Beyond Pseudo-Rotations in Pseudo-Euclidean Spaces ( Mathematical Analysis and its Applications )

Publication series ：Mathematical Analysis and its Applications

Author： Ungar Abraham

Publisher： Elsevier Science‎

Publication year： 2018

E-ISBN: 9780128117743

P-ISBN(Paperback): 9780128117736

Subject： O412.1 relativistic

Keyword：数值分析,数学

Language： ENG

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Description

Beyond Pseudo-Rotations in Pseudo-Euclidean Spaces presents for the first time a unified study of the Lorentz transformation group SO(m, n) of signature (m, n), m, n ∈ N, which is fully analogous to the Lorentz group SO(1, 3) of Einstein’s special theory of relativity. It is based on a novel parametric realization of pseudo-rotations by a vector-like parameter with two orientation parameters. The book is of interest to specialized researchers in the areas of algebra, geometry and mathematical physics, containing new results that suggest further exploration in these areas.

Introduces the study of generalized gyrogroups and gyrovector spaces
Develops new algebraic structures, bi-gyrogroups and bi-gyrovector spaces
Helps readers to surmount boundaries between algebra, geometry and physics
Assists readers to parametrize and describe the full set of generalized Lorentz transformations in a geometric way
Generalizes approaches from gyrogroups and gyrovector spaces to bi-gyrogroups and bi-gyrovector spaces with geometric entanglement

Chapter

Front Cover

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces: An Introduction to the Theory of Bi-gyrogroups and Bi-gyrovector Spaces Introduction to the Theory of Bi-gyrogroups and Bi-gyrovector Spaces

CHAPTER 1: Introduction

1.1. Introduction

1.2. Quantum Entanglement and Geometric Entanglement

1.3. From Galilei to Lorentz Transformations

1.4. Galilei and Lorentz Transformations of Particle Systems

1.5. Chapters of the Book

CHAPTER 2: Einstein Gyrogroups

2.1. Introduction

2.2. Einstein Velocity Addition

2.3. Einstein Addition with Respect to Cartesian Coordinates

2.4. Einstein Addition vs. Vector Addition

2.5. Gyrations

2.6. From Einstein Velocity Addition to Gyrogroups

2.7. Gyrogroup Cooperation (Coaddition)

2.8. First Gyrogroup Properties

2.9. Elements of Gyrogroup Theory

2.10. The Two Basic Gyrogroup Equations

2.11. The Basic Gyrogroup Cancellation Laws

2.12. Automorphisms and Gyroautomorphisms

2.13. Gyrosemidirect Product

2.14. Basic Gyration Properties

2.15. An Advanced Gyrogroup Equation

2.16. Gyrocommutative Gyrogroups

CHAPTER 3: Einstein Gyrovector Spaces

3.1. The Abstract Gyrovector Space

3.2. Einstein Special Relativistic Scalar Multiplication

3.3. Einstein Gyrovector Spaces

3.4. Einstein Addition and Differential Geometry

3.5. Euclidean Lines

3.6. Gyrolines – The Hyperbolic Lines

3.7. Gyroangles – The Hyperbolic Angles

3.8. The Parallelogram Law

3.9. Einstein Gyroparallelograms

3.10. The Gyroparallelogram Law

3.11. Euclidean Isometries

3.12. The Group of Euclidean Motions

3.13. Gyroisometries – The Hyperbolic Isometries

3.14. Gyromotions – The Motions of Hyperbolic Geometry

CHAPTER 4: Bi-gyrogroups and Bi-gyrovector Spaces – P

4.1. Introduction

4.2. Pseudo-Euclidean Spaces and Pseudo-Rotations

4.3. Matrix Representation of SO(m,n)

4.4. Parametric Realization of SO(m,n)

4.5. Bi-boosts

4.6. Lorentz Transformation Decomposition

4.7. Inverse Lorentz Transformation

4.8. Bi-boost Parameter Composition

4.9. On the Block Entries of the Bi-boost Product

4.10. Bi-gyration Exclusion Property

4.11. Automorphisms of the Parameter Bi-gyrogroupoid

4.12. Squared Bi-boosts

4.13. Commuting Relations Between Bi-gyrations and Bi-rotations

4.14. Product of Lorentz Transformations

4.15. The Bi-gyrocommutative Law in Bi-gyrogroupoids

4.16. The Bi-gyroassociative Law in Bi-gyrogroupoids

4.17. Bi-gyration Reduction Properties in Bi-gyrogroupoids

4.18. Bi-gyrogroups – P

4.19. Bi-gyration Decomposition and Polar Decomposition

4.20. The Bi-gyroassociative Law in Bi-gyrogroups

4.21. The Bi-gyrocommutative Law in Bi-gyrogroups

4.22. Bi-gyrogroup Gyrations

4.23. Bi-gyrogroups are Gyrocommutative Gyrogroups

4.24. Bi-gyrovector Spaces

4.25. On the Pseudo-inverse of a Matrix

4.26. Properties of Bi-gyrovector Space Scalar Multiplication

4.27. A Commuting Relation by SVD

4.28. Einstein Bi-gyrogroups and Bi-gyrovector Spaces – P

CHAPTER 5: Bi-gyrogroups and Bi-gyrovector Spaces – V

5.1. Introduction

5.2. Bi-boost Parameter Change, P→ V

5.3. Matrix Balls of the Parameter V

5.4. Reparametrizing the Bi-boost

5.5. Lorentz Transformation Decomposition

5.6. Examples

5.7. Inverse Lorentz Transformation

5.8. Bi-boosts

5.9. Bi-boost Product

5.10. Product of Bi-boosts

5.11. Product of Lorentz Transformations

5.12. Bi-gyroassociative Law in Bi-gyrogroupoids

5.13. Relationships Between the Bi-boost Parameters P and V

5.14. Properties of Bi-gyrations in the Ball

5.15. Bi-gyrogroups – V

5.16. Einstein Bi-gyroaddition and Bi-gyration

5.17. On the Special Relativistic Einstein Gyroaddition

5.18. Bi-gamma Identities for the Primed Binary Operation

5.19. Bi-gyrogroup Gyrations

5.20. Uniqueness of Left and Right Gyrations

5.21. Bi-gyrogroups Are Gyrocommutative Gyrogroups

5.22. Matrix Division Notation

5.23. Additive Decomposition of the Lorentz Bi-boost

5.24. Bi-gyrovector Spaces

5.25. On the Pseudo-inverse of a Matrix

5.26. Scalar Multiplication: The SVD Formula

5.27. Properties of Bi-gyrovector Space Scalar Multiplication

5.28. Scaling Property

5.29. Homogeneity Property

5.30. Bi-gyrohalf

5.31. A Commuting Relation for Scalar Multiplication and Rotations

5.32. P-V Scalar Multiplication Relationship

5.33. Bi-gyrovector Space Isomorphism

5.34. Group Isomorphism Between Lorentz Groups

5.35. Einstein Bi-gyrogroups and Bi-gyrovector Spaces – V

5.36. The Bi-gamma Norm

5.37. The Bi-gyrotriangle Inequality

CHAPTER 6: Applications to Time-Space of Signature (m,n)

6.1. Application of the Galilei Bi-boost of Signature (1,n)

6.2. Application of the Galilei Bi-boost of Signature (m,n)

6.3. Application of the Lorentz Bi-boost of Signature (1,n)

6.4. Application of the Lorentz Bi-boost of Signature (m,n)

6.5. Particles that are Systems of Subparticles

6.6. Einstein Addition of Signature (m,n)

6.7. On the Bi-boost Product

6.8. Relativistic Time-Space Bi-norm of Signature (m,n)

6.9. Relativistic Time-Space Bi-inner Product of Signature (m,n)

6.10. A Bi-gyrotriangle Bi-gamma Identity

6.11. (m,n)-Velocity Structures that Einstein Addition Preserves

6.12. (m,n)-Velocity Structures that Einstein Scalar Multiplication Preserves

6.13. Partially Integrated Particles: Example

6.14. Linking Einstein Addition of Signature (m,n) with the Special Relativistic Einstein Addition

6.15. Linking Einstein Scalar Multiplication of Signature (m,n) with the Special Relativistic Einstein Scalar Multiplication

6.16. Natural Sub-bi-gyrovector Spaces

6.17. Bi-gyrohyperbolic Matrix Funct

6.18. Bi-gyrohyperbolic Bi-boosts

CHAPTER 7: Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

7.1. Introduction

7.2. Bi-gyropoints in Bi-gyrovector Spaces

7.3. Bi-gyrodistance

7.4. Bi-gyrolines in Bi-gyrovector Spaces

7.5. Reduction of Bi-gyrolines to Gyrolines

7.6. Betweenness

7.7. The Bi-gyrotriangle Equality

7.8. A Bi-gyroruler for Bi-gyrolines

7.9. Bi-gyromidpoints

7.10. Reduction of Bi-gyrotriangles to Gyrotriangles

7.11. Bi-gyroparallelograms

7.12. Bi-gyroisometries: The Bi-hyperbolic Isometries

7.13. Bi-gyromotions: The Motions of Bi-hyperbolic Geometry

7.14. Bi-gyrosemidirect Product Groups

Notation and Special Symbols

References

Index

Back Cover

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