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New Insights in the Standard Model of Quantum Physics in Clifford Algebra Claude Daviau, Jacques Bertrand

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New Insights in the Standard Model of Quantum Physics in Clifford Algebra Claude Daviau, Jacques Bertrand New insights in the standard model of quantum physics in Clifford Algebra Claude Daviau, Jacques Bertrand To cite this version: Claude Daviau, Jacques Bertrand. New insights in the standard model of quantum physics in Clifford Algebra. 2013. hal-00907848v2 HAL Id: hal-00907848 https://hal.archives-ouvertes.fr/hal-00907848v2 Preprint submitted on 3 Dec 2013 (v2), last revised 2 Jun 2014 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. New insights in the standard model of quantum physics in Clifford Algebra Claude Daviau Le Moulin de la Lande 44522 Pouillé-les-coteaux France email: [email protected] Jacques Bertrand 15 avenue Danielle Casanova 95210 Saint-Gratien France email: [email protected] 6 décembre 2013 2 keywords : invariance group, Dirac equation, electromagnetism, weak interactions, strong interactions, Clifford algebras, magnetic monopoles. Contents 1 Clifford algebras 1 1.1 What is a Clifford algebra? . 2 1.2 Clifford algebra of an Euclidean plan: Cl2 ........... 3 1.3 Clifford algebra of the physical space: Cl3 ........... 3 1.3.1 Cross-product, orientation . 4 1.3.2 Pauli algebra . 5 1.3.3 Three conjugations are used: . 5 1.3.4 Gradient, divergence and curl . 6 1.3.5 Space-time in space algebra: . 7 1.3.6 Relativistic invariance: . 7 1.3.7 Restricted Lorentz group: . 9 1.4 Clifford algebra of the space-time: Cl1,3 . 10 1.4.1 Dirac matrices: . 11 1.5 Clifford Algebra Cl5,1 = Cl1,5 . 12 2 Dirac equation 15 2.1 With the Dirac matrices . 15 2.1.1 Second order equation . 16 2.1.2 Conservative current . 16 2.1.3 Tensors . 17 2.1.4 Gauge invariances . 18 2.1.5 Relativistic invariance . 19 2.2 The wave with the space algebra . 20 2.2.1 Relativistic invariance . 22 2.2.2 More tensors . 23 2.2.3 Plane waves . 25 2.3 The Dirac equation in space-time algebra . 26 2.4 Invariant Dirac equation . 27 2.4.1 Charge conjugation . 29 2.4.2 Link between the wave of the particle and the wave of the antiparticle . 30 2.5 About the Pauli algebra . 31 3 4 CONTENTS 3 The homogeneous nonlinear wave equation. 33 3.1 Gauge invariances . 36 3.2 Plane waves . 37 3.3 Relativistic invariance . 38 3.4 Wave normalization . 40 3.5 Charge conjugation . 41 3.6 The Hydrogen atom . 43 4 Invariance of electromagnetic laws 45 4.1 Maxwell-de Broglie electromagnetism . 45 4.1.1 Invariance under Cl3∗ ................... 47 4.2 Electromagnetism with monopoles . 50 4.3 Back to space-time . 51 4.3.1 From Cl3 to Cl1,3 .................... 51 4.3.2 Electromagnetism . 53 4.4 Four photons . 54 4.4.1 The electromagnetism of the photon . 57 4.4.2 Three other photons of Lochak . 59 4.5 Uniqueness of the electromagnetic field . 61 4.6 Remarks . 61 5 Consequences 63 5.1 Anisotropy . 63 5.1.1 Torsion . 65 5.2 Systems of electrons . 66 5.3 Equation without Lagrangian formalism . 67 5.3.1 Plane waves . 69 6 Electro-weak and strong interactions 71 6.1 The Weinberg-Salam model for the electron . 71 6.2 Invariances . 81 6.3 The quark sector . 82 6.4 Chromodynamics . 87 6.4.1 Three generations, four neutrinos . 89 6.5 Geometric transformation linked to the wave . 90 6.5.1 In space-time algebra . 91 6.5.2 Extension to the complete wave . 92 6.5.3 Invariance . 94 6.6 Existence of the inverse . 96 6.7 Wave equations . 98 7 Magnetic monopoles 101 7.1 Russian experimental works . 101 7.2 Works at E.C.N. 103 7.2.1 Results about powder and gas . 104 CONTENTS 5 7.2.2 Stains . 107 7.2.3 Traces . 107 7.3 Electrons and monopoles . 114 7.3.1 Charge conjugation . 115 7.3.2 The interaction electron-monopole . 116 7.3.3 Electro-weak interactions with monopoles . 117 7.3.4 Gauge invariant wave equation . 119 8 Conclusion 121 8.1 Old flaws . 121 8.2 Our work . 122 8.3 Principle of minimum . 125 8.4 Theory versus experiment . 126 8.5 Future applications . 127 8.6 Improved standard model . 128 A Calculations in Clifford algebras 131 A.1 Invariant equation and Lagrangian . 131 A.2 Calculation of the reverse in Cl5,1 . 137 B Calculations for the gauge invariance 141 C The hydrogen atom 145 C.1 Separating variables . 145 C.2 Angular momentum operators . 149 C.3 Resolution of the linear radial system . 152 C.4 Calculation of the Yvon-Takabayasi angle . 157 C.5 Radial polynomials with degree 0 . 160 6 CONTENTS Introduction To see from where comes the standard model that rules today quantum physics, we first return to its beginning. When the idea of a wave associ- ated to the move of a particle was found, Louis de Broglie was following consequences of the restricted relativity [24]. The first wave equation found by Schrödinger [48] was not relativistic, and could not be the true wave equation. In the same time the spin of the electron was discovered. This remains the main change from pre-quantum physics, since the spin 1/2 has none classical equivalent. Pauli gave a wave equation for a non-relativistic equation with spin. This equation was the starting point of the attempt made by Dirac [29] to get a relativistic wave equation for the electron. This Dirac equation was a very great success. Until now it is still considered as the wave equation for each particle with spin 1/2, electrons but also positrons, muons and anti-muons, neutrinos, quarks. This wave equation was intensively studied by Louis de Broglie and his students. He published a first book in 1934 [25] explaining how this equation gives in the case of the hydrogen atom the quantification of energy levels, awaited quantum numbers, the true number of quantum states, the true energy levels and the Landé factors. The main novelty in physics coming with the Dirac theory is the fact that the wave has not vector or tensor properties under a Lorentz rotation, the wave is a spinor and transforms very differently. It results from this transformation that the Dirac equa- tion is form invariant under Lorentz rotations. This form invariance is the departure of our study and is the central thread of this book. The Dirac equation was built from the Pauli equation and is based on 4 4 complex matrices, which were constructed from the Pauli matrices. Many× years after this first construction, D. Hestenes [31] used the Clifford algebra of space-time to get a different form of the same wave equation. Tensors which are constructed from the Dirac spinors appear differently and the relations between these tensors are more easily obtained. One of the parameters of the Dirac wave, the Yvon-Takabayasi angle [49], was completely different from all classical physics. G. Lochak under- stood that this angle allows a second gauge invariance and he found a wave equation for a magnetic monopole from this second gauge invariance [38]. He showed that a wave equation with a nonlinear mass term was possible i ii CONTENTS for his magnetic monopole. When this mass term is null, the wave is made of two independent Weyl spinors. This mass term is compatible with the electric gauge ruling the Dirac equation. So it can replace the linear term of the Dirac equation of the electron [7]. A nonlinear wave equation for the electron was awaited by de Broglie, because it was necessary to link the particle to the wave. But this do not explain how to choose the nonlinearity. And the nonlinearity is a formidable problem in quantum physics: quantum theory is a linear theory, it is by solving the linear wave equation that the quantification of energy levels and quantum numbers are obtained in the hydrogen atom. If you start from a nonlinear wave equation, usually you will not be even able to get quantification and quantum numbers. Nevertheless the study of this nonlinear wave equation began in the case where the Dirac equation is its linear approximation. In this case the wave equation is homogeneous. It is obtained from a Lagrangian density which differs from the Lagrangian of the linear theory only by the mass term. Therefore many results are similar. For instance the dynamics of the electron are the same, and the electron follows the Lorentz force. Two formalisms were available, the Dirac formalism with 4 4 complex matrices, and the real Clifford algebra of space-time. A matrix× representa- tion links these formalisms. Since the hydrogen case gave the main result, a first attempt was made to solve the nonlinear equation in this case. Heinz Krüger gave a precious tool [36] by finding a way to separate the spheri- cal coordinates. Moreover the beginning of this resolution by separation of variables was the same in the case of the linear Dirac equation and in the case of the nonlinear homogeneous equation. But then there was a great difficulty: The Yvon-Takabayasi angle is null in the x3 = 0 plane; This angle is a complicated function of an angular variable and of the radial variable; Moreover for any solution with a not constant radial polynomial, circles ex- ist where the Yvon-Takabayasi angle is not defined; In the vicinity of these circles this angle is not small and the solutions of the Dirac equation have no reason to be linear approximations of solutions of the nonlinear homoge- neous equation.
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