d [ − ] , (ˆ b m =1 † f =1 ) † = ˆ b m =1 f =1 = d − d [ − ] d − d − [ − ] · · ·
13 14 [ − ]
11 12 [ − ] ( − ) ( − ) [+] [+] ( − i ) , (8)the rest products in · · · d − d − [ − ] d − d [ − ] are assumed to be all projectors with k = −
1, [ − ]. All the restmembers of this irreducible representation are reachable by S ab .Let us see how do S ab ’s transform the ”basis vectors”. S ac ab ( k ) cd ( k ) = − i η aa η cc ab [ − k ] cd [ − k ] ,S ac ab [ k ] cd [ k ] = i ab ( − k ) cd ( − k ) ,S ac ab ( k ) cd [ k ] = − i η aa ab [ − k ] cd ( − k ) ,S ac ab [ k ] cd ( k ) = i η cc ab ( − k ) cd [ − k ] , (9)We learn from Eq. (A1) that S transforms ˆ b m =1 † f =1 into, let us call it ˆ b m =2 † f =1 ,ˆ b m =2 † f =1 = [ − i ] (+) [+] (+) (+)
11 12 [ − ]
13 14 [ − ] · · · d − d − [ − ] d − d [ − ] .Application of all possible S dg generates 2 d − members of each Clifford odd irreducible repre-sentation. To each irreducible representation the Hermitian conjugated irreducible representationbelongs.The Hermitian conjugated partner of the starting ”basis vector” of an odd product of nilpotentsobviously belong to another irreducible representation, since it is not reachable by S ab . Each S cd namely transforms a pair of projectors into a pair of nilpotents, a pair of nilpotents into a pairof 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 ˆ b m † f an odd number of nilpotents, each carrying its own k , into the same number of nilpotents, eachcarrying then − k [84].1From Eq. (A1) we learn that the starting member ˆ b m =1 † f =2 of the next irreducible representationcan be obtained from ˆ b m =1 † f =1 by replacing, for example, (+ i ) [+] in ˆ b m =1 † f =1 with [+ i ] (+). This new”basis vector” does not belong to either the starting irreducible representation, or to the Hermitianconjugated partners of the starting irreducible representation, due to the way how it is creating: S transforms (+ i ) [+] into [ − i ] ( − ), the Hermitian conjugation transforms ( − i ) [+].Exchanging all possible pairs in the starting ”basis vector” by keeping the same k ’s whiletransforming a pair of nilpotents into a pair of projectors, a pair of projectors into a pair ofnilpotents and a pair of a nilpotent and a projector into a pair of the projector and the nilpotent,we generate 2 d − irreducible representations with 2 d − members each.The Hemitian conjugation then generates 2 d − · d − partners to the 2 d − members of each ofthe 2 d − irreducible representations.One can find that the algebraic product of ˆ b mf ∗ A ˆ b m † f is the same for all m of a particu-lar irreducible representation f (since ˆ b mf (2 S ab ) † ∗ A (2 S ab )ˆ b m † f = ˆ b mf ∗ A ˆ b m † f , due to the relation( − iS ab ) † ( − iS ab ) = 1).Each irreducible representation contributes different algebraic product ˆ b mf ∗ A ˆ b m † f .For the representation of Eq. (8) the product ˆ b m =1 f =1 ∗ a ˆ b m =1 † f =1 is equal to | ψ oc > | f =1 = [ − i ] [+] [+] [ − ] [ − ]
11 12 [ − ]
13 14 [ − ] · · · d − d − [ − ] d − d [ − ] .This can be checked by using Eq. (7).Defining the vacuum state | ψ oc > for the vector space determined by γ a ’s as a sum of all differentproducts of (cid:80) d − f =1 ˆ b mf ∗ A ˆ b m † f , ∀ m , and for d = 2 n + 1, one obtains | ψ oc > = [ − i ] [ − ] [ − ] · · · d − d [ − ] + [+ i ] [+] [ − ] · · · d − d [ − ]+ [+ i ] [ − ] [+] · · · d − d [ − ] + · · · | > , for d = 2(2 n + 1) . (10)Let me add that the application of any member of the Cartan subalgebras on the vacuum state, S ab | ψ oc > = 0 , ˜ S ab | ψ oc > = 0 , ∀ S ab and ˜ S ab belonging to Cartan subalgeras of Eq. (5).One finds that all the members of all the irreducible representations fulfill together with their2Hermitian conjugated partners the relationsˆ b mf ∗ A | ψ oc > = 0 · | ψ oc > , ˆ b m † f ∗ A | ψ oc > = | ψ mf > , { ˆ b mf , ˆ b m (cid:48) f (cid:48) } ∗ A + | ψ oc > = 0 | ψ oc > , { ˆ b mf , ˆ b m (cid:48) † f } ∗ A + | ψ oc > = δ mm (cid:48) | ψ oc > , { ˆ b m † f , ˆ b m (cid:48) † f (cid:48) } ∗ A + | ψ oc > = 0 · | ψ oc > , (11)for each f . ∗ A represents the algebraic multiplication of ˆ b m † f ’s and ˆ b m (cid:48) f (cid:48) ’s among themselves andwith the vacuum state | ψ oc > of Eq.(10).The relations of Eq. (11) almost manifest the anticommutation relations for the second quantizedfermion fields postulated by Dirac [71]. It is pointed out almost , since the relation { ˆ b mf , ˆ b m (cid:48) † f (cid:48) } ∗ A + | ψ oc > = δ mm (cid:48) δ ff (cid:48) | ψ oc > (12)is not fulfilled. There are, namely, besides ˆ b mf , 2 d − − b mf does — when multiplying ˆ b m † f from the left hand side. ˆ b mf (cid:48) ∗ A ˆ b m † f (cid:54) = 0for 2 d − − f (cid:48) (cid:54) = f , while ˆ b mf ∗ A ˆ b m † f = 1 . Let me illustrate this on the example of ˆ b m =1 † f =1 of Eq. (8). Besides (ˆ b m =1 † f =1 ) † = ˆ b m =1 f =1 = d − d [ − ] d − d − [ − ] · · ·
13 14 [ − ]
11 12 [ − ] ( − ) ( − ) [+] [+] ( − i ) also d − d [ − ] d − d − [ − ] · · ·
13 14 [ − ]
11 12 [ − ] ( − ) ( − ) [+] ( − ) [+ i ] , d − d [ − ] d − d − [ − ] · · ·
13 14 [ − ]
11 12 [ − ] ( − ) ( − ) ( − ) [+] [+ i ] , etc (13)applied on ˆ b m =1 † f =1 , give a nonzero contributions.But index f determine different irreducible representations and we can not expect that the al-gebraic 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 γ a ’s, appearing after Eq. (7), can be as wellrepeated also for the Clifford algebra with ˜ γ a ’s.The Dirac’s postulates for the second quantized fermion fields include the infinite basis inmomentum space, while we treated so far the finite dimensional internal space of fermions. Beforeextending the vector basis space by making the tensor product of internal space and the momentum3space let us recognize that the observed quarks and leptons and antiquarks and antileptons do not atall suggest that there might be two different internal spaces, which could be described by two kindsof the Clifford algebra objects. Let us therefore first reduce the Clifford space by the postulate,which leave only γ a ’s as the algebra describing the internal degrees of freedom of fermions, while˜ γ a ’s are used to give quantum numbers to different irreducible representations.
1. Reduction of the Clifford space
It is needed to give to each irreducible representation of the Lorentz transformations in theinternal space of fermions the quantum number, which will distinguish among the 2 d − differentirreducible representations. If we keep the Clifford algebra with γ a ’s to describe the internal spaceof fermions, then ˜ γ a ’s, or rather ˜ S ab ’s, can be used to determine ”family” quantum number of eachirreducible representation of the Lorentz algebra in the Clifford space of γ a ’s.We want that all the relations among γ a ’s and ˜ γ a ’s, presented in Eq. (2), remain unchanged,while the eigenvalues of the Cartan subalgebra of ˜ S ab are expected to be changed.The postulate [2, 7, 9, 10, 12, 50] ˜ γ a B = ( − ) B i Bγ a , (14)with ( − ) B = −
1, if B is a function of an odd product of γ a ’s, otherwise ( − ) B = 1 [50], does justthat [85]. It is not difficult to check that the relations in Eq. (2), concerning ˜ γ a ’s are still valid: { γ a , γ b } + = 2 η ab = { ˜ γ a , ˜ γ b } + , { γ a , ˜ γ b } + = 0 , ( γ a ) † = η aa γ a , (˜ γ a ) † = η aa ˜ γ a .After this postulate the vector space of ˜ γ a ’s is ”frozen out”. And also the Grassmann algebraspace is now reduced to θ a = γ a and ∂∂θ a = 0 [86]. No vector space of ˜ γ a ’s exists any longer, whatis in agreement with the observed properties of fermions. While the anticommutation relationsamong γ a ’s and ˜ γ a ’s remain the same as in Eq. (2), it follows for the eigenvalues of ˜ S ab S ab ab ( k )= k ab ( k ) , ˜ S ab ab ( k )= k ab ( k ) ,S ab ab [ k ]= k ab [ k ] , ˜ S ab ab [ k ]= − k ab [ k ] , (15)showing that the eigenvalues of S ab on the nilpotents and projectors of γ a ’s differ from the eigenval-ues of ˜ S ab on the nilpotents and projectors of γ a ’s. The members of the Cartan subalgebra of ˜ S ab ,Eq. (5), can now be used to give to the irreducible representations of S ab the ”family” quantumnumbers.4Let me mention that if one arranges the space of odd products of γ a ’s with respect to S ab (= S ab + ˜ S ab ), these new ”basis vector” will form multiplets with integer spins and charges in adjointrepresentations. Like the ”basis vectors” expressed by Grassmann algebra do in Ref. [13], Table I,but this time with θ a ’s replaced by γ a ’s.It is useful to notice that γ a transform ab ( k ) into ab [ − k ], never to ab [ k ], while ˜ γ a transform ab ( k ) into ab [ k ], never to ab [ − k ] γ a ab ( k )= η aa ab [ − k ] , γ b ab ( k )= − ik ab [ − k ] ,γ a ab [ k ]= ab ( − k ) , γ b ab [ k ]= − ikη aa ab ( − k ) , ˜ γ a ab ( k )= − iη aa ab [ k ] , ˜ γ b ab ( k )= − k ab [ k ] , ˜ γ a ab [ k ]= i ab ( k ) , ˜ γ b ab [ k ]= − kη aa ab ( k ) . (16)Some additional applications of ˜ γ a ’s and ˜ S ab ’s on nilpotents and projectors expressed by the γ a ’s can be found in App. A.Each irreducible representation has now the ”family” quantum number, determined by ˜ S ab of the Cartan subalgebra of Eq. (5). Now we can replace the fourth equation in Eq. (11) — { ˆ b mf , ˆ b m (cid:48) † f } ∗ A + | ψ oc > = δ mm (cid:48) | ψ oc > — with the relation in Eq. (12) — { ˆ b mf , ˆ b m (cid:48) † f (cid:48) } ∗ A + | ψ oc > = δ mm (cid:48) δ ff (cid:48) | ψ oc > .Each family contributes in even dimensional spaces one summand of d projectors to the vacuumstate | ψ oc > of fermions.Correspondingly the ”basis vectors” and their Hermitian conjugated partners fulfill algebraicallythe anticommutation relations of Dirac’s second quantized fermions: Different irreducible represen-tations carry different ”family” quantum numbers and to each ”family” quantum number only oneHermitian conjugated partner with the same ”family” quantum number belongs. Each summandof the vacuum state, Eq. (10), belongs to a particular ”family”.One can easily check that each ”basis vector” ˆ b m † f , applied algebraically on | ψ oc > , gives nonzerocontribution on the summand in the odd number of γ a ’s, determined by ˆ b mf ˆ b m † f , which is the samefor all m of particular f , representing therefore the corresponding state | ψ fm > , while on all othersummands ˆ b m † f gives zero, ˆ b mf applying on | ψ oc > gives zero for all f and all m .To define creation and annihilation operators, which determine on the vacuum state the singlefermion states, we ought to make the tensor products of the 2 d − × d − ”basis vectors”, describingthe internal space of fermions and of infinite basis of momenta. The oddness of the products of the odd number of γ a ’s guarantees the anticommuting propertiesof all the objects which include an odd number of γ a ’s. γ a ’s.
2. Second quantized fermion fields
Since the nonrelativistic quantum theory is an approximation of the relativistic second quantizedfield theory — as the relativistic classical physics is an approximation of the quantum physics, andas the nonrelativistic classical physics, which we use the most of time, is the approximation of therelativistic classical physics — let us try to recognize what properties should the single particlestates 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 productsamong which form the Hilbert space, are discussed in details. In this talk I am presenting this topicfrom 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 wayhow does the symmetry of the space breaks from d = (13 + 1) to d = (3 + 1).In Table III one irreducible representation of SO (13 + 1) of one family (belonging to the oneof the two groups of four families which includes the so far observed three families) is presented.The first ”basis vector” describes the internal degrees of freedom of the right handed quark ˆ u c † R ,of the first family with ( ˜ S , ˜ S , ˜ S , ˜ S ) equal to ( , − , − , ), presented in Table IV as ˆ u c † R .The ”basis vector” ˆ b m =1 † f =1 , Eq. 8, represents for d = (13 + 1) just this ˆ u c † R quark, and ˆ b m =1 f =1 is itsHermitian conjugated partner.The ”basis vector” ˆ b m =1 † f =2 represents for d = (13 + 1) the right handed u -quark with all theproperties of ˆ u c † R except for the family quantum numbers, which are now equal to ( − , , − , ).One can read in Table III that the spin of this right handed quark ˆ u c † R is + , the weak SU (2)charge is zero, the colour charge is ( , √ ). It carries the additional SU (2) charge equal to andthe ”fermion” quantum number — τ charge — equal to .When solving the equations of motion for free massless fermions, which follow from the action,presented in Eq. (1), under the assumption, that at low energies the momentum of this right handed6quark is p a = ( p , p , p , p , , · · · , s = 1 is the superpositionˆ u sf =1 † R ( (cid:126)p ) = β (ˆ u c † R ↑ + p + ip | p | + | p | ˆ u c † R ↓ ) , (17)with | p | = | (cid:126)p | , with ↑ , ↓ denoting spin ± , respectively, and with β ∗ β = | p | + | p | | p | normalizing thestate.There are steps from the d = (13 + 1) dimensional space to the step where momentum inhigher dimensions do not contribute to dynamics in d = (3 + 1), while the break of symmetrymakes the internal degrees of freedom (spins and families) to manifest as the spin and charges aspresented in Table III and families as presented at Table IV. One finds the detailed presentationsin Ref.( [3–5, 9, 53, 56, 74] and the references therein).Let us here represent the general solutions of equations of motion for free massless fermions withthe internal space of fermions described by the ”basis vectors” ˆ b m † f , fulfilling the relations of Eq. (11),for each family f separately, and also with respect to different families, ˆ b mf ∗ A ˆ b m (cid:48) † f = δ mm (cid:48) δ ff (cid:48) ,ˆ b sf † ( (cid:126)p ) | p = | (cid:126)p | def = (cid:88) m c sf m ( (cid:126)p, | p | = | (cid:126)p | ) ˆ b m † f , ˆ b sf † tot ( (cid:126)p, (cid:126)x ) def = (ˆ b sf † ( (cid:126)p ) e − i ( p x − (cid:126)p · (cid:126)x ) ) | | p | = | (cid:126)p | , (cid:88) m ( c sf ∗ m ( (cid:126)p ) · c s (cid:48) f (cid:48) m ( (cid:126)p )) | | p | = | (cid:126)p | = δ ss (cid:48) δ ff (cid:48) , (18) s represents different orthonormalized solutions of the equations of motion, c sf m ( (cid:126)p, | p | = | (cid:126)p | ) arecoefficients, depending on momentum | (cid:126)p | with | p | = | (cid:126)p | . For the case of the right handed u -quarksof Eq. (17) the two nonzero coefficients are β and β p + ip | p | + | p | .Creation operators of an odd Clifford character ˆ b sf † tot ( (cid:126)p ) create the single particle states, < x | ψ sf ( ˜p , p ) > | p = | ˜p | , manifesting the oddness of the creation operators < x | ψ sf ( ˜p , p ) > | p = | ˜p | = (cid:90) dp δ ( p − | (cid:126)p | ) ˆ b sf † ( (cid:126)p ) e − ip a x a ∗ A | ψ oc > = (ˆ b sf † ( (cid:126)p ) · e − i ( p x − ε(cid:126)p · (cid:126)x ) ) | p = | (cid:126)p | ∗ A | ψ oc > , (19)with the property (cid:90) d d − x ( √ π ) d − < ψ s (cid:48) f (cid:48) ( (cid:126)p (cid:48) , p (cid:48) = | (cid:126)p (cid:48) | ) | x > < x || ψ sf ( (cid:126)p, p = | (cid:126)p | ) > = (cid:90) d d − x ( √ π ) d − e ip (cid:48) a x a | p (cid:48) = | (cid:126)p (cid:48) | e − ip a x a | p | = | (cid:126)p | · < ψ oc | (ˆ b s (cid:48) f (cid:48) ( (cid:126)p (cid:48) ) ˆ b sf † ( (cid:126)p )) ∗ A | ψ oc > = δss (cid:48) δ ff (cid:48) δ ( (cid:126)p − (cid:126)p (cid:48) ) . (20)7One further finds the single particle fermion states in the coordinate representation | ψ sf ( ˜x , x ) > = (cid:90) + ∞−∞ d d − p ( √ π ) d − ( ˆb sf † ( ˜p ) e − i ( p x − ε ˜p · ˜x ) | p = | ˜p | ∗ A | ψ oc > = (cid:88) m ˆ b m † f | ψ oc> (cid:90) + ∞−∞ d d − p ( √ π ) d − ( c sf m ( (cid:126)p ) e − i ( p x − ε(cid:126)p · (cid:126)x ) ) | p = | (cid:126)p | = (cid:88) m ˆ b m † f | ψ oc > c sf m ( − i ∂∂x a , | p | = | ( − i ∂∂x a | ) δ ( (cid:126)x ) , (21)where it is taken into account that ˆ b sf † ( (cid:126)p ) | p = | (cid:126)p | | ψ oc > = (cid:80) m c sf m ( (cid:126)p, | p | = | (cid:126)p | ) ˆ b m † f , Eq. (18), andthat (cid:82) d d − x ( √ π ) d − e ip (cid:48) a x a e − ip a x a = δ ( (cid:126)p − (cid:126)p (cid:48) ). ε = ±
1, depending on handedness and spin of solutions.Taking into account the above derivations, leading to (cid:82) dp δ ( p − | (cid:126)p | ) e i ( p x − p x ) = 1 and < ψ oc | ˆ b sf ( (cid:126)p, p ) ∗ A ˆ b s (cid:48) f (cid:48) † ( (cid:126)p, p ) | ψ oc > = δ ss (cid:48) δ ff (cid:48) , one finds < ψ sf ( (cid:126)x, x ) | ψ s (cid:48) f (cid:48) ( (cid:126)x (cid:48) , x ) > == (cid:90) + ∞−∞ d d − p ( √ π ) d − (cid:90) + ∞−∞ δ ( p − | (cid:126)p | ) < ψ sf ( (cid:126)x, x ) | (cid:126)p > < (cid:126)p | ψ s (cid:48) f (cid:48) ( (cid:126)x (cid:48) , x ) > = (cid:90) + ∞−∞ d d − p ( √ π ) d − e − i(cid:126)p · (cid:126)x e i(cid:126)p · (cid:126)x (cid:48) (cid:90) dp δ ( p − | (cid:126)p | ) < ψ oc | ˆ b sf ( (cid:126)p, p ) ∗ A ˆ b s (cid:48) f (cid:48) † ( (cid:126)p, p ) ∗ A | ψ oc > == δ ss (cid:48) δ ff (cid:48) δ ( (cid:126)x − (cid:126)x (cid:48) ) . (22)The scalar product < ψ sf ( (cid:126)x, x ) | ψ s (cid:48) f (cid:48) ( (cid:126)x (cid:48) , x ) > has obviously the desired properties of the secondquantized states.The new creation operators ˆ b sf † tot ( (cid:126)p, (cid:126)x ), which are generated on the tensor products of bothspaces, internal and momentum, fulfill obviously the below anticommutation relations when appliedon | ψ oc > { ˆ b sftot ( (cid:126)p, (cid:126)x ) , ˆ b sf † tot ( (cid:126)p (cid:48) , (cid:126)x ) } + ∗ T | ψ oc > = δ ss (cid:48) δ ff (cid:48) δ ( (cid:126)p − (cid:126)p (cid:48) ) | ψ oc > , { ˆ b sftot ( (cid:126)p, (cid:126)x ) , ˆ b s (cid:48) f (cid:48) tot ( (cid:126)p (cid:48) , (cid:126)x ) } + ∗ T | ψ oc > = 0 · | ψ oc > , { ˆ b sf † tot ( (cid:126)p, (cid:126)x ) , ˆ b s (cid:48) f (cid:48) † tot ( (cid:126)p (cid:48) , (cid:126)x ) } + ∗ T | ψ oc > = 0 · | ψ oc > , ˆ b sf † tot ( (cid:126)p, (cid:126)x ) ∗ T | ψ oc> = | ψ sf ( (cid:126)p, , (cid:126)x ) > , ˆ b sftot ( (cid:126)p, , (cid:126)x ) ∗ T | ψ oc > = 0 · | ψ oc > , | p | = | (cid:126)p | . (23)It is not difficult to show that ˆ b sftot ( (cid:126)p, (cid:126)x ) and ˆ b sf † tot ( (cid:126)p, (cid:126)x ) manifest the same anticommutation relationsalso on tensor products of an arbitrary chosen products of sets of single fermion states [13].8
3. Hilbert space of fermion fields
The tensor products of any number of any sets of the single fermion creation operators ˆ b sf † tot ( (cid:126)p, (cid:126)x )(fulfilling together with their Hermitian conjugated partners annihilation operators ˆ b sftot ( (cid:126)p, (cid:126)x ) theanticommutation relations of Eq. (23)) form the Hilbert space of the second quantized fermionfields. The number of the sets is infinite. The internal space, defined by ˆ b mf , contributes in d -dimensional space for each chosen momentum (cid:126)p (and for a parameter (cid:126)x ) the finite number,2 d − · d − , of such sets, the total Hilbert space has, due to the infinite basis in the momentum (orcoordinate) space, the infinite number of sets N H = ∞ (cid:89) (cid:126)p d − . (24)The number operator is defined as N sf(cid:126)p = ˆ b sf † tot ( (cid:126)p, (cid:126)x ) ∗ T ˆ b sftot ( (cid:126)p, (cid:126)x ) ,N sf(cid:126)p | ψ oc > = 0 · | ψ oc > . (25)The vacuum state contains no fermions.The Clifford odd objects ˆ b sf † tot ( (cid:126)p, (cid:126)x ) demonstrate their oddness also with respect to the wholeHilbert space H , that is with respect to any tensor product of members of any sets of creation op-erators ˆ b sf † tot ( (cid:126)p (cid:48) , (cid:126)x )). Correspondingly the anticommutation relations follow also for the applicationof ˆ b sf † tot ( (cid:126)p, (cid:126)x ) and ˆ b sftot ( (cid:126)p, (cid:126)x ) on H{ ˆ b sftot ( (cid:126)p, (cid:126)x ) , ˆ b s (cid:48) f (cid:48) † tot ( (cid:126)p (cid:48) , (cid:126)x ) } ∗ T + H = δ ss (cid:48) δ ff (cid:48) δ ( (cid:126)p − (cid:126)p (cid:48) ) H , { ˆ b sf † tot ( (cid:126)p, (cid:126)x ) , ˆ b s (cid:48) f (cid:48) † tot ( (cid:126)p (cid:48) , (cid:126)x ) } ∗ T + H = 0 · H , { ˆ b sf † tot ( (cid:126)p, (cid:126)x ) , ˆ b s (cid:48) f (cid:48) † tot ( (cid:126)p (cid:48) , (cid:126)x ) } ∗ T + H = 0 · H . (26)I presented in this talk the derivation of the creation and annihilation operator of the secondquantized fermion fields, which obey the Dirac’s postulates for the second quantized fermion fieldswithout postulating them, just by analyzing properties of creation and annihilation operatorsobtained as tensor products of the ”basis vectors” of an odd Clifford algebra and of the basisin either momentum or coordinate space. In Ref. [10–13] the relation between the creation andannihilation operators, postulated by Dirac and the ones presented in this talk are discussed.9
4. 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 succeededto improve much the part presented in this subsection. I have been working on the symmetries ofthe spin-charge-family theory and in particular on how can the theory, using the Clifford algebra todescribe all the internal properties of fermions — spins, charges and families — help to explain theassumptions of the second quantized fermion fields. I shall therefore review the other achievementsof the theory very briefly.In Eq. (1) the starting action is presented for fermion and boson fields in d = (13 + 1). In orderthat predictions of the spin-charge-family theory are in agreement with the observed propertiesof quarks and leptons and antiquarks and antileptons, of the vector gauge fields and of the scalargauge fields (manfesting as the higgs and Yukawa couplings), the manifold M (13+1) ought to breakfirst into M (7+1) × M (6) (which manifests as SO (7 , × SU (3) × U (1)), affecting fermions, vectorgauge fields and scalar gauge fields.This first break is caused by the scalar condensate of two right handed neutrinos, presented inTable V, Sect. B, which interact with all the scalar gauge fields (with the gauge fields with thespace index (5 , , , · · · , , , , ((7+1) / − families [72]. The rest of families getheavy masses [87].The fermion families are arranged into twice two groups of massless four families, with re-spect to family quantum numbers as presented in Table IV in Sect. B, each group manifesting SU (2) ⊂ SO (3 , × SU (2) ⊂ SO (4) symmetry. One group manifests the SU (2) L × SU (2) L symmetry,the other SU (2) R × SU (2) R symmetry.The nonzero vacuum expectation values of the scalar fields with the space index (7 , d = (3 + 1).The superposition of the Lorentz members of the Clifford algebra, manifesting in d = (3 + 1)the spins, Eq. (B1), charges, Eqs. (B2, B3) and families, Eqs (B4, B5). are presented in Sect. B.0Let me rewrite the fermion part of the action, Eq. (1), by taking into account the degrees offreedom the action manifests in d = (3 + 1) in the way that we can clearly see that the action doesmanifest in the low energy regime by the standard model required properties of fermions, of vectorgauge fields and of scalar gauge fields [1–3, 7, 9, 55–57, 75, 76]. L f = ¯ ψγ m ( p m − (cid:88) A,i g Ai τ Ai A Aim ) ψ + { (cid:88) s =7 , ¯ ψγ s p s ψ } + { (cid:88) t =5 , , ,..., ¯ ψγ t p t ψ } , (27)where p s = p s − S s (cid:48) s ” ω s (cid:48) s ” s − ˜ S ab ˜ ω abs , p t = p t − S t (cid:48) t ” ω t (cid:48) t ” t − ˜ S ab ˜ ω abt , with m ∈ (0 , , , s ∈ (7 , , ( s (cid:48) , s ”) ∈ (5 , , , a, b ) (appearing in ˜ S ab ) run within either (0 , , ,
3) or (5 , , , t runs ∈ (5 , . . . , t (cid:48) , t ”) run either ∈ (5 , , ,
8) or ∈ (9 , , . . . , ψ represents all family members of all the 2 − = 8 families. a. The first line of Eq. (27) determines in d = (3 + 1) the kinematics and dynamics of fermionfields, coupled to the vector gauge fields [3, 5, 9]. The vector gauge fields are the superposition ofthe spin connection fields ω stm , m = (0 , , , s, t ) = (5 , , · · · , , S st .They are shortly presented in Sect. 34.The operators τ Ai ( τ Ai = (cid:80) a,b c Aiab S ab , S ab are the generators of the Lorentz transformationsin the Clifford space of γ a ’s) are presented in Eqs. (B2, B3) of Sect. B. They represent the colourcharge, (cid:126)τ , the weak charge, (cid:126)τ , and the hyper charge, Y = τ + τ , τ is the fermion charge,originating in SO (6) ⊂ SO (13 , τ belongs together with (cid:126)τ of SU (2) weak to SO (4) group( ⊂ SO (13 + 1)). One fermion irreducible representation of the Lorentz group contains , as seen in Table III, quarks and leptons and antiquarks and antileptons , belonging to the first family in Table IV. Onecan notice that the SO (7 ,
1) subgroup content of the SO (13 ,
1) group is the same for the quarksand leptons and the same for the antiquarks and antileptons. Quarks distinguish from leptons, andantiquarks from antileptons, only in the SO (6) ⊂ SO (13 ,
1) part, that is in the colour ( τ , τ )part and in the fermion quantum number τ . The quarks distinguish from antiquarks, and leptonsfrom antileptons, in the handedness, in the colour part and in the τ part, explaining the relationbetween handedness and charges of fermions and antifermions [88].The vector gauge fields, which interact with the condensate, presented in Table V, becomemassive. 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 assumedgauge fields before the electroweak break of the mass protection.After the electroweak break, caused by the scalar fields, the only conserved charges are thecolour and the electromagnetic charge Q = τ + Y , Y = τ + τ . b. The second line of Eq. (27) is the mass term, responsible in d = (3 + 1) for the massesof fermions. The interaction of fermions with the superposition of the spin connection fields withthe space index s = (7 , ω s (cid:48) t (cid:48) s or ˜ ω abs . These scalar fields explain the appearance of the higgs andYukawa couplings of the standard model . Their properties are shortly presented in Subsect. II B 2.These scalar gauge fields split into two groups of four families, one group manifesting the symme-try — (cid:103) SU (2) ( (cid:103) SO (3 , ,L ) × (cid:103) SU (2) ( (cid:103) SO (4) ,L ) × U (1) — and the other the symmetry — (cid:103) SU (2) ( (cid:103) SO (3 , ,R ) × (cid:103) SU (2) ( (cid:103) SO (4) ,R ) × U (1), Eq. (37). The scalar gauge fields, manifesting SU (2) L,R × SU (2) L,R ,are the superposition of the gauge fields ˜ ω abs , s = (7 , , ( a, b ) = either (0 , , ,
3) or (5 , , , U (1) singlet scalar gauge fields are superposition of ω s (cid:48) t (cid:48) s , s = (7 , s (cid:48) , t (cid:48) ) = (5 , , , , , · · · , S s (cid:48) t (cid:48) arranged into superposition of τ , τ and τ .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 offree parameters .To one of the groups of four families the observed quarks and leptons belong [55, 58, 61, 62].We predict the mixing matrices for quarks, taking as the input the masses of the fourth family,since the elements for the 3 × × dark matter appearance andit is so far in agreement with experimental evidences of the dark matter [56, 65].I discuss predictions of the spin-charge-family theory for the properties of the lower four familiesand of the dark matter in Sect. III. c. The third line of Eq. (27) represents the scalar fields, which cause transitions fromantileptons and antiquarks into quarks and leptons and back, offering the explanation for the2matter/antimatter asymmetry in the expanding universe at non equilibrium conditions [4]. Theyare colour triplets with respect to the space index equal to (9 , , , , , S ab in adjoint representations, as canbe seen in Table II and in Fig. 1 of Subsect. II B 2. I discuss properties of these scalar fields, offeredby the spin-charge-family theory, in Sect. III. B. Properties of vector and scalar gauge fields in spin-charge-family theory
In the starting action, Eq. (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, A gf = (cid:90) d d x E ( α R + ˜ α ˜ R ) ,R = 12 { f α [ a f βb ] ( ω abα,β − ω caα ω cbβ ) } + h.c. , ˜ R = 12 { f α [ a f βb ] (˜ ω abα,β − ˜ ω caα ˜ ω cbβ ) } + h.c. , (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 f βa and e aα are vielbeinsand inverted vielbeins respectively e aα f βa = δ βα , e aα f αb = δ ab , (29) E = det ( e aα ).Varying the action of Eq. (28) with respect to the spin connection fields, the expression for thespin connection fields ω eab follows ω abe = 12 E { e eα ∂ β ( Ef α [ a f βb ] ) − e aα ∂ β ( Ef α [ b f βe ] ) − e bα ∂ β ( Ef α [ e f βa ] ) } + 14 { ¯Ψ( γ e S ab − γ [ a S b ] e )Ψ }− d − { δ ea [ 1 E e dα ∂ β ( Ef α [ d f βb ] ) + ¯Ψ γ d S db Ψ] − δ eb [ 1 E e dα ∂ β ( Ef α [ d f βa ] ) + ¯Ψ γ d S da Ψ] } . (30)If replacing S ab in Eq. (30) with ˜ S ab , the expression for the spin connection fields ˜ ω abe follows.In Ref. [5] it is proven that in spaces with the desired symmetry the vielbein can be expressedwith the gauge fields, if only one of the two spin connection fields are present f σm = (cid:88) A (cid:126)τ Aσ (cid:126) A Am , (31)3 with A Aim = (cid:88) st c Aist ω stm ,τ Aiσ = (cid:88) st c Aist ( e sτ f σt − e tτ f σs ) x τ ,τ Ai = (cid:88) st c Aist S st . (32)If fermions are not present them spin connections of both kinds are uniquely determined by viel-beins, as can be noticerd from Eq. (30). If fermions are present, carrying both — family membersand family quantum numbers — then vielbeins and both kinds of spin connections are influencedby the presence of fermions, which carry different family and family members quantum numbers.The scalar (gauge) fields, carrying the space index s = (5 , , . . . , d ), offer in the spin-charge-family for s = (7 ,
8) the explanation for the origin of the Higgs’s scalar and the Yukawa couplingsof the standard model , while scalars with the space index s = (9 , , . . . ,
14) offer the explanationfor the proton decay, as well as for the matter/antimatter asymmetry in the universe.We use the notation τ Ai = (cid:88) a,b c Aiab S ab , { τ Ai , τ Bj } − = iδ AB f Aijk τ Ak ,A Aia = (cid:88) s,t c Aist ω sta , (33) a = m = (0 , , ,
3) for vector gauge fields and a = s = (5 , , . . . ,
14) for scalar aguge fields.The explicit expressions for c Aiab , and correspondingly for τ Ai , and A Aia , are written in Sect. B.
1. 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) τ vector gauge fields. The U (1) τ vector gauge field is the vector gauge field of τ (= − ( S + S
11 12 + S
13 14 )) - the fermioncharge. The hyper charge vector gauge field of the standard model is the superposition of thethird component of the second SU (2) II vector gauge fields and the U (1) τ vector gauge field( A Ym = cos θ A τ m + sin θ A m , θ is the angle of the break of the SU (2) II × U (1) τ symmetry to U (1) Y at the scale ≥ or higher, [9] and references therein). After the appearance of thecondensate, presented in Table V, 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 components4of the second SU (2) II vector gauge fields and the superposition A Y (cid:48) m = − sin θ A τ m + cos θ A m ,which is the gauge field of Y (cid:48) (= − tan θ τ + τ ) gain high masses due to the interaction withthe condensate. All the vector gauge fields are expressible with the spin connection fields ω stm , A Aim = (cid:88) s,t c Aist ω stm . (34)Let me present expressions for the two SU (2) vector gauge fields, SU (2) I and SU (2) II (cid:126) A m = (cid:126)A m = ( ω m − ω m , ω m + ω m , ω m − ω m ) ,(cid:126) A m = (cid:126)A m = ( ω m + ω m , ω m − ω m , ω m + ω m ) . (35)The reader can similarly construct all the other vector gauge fields from the coefficients for thecorresponding 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 ,
2. Scalar gauge fields in d = (3 + 1) There are in the spin-charge-family theory scalar fields taking care of the masses of quarksand leptons: They have the space index s = (7 ,
8) and carry with respect to the space indexthe weak charge τ = ± and the hyper charge Y = ∓ . With respect to τ Ai = (cid:80) ab c Aiab S ab and ˜ τ Ai = (cid:80) ab c Aiab ˜ S ab they carry charges and family charges in adjoint representations, Table I,Eq. (39).There are scalar fields transforming antileptons and antiquarks into quarks and leptons andback. They carry space index s = (9 , , . . . , τ Ai and ˜ τ Ai in adjoint representations.The infinitesimal generators S ab , which apply on the spin connections ω bde (= f αe ω bdα ) and˜ ω ˜ b ˜ de (= f αe ˜ ω ˜ b ˜ dα ), on either the space index e or any of the indices ( b, d, ˜ b, ˜ d ), as follows S ab A d...e...g = i ( η ae A d...b...g − η be A d...a...g ) , (36)(see Section IV. and Appendix B in Ref. [9]). Scalar gauge fields determining scalar higgs and Yukawa couplings A Ais for all the scalar gauge fields with s = (7 , ω abs — in this case Ai = ( Q , Q (cid:48) , Y (cid:48) ) - or in ˜ ω ˜ a ˜ bs — in this caseall the family quantum numbers of all eight families contribute. All these gauge fields contributeto the masses of the quarks and leptons and the antiquarks and antileptons after gaining nonzerovacuum expectation values. A Ais represents ( A Qs , A Q (cid:48) s , A Y (cid:48) s , (cid:126) ˜ A ˜1 s , (cid:126) ˜ A ˜ N ˜ L s , (cid:126) ˜ A ˜2 s , (cid:126) ˜ A ˜ N ˜ R s ) ,τ Ai represents ( Q, Q (cid:48) , Y (cid:48) , (cid:126) ˜ τ , (cid:126) ˜ N L , (cid:126) ˜ τ , (cid:126) ˜ N R ) . (37)Here τ Ai represent all the operators, which apply on the fermions. These scalars, the gauge scalarfields of the generators τ Ai and ˜ τ Ai , 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 respectto this space index the weak and the hyper charge ( ∓ , ± ), respectively. This means that allthese scalars have properties as required for the Higgs in the standard model .We need to know the application of the operators τ (= ( S − S ), Y (= τ + τ ) and Q (= τ + Y ), Eq (B2, B3, B7), with S ab defined in Eq. (36), on the scalar fields with the spaceindex s = (7 , standardmodel let the scalar fields be eigenstates of τ = ( S − S ).I rewrite for this purpose the second line of Eq. (27) as follows, ignoring the momentum p s , s = (5 , , . . . , d ), since it is expected that solutions with nonzero momenta in higher dimensions donot contribute to the masses of fermion fields at low energies in d = (3+1). We pay correspondinglyno attention to the momentum p s , s ∈ (5 , . . . , (cid:88) s =(7 , ,A,i ¯ ψ γ s ( − τ Ai A Ais ) ψ = − ¯ ψ { (+) τ Ai ( A Ai − i A Ai ) + ( − ) ( τ Ai ( A Ai + i A Ai ) } ψ , ( ± )= 12 ( γ ± i γ ) , A Ai ± ) := ( A Ai ∓ i A Ai ) , (38)with the summation over A and i performed, since A Ais represent the scalar fields ( A Qs , A Q (cid:48) s , A Y (cid:48) s )determined by ω s (cid:48) ,s (cid:48)(cid:48) ,s and those determined by (˜ ω a,b,s ˜ A ˜4 s , (cid:126) ˜ A ˜1 s , (cid:126) ˜ A ˜2 s , (cid:126) ˜ A ˜ N R s and (cid:126) ˜ A ˜ N L s ).The application of the operators τ , Y ( Y = ( S + S ) − ( S + S
11 12 + S
13 14 )) and Q on the scalar fields ( A Ai ∓ i A Ai ) with respect to the space index s = (7 , TABLE I: The two scalar weak doublets, one with τ = − and the other with τ = + , both withthe ”fermion” quantum number τ = 0, are presented. In this table all the scalar fields carry besides thequantum numbers determined by the space index also the quantum numbers A and i from Eq. (37). Thetable is taken from Ref. [9]. name superposition τ τ spin τ QA Ai − ) A Ai + iA Ai + − A Ai − ) A Ai + iA Ai − − A Ai A Ai − iA Ai − + A Ai A Ai − iA Ai + + Eq. (36) to make the application of the generators S ab on the space indexes, gives τ ( A Ai ∓ i A Ai ) = ±
12 ( A Ai ∓ i A Ai ) ,Y ( A Ai ∓ i A Ai ) = ∓
12 ( A Ai ∓ i A Ai ) ,Q ( A Ai ∓ i A Ai ) = 0 . (39)Since τ , Y , τ and τ , τ − give zero if applied on ( A Qs , A Q (cid:48) s and A Y (cid:48) s ) with respect to thequantum numbers ( Q, Q (cid:48) , Y (cid:48) ), and since Y and τ commute with the family quantum numbers,one sees that the scalar fields A Ais ( =( A Qs , A Ys , A Y (cid:48) s , ˜ A ˜4 s , ˜ A ˜ Qs , (cid:126) ˜ A ˜1 s , (cid:126) ˜ A ˜2 s , (cid:126) ˜ A ˜ N R s , (cid:126) ˜ A ˜ N L s )), rewritten as A Ai ± ) = ( A Ai ∓ i A Ai ) , are eigenstates of τ and Y , having the quantum numbers of the standardmodel Higgs’ scalar.These superposition of A Ai ± ) are presented in Table I as two doublets with respect to the weakcharge τ , with the eigenvalue of τ (the second SU (2) II charge) equal to either − or + ,respectively. The operators τ ± = τ ± iτ τ ± = 12 [( S − S ) ∓ i ( S + S )] , (40)transform one member of a doublet from Table I into another member of the same doublet, keeping τ (= ( S + S )) unchanged, clarifying the above statement.It is not difficult to show that the scalar fields A Ai ± ) are triplets as the gauge fields of the family quantum numbers ( (cid:126) ˜ N R , (cid:126) ˜ N L , (cid:126) ˜ τ , (cid:126) ˜ τ ; Eqs. (B4, B5, 36)) or singlets as the gauge fields of Q = τ + Y, Q (cid:48) = − tan ϑ Y + τ and Y (cid:48) = − tan ϑ τ + τ .Let us do this for ˜ A N L i ± ) and for A Q ± ) , taking into account Eq. (B1) (where we replace S ab by7 S ab ) and Eq. (36), and recognizing that ˜ A N L ± ± ) = ˜ A N L ± ) ∓ i ˜ A N L ± ) .˜ A ˜ N L ± ± ) = { (˜ ω ± ) + i ˜ ω ± ) ) ∓ i (˜ ω ± ) + i ˜ ω ± ) ) } , ˜ A ˜ N L ± ) = (˜ ω ± ) + i ˜ ω ± ) ) ,A Q ± ) = ω ± ) − ( ω ± ) + ω
11 12 ± ) + ω
13 14 ± ) ) . One finds ˜ N L ˜ A ˜ N L ± ± ) = ± ˜ A ˜ N L ± ± ) , ˜ N L ˜ A ˜ N L ± ) = 0 ,Q A Q ± ) = 0 . (41)with Q = S + τ = S − ( S + S
11 12 + S
13 14 ), and with τ defined in Eq. (B3), if replacing S ab by S ab from Eq. (36). Similarly one finds properties with respect to the Ai quantum numbersfor all the scalar fields A Ai ± ) .After the appearance of the condensate (Table V), which breaks the SU (2) II symmetry andbrings masses to all the scalar fields, the weak (cid:126)τ and the hyper charge Y remain the conservedcharges.At the electroweak scale the scalar gauge fields with the space index (7 , L sg = E (cid:88) A,i { ( p m A Ais ) † ( p m A Ais ) − ( − λ Ai + ( m (cid:48) Ai ) )) A Ai † s A Ais + (cid:88) B,j Λ AiBj A Ai † s A Ais A Bj † s A Bjs } , (42)gain nonzero vacuum expectation values and cause the electroweak break [89]. The above Lagrangedensity needs to be studied. At this stage is just postulated.The two groups of four families became massive. The mass matrices manifest either (cid:103) SU (2) (cid:103) SO (3 , L × (cid:103) SU (2) (cid:102) SU (4) L × U (1) symmetry, this is the case for the lower four families of theeight families, presented in Table IV, or (cid:103) SU (2) (cid:103) SO (3 , R × (cid:103) SU (2) (cid:102) SU (4) R × U (1) symmetry, this isthe case for the higher four families, presented in Table IV. The same three U (1) singlet fieldscontribute to the masses of both groups, the two SU (2) triplet fields are for each of the two groupsdifferent, although manifesting the same symmetries.The mass matrix of family member — quarks and leptons — are 4 × × × M α = − a − a e d be − a − a b dd b a − a eb d e a − a α , (43) with α representing family members — quarks and leptons of left and right handedness [53–55, 57, 58, 62].The mass matrices of the upper four families have the same symmetry as the mass matricesof the lower four families, but the scalar fields determining the masses of the upper four familieshave different properties (nonzero vacuum expectation values, masses and coupling constants) thanthose of the lower four, giving to quarks and leptons of the upper four families much higher massesin comparison with the lower four families of quarks and leptons, what offers the explanation forthe appearance of the dark matter , studied at Refs. [56, 65]. Scalar fields transforming antiquarks and antileptons into quarks and leptons
I follow in this part to a great deal similar part in Ref. [3].To the matter-antimatter asymmetry the terms contribute, which cause transitions from antilep-tons into quarks and from antiquarks into quarks and back. These are terms included into the thirdline of Eq. (27). Let me rewrite this part of the fermion action L f (cid:48) = ψ † γ γ t { (cid:80) t =(9 , ,... [ p t − ( S s (cid:48) s ” ω s (cid:48) s ” t + S t (cid:48) t (cid:48)(cid:48) ω t (cid:48) t ” t + ˜ S ab ˜ ω abt )] } ψ, as follows L f ” = ψ † γ ( − ) { (cid:88) + , − (cid:88) ( t t (cid:48) ) tt (cid:48) ( ± ○ ) · [ τ A tt (cid:48) ( ± ○ ) + τ − A − tt (cid:48) ( ± ○ ) + τ A tt (cid:48) ( ± ○ ) + τ A tt (cid:48) ( ± ○ ) + τ − A − tt (cid:48) ( ± ○ ) + τ A tt (cid:48) ( ± ○ ) + ˜ τ ˜ A tt (cid:48) ( ± ○ ) + ˜ τ − ˜ A − tt (cid:48) ( ± ○ ) + ˜ τ ˜ A tt (cid:48) ( ± ○ ) + ˜ τ ˜ A tt (cid:48) ( ± ○ ) + ˜ τ − ˜ A − tt (cid:48) ( ± ○ ) + ˜ τ ˜ A tt (cid:48) ( ± ○ ) + ˜ N + R ˜ A N R + tt (cid:48) ( ± ○ ) + ˜ N − R ˜ A N R − tt (cid:48) ( ± ○ ) + ˜ N R ˜ A N R tt (cid:48) ( ± ○ ) + ˜ N + L ˜ A N L + tt (cid:48) ( ± ○ ) + ˜ N − L ˜ A N L − tt (cid:48) ( ± ○ ) + ˜ N L ˜ A N L tt (cid:48) ( ± ○ ) + (cid:88) i τ i A i tt (cid:48) ( ± ○ ) + τ A tt (cid:48) ( ± ○ ) + (cid:88) i ˜ τ i ˜ A i tt (cid:48) ( ± ○ ) + ˜ τ ˜ A tt (cid:48) ( ± ○ ) ] } ψ , (44)where ( t, t (cid:48) ) run in pairs over [(9 , , (11 , , (13 , − of tt (cid:48) ( ± ○ ) .9In Eq. (44) the relations below are used (cid:88) t,s (cid:48) ,s (cid:48)(cid:48) γ t S s (cid:48) s ” ω s (cid:48) s ” t = (cid:88) + , − (cid:88) ( t t (cid:48) ) tt (cid:48) ( ± ○ ) 12 S s (cid:48) s ” ω s ” s ” tt (cid:48) ( ± ○ ) ,ω s ” s ” tt (cid:48) ( ± ○ ) : = ω s ” s ” tt (cid:48) ( ± ) = ( ω s (cid:48) s ” t ∓ i ω s (cid:48) s ” t (cid:48) ) , tt (cid:48) ( ± ○ ): = = 12 ( γ t ± γ t (cid:48) ) , (cid:88) + , − (cid:88) ( t t (cid:48) ) tt (cid:48) ( ± ○ ) 12 S s (cid:48) s ” ω s ” s ” tt (cid:48) ( ± ○ ) = tt (cid:48) ( ± ○ ) { τ A tt (cid:48) ( ± ○ ) + τ − A − tt (cid:48) ( ± ○ ) + τ A tt (cid:48) ( ± ○ ) + τ A tt (cid:48) ( ± ○ ) + τ − A − tt (cid:48) ( ± ○ ) + τ A tt (cid:48) ( ± ○ ) } ,A ± tt (cid:48) ( ± ○ ) = ( ω tt (cid:48) ( ± ○ ) + ω tt (cid:48) ( ± ○ ) ) ∓ i ( ω tt (cid:48) ( ± ○ ) − ω tt (cid:48) ( ± ○ ) ) , A tt (cid:48) ( ± ○ ) = ( ω tt (cid:48) ( ± ○ ) + ω tt (cid:48) ( ± ○ ) ) ,A ± tt (cid:48) ( ± ○ ) = ( ω tt (cid:48) ( ± ○ ) − ω tt (cid:48) ( ± ○ ) ) ∓ i ( ω tt (cid:48) ( ± ○ ) + ω tt (cid:48) ( ± ○ ) ) , A tt (cid:48) ( ± ○ ) = ( ω tt (cid:48) ( ± ○ ) − ω tt (cid:48) ( ± ○ ) ) , ( t t (cid:48) ) ∈ ((9 10) , (11 12) , (13 14)) . (45)The rest of expressions in Eq. (45) are obtained in a similar way. They are presented in Eq. (C3).The scalar fields with the scalar index s = (9 , , · · · , S ab or in ˜ S ab [90].If the antiquark ¯ u ¯ c L , from the line 43 presented in Table III, with the ”fermion” charge τ = − ,the weak charge τ = 0, the second SU (2) II charge τ = − , the colour charge ( τ , τ ) =( , − √ ), the hyper charge Y (= τ + τ =) − and the electromagnetic charge Q ( = Y + τ =) − submits the A (cid:12) ⊕ ) scalar field, it transforms into u c R from the line 17 of Table III, carrying thequantum numbers τ = , τ = 0, τ = , ( τ , τ ) = (0 , − √ ), Y = and Q = . Thesetwo quarks, d c R and u c R can bind together with u c R from the 9 th line of the same table (at lowenough energy, after the electroweak transition, and if they belong to a superposition with the lefthanded partners to the first family) -into the colour chargeless baryon - a proton. This transitionis presented in Figure 1.The opposite transition at low energies would make the proton decay. III. ACHIEVEMENTS AND CONCLUSIONS
It remains to point out the achievements of the spin-charge-family theory so far and tell theopen problems.0 u c R τ = , τ =0 , τ = ( τ , τ )=( − , √ ) Y = ,Q = u c R ¯ u ¯ c L τ = − , τ =0 , τ = − ( τ , τ )=( , − √ ) Y = − ,Q = − u c R τ = , τ =0 , τ = ( τ , τ )=(0 , − √ ) Y = ,Q = ¯ e + L τ = , τ =0 , τ = ( τ , τ )=(0 , Y =1 ,Q =1 d c R τ = , τ =0 , τ = − ( τ , τ )=( , √ ) Y = − ,Q = − • A ! , τ =2 × ( − ) , τ =0 , τ = − τ , τ )=( , √ ) Y = − ,Q = − • FIG. 1: The birth of a ”right handed proton” out of an positron ¯ e + L , antiquark ¯ u ¯ c L and quark (spectator) u c R . The family quantum number can be any. Achievements : a. The simple starting action, Eq. (1), with the Clifford algebra used to describe the internalspace of fermions, which in d ≥ (13 + 1) interact with the vielbeins and the two kinds of spinconnection fields, offers a.i. that one irreducible representation of the Lorentz algebra in internalspace manifests in d = (3 + 1) all the fermions and antifermions with the spins and charges ofthe standard model , a.ii. that eight irreducible representations define in d = (3 + 1) (after thereduction 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 fieldsof the standard model , a.iv. that the scalar fields with respect to d = (3 + 1), carrying the weakand the hyper charge ± and ∓ , respectively, forming two groups of scalar fields manifestingeach the SU (2) × SU (2) × U (1) symmetry, offer the explanation for the Higgs’s scalar and Yukawacouplings of the standard model giving masses to two groups of four families — the lower fourfamilies predicting the fourth family of quarks and leptons to the observed three, the stable ofthe upper four families offering explanation for the dark matter , a.v. that both groups of fourfamilies together spread masses from almost zero to ≥ GeV, a.vi. that the scalar gaugefields manifesting as colour triplets and antitriplets offer the explanation for the matter/antimatter1asymmetry of the ordinary matter. b. The decision to describe the internal space of fermions with the Clifford odd algebra enablesto define the creation operators as tensor products of finite number of ”basis vectors” of internalspace and infinite basis in ordinary space applying on the vacuum state, which fulfill togetherthe their Hermitian conjugated annihilation operators the anticommutation relations postulatedby Dirac for the second quantized fields. The single fermion states have therefore by themselvesthe anticommuting character. Tensor products of any number and any kind of the single fermioncreation 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. I and II and alsoseveral predictions. A. Prediction of the fourth family to the observed three families, Subsect. II A 4.Taking into account the experimental data for masses of the observed families of quarks and thecorresponding mixing matrix we fit 6 parameters of the two quark mass matrices, presented inEq. (43), to twice 3 measured massess of quarks and to 6 measured parameters of the mixingmatrix.Althrough any accurate 3 × × × × m u = m d , Ref. [58], [arXiv:1412.5866]:1. m u = 700 GeV, m d = 700 GeV..... new m u = 1 200 GeV, m d = 1200 GeV..... new | V ( ud ) | = exp n . ± . . ± . . ± . new . . . . ( ) new . . . . [ ] exp n . ± .
008 0 . ± .
016 0 . ± . new . . . . ( ) new . . . . [ ] exp n . ± . . ± . . ± . new . . . . new . . . . [ ] new . ( ) . ( ) . . new . . . . [ ] . (46) One can see that the above results for the mixing matrices of the lower three families are in2agreement with what Ref. [59] requires, namely that V u d > V u d , V u d < V u d , and V u d < V u d .Since we have not yet fit the mass matrix of Eq. (43) to the newest experimental data [60],which appear after our Bled 2020, the evaluation for our 4 × B. The spin-charge-family theory predicts in the low energy regime (up to 10 GeV or higher)the existence of two decoupled groups of four families, which at the electroweak break becomemassive [56]. The stable family of the upper group of four families (with almost zero Yukawacouplings to the lower group of four families) is the candidate for the dark matter, Subsect. II A 4.I review here briefly the estimations done in Ref. [56]. We used the simple hydrogen-like modelto evaluate properties of the fifth family heavy baryons, taking into account that for masses ofthe order of a few TeV or larger the force among the constituents of the fifth family baryons isdetermined mostly by one gluon exchange. The fifth family neutron is estimated as the moststable nucleon. The ”nuclear interaction” among these baryons is found to have very interestingproperties. We studied scattering amplitudes among fifth family neutrons and with the ordinarymatter.We followed the behaviour of the fifth family quarks and antiquarks in the plasma of the expand-ing universe, through the freezing out procedure, solving the Boltzmann equations, through thecolour phase transition, while forming neutrons, up to the present dark matter, taking into accountthe cosmological evidences, the direct experimental evidences and all others known properties ofthe dark matter.The cosmological evolution suggested the limits for the masses of the fifth family quarks10 TeV < m q c < a few · TeV (47)and for the scattering cross sections10 − fm < σ c < − fm , (48)while the measured density of the dark matter does not put much limitation on the properties ofheavy enough clusters.3The direct measurements limit the fifth family quark mass toseveral 10 TeV < m q c < TeV . (49)We also find that our fifth family baryons of the mass of several 10 TeV/ c have for a factor morethan 100 times too small scattering amplitude with the ordinary matter to cause a measurableheat flux on the Earth’s surface. C. The spin-charge-family theory predicts several scalar fields with the weak and the hypercharge of the Higg’s scalar ( ± , ∓ ) — two triplets and three singlets — offering the explanationfor the existence of the Higgs’s scalar and Yukawa couplings, Subsect. II B 2.The additional two triplets and the same three singlets determine properties of the upper fourfamilies of quarks and leptons, Subsect. II B 2. D. The spin-charge-family theory predicts several scalar fields which are colour triplets or an-titriplets, offering the explanation for the matter/antimatter asymmetry in the (nonequilibrium)expanding universe as well as the proton decay [4], Subsect. II B 2. E. The mass matrices of the two fourth family groups are close to democratic one, causingspreading of the fermion masses from 10 − MeV to 10 GeV or even higher.I conclude by saying that there are still a lot of open problems to be solved. Some of them arecommon to the other theories, like the Kaluza-Klein-like theories, the others require to extract asmuch as possible from the offer of the theory. We need collaborators, since the more work is putinto the spin-charge-family theory the more explanations for the observed phenomena follow.
Appendix A: Some useful relations
From Eq. (16) it follows S ac ab ( k ) cd ( k ) = − i η aa η cc ab [ − k ] cd [ − k ] , ˜ S ac ab ( k ) cd ( k ) = i η aa η cc ab [ k ] cd [ k ] ,S ac ab [ k ] cd [ k ] = i ab ( − k ) cd ( − k ) , ˜ S ac ab [ k ] cd [ k ] = − i ab ( k ) cd ( k ) ,S ac ab ( k ) cd [ k ] = − i η aa ab [ − k ] cd ( − k ) , ˜ S ac ab ( k ) cd [ k ] = − i η aa ab [ k ] cd ( k ) ,S ac ab [ k ] cd ( k ) = i η cc ab ( − k ) cd [ − k ] , ˜ S ac ab [ k ] cd ( k ) = i η cc ab ( k ) cd [ k ] . (A1)4By using Eq. (14) one finds the relations ab ˜( k ) ab ( k ) = 0 , ab ˜( − k ) ab ( k )= − i η aa 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 ] . (A2) Appendix B: One irreducible representation of the internal space and families described bythe Clifford algebra γ a Below the subgroups of the starting groups SO (13 ,
1) and ˜ SO (13 ,
1) are presented, manifestingin d = (3 + 1) the spins, charges and families of fermions in the spin-charge-family theory.Table III, representing one SO (13 ,
1) irreducible representation of fermions — quarks and leptonsand antiquarks and antileptons — uses these expressions. a.i.
The generators of the two SU (2) ( ⊂ SO (3 , ⊂ SO (7 , ⊂ SO (13 , (cid:126)N ± (= (cid:126)N ( L,R ) ) : = 12 ( S ± iS , S ± iS , S ± iS ) , (B1)are presented. a.ii. The generators of the two SU (2) ( SU (2) ⊂ SO (4) ⊂ SO (7 , ⊂ SO (13 , (cid:126)τ : = 12 ( S − S , S + S , S − S ) ,(cid:126)τ : = 12 ( S + S , S − S , S + S ) , (B2)are presented. a.iii. The SU (3) and U (1) subgroups of SO (6) ⊂ SO (13 , (cid:126)τ := 12 { S − S
10 11 , S + S
10 12 , S − S
11 12 ,S − S
10 13 , S + S
10 14 , S
11 14 − S
12 13 ,S
11 13 + S
12 14 , √ S + S
11 12 − S
13 14 ) } ,τ := −
13 ( S + S
11 12 + S
13 14 ) , (B3)5are presented. b.i. The two (cid:103) SU (2) subgroups of ˜ SO (3 ,
1) ( ⊂ (cid:103) SO (7 , ⊂ (cid:103) SO (13 , (cid:126) ˜ N L,R : = 12 ( ˜ S ± i ˜ S , ˜ S ± i ˜ S , ˜ S ± i ˜ S ) , (B4)are presented. b.ii. The two (cid:103) SU (2) subgroups of (cid:103) SO (4) ( ⊂ (cid:103) SO (7 , ⊂ (cid:103) SO (13 , (cid:126) ˜ τ : = 12 ( ˜ S − ˜ S , ˜ S + ˜ S , ˜ S − ˜ S ) ,(cid:126) ˜ τ : = 12 ( ˜ S + ˜ S , ˜ S − ˜ S , ˜ S + ˜ S ) , (B5)are presented. b.iii. The group ˜ U (1), the subgroup of (cid:103) SO (6) ( ⊂ (cid:103) SO (13 , τ := −
13 ( ˜ S + ˜ S
11 12 + ˜ S
13 14 ) , (B6)are presented. c. Relations among the hyper, weak and the second SU (2) charges Y := τ + τ , Y (cid:48) := − τ tan ϑ + τ , Q := τ + Y , Q (cid:48) := − Y tan ϑ + τ , ˜ Y := ˜ τ + ˜ τ , ˜ Y (cid:48) := − ˜ τ tan ϑ + ˜ τ , ˜ Q := ˜ Y + ˜ τ , ˜ Q (cid:48) = − ˜ Y tan ϑ + ˜ τ , (B7)are presented.Below are some of the above expressions written in terms of nilpotents and projectors N ± + = N ± i N = − ( ∓ i ) ( ± ) , N ±− = N − ± i N − = ( ± i ) ( ± ) , ˜ N ± + = − ˜( ∓ i ) ˜( ± ) , ˜ N ±− = ˜( ± i ) ˜( ± ) ,τ ± = ( ∓ ) ( ± ) ( ∓ ) , τ ∓ = ( ∓ ) ( ∓ ) ( ∓ ) , ˜ τ ± = ( ∓ ) ˜( ± ) ˜( ∓ ) , ˜ τ ∓ = ( ∓ ) ˜( ∓ ) ˜( ∓ ) . (B8) i a ˆ b † i Γ (3+1) S τ τ τ τ τ Y Q (Anti)octet , Γ (7+1) = ( −
1) 1 , Γ (6) = (1) − u c † R (+ i ) [+] | [+] (+) || (+)
11 12 [ − ]
13 14 [ − ] 1
12 12 12 √ u c † R [ − i ] ( − ) | [+] (+) || (+)
11 12 [ − ]
13 14 [ − ] 1 −
12 12 12 √ Continued on next page i a ˆ b † i Γ (3+1) S τ τ τ τ τ Y Q (Anti)octet , Γ (7+1) = ( −
1) 1 , Γ (6) = (1) − d c † R (+ i ) [+] | ( − ) [ − ] || (+)
11 12 [ − ]
13 14 [ − ] 1 −
12 12 12 √ − − d c † R [ − i ] ( − ) | ( − ) [ − ] || (+)
11 12 [ − ]
13 14 [ − ] 1 − −
12 12 12 √ − − d c † L [ − i ] [+] | ( − ) (+) || (+)
11 12 [ − ]
13 14 [ − ] -1 −
12 12 √ − d c † L (+ i ) ( − ) | ( − ) (+) || (+)
11 12 [ − ]
13 14 [ − ] -1 − −
12 12 √ − u c † L [ − i ] [+] | [+] [ − ] || (+)
11 12 [ − ]
13 14 [ − ] -1
12 12
12 12 √ u c † L (+ i ) ( − ) | [+] [ − ] || (+)
11 12 [ − ]
13 14 [ − ] -1 −
12 12
12 12 √ u c † R (+ i ) [+] | [+] (+) || [ − ]
11 12 (+)
13 14 [ − ] 1 −
12 12 √
10 ˆ u c † R [ − i ] ( − ) | [+] (+) || [ − ]
11 12 (+)
13 14 [ − ] 1 − −
12 12 √
11 ˆ d c † R (+ i ) [+] | ( − ) [ − ] || [ − ]
11 12 (+)
13 14 [ − ] 1 − −
12 12 √ − −
12 ˆ d c † R [ − i ] ( − ) | ( − ) [ − ] || [ − ]
11 12 (+)
13 14 [ − ] 1 − − −
12 12 √ − −
13 ˆ d c † L [ − i ] [+] | ( − ) (+) || [ − ]
11 12 (+)
13 14 [ − ] -1 − −
12 12 √ −
14 ˆ d c † L (+ i ) ( − ) | ( − ) (+) || [ − ]
11 12 (+)
13 14 [ − ] -1 − − −
12 12 √ −
15 ˆ u c † L [ − i ] [+] | [+] [ − ] || [ − ]
11 12 (+)
13 14 [ − ] -1
12 12 −
12 12 √
16 ˆ u c † L (+ i ) ( − ) | [+] [ − ] || [ − ]
11 12 (+)
13 14 [ − ] -1 −
12 12 −
12 12 √
17 ˆ u c † R (+ i ) [+] | [+] (+) || [ − ]
11 12 [ − ]
13 14 (+) 1 − √
18 ˆ u c † R [ − i ] ( − ) | [+] (+) || [ − ]
11 12 [ − ]
13 14 (+) 1 − − √
19 ˆ d c † R (+ i ) [+] | ( − ) [ − ] || [ − ]
11 12 [ − ]
13 14 (+) 1 − − √ − −
20 ˆ d c † R [ − i ] ( − ) | ( − ) [ − ] || [ − ]
11 12 [ − ]
13 14 (+) 1 − − − √ − −
21 ˆ d c † L [ − i ] [+] | ( − ) (+) || [ − ]
11 12 [ − ]
13 14 (+) -1 − − √ −
22 ˆ d c † L (+ i ) ( − ) | ( − ) (+) || [ − ]
11 12 [ − ]
13 14 (+) -1 − − − √ −
23 ˆ u c † L [ − i ] [+] | [+] [ − ] || [ − ]
11 12 [ − ]
13 14 (+) -1
12 12 − √
24 ˆ u c † L (+ i ) ( − ) | [+] [ − ] || [ − ]
11 12 [ − ]
13 14 (+) -1 −
12 12 − √
25 ˆ ν † R (+ i ) [+] | [+] (+) || (+)
11 12 (+)
13 14 (+) 1 − ν † R [ − i ] ( − ) | [+] (+) || (+)
11 12 (+)
13 14 (+) 1 − − e † R (+ i ) [+] | ( − ) [ − ] || (+)
11 12 (+)
13 14 (+) 1 − − − −
128 ˆ e † R [ − i ] ( − ) | ( − ) [ − ] || (+)
11 12 (+)
13 14 (+) 1 − − − − −
129 ˆ e † L [ − i ] [+] | ( − ) (+) || (+)
11 12 (+)
13 14 (+) -1 − − − −
130 ˆ e † L (+ i ) ( − ) | ( − ) (+) || (+)
11 12 (+)
13 14 (+) -1 − − − − −
131 ˆ ν † L [ − i ] [+] | [+] [ − ] || (+)
11 12 (+)
13 14 (+) -1
12 12 − −
032 ˆ ν † L (+ i ) ( − ) | [+] [ − ] || (+)
11 12 (+)
13 14 (+) -1 −
12 12 − −
033 ˆ¯ d ¯ c † L [ − i ] [+] | [+] (+) || [ − ]
11 12 (+)
13 14 (+) -1 − − √ −
16 13 13
34 ˆ¯ d ¯ c † L (+ i ) ( − ) | [+] (+) || [ − ]
11 12 (+)
13 14 (+) -1 − − − √ −
16 13 13
35 ¯ u ¯ c † L [ − i ] [+] | ( − ) [ − ] || [ − ]
11 12 (+)
13 14 (+) -1 − − − √ − − −
36 ¯ u ¯ c † L (+ i ) ( − ) | ( − ) [ − ] || [ − ]
11 12 (+)
13 14 (+) -1 − − − − √ − − −
37 ˆ¯ d ¯ c † R (+ i ) [+] | [+] [ − ] || [ − ]
11 12 (+)
13 14 (+) 1
12 12 − − √ − −
16 13
38 ˆ¯ d ¯ c † R [ − i ] ( − ) | [+] [ − ] || [ − ]
11 12 (+)
13 14 (+) 1 −
12 12 − − √ − −
16 13
39 ˆ¯ u ¯ c † R (+ i ) [+] | ( − ) (+) || [ − ]
11 12 (+)
13 14 (+) 1 − − − √ − − −
40 ˆ¯ u ¯ c † R [ − i ] ( − ) | ( − ) (+) || [ − ]
11 12 (+)
13 14 (+) 1 − − − − √ − − − Continued on next page i a ˆ b † i Γ (3+1) S τ τ τ τ τ Y Q (Anti)octet , Γ (7+1) = ( −
1) 1 , Γ (6) = (1) − d ¯ c † L [ − i ] [+] | [+] (+) || (+)
11 12 [ − ]
13 14 (+) -1
12 12 − √ −
16 13 13
42 ˆ¯ d ¯ c † L (+ i ) ( − ) | [+] (+) || (+)
11 12 [ − ]
13 14 (+) -1 −
12 12 − √ −
16 13 13
43 ˆ¯ u ¯ c † L [ − i ] [+] | ( − ) [ − ] || (+)
11 12 [ − ]
13 14 (+) -1 −
12 12 − √ − − −
44 ˆ¯ u ¯ c † L (+ i ) ( − ) | ( − ) [ − ] || (+)
11 12 [ − ]
13 14 (+) -1 − −
12 12 − √ − − −
45 ˆ¯ d ¯ c † R (+ i ) [+] | [+] [ − ] || (+)
11 12 [ − ]
13 14 (+) 1
12 12 − √ − −
16 13
46 ˆ¯ d ¯ c † R [ − i ] ( − ) | [+] [ − ] || (+)
11 12 [ − ]
13 14 (+) 1 −
12 12 − √ − −
16 13
47 ˆ¯ u ¯ c † R (+ i ) [+] | ( − ) (+) || (+)
11 12 [ − ]
13 14 (+) 1 − − √ − − −
48 ˆ¯ u ¯ c † R [ − i ] ( − ) | ( − ) (+) || (+)
11 12 [ − ]
13 14 (+) 1 − − − √ − − −
49 ˆ¯ d ¯ c † L [ − i ] [+] | [+] (+) || (+)
11 12 (+)
13 14 [ − ] -1 √ −
16 13 13
50 ˆ¯ d ¯ c † L (+ i ) ( − ) | [+] (+) || (+)
11 12 (+)
13 14 [ − ] -1 − √ −
16 13 13
51 ˆ¯ u ¯ c † L [ − i ] [+] | ( − ) [ − ] || (+)
11 12 (+)
13 14 [ − ] -1 − √ − − −
52 ˆ¯ u ¯ c † L (+ i ) ( − ) | ( − ) [ − ] || (+)
11 12 (+)
13 14 [ − ] -1 − − √ − − −
53 ˆ¯ d ¯ c † R (+ i ) [+] | [+] [ − ] || (+)
11 12 (+)
13 14 [ − ] 1
12 12 √ − −
16 13
54 ˆ¯ d ¯ c † R [ − i ] ( − ) | [+] [ − ] || (+)
11 12 (+)
13 14 [ − ] 1 −
12 12 √ − −
16 13
55 ˆ¯ u ¯ c † R (+ i ) [+] | ( − ) (+) || (+)
11 12 (+)
13 14 [ − ] 1 − √ − − −
56 ˆ¯ u ¯ c † R [ − i ] ( − ) | ( − ) (+) || (+)
11 12 (+)
13 14 [ − ] 1 − − √ − − −
57 ˆ¯ e † L [ − i ] [+] | [+] (+) || [ − ]
11 12 [ − ]
13 14 [ − ] -1 e † L (+ i ) ( − ) | [+] (+) || [ − ]
11 12 [ − ]
13 14 [ − ] -1 − ν † L [ − i ] [+] | ( − ) [ − ] || [ − ]
11 12 [ − ]
13 14 [ − ] -1 − ν † L (+ i ) ( − ) | ( − ) [ − ] || [ − ]
11 12 [ − ]
13 14 [ − ] -1 − − ν † R (+ i ) [+] | ( − ) (+) || [ − ]
11 12 [ − ]
13 14 [ − ] 1 −
12 12
062 ˆ¯ ν † R [ − i ] ( − ) | ( − ) (+) || [ − ]
11 12 [ − ]
13 14 [ − ] 1 − −
12 12
063 ˆ¯ e † R (+ i ) [+] | [+] [ − ] || [ − ]
11 12 [ − ]
13 14 [ − ] 1
12 12
12 12
164 ˆ¯ e † R [ − i ] ( − ) | [+] [ − ] || [ − ]
11 12 [ − ]
13 14 [ − ] 1 −
12 12
12 12 TABLE III:
The left handed (Γ (13 , = − 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) = − SU (2) I )charged ( τ = ± , ( (cid:126)τ = ( S − S , S + S , S − S )) and SU (2) II chargeless ( τ = 0, (cid:126)τ = ( S + S , S − S , S + S ))quarks and leptons and the right handed (Γ (3+1) = 1), weak ( SU (2) I ) chargeless and SU (2) II charged ( τ = ± ) quarks and leptons, bothwith the spin S up and down ( ± , 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 ( ( τ , τ ) = [( , √ ) , ( − , √ ) , (0 , − √ )], ( (cid:126)τ = ( S − S
10 11 , S + S
10 12 , S − S
11 12 ,S − S
10 13 , S + S
10 14 , S
11 14 − S
12 13 , S
11 13 + S
12 14 , √ ( S + S
11 12 − S
13 14 )), carrying the ”fermion charge” ( τ = , = − ( S + S
11 12 + S
13 14 ). The colourless leptons carry the ”fermion charge” ( τ = − ). In the same multiplet of creation operators the lefthanded 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 areantitriplets, carrying the ”fermion charge” ( τ = − ). The anti-colourless antileptons carry the ”fermion” charge ( τ = ). Y = ( τ + τ ) isthe hyper charge, the electromagnetic charge is Q = ( τ + Y ). The creation operators of opposite charges (antifermion creation operators) arereachable from the particle ones besides by S ab also by the application of the discrete symmetry operator C N P N , presented in Refs. [69, 70].The reader can find this Weyl representation also in Refs. [4, 9, 75, 76] and in the references therein. Table III represents in the spin-charge-family theory the creation operators for the observed quarksand leptons and antiquarks and antileptons for a particular family, Table (IV). Hermitian conju-gation of the creation operators of Table III generates the corresponding annihilation operators,fulfilling together with the creation operators anticommutation relations for fermions of Eq. (23).The condensate of two right handed neutrinos with the family quantum numbers of the upper8four families, causing the break of the starting symmetry SO (13 ,
1) into SO (7 , × SU (3) × U (1),is presented in Table V. Appendix C: Expressions for scalar fields in term of ω s (cid:48) s (cid:48)(cid:48) s and ˜ omega abs 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, arepresented in the second line of Eq. (27). These scalar fields are included in the covariant derivativesas − S s (cid:48) s ” ω s (cid:48) s ” s − ˜ S ab ˜ ω abs , s ∈ (7 , a, b ), ∈ (0 , . . . , , (5 , . . . , ω abs (they contribute to mass matrices of quarks and leptons and tomasses of the heavy bosons), if taking into account Eqs. (B4, B5, B7), (cid:88) a,b −
12 ˜ S ab ˜ ω abs = − ( (cid:126) ˜ τ ˜1 (cid:126) ˜ A ˜1 s + (cid:126) ˜ N ˜ L (cid:126) ˜ A ˜ N ˜ L s + (cid:126) ˜ τ ˜2 (cid:126) ˜ A ˜2 s + (cid:126) ˜ N ˜ R (cid:126) ˜ A ˜ N ˜ R s ) ,(cid:126) ˜ A ˜1 s = (˜ ω s − ˜ ω s , ˜ ω s + ˜ ω s , ˜ ω s − ˜ ω s ) ,(cid:126) ˜ A ˜ N ˜ L s = (˜ ω s + i ˜ ω s , ˜ ω s + i ˜ ω s , ˜ ω s + i ˜ ω s ) ,(cid:126) ˜ A ˜2 s = (˜ ω s + ˜ ω s , ˜ ω s − ˜ ω s , ˜ ω s + ˜ ω s ) ,(cid:126) ˜ A ˜ N ˜ R s = (˜ ω s − i ˜ ω s , ˜ ω s − i ˜ ω s , ˜ ω s − i ˜ ω s ) , ( s ∈ (7 , . (C1)Scalars, expressed in terms of ω abc (contributing as well to the mass matrices of quarks and leptonsand to masses of the heavy bosons) follow, if using Eqs. (B2, B3, B7) (cid:88) s (cid:48) ,s (cid:48)(cid:48) − S s (cid:48) s ” ω s (cid:48) s ” s = − ( g τ A s + g τ A s + g τ A s ) ,g τ A s + g τ A s + g τ A s = g Q QA Qs + g Q (cid:48) Q (cid:48) A Q (cid:48) s + g Y (cid:48) Y (cid:48) A Y (cid:48) s ,A s = − ( ω s + ω
11 12 s + ω
13 14 s ) ,A s = ( ω s − ω s ) , A s = ( ω s + ω s ) ,A Qs = sin ϑ A s + cos ϑ A Ys , A Q (cid:48) s = cos ϑ A s − sin ϑ A Ys ,A Y (cid:48) s = cos ϑ A s − sin ϑ A s , ( s ∈ (7 , . (C2)Scalar fields from Eq. (C1) interact with quarks and leptons and antiquarks and antileptons throughthe family quantum numbers, while those from Eq. (C2) interact through the family members9quantum numbers. In Eq. (C2) the coupling constants are explicitly written in order to see theanalogy 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 ( (cid:126)A m ), the SU (2) II charge ( (cid:126)A m ), the weak SU (2) I charge( (cid:126)A m ) and the U (1) charge originating in SO (6) ( (cid:126)A m ), can be found by taking into account Eqs. (B2,B3). Equivalently one finds the vector gauge fields in the ”tilde” sector, or one just uses theexpressions from Eqs. (C2, C1), if replacing the scalar index s with the vector index m .The expression for (cid:80) tab γ t ˜ S ab ˜ ω abt , used in Eq. (45) ( ˜ S ab are the infinitesimal generators ofeither (cid:103) SO (3 ,
1) or (cid:103) SO (4), while ˜ ω abt belong to the corresponding gauge fields with t = (9 , . . . , (cid:88) abt γ t
12 ˜ S ab ˜ ω abt = (cid:88) + − tt (cid:48) ab tt (cid:48) ( ± ○ ) 12 ˜ S ab ˜ ω ab tt (cid:48) ( ± ○ ) = (cid:88) + − tt (cid:48) tt (cid:48) ( ± ○ ) { ˜ τ ˜ A tt (cid:48) ( ± ○ ) + ˜ τ − ˜ A − tt (cid:48) ( ± ○ ) + ˜ τ ˜ A tt (cid:48) ( ± ○ ) +˜ τ ˜ A tt (cid:48) ( ± ○ ) + ˜ τ − ˜ A − tt (cid:48) ( ± ○ ) + ˜ τ ˜ A tt (cid:48) ( ± ○ ) +˜ N + R ˜ A N R + tt (cid:48) ( ± ○ ) + ˜ N − R ˜ A N R − tt (cid:48) ( ± ○ ) + ˜ N R ˜ A N R tt (cid:48) ( ± ○ ) +˜ N + L ˜ A N L + tt (cid:48) ( ± ○ ) + ˜ N − L ˜ A N L − tt (cid:48) ( ± ○ ) + ˜ N L ˜ A N L tt (cid:48) ( ± ○ ) } , ˜ A N R ± tt (cid:48) ( ± ○ ) = (˜ ω tt (cid:48) ( ± ○ ) − i ˜ ω tt (cid:48) ( ± ○ ) ) ∓ i (˜ ω tt (cid:48) ( ± ○ ) − i ˜ ω tt (cid:48) ( ± ○ ) ) , ˜ A N R tt (cid:48) ( ± ○ ) = (˜ ω tt (cid:48) ( ± ○ ) − i ˜ ω tt (cid:48) ( ± ○ ) ) , ˜ A N L ± tt (cid:48) ( ± ○ ) = (˜ ω tt (cid:48) ( ± ○ ) + i ˜ ω tt (cid:48) ( ± ○ ) ) ∓ i (˜ ω tt (cid:48) ( ± ○ ) + i ˜ ω tt (cid:48) ( ± ○ ) ) , ˜ A N R tt (cid:48) ( ± ○ ) = (˜ ω tt (cid:48) ( ± ○ ) + i ˜ ω tt (cid:48) ( ± ○ ) ) . (C3)The term (cid:80) tt (cid:48) t (cid:48)(cid:48) γ t S t (cid:48) t ” ω t (cid:48) t ” t in Eq. (27) can be rewritten with respect to the generators S t (cid:48) t ” and the corresponding gauge fields ω s (cid:48) s ” t as one colour octet scalar field and one U (1) II singletscalar field (Eq. B3) (cid:88) tt (cid:48)(cid:48) t (cid:48)(cid:48)(cid:48) γ t S t ” t (cid:48) ” ω t ” t (cid:48) ” t = (cid:88) + , − (cid:88) ( t t (cid:48) ) tt (cid:48) ( ± ○ ) { (cid:126)τ · (cid:126)A tt (cid:48) ( ± ○ ) + τ · A tt (cid:48) ( ± ○ ) } , ( t t (cid:48) ) ∈ ((9 10) ,
11 12) ,
13 14)) . (C4) Acknowledgments
The author thanks Department of Physics, FMF, University of Ljubljana, Society of Mathemati-cians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory, and Matjaˇz Breskvar of Beyond Semiconductor for donations, in particular for sponsoring0the annual workshops entitled ”What comes beyond the standard models” at Bled. [1] N. Mankoˇc Borˇstnik, ”Spin connection as a superpartner of a vielbein”,
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New J. of Phys. (2008) 093002, hep-ph/0606159, hep/ph-07082846.[56] G. Bregar, N.S. Mankoˇc Borˇstnik, ”Does dark matter consist of baryons of new stable family quarks?”, Phys. Rev. D , 083534 (2009), 1-16[57] G. Bregar, N.S. Mankoˇc Borˇstnik, ”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. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman, DMFA Zaloˇzniˇstvo,Ljubljana December 2013, p. 31-51, http://arxiv.org/abs/1212.4055.[58] G. Bregar, N.S. Mankoˇc Borˇstnik, ”The new experimental data for the quarks mixing matrix are inbetter agreement with the spin-charge-family theory predictions”, Proceedings to the 17 t h Workshop”What comes beyond the standard models”, Bled, 20-28 of July, 2014, Ed. N.S. Mankoˇc Borˇstnik, H.B.Nielsen, D. Lukman, DMFA Zaloˇzniˇstvo, Ljubljana December 2014, p.20-45 [ arXiv:1502.06786v1][arXiv:1412.5866].[59] B. Belfatto, R. Beradze, Z. Berezhiani, ”The CKM unitarity problem: A trace of new physics at theTeV scale?”, [arXiv:1906.02714].[60] Review of Particle, Particle Data Group , P A Zyla, R M Barnett, J Beringer, O Dahl, D A Dwyer, D EGroom, C -J Lin, K S Lugovsky, E Pianori ...., Author Notes, Progress of Theoretical and ExperimentalPhysics, Volume 2020, Issue 8, August 2020, 083C01, https://doi.org/10.1093/ptep/ptaa104, 14 August2020.[61] A. Hernandez-Galeana and N.S. Mankoˇc Borˇstnik, ””The symmetry of 4 × spin-charge-family theory — SU (2) × SU (2) × U (1) — remains in all loop corrections”, Proceed-ings to the 21 st Workshop ”What comes beyond the standard models”, 23 of June - 1 of July, 2017,Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman, DMFA Zaloˇzniˇstvo, Ljubljana, December 2018[arXiv:1902.02691, arXiv:1902.10628].[62] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, ”Do the present experiments exclude the existence of the fourthfamily members?”, Proceedings to the 19 t h Workshop ”What comes beyond the standard models”,Bled, 11-19 of July, 2016, Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman, DMFA Zaloˇzniˇstvo,Ljubljana December 2016, p.128-146 [arXiv:1703.09699].[63] A. Ali in discussions and in private communication at the Singapore Conference on New Physics at theLarge Hadron Collider, 29 February - 4 March 2016.[64] M. Neubert, in duscussions at the Singapore Conference on New Physics at the Large Hadron Collider,29 February - 4 March 2016.[65] N.S. Mankoˇc Borˇstnik, M. Rosina, ”Are superheavy stable quark clusters viable candidates for the darkmatter?”, International Journal of Modern Physics D (IJMPD) (No. 13) (2015) 1545003.[66] D. Hestenes, G. Sobcyk, ”Clifford algebra to geometric calculus”, Reidel 1984. [67] P. Lounesto, P. Clifford algebras and spinors, Cambridge Univ. Press.2001.[68] M. Pavˇsiˇc, ”Quantized fields ´a la Clifford and unification” [arXiv:1707.05695].[69] N.S. Mankoˇc Borˇstnik and H.B.F. Nielsen, ”Discrete symmetries in the Kaluza-Klein theories”, JHEP spinortechnique ”, .Int. J Mod. Phys. A 29 , 1450124 (2014).[71] P.A.M. Dirac
Proc. Roy. Soc. (London) , A 117 (1928) 610.[72] D. Lukman, N.S. Mankoˇc Borˇstnik and H.B. Nielsen, ”An effective two dimensionality cases bring anew hope to the Kaluza-Klein-like theories”,
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J. Phys. A: Math. Theor. d = (1+5) with a zweibeinand two kinds of spin connection fields on an almost S ”, Proceedings to the 15 th Workshop ”Whatcomes beyond the standard models”, Bled, 9-19 of July, 2012, Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen,D. Lukman, DMFA Zaloˇzniˇstvo, Ljubljana December 2012, 157-166, [arXiv:1302.4305].[75] A.Borˇstnik Braˇciˇc, N. Mankoˇc Borˇstnik,“The approach Unifying Spins and Charges and Its Predic-tions“, Proceedings to the Euroconference on Symmetries Beyond the Standard Model”, Portoroˇz,July 12 - 17, 2003, Ed. by Norma Mankoˇc Borˇstnik, Holger Bech Nielsen, Colin Froggatt, DraganLukman, DMFA Zaloˇzniˇstvo, Ljubljana December 2003, p. 31-57, [arXiv:hep-ph/0401043, arXiv:hep-ph/0401055].[76] A. Borˇstnik Braˇciˇc, N. S. Mankoˇc Borˇstnik, ”On the origin of families of fermions and their massmatrices”, hep-ph/0512062, Phys Rev.
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B 644 (2007) 198-202 [arXiv:hep-th/0608006].[81] Correspondingly the spin-charge-family theory shares with the Kaluza-Klein like theories their weakpoints, at least: a. Not yet solved the quantization problem of the gravitational field. b. The spon-taneous break of the starting symmetry, which would at low energies manifest the observed almostmassless fermions [32]. Concerning this second point we proved on the toy model of d = (5 + 1) thatthe break of symmetry can lead to (almost) massless fermions [72–74].[82] f αa are inverted vielbeins to e aα with the properties e aα f αb = δ ab , e aα f βa = δ βα , E = det( e aα ). Latin in- dices 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 indicatea general index ( a, b, c, .. and α, β, γ, .. ), from the middle of both the alphabets the observed dimen-sions 0 , , , m, n, .. and µ, ν, .. ), indexes from the bottom of the alphabets indicate the compactifieddimensions ( s, t, .. and σ, τ, .. ). We assume the signature η ab = diag { , − , − , · · · , − } .[83] A toy model [72 ? ] was studied in d = (5 + 1) with the same action as in Eq. (1). The break from d = (5 + 1) to d = (3 + 1) × an almost S was studied. For a particular choice of vielbeins and for a classof spin connection fields the manifold M (5+1) breaks into M (3+1) times an almost S , while 2 ((3+1) / − families remain massless and mass protected. Equivalent assumption, although not yet proved how doesit really work, is made in the d = (13 + 1) case. This study is in progress quite some time.[84] The ”basis vectors” with an even number of nilpotents have in even dimensional spaces the propertythat there is one member of each representation which is self adjoint, the one which is the product ofonly projectors.[85] Eq. (14) requires that ˜ γ a ( a + a b γ b + a bc γ b γ c + · · · ) = ( ia γ a +( − i ) a b γ b γ a + ia bc γ b γ c γ a + · · · ), what meansthat the relation ˜ γ a a = ia γ a is only one of the relations included into Eq. (14). Another relation, forexample, is ˜ γ a γ a = ( − i ) γ a γ a = − iη aa . One correspondingly finds { ˜ γ a , ˜ γ b } + = 2 η ab = ˜ γ a ˜ γ b + ˜ γ b ˜ γ a =˜ γ a iγ b + ˜ γ b iγ a = iγ b ( − i ) γ a + iγ a ( − i ) γ b = 2 η ab . { ˜ γ a , γ b } + = 0 = ˜ γ a γ b + γ b ˜ γ a = γ b ( − i ) γ a + γ b iγ a = 0. { ˜ γ a , γ a } + = 0 = ˜ γ a γ a + γ a ˜ γ a = γ a ( − i ) γ a + γ a iγ a = 0.[86] Let me show how does the Grassmann space loose the Hermitian conjugated partners to θ a ’s, while θ a ’s become equal to γ a ’s. My statement that Eq. (14) requires θ a = γ a and ∂∂θ a = 0 can be provedas follows. There are only two requirements which have to be analyzed in details, ˜ γ a ( α ) = iαγ a , α isany constant and ˜ γ a γ a = − iγ a γ a . Both relations apply on | ψ oc > : In the Grassmann case the vacuumstate is identity | > , while in the Clifford algebra the vacuum state is the sum of even products of γ a ’s as seen in Eq. (10), which applies on identity. Let us express γ a ’s, ˜ γ a ’s and | ψ oc > in terms of θ a ’sand ∂∂θ a as written in Eq. (3). Eq. (3) requires that γ a = ( θ a + ∂∂θ a ), ˜ γ a = i ( θ a − ∂∂θ a ). Let us put theseexpressions into Eq. (14) and let | ψ oc > be expressed in terms of θ a ’s. Taking into account that θ a ’sapplying on identity gives θ a ’s back while ∂∂θ a applying on identity gives zero, it follows that | ψ oc > = a + a ab θ a θ b + · · · , the rest of expansion is irrelevant for the proof. The constant α can be skipped,since constants appear in | ψ oc > = a + a ab θ a θ b + · · · anyhow. The first relation [ ˜ γ a = iγ a ] | ψ o c > ,expressed with θ a ’s and ∂∂θ a , reads: i ( θ a − ∂∂θ a )( a + a ab θ a θ b + · · · ) = i ( θ a + ∂∂θ a )( a + a ab θ a θ b + · · · ).From this we find iθ a a = ia θ a and i ( − ∂∂θ a ) a ab θ a θ b = i ∂∂θ a a b θ a θ b , requiring that ∂∂θ a = 0 (as anoperator Hermitian conjugated to θ a for ∀ a ). These relation requires that the derivatives should notexist any longer, if the relation should hold. Then it follows from γ a = ( θ a + ∂∂θ a ) that θ a = γ a , whichmeans that the Grassmann space has no meaning any longer, the only remaining space is the space ofthe Clifford products of odd number of γ a ’s, on which γ a ’s and ˜ γ a ’s operate: [ ˜ γ a = iγ a ] | ψ o c > and[ ˜ γ a γ b = − iγ b γ a ] | ψ oc > . This complites the proof .[87] A toy model [72, 73] was studied in d = (5 + 1) with the same action as in Eq. (1). The break from d = (5 + 1) to d = (3 + 1) × an almost S was studied. For a particular choice of vielbeins and for a classof spin connection fields the manifold M (5+1) breaks into M (3+1) times an almost S , while 2 ((3+1) / − families remain massless and mass protected. Equivalent assumption, although not yet proved how doesit really work, is made in the d = (13 + 1) case. This study is in progress.[88] 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 standardmodel with no need to connect ”by hand” the handedness and charges of fermions and antifermions.[89] The expression for the Lagrange density of Eq. (42) is only estimated, more or less guessed, I have noestimate yet for the constants.[90] Although carrying the colour charge in one of the triplet or antitriplet states, these fields can not beinterpreted 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 notthose required by the supersymmetric partners to the family members. TABLE II: Quantum numbers of the scalar gauge fields carrying the space index t = (9 , , · · · , SU (2) charges, ( τ and (cid:126)τ ),and the two (cid:103) SU (2) charges, ( (cid:126) ˜ τ and (cid:126) ˜ τ ), triplets (that is in the adjoint representations of the correspondinggroups), and they all carry twice the ”fermion” number ( τ ) of the quarks. The quantum numbers of thetwo vector gauge fields, the colour and the U (1) II ones, are added. field prop. τ τ τ ( τ , τ ) Y Q ˜ τ ˜ τ ˜ τ ˜ N L ˜ N R A ± ± ○ ) scalar ∓ ○ ± ± ○ , ± ○ √ ) ∓ ○ ∓ ○ + ∓ A
139 10( ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ A ±
11 12( ± ○ ) scalar ∓ ○ ∓ ∓ ○ , ± ○ √ ) ∓ ○ ∓ ○ + ∓ A ± ○ ) scalar ∓ ○ ∓ ○ , ± ○ √ ) ∓ ○ ∓ ○ A ±
13 14( ± ○ ) scalar ∓ ○ ∓ , ∓ ○ √ ) ∓ ○ ∓ ○ + ∓ A ± ○ ) scalar ∓ ○ , ∓ ○ √ ) ∓ ○ ∓ ○ A ± ± ○ ) scalar ∓ ○ ± ± ○ , ± ○ √ ) ∓ ○ + ∓ ∓ ○ + ∓ A
239 10( ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ · · · ˜ A ± ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ ± A ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ · · · ˜ A ± ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ ± A ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ · · · ˜ A NL ± ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ ± A NL ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ · · · ˜ A NR ± ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ ± A NR ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ · · · A i ± ○ ) scalar ∓ ○ ± ± ○ , ± ○ √ ) ∓ ○ ∓ ○ · · · A
49 10( ± ○ ) scalar ∓ ○ ± ○ , ± ○ √ ) ∓ ○ ∓ ○ · · · (cid:126)A m vector 0 0 0 octet 0 0 0 0 0 0 0 A m vector 0 0 0 0 0 0 0 0 0 0 0 ˜ τ ˜ τ ˜ N L ˜ N R ˜ τ I ˆ u c † R (+ i ) [+] | [+] (+) || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R (+ i ) [+] | [+] (+) || (+)
11 12 (+)
13 14 (+) − − − I ˆ u c † R [+ i ] (+) | [+] (+) || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R [+ i ] (+) | [+] (+) || (+)
11 12 (+)
13 14 (+) − − I ˆ u c † R (+ i ) [+] | (+) [+] || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R (+ i ) [+] | (+) [+] || (+)
11 12 (+)
13 14 (+) − − I ˆ u c † R [+ i ] (+) | (+) [+] || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R [+ i ] (+) | (+) [+] || (+)
11 12 (+)
13 14 (+) − II ˆ u c † R [+ i ] [+] | [+] [+] || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R [+ i ] [+] | [+] [+] || (+)
11 12 (+)
13 14 (+) 0 − − − II ˆ u c † R (+ i ) (+) | [+] [+] || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R (+ i ) (+) | [+] [+] || (+)
11 12 (+)
13 14 (+) 0 − − II ˆ u c † R [+ i ] [+] | (+) (+) || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R [+ i ] [+] | (+) (+) || (+)
11 12 (+)
13 14 (+) 0 − − II ˆ u c † R (+ i ) (+) | (+) (+) || (+)
11 12 [ − ]
13 14 [ − ] ˆ ν † R (+ i ) (+) | (+) (+) || (+)
11 12 (+)
13 14 (+) 0 − TABLE IV: Eight families of creation operators of ˆ u c † R — the right handed u -quark with spin and thecolour charge ( τ = 1 / τ = 1 / (2 √ ν † R — of spin , appearing in the 25 th line of Table III — are presented in the leftand in the right column, respectively. Table is taken from [9]. Families belong to two groups of four families,one ( I ) is a doublet with respect to ( (cid:126) ˜ N L and (cid:126) ˜ τ (1) ) and a singlet with respect to ( (cid:126) ˜ N R and (cid:126) ˜ τ (2) ), the other( II ) is a singlet with respect to ( (cid:126) ˜ N L and (cid:126) ˜ τ (1) ) and a doublet with respect to ( (cid:126) ˜ N R and (cid:126) ˜ τ (2) ), Eq. (B1). Allthe families follow from the starting one by the application of the operators ( ˜ N ± R,L , ˜ τ (2 , ± ), Eq. (B8). Thegenerators ( N ± R,L , τ (2 , ± ), Eq. (B8), transform ˆ u † R to all the members of one family of the same colour.The same generators transform equivalently the right handed neutrino ˆ ν † R to all the colourless members ofthe same family. state S S τ τ τ Y Q ˜ τ ˜ τ ˜ τ ˜ Y ˜ Q ˜ N L ˜ N R ( | ν VIII1R > | ν VIII2R > ) 0 0 0 1 − − | ν V III R > | e V III R > ) 0 0 0 0 − − − − | e V III R > | e V III R > ) 0 0 0 − − − − − ν R , with the quantum numbers of the V III th family, coupled to spin zero and belonging to a triplet with respect to the generators τ i , is presented,together with its two partners. The condensate carries (cid:126)τ = 0, τ = 1, τ = − Q = 0 = Y . Thetriplet carries ˜ τ = −