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SPINORS and SPACE–TIME ANISOTROPY Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE{TIME ANISOTROPY University of Athens ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii - i ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. ii - iii ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics. Panayiotis Stavrinos is Assistant Professor in the University of Athens, where he obtained his Ph. D in 1990 and hold lecturer positions during 1990-1999. He is a Founding Member and Vice President of Balkan Soci- ety of Geometers , Member of the Editorial Board of the Journal of Balkan Society. Honorary Member to The Research Board of Advisors, American Biographical Institute (U.S.A.), 1996. Member of Tensor Society, (Japan), 1981. Dr. Stavrinos has published over 40 research works in different in- ternational Journals in the topics of local differential geometry, Finsler and Lagrange Geometry, applications of Finsler and Lagrange geometry to grav- itation, gauge and spinor theory as well as Einstein equations, deviation of geodesics, tidal forces, weak gravitational fields, gravitational waves. He is co-author in the monograph ”Introduction to the Physical Principles of Dif- ferential Geometry”, in Russian, published in St. Petersburg in 1996 (second edition in English, University of Athens Press, 2000). He has published two monographs in Greek for undergraduate and graduate students in the Department of Mathematics and Physics : ”Differential Geometry and its Applications Vol. I, II (University of Athens Press, 2000). iv Contents 0.1Preface............................... vii 0.1.1 Historicalremarksonspinortheory........... vii 0.1.2 Metric Spaces depending on Spinor Variables and Gauge FieldTheories....................... ix 0.1.3 Nonlinearconnectiongeometryandphysics....... x 0.1.4 Anholonomic frames and nonlinear connections in Ein- steingravity........................xiv 0.1.5 Thelayoutofthebook..................xvi 0.1.6 Acknowledgments....................xvii 0.2Notation..............................xix I Space{Time Anisotropy 1 1 Vector Bundles and N{Connections 3 1.1 Vector and Covector Bundles .................. 4 1.1.1 Vector and tangent bundles ............... 4 1.1.2 Covector and cotangent bundles ............. 5 1.1.3 Higher order vector/covector bundles . ......... 6 1.2NonlinearConnections...................... 10 1.2.1 N–connections in vector bundles ............. 10 1.2.2 N–connections in covector bundles: . ......... 11 1.2.3 N–connections in higher order bundles ......... 12 1.2.4 AnholonomicframesandN–connections........ 13 1.3Distinguishedconnectionsandmetrics............. 19 1.3.1 D–connections....................... 19 1.3.2 Metricstructure...................... 22 1.3.3 Someremarkabled–connections............. 25 1.3.4 AmostHermitiananisotropicspaces........... 27 1.4 TorsionsandCurvatures..................... 29 1.4.1 N–connectioncurvature................. 29 1.4.2 d–Torsions in v- and cv–bundles ............. 30 v vi CONTENTS 1.4.3 d–Curvatures in v- and cv–bundles ........... 31 1.5GeneralizationsofFinslerSpaces................ 32 1.5.1 FinslerSpaces....................... 32 1.5.2 LagrangeandGeneralizedLagrangeSpaces....... 34 1.5.3 CartanSpaces....................... 35 1.5.4 GeneralizedHamiltonandHamiltonSpaces...... 37 1.6 Gravity on Vector Bundles . ................. 38 2 Anholonomic Einstein and Gauge Gravity 41 2.1Introduction............................ 41 2.2AnholonomicFrames....................... 42 2.3HigherOrderAnisotropicStructures.............. 48 2.3.1 Ha–frames and corresponding N–connections ...... 48 2.3.2 Distinguishedlinearconnections............. 52 2.3.3 Ha–torsionsandha–curvatures............. 54 2.3.4 Einstein equations with respect to ha–frames ...... 55 2.4GaugeFieldsonHa–Spaces................... 56 2.4.1 Bundles on ha–spaces . ................. 57 2.4.2 Yang-Mills equations on ha-spaces ............ 60 2.5GaugeHa-gravity......................... 63 2.5.1 Bundles of linear ha–frames ............... 64 2.5.2 Bundles of affine ha–frames and Einstein equations . 65 2.6NonlinearDeSitterGaugeHa–Gravity............. 66 2.6.1 NonlineargaugetheoriesofdeSittergroup....... 67 2.6.2 Dynamics of the nonlinear de Sitter ha–gravity .... 69 2.7AnAnsatzfor4Dd–Metrics................... 72 2.7.1 Theh–equations..................... 74 2.7.2 Thev–equations..................... 75 2.7.3 H–vequations....................... 76 2.8AnisotropicCosmologicalSolutions............... 77 2.8.1 Rotation ellipsoid FRW universes ............ 77 2.8.2 ToroidalFRWuniverses................. 79 2.9 ConcludingRemarks....................... 80 3 Anisotropic Taub NUT { Dirac Spaces 85 3.1N–connectionsinGeneralRelativity............... 85 3.1.1 AnholonomicEinstein–DiracEquations......... 88 3.1.2 AnisotropicTaubNUT–DiracSpinorSolutions.... 94 3.2AnisotropicTaubNUTSolutions................ 96 3.2.1 AconformaltransformoftheTaubNUTmetric.... 97 CONTENTS vii 3.2.2 Anisotropic Taub NUT solutions with magnetic polar- ization........................... 99 3.3AnisotropicTaubNUT–DiracFields..............101 3.3.1 Diracfieldsandangularpolarizations..........101 3.3.2 Diracfieldsandextradimensionpolarizations.....103 3.4AnholonomicDirac–TaubNUTSolitons............104 3.4.1 Kadomtsev–Petviashvilitypesolitons..........105 3.4.2 (2+1)sine–Gordontypesolitons.............106 II Anisotropic Spinors 109 4 Anisotropic Clifford Structures 113 4.1DistinguishedCliffordAlgebras.................113 4.2 Anisotropic Clifford Bundles ...................118 4.2.1 Cliffordd-modulestructure...............118 4.2.2 AnisotropicCliffordfibration..............120 4.3AlmostComplexSpinors.....................121 5 Spinors and Anisotropic Spaces 127 5.1AnisotropicSpinorsandTwistors................128 5.2 MutualTransformsofTensorsandSpinors...........133 5.2.1 Transformationofd-tensorsintod-spinors.......133 5.2.2 Fundamental d–spinors .................134 5.3AnisotropicSpinorDifferentialGeometry............135 5.4 D-covariantderivation......................136 5.5Infeld-vanderWaerdencoefficients..............138 5.6 D-spinorsofAnisotropicCurvatureandTorsion........140 6 Anisotropic Spinors and Field Equations 143 6.1 AnisotropicScalarFieldInteractions..............143 6.2 AnisotropicProcaequations...................145 6.3 Anisotropic Gravitons and Backgrounds . .........146 6.4 AnisotropicDiracEquations..................146 6.5 Yang-Mills Equations in Anisotropic Spinor Form .......147 III Higher Order Anisotropic Spinors 149 7 Clifford Ha{Structures 153 7.1DistinguishedCliffordAlgebras.................153 7.2 Clifford Ha–Bundles . ....................158 viii CONTENTS 7.2.1 Clifford d–module structure in dv–bundles . ......158 7.2.2 Cliffordfibration.....................160 7.3AlmostComplexSpinorStructures...............161 8 Spinors and Ha{Spaces 165 8.1 D–SpinorTechniques.......................165 8.1.1 Cliffordd–algebra,d–spinorsandd–twistors......166 8.1.2 Mutualtransformsofd-tensorsandd-spinors.....169 8.1.3 Transformationofd-tensorsintod-spinors.......169 8.1.4 Fundamental d–spinors .................170 8.2 DifferentialGeometryofHa–Spinors..............171 8.2.1 D-covariantderivationonha–spaces..........172 8.2.2 Infeld–vanderWaerdencoefficients...........174 8.2.3 D–spinorsofha–spacecurvatureandtorsion......176 9 Ha-Spinors and Field Interactions 179 9.1Scalarfieldha–interactions....................179 9.2 Procaequationsonha–spaces..................181 9.3 HigherorderanisotropicDiracequations............182 9.4 D–spinor Yang–Mills fields . .................183 9.5D–spinorEinstein–CartanTheory................184 9.5.1 Einsteinha–equations..................184 9.5.2 Einstein–Cartand–equations...............185 9.5.3 Higherorderanisotropicgravitons............185 IV Finsler Geometry and Spinor Variables 187 10 Metrics Depending on Spinor Variables 189 10.1LorentzTransformation......................189 10.2Curvature.............................193 11 Field Equations in Spinor Variables 199 11.1Introduction............................199 11.2Derivationofthefieldequations.................201
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