Group Theory in Non-Linear Problems

This is the second volume of a series of books in various aspects of Mathematical Physics. Mathematical Physics has made great strides in recent years, and is rapidly becoming an important dis­ cipline in its own right. The fact that physical ideas can help create new mathematical theories, and rigorous mathematical theo­ rems can help to push the limits of physical theories and solve problems is generally acknowledged. We believe that continuous con­ tacts between mathematicians and physicists and the resulting dialogue and the cross fertilization of ideas is a good thing. This series of studies is published with this goal in mind. The present volume contains contributions which were original­ ly presented at the Second NATO Advanced Study Institute on Mathe­ matical Physics held in Istanbul in the Summer of 1972. The main theme was the application of group theoretical methods in general relativity and in particle physics. Modern group theory, in par­ ticular, the theory of unitary irreducibl~ infinite-dimensional representations of Lie groups is being increasingly important in the formulation and solution of dynamical problems in various bran­ ches of physics. There is moreover a general trend of approchement of the methods of general relativity and elementary particle physics. We hope it will be useful to present these investigations to a larger audience.

Contenu

Relativistic Symmetry Groups.- 1. Orthogonal and Conformal Groups.- 2. Asymptotically Simple Space-Times.- 3. The B.M.S. Group.- 4. Twistor Theory.- SL(2,C) Symmetry of the Gravitational Field.- 1. Spinor Representation of the Group SL(2,C).- 2. Connection between Spinors and Tensors.- 3. Maxwell, Weyl, and Riemann Spinors.- 4. Classification of Maxwell Spinor.- 5. Classification of Weyl Spinor.- 6. Isotopic Spin and Gauge Fields.- 7. Lorentz Invariance and the Gravitational Field.- 8. SL(2,C) Invariance and the Gravitational Field.- 9. Gravitational Field Equations.- Problems.- Coordinate Systems in Riemannian Space-Time: Classifications and Transformations; Generalization of the Poincaré Group.- 1. Introduction.- 2. Riemannian and Normal Coordinates.- 3. Transformations between Normal Coordinate Systems.- 4. Generalization of Inertial Frames to Curved Space-Time.- 5. Analytic Characterization of Geodesic Fermi Frames.- 6. Classifications of Coordinate Systems.- 7. The Principle of the Pre-Assigned Measurements.- 8. The Degree of Invariance of the Laws of Nature.- 9. Generalization of the Poincaré Group to Curved Space-Time; and Concluding Remarks.- Symmetric Spaces in Relativity and Quantum Theories.- 1. Introduction.- 2. Lie Transformation Groups, Lie Algebras, Covering and Pseudo-Orthogonal Groups.- I / Symmetric Spaces and Lie Triple Systems.- 3. Symmetric Spaces.- 4. Symmetric Spaces as Homogeneous Spaces of Groups.- 5. Lie Triple Systems as the Local Algebraic Structures of Symmetric Spaces.- 6. On Symmetric Spaces of Pseudo-Orthogonal Groups.- 7. Conformal Groups of Pseudo-Orthogonal Vector Spaces.- 8. Light Cones as Homogeneous but not Symmetric Spaces of the Pseudo-Orthogonal Groups.- 9. Applications in General Relativity.- II / Domains of Positivity and Formal Real Jordan Algebras.- 10. Domains of Positivity or Self-Dual Convex Cones.- 11. Jordan Algebras.- 12. The Relation between Domains of Positivity and Symmetric Spaces.- 13. The Jordan Algebra of Minkowski Space.- 14. The Jordan Algebra of Non-Relativistic Spin Observables.- III / Halfspaces of Jordan Algebras and Bounded Symmetric Domains.- 15. The Siegel Half Space.- 16. Halfspaces and Bounded Symmetric Domains.- 17. The Halfspace of Minkowski Space.- Boundary Values of Holomorphic Functions that Belong to Hilbert Spaces Carrying Analytic Representations of Semisimple Lie Groups.- 0. Preliminaries.- 1. The Discrete Series of SU(1,1).- 2. The Discrete Series of SU(2,2).- The Semisimp1e Subalgebras of the Algebra B3(SO(7)) and Their Inclusion Relations.- 1. Introduction.- 2. Classification Scheme.- 3. Actual Classification.- 4. Index of Embedding; Defining Matrix.- 5. Classification of Semisimple Subalgebras of B3.- External (Kinematical) and Internal (Dynamical) Conformal Symmetry and Discrete Mass Spectrum.- 1. Introduction.- 2. Conformal Transformations on External Co-ordinates.- 3. Conformal Transformations on Internal Co-ordinates.- 4. The Connection between the External and Internal Conformal Algebras. Discrete Mass Spectrum.- Non-Linear Problems in Transport Theory.- 1. A Non-Linear Transport Equation.- 2. General Properties of the Solution.- 3. Solution of the Milne Problem.- 4. Explicit Evaluation of the Milne Problem Solution.