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Quantum Mechanics in the Geometry of Space-Time
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1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Part I The Real Geometrical Algebra or Space–Time Algebra. Comparison with the Language of the Complex Matrices and Spinors 2 The Clifford Algebra Associated with the Minkowski Space–Time M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 The Clifford Algebra Associated with an Euclidean Space . . . 7 2.2 The Clifford Algebras and the ‘‘Imaginary Number’’ ffiffiffiffiffiffiffi p1 . . . . 9 2.3 The Field of the Hamilton Quaternions and the Ring of the Biquaternion as Clþð3; 0Þ and Clð3; 0Þ ’ Clþð1; 3Þ. . . . . 10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Comparison Between the Real and the Complex Language . . . . . 13 3.1 The Space–Time Algebra and the Wave Function Associated with a Particle: The Hestenes Spinor . . . . . . . . . . 13 3.2 The Takabayasi–Hestenes Moving Frame . . . . . . . . . . . . . . . 15 3.3 Equivalences Between the Hestenes and the Dirac Spinors . . . 15 3.4 Comparison Between the Dirac and the Hestenes Spinors . . . . 16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 vii Part II 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 4 Geometrical Properties of the U(1) Gauge . . . . . . . . . . . . . . . . . . 21 4.1 The Definition of the Gauge and the Invariance of a Change of Gauge in the U(1) Gauge . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.1 The U(1) Gauge in Complex Language. . . . . . . . . . . 21 4.1.2 The U(1) Gauge Invariance in Complex Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.3 A Paradox of the U(1) Gauge in Complex Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 The U(1) Gauge in Real Language . . . . . . . . . . . . . . . . . . . . 22 4.2.1 The Definition of the U(1) Gauge in Real Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.2 The U(1) Gauge Invariance in Real Language . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Relation Between the U(1) Gauge, the Spin and the Energy of a Particle of Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.1 Relation Between the U(1) Gauge and the Bivector Spin . . . . 25 5.2 Relation Between the U(1) Gauge and the Momentum–Energy Tensor Associated with the Particle . . . . . 25 5.3 Relation Between the U(1) Gauge and the Energy of the Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Part III Geometrical Properties of the Dirac Theory of the Electron 6 The Dirac Theory of the Electron in Real Language . . . . . . . . . . 29 6.1 The Hestenes Real form of the Dirac Equation . . . . . . . . . . . 29 6.2 The Probability Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.3 Conservation of the Probability Current. . . . . . . . . . . . . . . . . 30 6.4 The Proper (Bivector Spin) and the Total Angular–Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.5 The Tetrode Energy–Momentum Tensor . . . . . . . . . . . . . . . . 31 6.6 Relation Between the Energy of the Electron and the Infinitesimal Rotation of the ‘‘Spin Plane’’ . . . . . . . . . . . . 32 6.7 The Tetrode Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.8 The Lagrangian of the Dirac Electron . . . . . . . . . . . . . . . . . . 33 6.9 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 viii Contents 7 The Invariant Form of the Dirac Equation and Invariant Properties of the Dirac Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.1 The Invariant Form of the Dirac Equation . . . . . . . . . . . . . . . 35 7.2 The Passage from the Equation of the Electron to the One of the Positron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.3 The Free Dirac Electron, the Frequency and the Clock of L. de Broglie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.4 The Dirac Electron, the Einstein Formula of the Photoeffect and the L. de Broglie Frequency . . . . . . . . . . . . . . . . . . . . . 39 7.5 The Equation of the Lorentz Force Deduced from the Dirac Theory of the Electron . . . . . . . . . . . . . . . . . . . . . 40 7.6 On the Passages of the Dirac Theory to the Classical Theory of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Part IV The SU(2) Gauge and the Yang–Mills Theory in Complex and Real Languages 8 Geometrical Properties of the SU(2) Gauge and the Associated Momentum–Energy Tensor . . . . . . . . . . . . . . . . . . . . 45 8.1 The SU(2) Gauge in the General Yang–Mills Field Theory in Complex Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 8.2 The SU(2) Gauge and the Y.M. Theory in STA. . . . . . . . . . . 46 8.2.1 The SU(2) Gauge and the Gauge Invariance in STA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 8.2.2 A Momentum–Energy Tensor Associated with the Y.M. Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8.2.3 The STA Form of the Y.M. Theory Lagrangian . . . . . 49 8.3 Conclusions About the SU(2) Gauge and the Y.M. Theory . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Part V The SU(2) 3 U(1) Gauge in Complex and Real Languages 9 Geometrical Properties of the SU(2) 3 U(1) Gauge . . . . . . . . . . . 53 9.1 Left and Right Parts of a Wave Function . . . . . . . . . . . . . . . 53 9.2 Left and Right Doublets Associated with Two Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9.3 The Part SU(2) of the SU(2) 9 U(1) Gauge. . . . . . . . . . . . . . 56 9.4 The Part U(1) of the SU(2) 9 U(1) Gauge . . . . . . . . . . . . . . 56 9.5 Geometrical Interpretation of the SU(2) 9 U(1) Gauge of a Left or Right Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Contents ix 9.6 The Lagrangian in the SU(2) 9 U(1) Gauge. . . . . . . . . . . . . 57 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Part VI The Glashow–Salam–Weinberg Electroweak Theory 10 The Electroweak Theory in STA: Global Presentation. . . . . . . . . 61 10.1 General Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.2. The Particles and Their Wave Functions . . . . . . . . . . . . . . . . 62 10.2.1 The Right and Left Parts of the Wave Functions of the Neutrino and the Electron . . . . . . . . . . . . . . . 62 10.2.2 A Left Doublet and Two Singlets . . . . . . . . . . . . . . . 62 10.3 The Currents Associated with the Wave Functions . . . . . . . . . 62 10.3.1 The Current Associated with the Right and Left Parts of the Electron and Neutrino . . . . . . . . . . . . . . 63 10.3.2 The Currents Associated with the Left Doublet . . . . . 63 10.3.3 The Charge Currents . . . . . . . . . . . . . . . . . . . . . . . . 64 10.4 The Bosons and the Physical Constants. . . . . . . . . . . . . . . . . 65 10.4.1 The Physical Constants . . . . . . . . . . . . . . . . . . . . . . 65 10.4.2 The Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 10.5 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 11 The Electroweak Theory in STA: Local Presentation. . . . . . . . . . 67 11.1 The Two Equivalent Decompositions of the Part LI of the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 11.2 The Decomposition of the Part LII of the Lagrangian into a Charged and a Neutral Contribution . . . . . . . . . . . . . . . . . . 68 11.2.1 The Charged Contribution . . . . . . . . . . . . . . . . . . . . 69 11.2.2 The Neutral Contribution. . . . . . . . . . . . . . . . . . . . . 69 11.3 The Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 11.3.1 The Part U(1) of the SU(2) 9 U(1) Gauge . . . . . . . . 70 11.3.2 The Part SU(2) of the SU(2) 9 U(1) Gauge . . . . . . . 71 11.3.3 Zitterbewegung and Electroweak Currents in Dirac Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Part VII On a Change of SU(3) into Three SU(2) 3 U(1) 12 On a Change of SU(3) into Three SU(2) 3 U(1) . . . . . . . . . . . . . 75 12.1 The Lie Group SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 12.1.1 The Gell–Mann Matrices ka. . . . . . . . . . . . . . . . . . . 75 x Contents 12.1.2 The Column W on which the Gell–Mann Matrices Act . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 12.1.3 Eight Vectors Ga . . . . . . . . . . . . . . . . . . . . . . . . . . 76 12.1.4 A Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 12.1.5 On the Algebraic Nature of the Wk . . . . . . . . . . . . . . 76 12.1.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 12.2 A passage From SU(3) to Three SU(2) 9 U(1) . . . . . . . . . . . 77 12.3 An Alternative to the Use of SU(3) in Quantum Chromodynamics Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . 79 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Part VIII Addendum 13 A Real Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 83 13.1 General Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 13.2 Electromagnetism: The Electromagnetic Potential. . . . . . . . . . 84 13.2.1 Principles on the Potential . . . . . . . . . . . . . . . . . . . . 84 13.2.2 The Potential Created by a Population of Charges . . . 85 13.2.3 Notion of Charge Current . . . . . . . . . . . . . . . . . . . . 86 13.2.4 The Lorentz Formula of the Retarded Potentials. . . . . 87 13.2.5 On the Invariances in the Formula of the Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 88 13.3 Electrodynamics: The Electromagnetic Field, the Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 13.3.1 General Definition . . . . . . . . . . . . . . . . . . . . . . . . . 89 13.3.2 Case of Two Punctual Charges: The Coulomb Law . . 89 13.3.3 Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . 90 13.3.4 Electric and Magnetic Fields Deduced from the Lorentz Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 91 13.3.5 The Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . 93 13.4 Electrodynamics in the Dirac Theory of the Electron . . . . . . . 93 13.4.1 The Dirac Probability Currents. . . . . . . . . . . . . . . . . 94 13.4.2 Current Associated with a Level E of Energy . . . . . . 94 13.4.3 Emission of an Electromagnetic Field . . . . . . . . . . . . 95 13.4.4 Spontaneous Emission. . . . . . . . . . . . . . . . . . . . . . . 95 13.4.5 Interaction with a Plane Wave . . . . . . . . . . . . . . . . . 96 13.4.6 The Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Contents xi Part IX Appendices 14 Real Algebras Associated with an Euclidean Space . . . . . . . . . . . 105 14.1 The Grassmann (or Exterior) Algebra of Rn . . . . . . . . . . . . . 105 14.2 The Inner Products of an Euclidean Space E ¼ Rq;nq . . . . . . 105 14.3 The Clifford Algebra CIðEÞ Associated with an Euclidean Space E ¼ Rp;np . . . . . . . . . . . . . . . . . . . . . . 106 14.4 A Construction of the Clifford Algebra . . . . . . . . . . . . . . . . . 108 14.5 The Group OðEÞ in CIðEÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 15 Relation Between the Dirac Spinor and the Hestenes Spinor . . . . 111 15.1 The Pauli Spinor and Matrices . . . . . . . . . . . . . . . . . . . . . . . 111 15.2 The Dirac spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 15.3 The Quaternion as a Real Form of the Pauli spinor . . . . . . . . 113 15.4 The Biquaternion as a Real Form of the Dirac spinor . . . . . . . 114 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 16 The Movement in Space–Time of a Local Orthonormal Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 16.1 C.1 The Group SOþðEÞ and the Infinitesimal Rotations in ClðEÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 16.2 Study on Properties of Local Moving Frames . . . . . . . . . . . . 116 16.3 Infinitesimal Rotation of a Local Frame . . . . . . . . . . . . . . . . 116 16.4 Infinitesimal Rotation of Local Sub-Frames . . . . . . . . . . . . . . 117 16.5 Effect of a Local Finite Rotation of a Local Sub-Frame . . . . . 118 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 17 Incompatibilities in the Use of the Isospin Matrices . . . . . . . . . . . 121 17.1 W is an ‘‘Ordinary’’ Dirac Spinor . . . . . . . . . . . . . . . . . . . . . 121 17.2 W is a Couple (Wa; Wb) of Dirac Spinors . . . . . . . . . . . . . . . 121 17.3 W is a Right or a Left Doublet. . . . . . . . . . . . . . . . . . . . . . . 122 17.4 Questions about the Nature of the Wave Function . . . . . . . . . 122 18 A Proof of the Tetrode Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 123 19 About the Quantum Fields Theory . . . . . . . . . . . . . . . . . . . . . . . 125 19.1 On the Construction of the QFT. . . . . . . . . . . . . . . . . . . . . . 125 19.2 Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 19.3 An Artifice in the Lamb Shift Calculation . . . . . . . . . . . . . . . 127 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 |
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