Fields of Particles with Spin, Theory and Applications


Alina Ivashkevich, PhD – Researcher, B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Republic of Belarus
Nina Krylova, PhD  – Assistant Professor, Physical Department, Belarusian State University, Republic of Belarus
Elena Ovsiyuk, PhD  – Assistant Professor, Physical Department, Mozyr State Pedagogical University, Republic of Belarus
Vasiiliy Kisel, PhD – Assistant Professor, Belarusian State University of Informatics and Radio-Electronics, Republic of Belarus
Vladimir Balan, PhD – Department of Mathematics-Informatics, University Politehnica of Bucharest, Romania
Viktor Red’kov, PhD – Fundamental Interactions and Astrophysics Department, B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Republic of Belarus

Series: Physics Research and Technology
BISAC: SCI055000

The present book is devoted to the study of particles with spins in external fields and non-Euclidean space-time background. The key problems are: Coulomb task for a spin 1/2 particle and Heun equation; the hydrogen atom in de Sitter space; the fermion doublet in the non-Abelian monopole field and Pauli approximation; Pauli approximation for spin 1/2 and 1 particles in de Sitter space; the Dirac and Majorana particles in Schwarzschild space; the Dirac – Maxwell fields and spinor space structure; particles with spin 3/2, solutions with different symmetries and eliminating the gauge degrees of freedom in massless case; the matrix 30-component equation for a spin 2 field in Riemannian space-time; Finslerian geometzization of physical problems. The book may be of interest to researchers. It may serve as a pedagogical tool for either self-study or in courses at both the undergraduate and graduate level.

Table of Contents



1. Confluent Heun Functions and the Coulomb Problem for a Spin 1/2 Particle
1.1. The Coulomb Problem: Solutions Constructed by Hypergeometric and Partially by Heun Functions
1.2. Standard Treatment of the Coulomb Problem
1.3. Solutions Constructed Completely in Terms of Heun Functions

2. A Spin 1/2 Particle in 2D Spaces of Constant Curvature, in the Presence of a Magnetic field
2.1. Cylindric and Conformal Coordinates in Lobachevsky Plane H2
2.2. Landau Problem for a Scalar Particle in the Plane H2
2.3. Dirac Particle in (x, y) Coordinates, Model H2
2.4. Landau Problem in the Spherical Model S2, Coordinates (r,ϕ)
2.5. Complex Poincar´e Half-Plane for Spherical 2-Space

3. Hydrogen Atom in Static de Sitter Spaces
3.1. Separation of the Variables in dS Space
3.2. Qualitative Discussion
3.3. Reducing Radial Equation to the General Heun Equation
3.4. Semi-Classical Study
3.5. The Hydrogen Atom in AdS Space
3.6. Qualitative Study of the Problem in AdS Space
3.7. Semi-Classical Study for AdS Space
3.8. A Spin 1/2 Particle in dS and AdS Spaces

4. Scalar Particle in Non-Static de Sitter Spaces
4.1. Nonrelativistic Approximation, and Riemann Geometry
4.2. Schr ö dinger Equation in de Sitter Non-Static Models
4.3. Solving Equations
4.4. Klein–Gordon–Fock Equation in Curved Space-Time
4.5. Solving Equation in Expanding de Sitter Metric
4.6. Solving Equation in Oscillating de Sitter Metric

5. A Spin 1/2 Particle in Nonstatic de Sitter Spaces, Spherical Coordinates
5.1. Particle in Expanding de Sitter Model
5.2. Neutrino in Expanding de Sitter Space
5.3. Pauli Equation in Expanding de Sitter Space
5.4. A Spin 1/2 Particle in Oscillating de Sitter Model
5.5. Pauli Equation in Oscillating de Sitter Space

6. A Spin 1/2 Particle in Non-static de Sitter Models, Quasi-Cartesian Coordinates
6.1. Separation of the Variables in the Dirac Equation
6.2. Solving Equations in the Time Variable t
6.3. Behavior of Solutions in the Variable t near the Points cos(t) = 0
6.4. Constructing Solutions in the Variable z
6.5. Majorana Spinor Field
6.6. Independent Majorana Components

7. The Fermion Doublet in Non-Abelian Monopole Field, Pauli Approximation, Geometry
7.1. Pauli Equation for Fermion Doublet, GeneralAnalysis
7.2. Non-Abelian Monopole in Schwinger Gauge
7.3. Separating the Variables
7.4. Nonrelativistic Approximation, the Case j = 0
7.5. Nonrelativistic Approximation, the Case j > 0
7.6. The Doublet in the Spaces of Constant Curvature
7.7. Geometrization of the Monopole Problem, KCC-Invariants
7.8. The Euclidean Space
7.9. Riemannian Space
7.10. Lobachevsky Space
7.11. Pure Monopole BPZ-Solution, Euclidean Space
7.12. Geometrizing the Doublet Problem, the Case j = 0
7.13. Non-Relativistic Approximation, the Case j > 0

8. Analysis of the Dirac and Majorana Particle in a Schwarzschild Field
8.1. Dirac and Weyl Equations in an External Gravitational Field
8.2. Majorana Spinor Fields
8.3. A Spin 1/2 Particle in the Schwarzschild Field
8.4. Separation of the Variables
8.5. The Case of Majorana Particle
8.6. Qualitative Study
8.7. Analytical Treatment
8.8. Structure of the Power Series
8.9. General Study of the Tunneling Effect
8.10. Geometrization of the Maxwell and Dirac Theories in Schwarzschield Space-Time

9. Dirac Particle in Cylindric Parabolic Coordinates and Spinor Space Structure
9.1. Spinor Structure and Solutions of the Klein–Gordon–Fock Equation
9.2. Solutions of the Klein–Gordon–Fock Equation and Spinors
9.3. The Dirac Particle and the Space with Spinor Structure

10. Maxwell Equations in Space with Spinor Structure
10.1. Spinor Form of Maxwell Equations
10.2. Cylindrical Parabolic Coordinates
10.3. Continuity and Spinor Space Structure
10.4. Helicity Operator

11. Geometrization of Maxwell Electrodynamics
11.1. Optics and Lagrange Formalism
11.2. The Euler–Lagrange Equations

12. Finslerian Geometrization for the Problem of a Vector Particle in an External Coulomb Field
12.1. Setting the Problem
12.2. KCC-invariants
12.3. Second KCC-Invariant
12.4. Natural Splitting 4+4
12.5. Natural splitting, real-valued representation
12.6. Projections of 8 Equations on Different Planes

13. The Study of a Spin 1 Particle with Anomalous Magnetic Moment in the Coulomb Field
13.1. Separation of the Variables
13.2. The Case of Minimal j = 0
13.3. The Non-Relativistic Approximation at j = 0
13.4. Non-Relativistic Equations, j = 1, 2, 3,
13.5. KCC-Geometrical Approach to the Problem

14. Vector Particle with Electric Quadruple Moment in the Coulomb Field
14.1. Initial Equation
14.2. Separating the Variables in the Relativistic Equation
14.3. States with Parity P = (−1)j+1
14.4. The Case of Minimal j = 0
14.5. Non-Relativistic Approximation, P = (−1)j+1, j = 1, 2, 3,
14.6. Non-Relativistic Radial Equations, the Case of j = 1, 2, 3,
14.7. KCC-Geometrical Approach

15. Massive and Massless Fields with Spin 3/2, Solutions and Helicity Operator
15.1. Massive and Massless Spin 3/2 Fields
15.2. Separating the Variables
15.3. Helicity Operator
15.4. Helicity Operator and Solutions of the Wave Equation
15.5. The Plane Wave Solutions in Massless Case
15.6. Relation to Initial Basis
15.7. Helicity Operator

16. Solutions with Spherical Symmetry for a Massive Spin 3/2 Particle
16.1. System of Equations and Spherical Symmetry
16.2. Separating the Variables
16.3. Separating the Variables and Additional Constraints
16.4. Solving Equations for Functions f0, g0
16.5. The Matrix Form of the Main System
16.6. The Case of Minimal Value j = 1/2
16.7. Studying General Case j = 3/2, 5/2,
16.8. Further Study of the Solutions
16.9. Accounting for Algebraic and Differential Constraints

17. Massless Spin 3/2 Field, Spherical Solutions, Exclusion of the Gauge Degrees of Freedom
17.1. Massless Spin 3/2 Particle, General Theory
17.2. Separation of the Variables
17.3. Gradient Type Solutions
17.4. Solving the System of Radial Equations
17.5. Solving the Homogeneous Equations

18. Spin 3/2 Massless Field, Cylindric Symmetry, Eliminating the Gauge Degrees of Freedom
18.1. Separating the Variables
18.2. Massless Field
18.3. Gauge Solutions
18.4. Solving the Second Order Equations, the First Order Constraints

19. On the Matrix Equation for a Spin 2 Particle in Riemannian Space-Time, Tetrad Method
19.1. The Spin 2 Particle in Minkowski space
19.2. Structure of the Matrices of the First Order System for a Spin 2 Field
19.3. Extension to Riemannian Space-Time Geometry
19.4. The Spin 2 Field in Cylindrical Coordinates
19.5. The Equation in Spherical Coordinates
19.6. The Structure of the Lorentzian Generators
19.7. Relativistic Invariance, Additional Checking
19.8. Matrix Blocks in the Theory of a Spin 2 Particle



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