# Linear Representations of the Lorentz Group

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Linear Representations of the Lorentz Group is a systematic exposition of the theory of linear representations of the proper Lorentz group and the complete Lorentz group. This book consists of four chapters. The first two chapters deal with the basic material on the three-dimensional rotation group, on the complete Lorentz group and the proper Lorentz group, as well as the theory of representations of the three-dimensional rotation group. These chapters also provide the necessary basic information from the general theory of group representations. The third chapter is devoted to the representations of the proper Lorentz group and the complete Lorentz group, while the fourth chapter examines the theory of invariant equations. This book will prove useful to mathematicians and students.

## Table of Contents

Preface

Chapter I. The Three-Dimensional Rotation Group and the Lorentz Group

1. The Three-Dimensional Rotation Group

1. General Definition of a Group

2. Definition of the Three-Dimensional Rotation Group

3. Description of Rotations by means of Orthogonal Matrices

4. Eulerian Angles

5. The Description of Rotation by means of Unitary Matrices

6. The Invariant Integral Over the Rotation Group

7. The Invariant Integral on the Unitary Group

2. The Lorentz Group

1. The General Lorentz Group

2. The Complete Lorentz Group and the Proper Lorentz Group

Chapter II. The Representations of the Three-Dimensional Rotation Group

3. The Basic Concepts of the Theory of Finite-Dimensional Representations

1. Linear Spaces

2. Linear Operators

3. Definition of a Finite-Dimensional Representation of a Group

4. Continuous Finite-Dimensional Representations of the Three-Dimensional Rotation Group

5. Unitary Representations

4. Irreducible Representations of the Three-Dimensional Rotation Group in Infinitesimal Form

1. Differentiability of Representations of the Group G0

2. Basic Infinitesimal Matrices of the Group G0

3. Basic Infinitesimal Operators of a Representation of the Group G0

4. Relations Between the Basic Infinitesimal Operators of a Representation of the Group G0

5. The Condition for a Representation to be Unitary

6. General Form of the Basic Infinitesimal Operators of the Irreducible Representations of the Group G0

5. The Realization of Finite-Dimensional Irreducible Representations of the Three-Rimensional Rotation Group

1. The Connection Between the Representations of the Group G0 and the Representations of the Unitary Group U

2. Spinor Representations of the Group U

3. Realization of the Representations Gm in a Space of Polynomials

4. Basic Infinitesimal Operators of the Representation Gm

5. Orthogonality Relations

6. The Decomposition of a given Representation of the Three-Dimensional Rotation Group into Irreducible Representations

1. The Case of a Finite-Dimensional Unitary Representation

2. The Theorem of Completeness

3. General Definition of a Representation

4. Continuous Representations

5. The Integrals of Vector and Operator Functions

6. Decomposition of a Representation of the Group U into Irreducible Representations

7. The Case of a Unitary Representation

Chapter III. Irreducible Linear Representations of the Proper and Complete Lorentz Groups

7. The Infinitesimal Operators of a Linear Representation of the Proper Lorentz Group

1. The Infinitesimal Lorentz Matrices

2. Relations Between the Infinitesimal Lorentz Matrices

3. The Infinitesimal Operators of a Representation of the Proper Lorentz Group

4. Relations Between the Basic Infinitesimal Operators of a Representation

8. Determination of the Infinitesimal Operators of a Representation of the Group ℭ+

1. Statement of the Problem

2. Determination of the Operators H+, H_, H3

3. Determination of the Operators F+, F_, F3

4. The Conditions of Being Unitary

9. The Finite-Dimesional Representations of the Proper Lorentz Group

1. The Spinor Description of the Proper Lorentz Group

2. The Relation Between the Representations of the Groups ℭ+ and U

3. The Spinor Representations of the Group U

4. The Infinitesimal Operators of a Spinor Representation

5. The Irreducibility of a Spinor Representation

6. The Infinitesimal Operators of a Spinor Representation with Respect to a Canonical Basis

10. Principal Series of Representations of the Group U

1. Some Subgroups of the Group U

2. Canonical Decomposition of the Elements of the Group U

3. Residue Classes with Respect to K

4. Parametrization of the Space Z

5. Invariant Integral on the Group Z

6. The Definition of the Representations of the Principal Series

7. Irreducibility of the Representations of the Principal Series

11. Description of the Representations of the Principal Series and of Spinor Representations by means of the Unitary Group

1. A Description of the Space Z in Terms of the Unitary Subgroup

2. The Space Lm2(U)

3. The Realization of the Representation of the Principal Series in the Space Lm2(U)

4. The Representations Sk, Contained in Gm,p

5. Elementary Spherical Functions

6. Infinitesimal Operators of the Representation Gm,p in a Canonical Basis

7. The Case of Spinor Representations

12. Complementary Series of Representations of the Group U

1. Statement of the Problem of Complementary Series

2. The Condition for Positive Definiteness

3. The Spaces ℌσ and Hσ

4. A Description of the Representations of the Complementary Series in the Space ℌσ

5. A Description of the Representations of the Complementary Series with the Aid of the Unitary Subgroup

6. The Representations Sk, Contained in Dσ

7. The Elementary Spherical Functions of the Representations of the Complementary Series

8. The Infinitesimal Operators of the Representations Dσ in a Canonical Basis

13. The Trace of a Representation of the Principal or Complementary Series

1. An Invariant Integral on the Group U

2. Invariant Integrals on the Group K

3. Some Integral Relations

4. The Group Ring of the Group U

5. The Relation Between the Representations of the Group U and its Group Ring

6. The Case of a Unitary Representation of the Group U

7. The Trace of a Representation of the Principal Series

8. The Trace of a Representation of the Complementary Series

14. An Analogue of Plancherel's Formula

1. Statement of the Problem

2. Some Subgroups of the Group K

3. Canonical Decomposition of the Elements of the Group K

4. Some Integral Relations

5. Some Auxiliary Functions and Relations Between them

6. The Derivation of an Analogue of Plancherel's Formula

7. The Inverse Formulae

8. The Decomposition of the Regular Representation of the Group U into Irreducible Representations

15. A Description of all the Completely Irreducible Representations of the Proper Lorentz Group

1. Conjugate Representations

2. The Operators Ekjl

3. Equivalence of Representations

4. Completely Irreducible Representations

5. The Operators ekjl

6. The Ring Xkj

7. The Relation Between the Representations of the Rings X and Xkj

8. The Commutativity of the Rings Xkj

9. A Criterion of Equivalence

10. The Functional λ (x) in the Case of an Irreducible Representation of the Principal Series

11. The Functions Βν(ε)

12. The Ring Bkj

13. The General Form of the Functional λ (b)

14. The General Form of the Linear Multiplicative Functional λ(Β) in the Ring Bkj

15. The Complete Series of Completely Irreducible Representations of the Group U

16. A Fundamental Theorem

16. Description of all the Completely Irreducible Representations of the Complete Lorentz Group

1. Statement of the Problem

2. The Fundamental Properties of the Operator S

3. The Group Ring of the Group ℭ0

4. Induced Representations

5. Description of the Completely Irreducible Representations of the Ring ℭkj

6. Realizations of the Completely Irreducible Representations of the Group ℭ0

7. A Fundamental Theorem

Chapter IV. Invariant Equations

17. Equations Invariant with Respect to Rotations of Three-Dimensional Space

1. A General Definition of Quantities

2. The Concept of an Equation Invariant with Respect to a Representation of the Group G0

3. Conditions of Invariance

4. Conditions of Invariance in Infinitesimal Form

5. General Form of the Operators L1, L2, L3

18. Equations Invariant with Respect to Proper Lorentz Transformations

1. General Linear Representations of the Proper Lorentz Group in Infinitesimal Form

2. Some Special Cases of Representations of the Group ℭ+

3. The Concept of an Equation Invariant with Respect to Proper Lorentz Transformations

4. The General Form of an Equation Invariant with Respect to the Transformations of the Group ℭ+

19. Equations Invariant with Respect to Transformations of the Complete Lorentz Group

1. General Linear Representations of the Complete Lorentz Group in Infinitesimal Form

2. A Description of the Equations Invariant with Respect to the Complete Lorentz Group

20. Equations Derived from an Invariant Lagrangian Function

1. Invariant Bilinear Forms

2. Lagrangian Functions

3. The Definition of Rest Mass and Spin

4. Conditions of Definiteness of Density of Charge and Energy

5. The Case of Finite-Dimensional Equations

6. Examples of Invariant Equations

Appendix

References

Index

Volumes Published in the Series in Pure and Applied Mathematics

## Product details

- No. of pages: 464
- Language: English
- Copyright: © Pergamon 1964
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483184982

## About the Author

### M. A. Naimark

## About the Editor

### H. K. Farahat

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