Kinematical Theory of Spinning Particles - Classical and Quantum - M. Rivas




Contains:



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
10.1
Elementary 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

viii

KINEMATICAL THEORY OF SPINNING PARTICLES

Spinning particle in a uniform magnetic field
4.3
Spinning particle in a uniform electric field
4.4
4.5
Circular zitterbewegung
Spinning Galilei particle with orientation
General nonrelativistic spinning particle
6.1
Circular zitterbewegung
6.2
Classical non-relativistic gyromagnetic ratio
Interaction with an external field
Two-particle systems
8.1
Synchronous description
Two interacting spinning particles

5.
6.

7.
8.

9.

3. RELATIVISTIC
ELEMENTARY PARTICLES
1 . Poincaré group
1.1
Lorentz group
2 . Relativistic point particle
3 . Relativistic spinning particles
3.1
Bradyons
3.2
Relativistic particles with (anti)orbital spin
3.3
Canonical analysis
3.4
Circular zitterbewegung
4 . Luxons
4.1
Massless particles. (The photon)
4.2
Massive 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
1.1
Representation of Observables
2 . Nonrelativistic particles
2.1
Nonrelativistic point particle
2.2
Nonrelativistic spinning particles. Bosons
2.3
Nonrelativistic spinning particles. Fermions
2.4
General nonrelativistic spinning particle
3. Spinors
3.1
Spinor representation on SU(2)
3.2
Matrix representation of internal observables
3.3
Peter-Weyl theorem for compact groups
3.4
General spinors
4 . Relativistic particles
4.1
Relativistic point particle
4.2
General relativistic spinning particle
4.3
Dirac’s equation

Contents i x

4.4
4.5
4.6

5. OTHER SPINNING PARTICLE
MODELS
1 . Group theoretical models
1.1
Hanson and Regge spinning top
1.2
Kirillov-Kostant-Souriau model
Bilocal model
1.3
2 . Non-group based models
2.1
Spherically symmetric rigid body
2.2
Weyssenhoff-Raabe model
Bhabha-Corben model
2.3
2.4
Bargmann-Michel-Telegdi model
Barut-Zanghi model
2.5
Entralgo-Kuryshkin model
2.6

6. SPIN FEATURES
AND RELATED EFFECTS
1 . Electromagnetic structure of the electron
The time average electric and magnetic field
1.1
1.2
Gyromagnetic ratio
Instantaneous electric dipole
1.3
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
Examples
4.3
5 . Extending the kinematical group
Space-time dilations
5.1
Local rotations
5.2
Local Lorentz transformations
5.3
6 . Conformal invariance
Conformal group
6.1
Conformal group of Minkowski space
6.2
Conformal observables of the photon
6.3
Conformal observables of the electron
6.4
7 . Classical Limit of quantum mechanics

Dirac’s algebra
Photon quantization
Quantization of tachyons



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