gauge group of the standard model in - core · 2017. 4. 11. · vol. 27 (2017) gauge group of the...

Adv. Appl. Clifford Algebras 27 (2017), 279–290c© 2015 The Author(s).This article is published with open access at Springerlink.com0188-7009/010279-12published online May 20, 2015DOI 10.1007/s00006-015-0566-5

Advances inApplied Clifford Algebras

Gauge Group of the Standard Model in Cl1,5

Claude Daviau

Abstract. Describing a wave with spin 1/2, the Dirac equation is forminvariant under SL(2,C), subgroup of Cl∗3 = GL(2,C) which is the truegroup of form invariance of the Dirac equation. Firstly we use the Cl3 al-gebra to read all features of the Dirac equation for a wave with spin 1/2.We extend this to electromagnetic laws. Next we get the gauge group ofelectro-weak interactions, first in the leptonic case, electron+neutrino,next in the quark case. The complete wave for all objects of the firstgeneration uses the Clifford algebra Cl1,5. The gauge group is then en-larged into a U(1) × SU(2) × SU(3) Lie group. We consolidate boththe standard model and the use of Clifford algebras, true mathematicalframe of quantum physics.

Mathematics Subject Classification. 15A66; 35Q41; 81T13; 83E15.

Keywords. Invariance group, Dirac equation, Electromagnetism, Weakinteractions, Strong interactions, Standard model, Clifford algebra.

1. Form Invariance of the Dirac Equation

The standard model uses fermions and bosons. All fermions are describedwith a Dirac equation. The Dirac wave ψe of the electron is made of twoPauli waves

ψe =(

ξe

ηe

); ξe =

(ξ1e

ξ2e

); ηe =

(η1e

η2e

)(1.1)

where ξ1e, ξ2e, η1e, η2e are four functions of space and time with value in thecomplex field. In its first complex frame the Dirac equation reads

0 = [γμ(∂μ + iqAμ) + im]ψe; q =e

�c; m =

m0c

�(1.2)

where γμ, μ = 0, 1, 2, 3 are four complex matrices. Relativistic theory uses:

γ0 = γ0 =(

0 II 0

); γj = −γj =

(0 σj

−σj 0

);

I = σ0 = σ0 =(

1 00 1

)(1.3)

All my thanks go to Jacques Bertrand who helped me to develop the present work.

http://crossmark.crossref.org/dialog/?doi=10.1007/s00006-015-0566-5&domain=pdf

280 C. Daviau Adv. Appl. Clifford Algebras

σ1 = −σ1 =(

0 11 0

); σ2 = −σ2 =

(0 −ii 0

);

σ3 = −σ3 =(

1 00 −1

). (1.4)

The electro-weak theory uses also

γ5 =(

I 00 −I

); ψR = ψe

12(1 + γ5) =

(ξe

0

);

ψL = ψe12(1 − γ5) =

(0ηe

)(1.5)

To get the relativistic invariance the Dirac theory uses x = xμσμ and x′

satisfying

x =(

x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

); x′ = x′μσμ =

(x′0 + x′3 x′1 − ix′2

x′1 + ix′2 x′0 − x′3

)

(1.6)This is equivalent to identify the space algebra Cl3 to the matrix algebraM2(C)1. Noting a∗ the conjugate of a and with

M =(

a bc d

); M† =

(a∗ c∗

b∗ d∗

); M =

(d −b

−c a

); M = M

†(1.7)

det(M) = MM = MM = ad − bc = reiθ (1.8)

the Dirac theory associates to M the R transformation satisfying

x′ = R(x) = MxM† (1.9)

because we get, for any M :

(x′0)2 − (x′1)2 − (x′2)2 − (x′3)2 = det(x′) = det(M) det(x) det(M†)

= reiθ det(x)re−iθ = r2[(x0)2 − (x1)2 − (x2)2 − (x3)2]. (1.10)

Then R is a Lorentz dilation, product of a Lorentz rotation and an homothetywith ratio r. Moreover with

x′μ = Rμν xν ; N =

(M 00 M

); N =

(M 00 M†

)(1.11)

we get [5,10]:

R00 =

12(aa∗ + bb∗ + cc∗ + dd∗) > 0 (1.12)

det(Rμν ) = r4 (1.13)

Rμν γν = NγμN, μ = 0, 1, 2, 3. (1.14)

The form invariance of the Dirac equation comes from

0 = [γν(∂ν + iqAν) + im]ψe = [γνRμν (∂′

μ + iqA′μ) + im]ψe

= [Nγμ(∂′μ + iqA′

μ)N + im]ψe. (1.15)

1 This identification is not mine, it has been made 88 years ago. It is necessary to get

the form invariance of the Dirac equation. This form invariance is incorrectly known as

“relativistic invariance”. I enlarged the invariance group into Cl∗3 .

Vol. 27 (2017) Gauge Group of the Standard Model in Cl1,5 281

If det(M) = 1, N satisfies

MM = MM = 1; M = M−1; N = N−1 (1.16)

where N is the reverse of N . This gives

0 = [γμ(∂μ + iqAμ) + im]ψe = N−1[γμ(∂′μ + iqA′

μ) + im]Nψe. (1.17)

The Dirac theory lets then

ψ′e(x

′) = Nψe(x) (1.18)

and since

0 = [γμ(∂μ + iqAμ) + im]ψe ⇐⇒ 0 = γμ(∂′μ + iqA′

μ) + im]ψ′e (1.19)

the Dirac equation is said “form invariant”. Ten years ago, I noticed thatrelations (1.12), (1.13) and (1.14) are true even if det(M) �= 1. Then thefundamental group of form invariance of the Dirac wave is the group of theM , usually named GL(2, C). This group is also the multiplicative group Cl∗3of the invertible elements in Cl3.

2. Dirac Equation and Electromagnetism in Cl3

We have previously [3,5,10] let

φe =√

2(ξe − iσ2η∗e) =

√2

(ξ1e −η∗

2e

ξ2e η∗1e

)(2.1)

which implies

det(φe) = 2(ξ1eη∗1e + ξ2eη

∗2e) = Ω1 + iΩ2 = ρeiβ (2.2)

The φe wave is then a function of space-time into Cl3 = M2(C). We get

φe =√

2(ηe − iσ2ξ∗e ) =

√2

(η1e −ξ∗

2e

η2e ξ∗1e

); φe = φ†

e (2.3)

φeφe = φeφe = det(φe) = ρeiβ (2.4)

where β is the Yvon-Takabayasi angle. φ†e is the reverse of φe. (1.18) reads(

ξ′e

η′e

)=

(M 00 M

) (ξe

ηe

); ξ′

e = Mξe ; η′e = Mηe. (2.5)

It happens that we get, with any M and any η:

(−iσ2)η′∗ = (−iσ2)M∗η∗ = M(−iσ2)η∗ (2.6)

The link that I made in (2.1) between φe and the Weyl spinors ξe, ηe istherefore invariant under Cl∗3: if

φ′e(x

′) =√

2(ξ′e (−iσ2)η′

e∗) =

√2

(ξ′1e −η′

2e∗

ξ′2e η′

1e∗

)(2.7)

we simply getφ′

e = Mφe ; φ′e = Mφe. (2.8)

We have explained [5,10] how the Dirac equation reads in Cl3:

∇φeσ21 + qAφe + mφe = 0 ; ∇ = σμ∂μ ; A = σμAμ ; σ21 = σ2σ1. (2.9)


Multiplying on the left by φe, I got [5,6]:

φe(∇φe)σ21 + φeqAφe + mρeiβ = 0. (2.10)

The first term is form invariant because, for any M in Cl3, with

∇′ = σμ∂′μ (2.11)

I got the following general relation [3]:

∇ = M∇′M (2.12)

which gives with (2.8):

φe(∇φe)σ21 = φe M∇′Mφeσ21 = φ′e(∇′φ′

e)σ21. (2.13)

Next the form invariance of φeqAφe is necessary to satisfy both the forminvariance and the electric gauge invariance of the Dirac equation. This means

φeqAφe = φe Mq′A′Mφe

qA = Mq′A′M. (2.14)

Then qA which transforms like ∇ is named a “covariant vector” (in space-time), while vectors transforming like x, for instance J = φeφ

†e, are named

“contravariant”. This is now a physical distinction because, if det(M) �= 1then M �= M−1. The non-linear homogeneous equation studied in my thesis[2] reads in Cl3:

∇φeσ21 + qAφe + me−iβφe = 0 (2.15)and is then equivalent to the form invariant equation

φe(∇φe)σ21 + φeqAφe + mρ = 0. (2.16)

This equation is the starting point to get the gauge group of the standardmodel in the frame of Clifford algebra. Two of the eight numeric equationsequivalent to the form invariant equation are remarkable and well known:the law of conservation of the current of probability ∂μJμ = 0 and, still moreimportant, the scalar part of (2.16) or (2.10) reads simply: L = 0 where L isthe Lagrangian density of the Dirac equation. This Lagrangian density andthe whole equation (2.16) are form invariant under Cl∗3 because (2.8) implies

ρ′eiβ′= det(φ′

e) = det(Mφe) = det(M) det(φe) = rρei(β+θ) (2.17)ρ′ = rρ ; β′ = β + θ mod 2π. (2.18)

The form invariance of the wave equation is satisfied if and only if the massterm satisfies

mρ = m′ρ′ = m′rρ; m = rm′. (2.19)

I replace the L↑+ group (but actually the SL(2, C) group) by Cl∗3 = GL(2, C).

Therefore m and ρ are no more invariant. Only the mρ product is invariant,this is linked to the existence of the Planck constant [10].

The electromagnetism uses

F = E + i H, A = A0 + A; B = B0 + B (2.20)

j = j0 + j; k = k0 + k (2.21)


where E is the electric field, H is the magnetic field, A the space-time vectorpotential, B the magnetic potential, j the electric density of charge and cur-rent, k the magnetic density of charge and current. Laws of electromagnetismin the void with magnetic monopoles read (See [10], chap. 4)

F = ∇A + iB; ∇F =4π

cj + ik. (2.22)

The electromagnetic field F satisfies [6]:

F ′(x′) = MF (x)M−1 (2.23)

A′(x′) = MA(x)M† ; B′(x′) = MB(x)M† (2.24)

j(x) = Mj′(x′)M ; k(x) = Mk′(x′)M (2.25)

The contravariance (2.24) means that potentials move with sources. The co-variance of qA, j and k is compatible with all laws of electromagnetism andrelativistic mechanics [5,6,10]. The transformation (2.23) explains by itselfwhy only the SL(2, C) part of Cl∗3 was previously seen. The P rotor whichplays a central role in the Hestenes’ work [11] and in the Boudet’s work [1],actually an element of SL(2, C), is defined such as

M =√

reiθ/2P. (2.26)

This gives

M =√

reiθ/2P =√

reiθ/2P−1 (2.27)

M−1 =1√re−iθ/2P (2.28)

F ′ =√

reiθ/2PF1√re−iθ/2P = PFP (2.29)

and F transforms as if r = 1 and θ = 0. The electromagnetic field (andmore generally all gauge fields) transforms in such a way that we see onlythe relativistic Lorentz rotation induced by the P term. Velocities are alsoindependent from r. Then under the dilation R, with ratio r induced by anyM in Cl∗3, I got [5]:

e′ = r2e; �′ = r4�; m′

0 = r3m0; m′ = r−1m. (2.30)

Electric charge, proper mass and Planck “constant” are changed in the dila-tion R induced by a M in Cl∗3 if r �= 1. The dilation is the composition of theLorentz rotation induced by P and an homothety with ratio r, in any order.We evidently get the results of restricted relativity if r = 1.

3. Electro-weak Interactions

The electro-weak theory needs three spinorial waves in the electron-neutrinocase: the right ξe and the left ηe of the electron and the left spinor ηn of theelectronic neutrino. The form invariance of the Dirac theory imposes to usea wave Ψl satisfying


Ψl =(

φe φn

φn φe

); φe =

√2

(ξ1e −η∗

2e

ξ2e η∗1e

); φn =

√2

(0 −η∗

2n

0 η∗1n

)(3.1)

φe =√

2(

η1e −ξ∗2e

η2e ξ∗1e

); φn =

√2

(η1n 0η2n 0

); ηn =

(η1n

η2n

). (3.2)

The wave is a function of space and time with value into the space-timealgebra Cl1,3. The standard model uses only a left wave for the neutrino, thismay be seen in (3.1)2. I use the old matrix representation (1.3). Under thedilation R with ratio r induced by M we have

ξ′ = Mξ; η′ = Mη; η′n = Mηn; φ′

e = Mφe; φ′n = Mφn (3.3)

Ψ′l =

(φ′

e φ′n

φ′n φ′

e

)=

(M 00 M

)(φe φn

φn φe

)= NΨl (3.4)

The form (3.1) of the wave is compatible with the charge conjugation usedin the standard model: the positron wave ψe satisfies

ψe = iγ2ψ∗e ⇔ φe = φeσ1 (3.5)

Then Ψl contains the electron wave φe, the neutrino wave φn and also thepositron wave φe and the antineutrino wave φn:

Ψl =(

φe φn

φnσ1 φeσ1

); φe =

√2

(ξ1e −η∗

2e

ξ2e η∗1e

); φn =

√2

(ξ1n 0ξ2n 0

)(3.6)

And the antineutrino has consequently only a right wave. With:

a1 = det(φe) = φeφe = 2(ξ1eη∗1e + ξ2eη

∗2e) (3.7)

a2 = 2(ξ1eη∗1n + ξ2eη

∗2n) = 2(η∗

2eη∗1n − η∗

1eη∗2n) (3.8)

a3 = 2(ξ1eη∗1n + ξ2eη

∗2n) (3.9)

Ψl satisfies:det(Ψl) = a1a

∗1 + a2a

∗2. (3.10)

and the Ψl(x) matrix is usually invertible. To get the gauge group of electro-weak interactions I used [6] two projectors P± and four operators Pμ, whereμ = 0, 1, 2, 3, satisfying:

P±(Ψ) =12(Ψ ± iΨγ21) ; i = γ0γ1γ2γ3 ; γ21 = γ2γ1 (3.11)

P0(Ψ) = Ψγ21 + P−(Ψ)i (3.12)P1(Ψ) = P+(Ψ)γ3i (3.13)P2(Ψ) = P+(Ψ)γ3 (3.14)P3(Ψ) = P+(Ψ)(−i) (3.15)

These operators generate the Lie algebra of U(1) × SU(2). The covariantderivative of the Weinberg-Salam model:

Dμ = ∂μ − ig1Y

2Bμ + ig2TjW

jμ (3.16)

2 Then only a 12-dimensional subspace of Cl1,3 is used. The reward is the remarkable iden-tity det(Ψl) = |a1|2 + |a2|2. The isomorphism between Cl3 and M2(C) is an isomorphism

of real algebras. All dimensions of linear spaces are here dimensions on R, not dimensionson C. Quantum physics only needs real Clifford algebras.


where Y is the weak hypercharge (YL = −1, YR = −2 for the electron), hasa very simple translation in the Cl1,3 frame:

D = ∂∂∂ +g12BP0 +

g22

(W1P1 + W2P2 + W3P3) ; D = ND′N (3.17)

D = γμDμ; ∂∂∂ = γμ∂μ; B = γμBμ; Wj = γμW jμ (3.18)

Because we get from these definitions:

Dμξe = ∂μξe + ig1Bμξe (3.19)

Dμηe = ∂μηe + ig12

Bμηe − ig22

[(W 1μ + iW 2

μ)ηn − W 3μηe] (3.20)

Dμηn = ∂μηn + ig12

Bμηn − ig22

[(W 1μ − iW 2

μ)ηe + W 3μηn] (3.21)

which is equivalent to (3.16). The Weinberg–Salam θW angle satisfies

B + iW 3 = eiθW (A + iZ0). (3.22)

The U(1) × SU(2) gauge group is obtained by exponentiation. If aμ are fourreal parameters we use the gauge transformation

Ψ′l = [exp(aμPμ)](Ψl). (3.23)

The wave Eq. [8] reads

DΨlγ012 + mρ1χl = 0; γ012 = γ0γ1γ2 (3.24)

where

ρ1 =√

a1a∗1 + a2a∗

2 + a3a∗3 (3.25)

χl =1ρ21

(a∗1φe + a∗

2φnσ1 + a∗3φn −a∗

2φeLσ1 + a∗3φeR

a2φeLσ1 + a3φeR a1φe − a2φnσ1 + a3φn

)(3.26)

φeR = φe1 + σ3

2; φeL = φe

1 − σ3

2. (3.27)

This wave equation is equivalent to the invariant equation:

Ψl(DΨl)γ012 + mρ1Ψlχl = 0; Ψl =(

φe φ†n

φn φ†e

). (3.28)

The form invariance under Cl∗3 of this equation results from (1.11), (2.8),(2.11), (2.12), (3.4) and (3.17). We get:

Ψ′l(D

′Ψ′l)γ012 = Ψl(DΨl)γ012 (3.29)

and also [8]

mρ1 = m′rρ1 = m′ρ′1 ; Ψ′

lχ′l = Ψlχl. (3.30)


The wave equation is also gauge invariant under the gauge transformation(3.17) [8], becoming:

0 = Ψ′l(D

′Ψ′l)γ012 + mρΨ′

lχ′ (3.31)

D′ = ∂∂∂ +g12B′P0 +

g22

(W′1P1 + W′2P2 + W′3P3) (3.32)

B′μ = Bμ − 2

g1∂μa0 ; B′ = γμB′

μ (3.33)

W ′jμPj =

[exp(akPk)W j

μPj − 2g2

∂μ[exp(akPk)]]

exp(−akPk) (3.34)

W′j = γμW ′jμ. (3.35)

Two amongst the fourteen numeric equations equivalent to (3.29) are remark-able: the real part of (3.29) reads:

L = 0 ; L =< Ψl(DΨl)γ012 > +mρ1 (3.36)

We have then, for the pair electron-neutrino as for the alone electron, a dou-ble link between wave equation and Lagrangian formalism: the wave equationmay be obtained by the Lagrange equations from a Lagrangian density. Recip-rocal relation: the Lagrangian density comes as real part of the invariant waveequation. This explains why there is a principle of minimum.

The other remarkable numeric equation reads

∂μ(Dμ0 + Dμ

n) = 0 ; D0 = φeφ†e ; Dn = φnφ†

n. (3.37)

A conservative current exists, the total current D0 + Dn3.

4. Electro-weak and Strong Interactions

The standard model adds to the leptons (electron and its neutrino) in the first“generation” two quarks u and d with three states each. Weak interactionsacting only on left waves of quarks (and right waves of antiquarks) we have8 left spinors instead of 2. It is enough to add 2 dimensions to the space4.With our matrix representation it is enough to work with 8 × 8 matrices. SoI read the wave of all fermions of the first generation as follows:

Ψ =(

Ψl Ψr

Ψg Ψb

); Ψr =

(φdr φur

φur φdr

)=

(φdr φur

φurσ1 φdrσ1

)(4.1)

Ψg =(

φdg φug

φug φdg

); Ψb =

(φdb φub

φub φdb

). (4.2)

The Ψ wave is now a function of space and time with value into Cl1,5 = Cl5,1

which is a sub-algebra (on the real field) of Cl5,2 = M8(C). The covariantderivative (3.16) becomes

3 This current generalizes the probability current.4Needing a link between the reversion in space algebra, in space-time algebra and in theextended algebra, we cannot use other algebras, only Cl3, Cl1,3 and Cl1,5 = Cl5.1. Cal-culations are simpler in Cl1,5 than in Cl5,1. The signature of space-time is physically

determined in General Relativity.


D = ∂∂∂ +g12BP0 +

g22

(W1P1 + W2P2 + W3P3) (4.3)

D =3∑

μ=0

LμDμ; ∂∂∂ =3∑

μ=0

Lμ∂μ; B =3∑

μ=0

LμBμ; Wj =3∑

μ=0

LμW jμ (4.4)

Lμ =(

0 γμ

γμ 0

), μ = 0, 1, 2, 3; L4 =

(0 −I4I4 0

); L5 =

(0 ii 0

)(4.5)

We use two projectors satisfying

P±(Ψ) =12(Ψ ± iΨL21); i = L0123 (4.6)

Three operators act on quarks like on leptons:

P 1(Ψ) = P+(Ψ)L35; P 2(Ψ) = P+(Ψ)L5012; P 3(Ψ) = P+(Ψ)(−i). (4.7)

The fourth operator acts differently on the lepton and on the quark sector:

P 0(Ψ) =(

P0(Ψl) P ′0(Ψr)

P ′0(Ψg) P ′

0(Ψb)

); P0(Ψl) = Ψlγ21 + P−(Ψl)i (4.8)

P ′0(Ψc) = −1

3Ψcγ21 + P−(Ψc)i , c = r, g, b. (4.9)

These definitions are absolutely all that you have to change to go from thelepton case into the quark case, to get the gauge group of electro-weak inter-actions.

To get the generators of the SU(3) gauge group of chromodynamics Iconsider two new projectors:

P+ =12(I8 + L012345); P− =

12(I8 − L012345) (4.10)

and eight operators Γk, k = 1, 2, . . . , 8 so defined (shortening Ψc into c):

Γ1(Ψ) =12(L4ΨL4 + L01235ΨL01235) =

(0 gr 0

)(4.11)

Γ2(Ψ) =12(L5ΨL4 − L01234ΨL01235) =

(0 −igir 0

)(4.12)

Γ3(Ψ) = P+ΨP− − P−ΨP+ =(

0 r−g 0

)(4.13)

Γ4(Ψ) = L01253ΨP− =(

0 b0 r

); Γ5(Ψ) = L01234ΨP− =

(0 −ib0 ir

)(4.14)

Γ6(Ψ) = P−ΨL01253 =(

0 0b g

); Γ7(Ψ) = −iP−ΨL4 =

(0 0

−ib ig

)(4.15)

Γ8(Ψ) =1√3(P−ΨL012345 + L012345ΨP−) =

1√3

(0 rg −2b

). (4.16)

We explained in [6] how this is equivalent to the eight generators λk of SU(3).Everywhere in (4.11) to (4.16) the eight matrices Γk(Ψ) have a zero left upterm, therefore all Γk project the wave on its quark sector. The physical


translation is: leptons do not interact by strong interactions, this comes fromthe structure itself of the quantum wave. Now with

D =3∑

μ=0

LμDμ; ∂∂∂ =3∑

μ=0

Lμ∂μ; B =3∑

μ=0

LμBμ

Wj =3∑

μ=0

LμW jμ; Gk =

3∑μ=0

LμGkμ (4.17)

where the eight Gk are named “gluons”, the covariant derivative reads

D = ∂∂∂ +g12B P 0 +

g22

3∑j=1

WjP j +g32

8∑k=1

GkiΓk. (4.18)

The gauge group is obtained by exponentiation. We use four numbers aμ andeight numbers bk. We let

S = S0 + S1 + S2; S0 = a0P 0; S1 =3∑

j=1

ajP j ; S2 =8∑

k=1

bkiΓk. (4.19)

We get

exp(S) = exp(S0) exp(S1) exp(S2) = exp(S0) exp(S2) exp(S1)= exp(S2) exp(S1) exp(S0) = . . . (4.20)

in any order, because:

P 0P j = P jP 0, j = 1, 2, 3 (4.21)

PμiΓk = iΓkPμ, μ = 0, 1, 2, 3, k = 1, 2 . . . 8. (4.22)

Therefore the set G = {exp(S)} is a U(1) × SU(2) × SU(3) Lie group. Thegauge transformation reads

Ψ′ = [exp(S)](Ψ); D = LμDμ ; D′ = LμD′μ; B′

μ = Bμ − 2g1

∂μa0 (4.23)

D′ = ∂∂∂ +g12B′P 0 +

g22

j=3∑j=1

W′jP j +g32

k=8∑k=1

G′kiΓk (4.24)

W ′jμP j =

[exp(S1)W j

μP j − 2g2

∂μ[exp(S1)]]exp(−S1) (4.25)

G′kμiΓk =

[exp(S2)Gk

μiΓk − 2g3

∂μ[exp(S2)]]exp(−S2). (4.26)

We then get the gauge group of the standard model, automatically, and notanother group. It is possible to get operators exchanging Ψl and Ψc, c = r, g, blike Γ1 exchanging Ψr and Ψg but the difference between P0 and P ′

0 forbidsthe commutativity. Then we cannot get a greater group than the precedingU(1) × SU(2) × SU(3) gauge group.

I got also a remarkable identity [10] allowing det(Ψ) �= 0 and Ψ(x) isusually invertible. The existence of the inverse allows the construction of thewave of systems of fermions (See [4] and [6] 4.4.1). We got the wave equationfor electron+neutrino+quarks u and d [9].


We know three generations of leptons and quarks and the standardmodel study separately these three generations. The reason is simply thatour physical space is 3-dimensional, and we get the wave equation of leptonsthree times. One of the three is (3.28) that reads:

0 = Ψ3(D3Ψ3)γ012 + m3ρΨ3χ3

D3 = D; Ψ3 = Ψl ; χ3 = χl; m3 = m; ρ = ρ1 (4.27)

0 = Ψ1(D1Ψ1)γ023 + m1ρΨ1χ1 (4.28)

0 = Ψ2(D2Ψ3)γ031 + m2ρΨ2χ2. (4.29)

To go from one generation to another one is simple: I permute indices 1,2,3of σj everywhere in all preceding formulas with the circular permutation por p2: p : 1 �→ 2 �→ 3 �→ 1 ; p2 : 1 �→ 3 �→ 2 �→ 1. If p gives the muon, thewave of the pair muon-muonic neutrino follows (4.28) and this explains whya muon is like an electron, generally. But the covariant derivative is different,because in the place of (3.11) to (3.15) we must use

P 1±(Ψ) =

12(Ψ ± iΨγ32) (4.30)

P 10 (Ψ) = Ψγ32 + P−(Ψ)i; P 1

1 (Ψ) = P+(Ψ)γ1i (4.31)P 12 (Ψ) = P+(Ψ)γ1; P 1

3 (Ψ) = P+(Ψ)(−i). (4.32)

To add two quarks with three colors each we need

P ′10(Ψc) = −1

3Ψcγ32 + P 1

−(Ψc)i ; c = r, g, b. (4.33)

We must also change the link (3.5) between the wave of the particle and thewave of the antiparticle. The wave of the anti-muon must satisfy φμ = φμσ2

and we shall have a 3 index in the case of the third generation. We must alsochange the definition of left and right wave. For the second generation thisbecomes φμL = φμ

12 (1−σ1) ; φμR = φμ

12 (1+σ1) and so on. The Lagrangian

density, which is the scalar part of the invariant equation, must be calculatedseparately. Now since the Γk operators, generators of the SU(3) group ofchromodynamics, are unchanged by the circular permutation p used to passfrom one generation to another one, strong interactions are unperturbed bythe change of generation. This allows physical quarks composing particles tomix the generations. The mixing of waves of different generations, and thedifference between what we call “left” and “right” in each generation, inducethe wave of physical quarks to have both a left and a right wave.

If there are only three objects like σ21, there is one other term withsquare −1 in Cl3, i = σ1σ2σ3. This fourth term allows a fourth neutrino [7].More explanations shall be available soon in [12], where we explain also howinertia and gravitation take place aside electro-weak and strong interactions.The form invariance under Cl∗3 rules all physical interactions.

Open Access. This article is distributed under the terms of the Creative CommonsAttribution License which permits any use, distribution, and reproduction in anymedium, provided the original author(s) and the source are credited.


References

[1] Boudet, R.: Quantum Mechanics in the Geometry of Space-Time. Springer,Heidelberg Dordrecht London New York (2011)

[2] Daviau, C.: Equation de Dirac non lineaire. PhD thesis, Universite de Nantes(1993).

[3] Daviau, C: Interpretation cinematique de l’onde de l’electron. Ann. Fond. L.de Broglie 30, 3–4 (2005)

[4] Daviau, C: Cl∗3 invariance of the Dirac equation and of electromagnetism. Adv.Appl. Clifford Algebr. 22(3), 611–623 (2012)

[5] Daviau, C.: Double Space-Time and More. JePublie, Pouille-les-coteaux (2012)

[6] Daviau, C: Invariant quantum wave equations and double space-time. Adv.Imaging Elect. Phys. 179(1), 1–137 (2013)

[7] Daviau, C, Bertrand, J: A lepton Dirac equation with additional mass termand a wave equation for a fourth neutrino. Ann. Fond. Louis de Broglie 38, 57–81 (2013)

[8] Daviau, C., Bertrand, J.: Relativistic gauge invariant wave equation of theelectron-neutrino. J. Modern Phys. 5, 1001–1022 (2014). doi:10.4236/jmp.2014.511102

[9] Daviau, C., Bertrand, J.: A wave equation including leptons and quarks forthe standard model of quantum physics in Clifford algebra. J. Modern Phys.5, 2149–2173 (2014). doi:10.4236/jmp.2014.518210

[10] Daviau, C.; Bertrand. J.: New Insights in the Standard Model of Quan-tum Physics in Clifford Algebra. Je Publie, Pouille-les-coteaux. http://hal.archives-ouvertes.fr/hal-00907848 (2014)

[11] Hestenes, D.: A unified language for mathematics and physics and CliffordAlgebra and the interpretation of quantum mechanics. In: Chisholm, J.S.R. andCommon, A.K. (eds.) Clifford Algebras and Their Applications in Mathematicsand Physics. Reidel, Dordrecht (1986)

[12] Daviau, C., Bertrand, J.: Additional insights in the standard model of quantumphysics in clifford algebra. (To be published)

C. DaviauLe Moulin de la Lande44522 Pouille-les-coteauxFrancee-mail: [email protected]

Revised: October 2, 2014.

Accepted: May 9, 2015.

http://dx.doi.org/10.4236/jmp.2014.511102



http://hal.archives-ouvertes.fr/hal-00907848

http://hal.archives-ouvertes.fr/hal-00907848

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