Kinematical Theory of Spinning Particles: Classical and Quantum Mechanical Formalism of Elementary Particles

Kinematical Theory of Spinning Particles: Classical and Quantum Mechanical Formalism of Elementary Particles

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English

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Classical spin is described in terms of velocities and acceleration so that knowledge of advanced mathematics is not required. Written in the three-dimensional notation of vector calculus, it can be followed by undergraduate physics students, although some notions of Lagrangian dynamics and group theory are required. It is intended as a general course at a postgraduate level for all-purpose physicists. This book presents a unified approach to classical and quantum mechanics of spinning particles, with symmetry principles as the starting point. A classical concept of an elementary particle is presented. The variational statements to deal with spinning particles are revisited. It is shown that, by explicitly constructing different models, symmetry principles are sufficient for the description of either classical or quantum-mechanical elementary particles. Several spin effects are analyzed.

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Published 01 January 2001
Reads 4
EAN13 0306471337
License: All rights reserved
Language English

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Contents
List of Figures Acknowledgment Introduction
1. GENERAL FORMALISM 1 . Introduction 1 . 1 Kinematics and dynamics 2 . Variational versus Newtonian formalism 3 . Generalized Lagrangian formalism 4 . Kinematical variables 4 . 1 Examples 5 . Canonical formalism 6 . Lie groups of transformations 6 . 1 Casimir operators 6 . 2 Exponents of a group 6 . 3 Homogeneous space of a group 7 . Generalized Noether’s theorem 8 . Lagrangian gauge functions 9 . Relativity principle. Kinematical groups 10 .Elementary systems 10.1Elementary Lagrangian systems 11 .The formalism with the simplest kinematical groups 2. NONRELATIVISTIC ELEMENTARY PARTICLES 1 . Galilei group 2 . Nonrelativistic point particle 3 . Galilei spinning particles 4 . Galilei free particle with (anti)orbital spin 4 . 1 Interacting with an external electromagnetic field 4 . 2 Canonical analysis of the system vii
x i x v i i x i x
1 1 3 4 8 1 0 1 6 1 7 1 9 2 3 2 3 2 5 2 6 3 1 3 3 3 4 3 9 4 0
4 7 4 8 5 1 5 5 6 3 6 7 6 9
viii
KINEMATICAL THEORY OF SPINNING PARTICLES
4.3Spinning particle in a uniform magnetic field 4.4Spinning particle in a uniform electric field 4.5Circular zitterbewegung 5 .Spinning Galilei particle with orientation 6 .General nonrelativistic spinning particle 6.1Circular zitterbewegung 6.2Classical non-relativistic gyromagnetic ratio 7 .Interaction with an external field 8 . Two-particle systems 8.1Synchronous description 9 . Two interacting spinning particles 3. RELATIVISTIC ELEMENTARY PARTICLES 1 .Poincaré group 1.1Lorentz group 2 .Relativistic point particle 3 .Relativistic spinning particles 3.1Bradyons 3.2Relativistic particles with (anti)orbital spin 3.3Canonical analysis 3.4Circular zitterbewegung 4 .Luxons 4.1Massless particles. (The photon) 4.2Massive particles. (The electron) 5 .Tachyons 6 .Inversions 7 .Interaction with an external field 4. QUANTIZATION OF LAGRANGIAN SYSTEMS 1 .Feynman’s quantization of Lagrangian systems 1.1Representation of Observables 2 . Nonrelativistic particles 2.1Nonrelativistic point particle 2.2Nonrelativistic spinning particles. Bosons 2.3Nonrelativistic spinning particles. Fermions 2.4General nonrelativistic spinning particle 3 . S p i n o r s 3.1Spinor representation on SU(2) 3.2Matrix representation of internal observables 3.3Peter-Weyl theorem for compact groups 3.4General spinors 4 . Relativistic particles 4.1Relativistic point particle 4.2General relativistic spinning particle 4.3Dirac’s equation
72 84 85 86 87 90 92 93 98 99 103
109 110 113 118 121 121 137 142 145 147 148 151 160 162 163
169 170 173 177 177 179 182 184 185 189 196 196 200 203 203 204 206
C o n t e n t si x
4 . 4 Dirac’s algebra 4 . 5 Photon quantization 4 . 6 Quantization of tachyons 5. OTHER SPINNING PARTICLE MODELS 1 . Group theoretical models 1 . 1 Hanson and Regge spinning top 1 . 2 Kirillov-Kostant-Souriau model 1 . 3 Bilocal model 2 . Non-group based models 2 . 1 Spherically symmetric rigid body 2 . 2 Weyssenhoff-Raabe model 2 . 3 Bhabha-Corben model 2 . 4 Bargmann-Michel-Telegdi model 2 . 5 Barut-Zanghi model 2 . 6 Entralgo-Kuryshkin model 6. SPIN FEATURES AND RELATED EFFECTS 1 .Electromagnetic structure of the electron 1 . 1 The time average electric and magnetic field 1 . 2 Gyromagnetic ratio 1 . 3 Instantaneous electric dipole 1 . 4 Darwin term of Dirac’s Hamiltonian 2 .Classical spin contribution to the tunnel effect 2 . 1 Spin polarized tunneling 3 .Quantum mechanical position operator 4 . Finsler structure of kinematical space 4 . 1 Properties of the metric 4 . 2 Geodesics on Finsler space 4 . 3 Examples 5 .Extending the kinematical group 5 . 1 Space-time dilations 5 . 2 Local rotations 5 . 3 Local Lorentz transformations 6 .Conformal invariance 6 . 1 Conformal group 6 . 2 Conformal group of Minkowski space 6 . 3 Conformal observables of the photon 6 . 4 Conformal observables of the electron 7 .Classical Limit of quantum mechanics Epilogue References I n d e x
2 1 5 2 1 7 2 1 8
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 3 3 3 4 4 4 4
1 1 1 5 8 2 2 4 0 3 5 8
2 5 3 2 5 4 2 5 4 2 6 4 2 6 6 2 7 0 2 7 0 2 7 8 2 7 9 2 8 5 2 8 7 2 8 8 2 9 0 2 9 1 2 9 2 2 9 3 2 9 4 2 9 5 2 9 5 2 9 8 3 0 4 3 0 5 3 0 6 3 1 1 3 1 5 3 2 9