2009/10/31 by Peter J. Mohr
Physics and Astronomy · #Classical mechanics #Dirac equation #Eigenfunction #Eigenvalues and eigenvectors #Electromagnetic field #Electromagnetic radiation #Inhomogeneous electromagnetic wave equation #Lorentz transformation #Matrix representation of Maxwell's equations #Maxwell's equations #Mechanical and Optical Resonators #Optical field #Orbital Angular Momentum in Optics #Physics #Plane wave #Quantum and Classical Electrodynamics #Quantum electrodynamics #Quantum mechanics #Two-body Dirac equations #Wave equation #Wave function #Wave packet #physics.gen-ph #physics.optics
paper · pdf · doi:10.1016/j.aop.2009.11.007
published as Annals of Physics 325, 607-663 (2010) · 75 pages; typo corrected, slight text change
arxiv created 2009/11/18 · openalex publication_date 2009/11/27 · arxiv updated 2014/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Properties of six-component electromagnetic field solutions of a matrix form of the Maxwell equations, analogous to the four-component solutions of the Dirac equation, are described. It is shown that the six-component equation, including sources, is invariant under Lorentz transformations. Complete sets of eigenfunctions of the Hamiltonian for the electromagnetic fields, which may be interpreted as photon wave functions, are given both for plane waves and for angular-momentum eigenstates. Rotationally invariant projection operators are used to identify transverse or longitudinal electric and magnetic fields. For plane waves, the velocity transformed transverse wave functions are also transverse, and the velocity transformed longitudinal wave functions include both longitudinal and transverse components. A suitable sum over these eigenfunctions provides a Green function for the matrix Maxwell equation, which can be expressed in the same covariant form as the Green function for the Dirac equation. Radiation from a dipole source and from a Dirac atomic transition current are calculated to illustrate applications of the Maxwell Green function.