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Relativistic Elementary Particles 47 1 Kinematical Theory of Spinning Particles Fundamental Theories of Physics An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A. Editorial Advisory Board: JAMES T. CUSHING, University of Notre Dame, U.S.A. GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada TONY SUDBURY, University of York, U.K. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschaften, Germany Volume 116 Kinematical Theory of Spinning Particles Classical and Quantum Mechanical Formalism of Elementary Particles by Martin Rivas The University of the Basque Country, Bilbao, Spain KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT,LONDON, MOSCOW eBook ISBN 0-306-47133-7 Print ISBN 0-792-36824-X ©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://www.kluweronline.com and Kluwer's eBookstore at: http://www.ebooks.kluweronline.com To Merche This page intentionally left blank. Contents List of Figures xi Acknowledgment xvii Introduction xix 1. GENERAL FORMALISM 1 1. Introduction 1 1 . 1 Kinematics and dynamics 3 2 . Variational versus Newtonian formalism 4 3. Generalized Lagrangian formalism 8 4 . Kinematical variables 10 4.1 Examples 16 5 . Canonical formalism 17 6. Lie groups of transformations 19 6.1 Casimir operators 23 6.2 Exponents of a group 23 6 . 3 Homogeneous space of a group 25 7 . Generalized Noether’s theorem 26 8 . Lagrangian gauge functions 31 9 . Relativity principle. Kinematical groups 33 10 . Elementary systems 34 10.1 Elementary Lagrangian systems 39 11 . The formalism with the simplest kinematical groups 40 2. NONRELATIVISTIC ELEMENTARY PARTICLES 47 1 . Galilei group 48 2 . Nonrelativistic point particle 51 3 . Galilei spinning particles 55 4 . Galilei free particle with (anti)orbital spin 63 4 . 1 Interacting with an external electromagnetic field 67 4.2 Canonical analysis of the system 69 vii viii KINEMATICAL THEORY OF SPINNING PARTICLES 4.3 Spinning particle in a uniform magnetic field 72 4.4 Spinning particle in a uniform electric field 84 4.5 Circular zitterbewegung 85 5 . Spinning Galilei particle with orientation 86 6 . General nonrelativistic spinning particle 87 6.1 Circular zitterbewegung 90 6.2 Classical non-relativistic gyromagnetic ratio 92 7 . Interaction with an external field 93 8. Two-particle systems 98 8.1 Synchronous description 99 9. Two interacting spinning particles 103 3. RELATIVISTIC ELEMENTARY PARTICLES 109 1 . Poincaré group 110 1.1 Lorentz group 113 2 . Relativistic point particle 118 3 . Relativistic spinning particles 121 3.1 Bradyons 121 3.2 Relativistic particles with (anti)orbital spin 137 3.3 Canonical analysis 142 3.4 Circular zitterbewegung 145 4 . Luxons 147 4.1 Massless particles. (The photon) 148 4.2 Massive particles. (The electron) 151 5 . Tachyons 160 6 . Inversions 162 7 . Interaction with an external field 163 4. QUANTIZATION OF LAGRANGIAN SYSTEMS 169 1 . Feynman’s quantization of Lagrangian systems 170 1.1 Representation of Observables 173 2. Nonrelativistic particles 177 2.1 Nonrelativistic point particle 177 2.2 Nonrelativistic spinning particles. Bosons 179 2.3 Nonrelativistic spinning particles. Fermions 182 2.4 General nonrelativistic spinning particle 184 3. Spinors 185 3.1 Spinor representation on SU(2) 189 3.2 Matrix representation of internal observables 196 3.3 Peter-Weyl theorem for compact groups 196 3.4 General spinors 200 4. Relativistic particles 203 4.1 Relativistic point particle 203 4.2 General relativistic spinning particle 204 4.3 Dirac’s equation 206 Contents ix 4 . 4 Dirac’s algebra 215 4.5 Photon quantization 2 1 7 4.6 Quantization of tachyons 2 1 8 5. OTHER SPINNING PARTICLE MODELS 221 1 . Group theoretical models 221 1 . 1 Hanson and Regge spinning top 221 1 . 2 Kirillov-Kostant-Souriau model 225 1.3 Bilocal model 228 2 . Non-group based models 232 2 . 1 Spherically symmetric rigid body 232 2 . 2 Weyssenhoff-Raabe model 234 2.3 Bhabha-Corben model 240 2 . 4 Bargmann-Michel-Telegdi model 243 2.5 Barut-Zanghi model 245 2.6 Entralgo-Kuryshkin model 248 6. SPIN FEATURES AND RELATED EFFECTS 253 1 . Electromagnetic structure of the electron 254 1.1 The time average electric and magnetic field 254 1 . 2 Gyromagnetic ratio 264 1.3 Instantaneous electric dipole 266 1 . 4 Darwin term of Dirac’s Hamiltonian 270 2 . Classical spin contribution to the tunnel effect 270 2.1 Spin polarized tunneling 278 3 . Quantum mechanical position operator 279 4. Finsler structure of kinematical space 285 4.1 Properties of the metric 287 4 . 2 Geodesics on Finsler space 288 4.3 Examples 290 5 . Extending the kinematical group 291 5.1 Space-time dilations 292 5.2 Local rotations 293 5.3 Local Lorentz transformations 294 6 . Conformal invariance 295 6.1 Conformal group 295 6.2 Conformal group of Minkowski space 298 6.3 Conformal observables of the photon 304 6.4 Conformal observables of the electron 305 7 . Classical Limit of quantum mechanics 306 Epilogue 311 References 315 Index 329 This page intentionally left blank. List of Figures 1.1 Evolution in {t , r } space. 5 1.2 Evolution in kinematical {t, r , α } space. 43 2.1 Charge motion in the C.M. frame. 69 2.2 Initial phase of the charge and initial orientation (θ, φ ) of angular velocity ω. 76 2.3 Motion of the center of charge and center of mass of a negative charged particle in a uniform mag- netic field. The velocity of the center of mass is orthogonal to the field. 79 2.4 Precession of spin around the OZ axis. 81 2.5 Representation of observables U , W and Z in the center of mass frame. 89 2.6 Zitterbewegung of system described by Lagrangian (2.137), showing the relation between the spin ob- servables S , Y and W 90 2.7 Positive charged particles with parallel (a ) and an- tiparallel (b), spin and magnetic moment. 93 2.8 Representation in the C.M. frame of the motion of the center of mass of two opposite charged particles with antiparallel spins. For particle 2, the motion of its center of charge is also represented. The interaction produces a chaotic scattering. 105 2.9 Representation in the C.M. frame of the motion of the center of mass of an electron-positron pair, producing a bound state. 106 3.1 Motion of the center of charge of the system (3.97) around the C.M. 135 3.2 Motion of the center of charge of the system (3.98) around the C.M. 136 xi xii KINEMATICAL THEORY OF SPINNING PARTICLES 3.3 Possible zitterbewegung trajectories of spinning rel- ativistic particles. 141 3.4 System of three orthogonal vectors linked to the motion of the charge of the electron with αe = ω e = y. 156 3.5 Center of charge motion of system (3.178) in the C.M. frame. 158 4.1 Motion of charge in the C.M. frame. 207 4.2 Orientation in the Pauli-Dirac representation. 211 4.3 Orientation in the Weyl representation. 213 4.4 Space reversal of the electron is equivalent to a rotation of value π along S . 214 4.5 Time reversal of the electron produces a particle of negative energy. 214 5.1 Motion of the particle in the C.M. frame. 239 6.1 Instantaneous electric field of the electron at point P has a component along – a ⊥ and – β. 255 6.2 Average retarded (or advanced) electric field (6.3) and Coulomb field along the OZ axis. 256 6.3 Charge motion and observation point P. 257 6.4 Time average < Er ( r ) > of the radial component of the retarded electric field in the directions θ = 0,π /3, π /4 and π /6. 259 6.5 Time average of the component < Eθ ( r ) > of the retarded electric field in the directions θ = 0, π /3, π /4 andπ /6. It goes to zero very quickly. For θ = π /2 it vanishes everywhere. 260 6.6 Radial components of the dipole field B0r (r ) and the time average retarded magnetic field < Br ( r ) >, along the direction θ = π /6. 261 6.7 Radial components of the dipole field B0r ( r ) and the time average retarded magnetic field < Br ( r ) >, along the direction θ = π /4. 261 6.8 Radial components of the dipole field B0r ( r ) and the time average retarded magnetic field < Br (r ) >, along the direction θ = π /3. 261 6.9 Time average retarded magnetic field < B θ ( r ) > and the dipole field B 0 θ(r ), along the direction θ = π /6. 262 6.10 Time average retarded magnetic field < B θ ( r ) > and the dipole field B 0θ ( r ), along the direction θ = π /4. 262 6.11 Time average retarded magnetic field < Bθ ( r ) > and the dipole field B 0θ ( r ), along the direction θ = π /3. 262 List of Figures xiii 6.12 Time average retarded magnetic field < Br (r ) > along the directions θ = 0, π/3, π/4 and π/6 and its behavior at r = 0. For θ = π /2 it vanishes everywhere. 263 6.13 Time average retarded magnetic field < Bθ( r)> along the directions θ = 0, π/3, π/4 and π/6 and its behavior at r = 0.
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