Symmetries of Maxwell’s Equations

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 ofMaxwell’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 Equationsfor 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.