Symmetries of Maxwell's Equations

ISBN-13: 9789401081665 / Angielski / Miękka / 2011 / 214 str.

W. I. Fushchich; A. G. Nikitin

Opis

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the fina question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Guik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Spis treści

1. Various Formulations of Maxwell’s Equations.- 1. Maxwell’s Equations in Vector Notation.- 2. Maxwell’s Equations in Silberstein-Bateman-Majorana Form.- 3. Maxwell’s Equations in Dirac Form.- 4. The Equations in Kemmer-Duffin-Petiau Form.- 5. The Equation for the Potential.- 6. Maxwell’s Equations in the Momentum Representation.- 2. Relativistic Invariance of Maxwell’s Equations.- 7. Basic Definitions.- 8. The IA of Maxwell’s Equations in a Class of First-Order Differential Operators.- 9. Invariance of the Equations of the Electromagnetic Field in Vacuum Under the Algebra C(1, 3)?H.- 10. Lorentz Transformations.- 11. Discrete Symmetry Transformations.- 12. IA of Different Formulations of Maxwell’s Equations.- 3. Representations of the Poincaré Algebra.- 13. Classification of Irreducible Representations.- 14. The Explicit Form of the Lubanski-Pauli Vector.- 15. The Explicit Form of the Basis Elements of the Poincaré Algebra.- 16. Covariant Representations. Finite-Dimensional Representations of the Lorentz Group.- 17. Reduction of Solutions of Maxwell’s Equations by the Irreducible Representations of the Poincaré Group.- 4. Conformal Invariance of Maxwell’s Equations.- 18. Manifestly Hermitian Representation of the Conformal Algebra.- 19. The Generators of the Conformal Group on the Set of Solutions of Maxwell’s Equations.- 20. Transformations of the Conformal Group for E, H and j.- 21. Integration of Representations of the Conformal Algebra Corresponding to Arbitrary Spin.- 5. Nongeometric Symmetry of Maxwell’s Equations.- 22. Invariance of Maxwell’s Equations Under the Eight-Dimensional Lie Algebra A8.- 23. Another Proof of Theorem 6. The Finite Transformations of the Vectors E and H Generated by the Nongeometric IA.- 24. Invariance of Maxwell’s Equations Under a 23-dimensional Lie Algebra.- 25. Symmetry Relative to Transformations not Changing Time.- 26. Non-Lie Symmetry of Maxwell’s Equations in a Conducting Medium.- 6. Symmetry of the Dirac and Kemmer-Duffin-Petiau Equations.- 27. The IA of the Dirac Equation in the Class of Differential Operators.- 28. The IA of the Dirac Equation in the Class of Integro-Differential Operators.- 29. The Symmetry of the Eight-Component Dirac Equation.- 30. Symmetry of the Dirac Equation for a Massless Particle.- 31. Symmetry of the Kemmer-Duffin-Petiau Equation.- 32. Nongeometric Symmetry of the Dirac and KDP Equations for Particles Interacting with an External Field.- 7. Constants of Motion.- 33. Bilinear forms Conserved in Time.- 34. Constants of Motion for the Dirac Field.- 35. Classical Constants of Motion of the Electromagnetic Field.- 36. Constants of Motion Connected with Nongeometric Symmetry of Maxwell’s Equations.- 37. Formulation of Conservation Laws Using the Equation of Continuity.- 8. Symmetry of Subsystems of Maxwell’s Equation.- 38. Invariance of the First Pair of Maxwell’s Equations Under Galilean Transformations.- 39. Invariance Under the Group IGL (4, R).- 40. Symmetry of the Second Pair of the Maxwell’s Equations and the Equation of Continuity.- 41. Symmetry Relative to Nonlinear Coordinate Transformations.- 42. Symmetry of Subsystems of Maxwell’s Equations Invariant Under the Group O(3).- 43. Nongeometric Symmetry.- 44. Symmetry of the Equations for the Potential.- 9.Equations for the Electromagnetic Field Invariant under the Gailean Group.- 45. Two Types of Galilean-Invariant Equations for the Electromagnetic Field.- 46. Symmetry of Equations (45.1)?(45.4) and (45.7)?(45.10).- 47. Other Types of Galilean-Invariant Equations for the Electromagnetic Field.- 48. Irreducible Representations of the Lie Algebra of the Extended Galilean Group.- 10. Relativistic Equations for a Vector and Spinor Massless field.- 49. A Group-Theoretic Derivation of Maxwell’s Equations.- 50. Uniqueness of Maxwell’s Equations.- 51. Five Types of Inequivalent Equations for Massless Fields.- 52. Inequivalent Equations for a Massless Vector Field.- 11. Poincaré-Invariant Equations for a Massless Field with Arbitrary Spin.- 53. Covariant Equations for Massless Fields with Arbitrary Helicity.- 54. Equations in Dirac Form for Fields with Arbitrary Spin.- 55. Invariant Equations Without Superfluous Components.- 56. Inequivalent Equations for a Massless Field with Arbitrary Spin.- Conclusion.- Appendix 1.- On Complete Sets of Symmetry Operators for the Dirac and Maxwell Equations and Invariance Algebras of Relativistic Wave Equations for Particles of Arbitrary Spin.- Appendix 2.- Symmetry of Nonlinear Equations of Electrodynamics.- Appendix 3.- On Ansätze and Exact Solutions of the Nonlinear Dirac and Maxwell-Dirac Equations.- Appendix 4.- How to Extend the Symmetry of Equations?.- List of Additional References.