Linear Representations of the Lorentz Group

PrefaceChapter 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 GroupChapter 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 RepresentationChapter 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.