## Table of Contents

**Preface**

**Introduction**

**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

**Conclusions**

**References**