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SPINORS and GRAVITATIONAL FIELD Jean Claude Dutailly SPINORS AND GRAVITATIONAL FIELD Jean Claude Dutailly To cite this version: Jean Claude Dutailly. SPINORS AND GRAVITATIONAL FIELD. 2015. hal-01171507v2 HAL Id: hal-01171507 https://hal.archives-ouvertes.fr/hal-01171507v2 Preprint submitted on 4 Jul 2015 (v2), last revised 16 Nov 2016 (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. SPINORS AND GRAVITATIONAL FIELD Jean Claude Dutailly Paris (France) 4 July 2015 Abstract Spinors have been used in Particle Physics since the Dirac’s equation. However their physical meaning is still obscure. In this paper we show that Spinors, vectors of a 4 dimensional complex vector space, can be used to represent the kinematic characteristics of particles,, encompassing both their transversal and their rotational parts. The framework used is the geometry of General Relativity, which is presented in a comprehensive and consistent way, by the use of fiber bundles. Spinors can be can be used at any scale, and the definition of a deformable solid body is introduced. The gravitational field is treated as a gauge field, through a connection on a fiber bundle. Using the spinor representation the action of the gravitation field on a material body, with the covariant derivative, takes then a sim- ple form. The propagation of the field is studied by a two form valued in the Lie algebra, similar to the Riemann tensor. In this formulation, more general but fully compatible with the more traditional approach starting from the metric and the scalar curvature, , the structure of the gravita- tional field can be explored, showing the existence of a rotational and a transversal component, and quantized, the spin being similar to the usual 3 dimensional gravity. 1 Contents 1 GEOMETRY OF GENERAL RELATIVITY 9 1.1 Manifoldstructure .......................... 10 1.1.1 The Universehasthe structureofamanifold . 10 1.1.2 What is a manifold ? . 10 1.1.3 ThemanifoldstructureoftheUniverse. 11 1.2 Thetangentvectorspace . 12 1.3 Vectorfields.............................. 13 1.4 Fundamentalsymmetrybreakdown . 14 1.5 Trajectoriesofmaterialbodies . 16 1.5.1 Material bodies and particles . 16 1.5.2 World line and proper time . 16 1.6 Metriconthemanifold........................ 17 1.6.1 Lorentzmetric ........................ 17 1.6.2 Gaugegroup ......................... 18 1.6.3 Orientation .......................... 20 1.7 VelocitieshaveaconstantLorentznorm . 21 1.8 Standardchartofanobserver . 22 1.9 Trajectoryandspeedofaparticle . 23 1.10Fiberbundles............................. 25 1.10.1 Generalfiberbundle . 25 1.10.2 Principal bundle . 25 1.10.3 Vectorbundle......................... 26 1.10.4 Associatedfiberbundle . 26 1.10.5 Scalarproductandnorm . 28 1.11 Standardgaugesassociatedto anobserver . 29 1.12 Formulasforachangeofobserver. 30 1.13TheTetrad .............................. 31 1.13.1 The principal fiber bundle . 31 1.13.2 Tetrad............................. 32 1.13.3 Metric............................. 34 1.13.4 Inducedmetric ........................ 34 1.14 Fromparticlestomaterialbodies . 35 1.15 Special Relativity . 37 1.16 Someissuesaboutrelativity . 40 1.16.1 Preferredframes . 40 1.16.2 Timetravel .......................... 41 1.16.3 Twinsparadox ........................ 41 1.16.4 TheexpansionoftheUniverse . 42 2 KINEMATICS 44 2.1 Translational Momentum in the relativist context . 44 2.2 Theissuesoftheconceptofrotation . 46 2.2.1 Rotation in Galilean Geometry . 46 2.2.2 The group of displacements in Relativist Geometry . 48 2 2.3 Momentainthefiberbundlerepresentation . 49 3 CLIFFORD ALGEBRAS AND SPINORS 51 3.1 Clifford algebra and Spin groups . 51 3.1.1 Clifford Algebras . 51 3.1.2 Spingroup .......................... 51 3.1.3 Adjointmap ......................... 52 3.1.4 Lie algebra of the Spin group . 52 3.1.5 Expressionofelementsofthespingroup . 54 3.2 ScalarproductandNorm .. .. .. .. .. .. .. .. .. 55 3.2.1 Scalar product on the Clifford algebra . 55 3.2.2 NormontheLiealgebra. 55 3.3 Symmetrybreakdown ........................ 57 3.3.1 Clifford algebra Cl(3) . 57 3.3.2 Decomposition of the elements of the Spin group . 58 3.4 RepresentationofCliffordalgebras . 60 3.4.1 Complexification of real Clifford algebras . 60 3.4.2 Algebraic and geometric representations . 60 3.4.3 Chirality . 61 3.4.4 Thechoiceoftherepresentation . 62 3.4.5 Expression of the matrices for the Lie algebra and the Spin groups 64 3.5 ScalarproductofSpinors .. .. .. .. .. .. .. .. .. 65 3.6 NormonthespaceEofspinors . 67 4 THESPINORMODELOFKINEMATICS 69 4.1 Descriptionofthefiberbundles . 69 4.1.1 Thegeometricfiberbundles. 69 4.1.2 The kinematic bundle . 70 4.1.3 Fundamentalsymmetrybreakdown. 71 4.2 TrajectoriesandtheSpinGroup . 72 4.3 Spatialspinor............................. 74 4.4 Inertialspinor............................. 75 4.5 Spaceandtimereversal . .. .. .. .. .. .. .. .. .. 78 4.5.1 Timereversal......................... 79 4.5.2 Spacereversal:........................ 79 4.6 TotalSpinor.............................. 80 5 SPINOR FIELDS 82 5.1 Definition ............................... 82 5.2 More on the theory of the representationsof groups . 83 5.2.1 FunctionalRepresentations . 83 5.2.2 Isomorphismsofgroups . 84 5.2.3 Representations of Spin(3,1), Spin(3) and SO(3) . 84 5.2.4 Casimir element . 85 5.3 TheSpinofaparticle ........................ 86 5.3.1 Definition ........................... 86 3 5.3.2 Quantization of the Spinor . 87 5.3.3 Measure of the spatial spin of a particle . 89 5.3.4 Atomsandelectrons . .. .. .. .. .. .. .. .. 90 5.4 Materialbodiesandspinors . 91 5.4.1 Representation of a material body by sections of PG ... 91 5.4.2 Spinors representing a solid . 91 5.4.3 Aggregatingmatterfields . 92 5.4.4 Continuity equation . 94 5.4.5 Symmetries of a solid . 94 5.5 RelativistMomentum,SpinandEnergy . 95 6 GRAVITATIONAL FIELD 97 6.1 TheLawofEquivalence ....................... 98 6.2 Representation of the charges for the other fields . 99 7 CONNECTIONS 102 7.1 ConnectionsonFiberbundles . 102 7.2 Theconnectionofthegravitationalfield . 102 7.2.1 Potential ........................... 102 7.2.2 Covariant derivative on PG .................104 7.2.3 Covariant derivative for spinors . 104 7.2.4 Covariant derivatives for vector fields on M . 105 7.3 Kineticandpotentialenergy. 106 7.3.1 Kineticenergy ........................ 106 7.3.2 Inertialtensor. .. .. .. .. .. .. .. .. .. .. 107 7.3.3 The electromagnetic field . 109 7.4 Geodesics ...............................110 7.5 TheLeviCivitaconnection . 112 7.6 Theinertialobserver . 114 8 THE PROPAGATION OF FIELDS 116 8.1 Thestrengthoftheconnection . 116 8.1.1 Definition ........................... 116 8.1.2 Adjointbundle ........................ 118 8.2 Scalarcurvature ........................... 119 8.2.1 Riemann curvature of a principal connection . 119 8.2.2 Riemanntensorofanaffineconnection. 120 8.2.3 Riccitensorandscalarcurvature . 121 9 THE RELATIVIST MOMENTUM OF THE GRAVITATIONAL FIELD124 9.1 Polarizationofthefields . 124 9.2 From the holonomic basis of a chart to the orthonormal basis . 124 9.3 ThedualCliffordbundle. 125 9.4 2formsexpressedintheCliffordbundle . 126 9.5 Strength of the gravitational field in the Clifford algebras . 127 4 10 ENERGY OF THE GRAVITATIONAL FIELD 129 10.1 ScalarproductofformsoverM . 129 10.2 ScalarproductsontheLiealgebras. 130 10.3 Scalarproductforthestrengthofthefield. 131 10.4Energyofthefield .......................... 132 10.4.1 Identity ............................ 133 10.5 Norm on the spaces of the relativist momentum of the fields . 133 11 STRUCTURE OF THE GRAVITATIONAL FIELD 135 11.1 Quantization of the gravitational field . 135 11.2 Spin of the Gravitational Field . 135 11.3 Scalarcurvature ........................... 136 11.3.1 Symmetryoftheforcefield . 136 12 CONCLUSION 138 13 BIBLIOGRAPHY 140 14ANNEX1:CLIFFORDALGEBRAS 141 14.1 ProductsintheCliffordalgebra. 141 14.1.1 Product υ (r, w) υ (r′, w′)..................141 14.1.2 Product on Spin·(3, 1) ....................142 14.2 Characterization of the elements of the Spin group . 143 14.2.1 Inverse............................. 143 14.2.2 Relation between a,b, r, w . 143 14.3 HomogeneousSpace . 144 14.3.1 The sets isomorphic to Cl (3) ................144 14.3.2 Decomposition of the Lie algebra . 145 14.3.3 Homogeneousspace . 146 14.4Adjointmap.............................. 147 14.5Derivatives .............................. 148 14.6 Exponential on T1Spin ........................149 15 ANNEX 2 : FORMULAS 152 15.1 RelativistGeometry . 152 15.2Operatorj............................... 152 15.2.1 eigenvectors: . .. .. .. .. .. .. .. .. .. .. 152 15.2.2 Identities ........................... 153 15.2.3 Polynomials . 153 15.3 Dirac’smatrices............................ 153 15.4 γ matrices...............................154 15.5Cliffordalgebra............................ 155 15.5.1 LieAlgebras ......................... 155 15.5.2 Spingroups.......................... 155 5 Mechanics is the mother of Physics : it states the rules linking the motion of material bodies to the forces exercised by the fields. Material bodies oppose a resistance in the change of their
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