| Introduction | p. 1 |
| References | p. 3 |
| The Real Geometrical Algebra or Space-Time Algebra. Comparison with the Language of the Complex Matrices and Spinors and Spinors | |
| The Clifford Algebra Associated with the Minkowski Space-Time M | p. 7 |
| The Clifford Algebra Associated with 'an Euclidean Space | p. 7 |
| The Clifford Algebras and the "Imaginary Number" | p. 9 |
| The Field of the Hamilton Quaternions and the Ring of the Biquatemion as Cl+(3,0)and Cl(3,O) Cl+(1 3) | p. 10 |
| References | p. 11 |
| Comparison Between the Real and the Complex Language | p. 13 |
| The Space-Time Algebra and the Wave Function Associated with a Particle: The Hestenes Spinor | p. 13 |
| The Takabayasi-Hestenes Moving Frame | p. 15 |
| Equivalences Between the Hestenes and the Dirac Spinors | p. 15 |
| Comparison Between the Dirac and the Hestenes Spinors | p. 16 |
| References | p. 16 |
| The U{1) Gauge in Complex and Real Languages. Geometrical Properties and Relation with the Spin and the Energy of a Particle of Spin 1/2 | |
| Geometrical Properties of the U(1) Gauge | p. 21 |
| The Definition of the Gauge and the Invariance of a Change of Gauge in the U(l) Gauge | p. 21 |
| The U(l) Gauge in Complex Language | p. 21 |
| The U(l) Gauge Invariance in Complex Language | p. 21 |
| A Paradox of the U(l) Gauge in Complex Language | p. 22 |
| The U(l) Gauge in Real Language | p. 22 |
| The Definition of the U(l) Gauge in Real Language | p. 23 |
| The U(l) Gauge Invariance in Real Language | p. 23 |
| References | p. 24 |
| Relation Between the U(l) Gauge, the Spin and the Energy of a Particle of Spin 1/2 | p. 25 |
| Relation Between the U(l) Gauge and the Bivector Spin | p. 25 |
| Relation Between the U(l) Gauge and the Momentum-Energy Tensor Associated with the Particle | p. 25 |
| Relation Between the U(l) Gauge and the Energy of the Particle | p. 26 |
| Geometrical Properties of the Dirac Theory of the Electron | |
| The Dirac Theory of the Electron in Real Language | p. 29 |
| The Hestenes Real form of the Dirac Equation | p. 29 |
| The Probability Current | p. 30 |
| Conservation of the Probability Current | p. 30 |
| The Proper (Bivector Spin) and the Total Angular-Momenta | p. 31 |
| The Tetrode Energy-Momentum Tensor | p. 31 |
| Relation Between the Energy of the Electron and The Infinitesimal Rotation of the "Spin Plane" | p. 32 |
| The Tetrode Theorem | p. 33 |
| The Lagrangian of the Dirac Electron | p. 33 |
| Units | p. 33 |
| References | p. 34 |
| The Invariant Form of the Dirac Equation and Invariant Properties of the Dirac Theory | p. 35 |
| The Invariant Form of the Dirac Equation | p. 35 |
| The Passage from the Equation of the Electron to the One of the Positron | p. 36 |
| The Free Dirac Electron, the Frequency and the Clock of L. de Broglie | p. 37 |
| The Dirac Electron, the Einstein Formula of the Photoeffect and the L. de Broglie Frequency | p. 39 |
| The Equation of the Lorentz Force Deduced from the Dirac Theory of the Electron | p. 40 |
| On the Passages of the Dirac Theory to the Classical Theory of the Electron | p. 41 |
| References | p. 41 |
| The SU(2) Gauge and the Yang-Mills Theory in Complex and Real Languages | |
| Geometrical Properties of the SU(2) Gauge and the Associated Momentum-Energy Tensor | p. 45 |
| The SU(2) Gauge in the General Yang-Mills Field Theory in Complex Language | p. 45 |
| The SU(2) Gauge and the Y.M. Theory in STA | p. 46 |
| The SU(2) Gauge and the Gauge Invariance in STA | p. 46 |
| A Momentum-Energy Tensor Associated with the Y.M. Theory | p. 48 |
| The STA Form of the Y.M. Theory Lagrangian | p. 49 |
| Conclusions About the SU(2) Gauge and the Y.M. Theory | p. 49 |
| References | p. 50 |
| The SU(2) x U(l) Gauge in Complex and Real Languages | |
| Geometrical Properties of the SU(2) x U(l) Gauge | p. 53 |
| Left and Right Parts of a Wave Function | p. 53 |
| Left and Right Doublets Associated with Two Wave Functions | p. 54 |
| The Part SU(2) of the SU(2) x U(l) Gauge | p. 56 |
| The Part U(1) of the SU(2) x (7(1) Gauge | p. 56 |
| Geometrical Interpretation of the SU(2) x U(1) Gauge of a Left or Right Doublet | p. 56 |
| The Lagrangian in the SU(2) x U(l) Gauge | p. 57 |
| References | p. 57 |
| The Glashow-Salam-Weinberg Electroweak Theory | |
| The Electroweak Theory in STA: Global Presentation | p. 61 |
| General Approach | p. 61 |
| The Particles and Their Wave Functions | p. 62 |
| The Right and Left Parts of the Wave Functions of the Neutrino and the Electron | p. 62 |
| A Left Doublet and Two Singlets | p. 62 |
| The Currents Associated with the Wave Functions | p. 62 |
| The Current Associated with the Right and Left Parts of the Electron and Neutrino | p. 63 |
| The Currents Associated with the Left Doublet | p. 63 |
| The Charge Currents | p. 64 |
| The Bosons and the Physical Constants | p. 65 |
| The Physical Constants | p. 65 |
| The Bosons | p. 65 |
| The Lagrangian | p. 65 |
| References | p. 66 |
| The Electroweak Theory in STA: Local Presentation | p. 67 |
| The Two Equivalent Decompositions of the Part LI of the Lagrangian | p. 67 |
| The Decomposition of the Part LII of the Lagrangian into a Charged and a Neutral Contribution | p. 68 |
| The Charged Contribution | p. 69 |
| The Neutral Contribution | p. 69 |
| The Gauges | p. 70 |
| The Part U(l) of the SU(2) x U(l) Gauge | p. 70 |
| The Part SU(2) of the SU(2) x U(l) Gauge | p. 71 |
| Zitterbewegung and Electroweak Currents in Dirac Theory | p. 71 |
| References | p. 72 |
| On a Change of SU(3) into Three SU(2) x U(l) | |
| On a Change of SU{3) into Three SU(2) X U(l) | p. 75 |
| The Lie Group SU(3) | p. 75 |
| The Gell-Mann Matrices | p. 75 |
| The Column $$$ on which the Gell-Mann Matrices Act | p. 76 |
| Eight Vectors Ga | p. 76 |
| A Lagrangian | p. 76 |
| On the Algebraic Nature of the $$$ | p. 76 |
| Comments | p. 77 |
| A passage From SU(3) to Three SU(2) x U(l) | p. 77 |
| An Alternative to the Use of SU(3) in Quantum Chromodynamics Theory? | p. 79 |
| References | p. 79 |
| p. Addendum |
| A Real Quantum Electrodynamics | p. 83 |
| General Approach | p. 83 |
| Electromagnetism: The Electromagnetic Potential | p. 84 |
| Principles on the Potential | p. 84 |
| The Potential Created by a Population of Charges | p. 85 |
| Notion of Charge Current | p. 86 |
| The Lorentz Formula of the Retarded Potentials | p. 87 |
| On the Invariances in the Formula of the Retarded Potentials | p. 88 |
| Electrodynamics: The Electromagnetic Field, the Lorentz Force | p. 89 |
| General Definition | p. 89 |
| Case of Two Punctual Charges: The Coulomb Law | p. 89 |
| Electric and Magnetic Fields | p. 90 |
| Electric and Magnetic Fields Deduced from the Lorentz Potential | p. 91 |
| The Poynting Vector | p. 93 |
| Electrodynamics in the Dirac Theory of the Electron | p. 93 |
| The Dirac Probability Currents | p. 94 |
| Current Associated with a Level E of Energy | p. 95 |
| Emission of an Electromagnetic Field | p. 95 |
| Spontaneous Emission | p. 95 |
| Interaction with a Plane Wave | p. 96 |
| The Lamb Shift | p. 100 |
| References | p. 100 |
| Appendices | |
| Real Algebras Associated with an Euclidean Space | p. 105 |
| The Grassmann (or Exterior) Algebra of Rn | p. 105 |
| The Inner Products of an Euclidean Space E = Rq,n-q | p. 105 |
| The Clifford Algebra CI(E) Associated with an Euclidean Space E = Rp,n-p | p. 106 |
| A Construction of the Clifford Algebra | p. 108 |
| The Group 0{E) in CI(E) | p. 109 |
| References | p. 110 |
| Relation Between the Dirac Spinor and the Hestenes Spinor | p. 111 |
| The Pauli Spinor and Matrices | p. 111 |
| The Dirac spinor | p. 111 |
| The Quaternion as a Real Form of the Pauli spinor | p. 113 |
| The Biquaternion as a Real Form of the Dirac spinor | p. 114 |
| References | p. 114 |
| The Movement in Space-Time of a Local Orthonormal Frame | p. 115 |
| C.l The Group SO+(E) and the Infinitesimal Rotations in Cl(E) | p. 115 |
| Study on Properties of Local Moving Frames | p. 116 |
| Infinitesimal Rotation of a Local Frame | p. 116 |
| Infinitesimal Rotation of Local Sub-Frames | p. 117 |
| Effect of a Local Finite Rotation of a Local Sub-Frame | p. 118 |
| References | p. 119 |
| Incompatibilities in the Use of the Isospin Matrices | p. 121 |
| $$$ is an "Ordinary" Dirac Spinor | p. 121 |
| $$$ is a Couple (*Ña, Ñb) of Dirac Spinors | p. 121 |
| $$$ is a Right or a Left Doublet | p. 122 |
| Questions about the Nature of the Wave Function | p. 122 |
| A Proof of the Tetrode Theorem | p. 123 |
| About the Quantum Fields Theory | p. 125 |
| On the Construction of the QFT | p. 125 |
| Questions | p. 126 |
| An Artifice in the Lamb Shift Calculation | p. 127 |
| References | p. 128 |
| Index | p. 129 |
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