Gauge Group of the Standard Model in Cl1,5

Describing a wave with spin 1/2, the Dirac equation is form invariant under SL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${SL(2,\mathbb{C})}$$\end{document}, subgroup of Cl3∗=GL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Cl_3^*=GL(2,\mathbb{C})}$$\end{document} which is the true group of form invariance of the Dirac equation. Firstly we use the Cl3 algebra 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 of electro-weak interactions, first in the leptonic case, electron+neutrino, next in the quark case. The complete wave for all objects of the first generation uses the Clifford algebra Cl1,5. The gauge group is then enlarged into a U(1)×SU(2)×SU(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U(1)\times SU(2)\times SU(3)}$$\end{document} Lie group. We consolidate both the standard model and the use of Clifford algebras, true mathematical frame of quantum physics.


Form Invariance of the Dirac Equation
The standard model uses fermions and bosons. All fermions are described with a Dirac equation. The Dirac wave ψ e of the electron is made of two Pauli waves where ξ 1e , ξ 2e , η 1e , η 2e are four functions of space and time with value in the complex field. In its first complex frame the Dirac equation reads where γ μ , μ = 0, 1, 2, 3 are four complex matrices. Relativistic theory uses: All my thanks go to Jacques Bertrand who helped me to develop the present work.

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C. Daviau Adv. Appl. Clifford Algebras The electro-weak theory uses also To get the relativistic invariance the Dirac theory uses x = x μ σ μ and x satisfying (1.6) This is equivalent to identify the space algebra Cl 3 to the matrix algebra M 2 (C) 1  because we get, for any M : Then R is a Lorentz dilation, product of a Lorentz rotation and an homothety with ratio r. Moreover with we get [5,10]: (1.14) The form invariance of the Dirac equation comes from  (1.16) where N is the reverse of N . This gives (1.17) The Dirac theory lets then ψ e (x ) = Nψ e (x) (1.18) and since the Dirac equation is said "form invariant". Ten years ago, I noticed that relations (1.12), (1.13) and (1.14) are true even if det(M ) = 1. Then the fundamental group of form invariance of the Dirac wave is the group of the M , usually named GL (2, C). This group is also the multiplicative group Cl * 3 of the invertible elements in Cl 3 .

Dirac Equation and Electromagnetism in Cl 3
We have previously [3,5,10] let The φ e wave is then a function of space-time into Cl 3 = M 2 (C). We get where β is the Yvon-Takabayasi angle. It happens that we get, with any M and any η: The link that I made in (2.1) between φ e and the Weyl spinors ξ e , η e is therefore invariant under Cl * 3 : if we simply get φ e = Mφ e ; φ e = M φ e .
(2.8) We have explained [5,10] how the Dirac equation reads in Cl 3 : Adv. Appl. Clifford Algebras Multiplying on the left by φ e , I got [5,6]: The first term is form invariant because, for any M in Cl 3 , with I got the following general relation [3]: which gives with (2.8): 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 Vol. 27 (2017) Gauge Group of the Standard Model in Cl 1,5 283 where E is the electric field, H is the magnetic field, A the space-time vector potential, B the magnetic potential, j the electric density of charge and current, k the magnetic density of charge and current. Laws of electromagnetism in the void with magnetic monopoles read (See [10], chap. 4) The electromagnetic field F satisfies [6]: The contravariance (2.24) means that potentials move with sources. The covariance of qA, j and k is compatible with all laws of electromagnetism and relativistic mechanics [5,6,10]. The transformation (2.23) explains by itself why only the SL(2, C) part of Cl * 3 was previously seen. The P rotor which plays 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 This gives and F transforms as if r = 1 and θ = 0. The electromagnetic field (and more generally all gauge fields) transforms in such a way that we see only the relativistic Lorentz rotation induced by the P term. Velocities are also independent from r. Then under the dilation R, with ratio r induced by any M in Cl * 3 , I got [5]: Electric charge, proper mass and Planck "constant" are changed in the dilation R induced by a M in Cl * 3 if r = 1. The dilation is the composition of the Lorentz rotation induced by P and an homothety with ratio r, in any order. We evidently get the results of restricted relativity if r = 1.

Electro-weak Interactions
The electro-weak theory needs three spinorial waves in the electron-neutrino case: the right ξ e and the left η e of the electron and the left spinor η n of the electronic neutrino. The form invariance of the Dirac theory imposes to use a wave Ψ l satisfying 284 C. Daviau Adv. Appl. Clifford Algebras The wave is a function of space and time with value into the space-time algebra The form (3.1) of the wave is compatible with the charge conjugation used in the standard model: the positron wave ψ e satisfies Then Ψ l contains the electron wave φ e , the neutrino wave φ n and also the positron wave φ e and the antineutrino wave φ n : And the antineutrino has consequently only a right wave. With: Ψ l satisfies: det(Ψ l ) = a 1 a * 1 + a 2 a * 2 .
(3.10) and the Ψ l (x) matrix is usually invertible. To get the gauge group of electroweak interactions I used [6] two projectors P ± and four operators P μ , where μ = 0, 1, 2, 3, satisfying: (3.14) These operators generate the Lie algebra of U (1) × SU (2). The covariant derivative of the Weinberg-Salam model: where Y is the weak hypercharge (Y L = −1, Y R = −2 for the electron), has a very simple translation in the Cl 1,3 frame: Because we get from these definitions: which is equivalent to (3.16). The Weinberg-Salam θ W angle satisfies The U (1) × SU (2) gauge group is obtained by exponentiation. If a μ are four real parameters we use the gauge transformation where ρ 1 = a 1 a * 1 + a 2 a * 2 + a 3 a * 3 (3.25) This wave equation is equivalent to the invariant equation: (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 )γ 012 = Ψ l (DΨ l )γ 012 (3.29) and also [8] The wave equation is also gauge invariant under the gauge transformation (3.17) [8], becoming: Two amongst the fourteen numeric equations equivalent to (3.29) are remarkable: the real part of (3.29) reads: A conservative current exists, the total current D 0 + D n 3 .

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 interactions acting only on left waves of quarks (and right waves of antiquarks) we have 8 left spinors instead of 2. It is enough to add 2 dimensions to the space 4 . With our matrix representation it is enough to work with 8 × 8 matrices. So I read the wave of all fermions of the first generation as follows: The Ψ wave is now a function of space and time with value into Cl 1,5 = Cl 5,1 which is a sub-algebra (on the real field) of We use two projectors satisfying Three operators act on quarks like on leptons: The fourth operator acts differently on the lepton and on the quark sector: (4.9) These definitions are absolutely all that you have to change to go from the lepton case into the quark case, to get the gauge group of electro-weak interactions.
To get the generators of the SU (3) gauge group of chromodynamics I consider two new projectors: and eight operators Γ k , k = 1, 2, . . . , 8 so defined (shortening Ψ c into c): (4.16) We explained in [6] how this is equivalent to the eight generators λ k of SU (3). Everywhere in (4.11) to (4.16) the eight matrices Γ k (Ψ) have a zero left up term, therefore all Γ k project the wave on its quark sector. The physical 288 C. Daviau Adv. Appl. Clifford Algebras translation is: leptons do not interact by strong interactions, this comes from the structure itself of the quantum wave. Now with where the eight G k are named "gluons", the covariant derivative reads The gauge group is obtained by exponentiation. We use four numbers a μ and eight numbers b k . We let We get . . (4.20) in any order, because: P 0 P j = P j P 0 , j = 1, 2, 3 (4.21) P μ iΓ k = iΓ k P μ , μ = 0, 1, 2, 3, k = 1, 2 . . . 8.
We then get the gauge group of the standard model, automatically, and not another group. It is possible to get operators exchanging Ψ l and Ψ c , c = r, g, b like Γ 1 exchanging Ψ r and Ψ g but the difference between P 0 and P 0 forbids the commutativity. Then we cannot get a greater group than the preceding U (1) × SU (2) × SU (3) gauge group. I got also a remarkable identity [10] allowing det(Ψ) = 0 and Ψ(x) is usually invertible. The existence of the inverse allows the construction of the wave of systems of fermions (See [4] and [6] 4.4.1). We got the wave equation for electron+neutrino+quarks u and d [9]. We know three generations of leptons and quarks and the standard model study separately these three generations. The reason is simply that our physical space is 3-dimensional, and we get the wave equation of leptons three times. One of the three is (3.28) that reads: To go from one generation to another one is simple: I permute indices 1,2,3 of σ j everywhere in all preceding formulas with the circular permutation p or p 2 : p : 1 → 2 → 3 → 1 ; p 2 : 1 → 3 → 2 → 1. If p gives the muon, the wave of the pair muon-muonic neutrino follows (4.28) and this explains why a 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 To add two quarks with three colors each we need (4.33) We must also change the link (3.5) between the wave of the particle and the wave of the antiparticle. The wave of the anti-muon must satisfy φ μ = φ μ σ 2 and we shall have a 3 index in the case of the third generation. We must also change the definition of left and right wave. For the second generation this becomes φ μL = φ μ 1 2 (1 − σ 1 ) ; φ μR = φ μ 1 2 (1 + σ 1 ) and so on. The Lagrangian density, which is the scalar part of the invariant equation, must be calculated separately. Now since the Γ k operators, generators of the SU (3) group of chromodynamics, are unchanged by the circular permutation p used to pass from one generation to another one, strong interactions are unperturbed by the change of generation. This allows physical quarks composing particles to mix the generations. The mixing of waves of different generations, and the difference between what we call "left" and "right" in each generation, induce the wave of physical quarks to have both a left and a right wave.
If there are only three objects like σ 21 , there is one other term with square −1 in Cl 3 , i = σ 1 σ 2 σ 3 . This fourth term allows a fourth neutrino [7]. More explanations shall be available soon in [12], where we explain also how inertia and gravitation take place aside electro-weak and strong interactions. The form invariance under Cl * 3 rules all physical interactions.
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