Equivalence of Maxwell's source-free equations to the time-dependent Schroedinger equation for a solitary particle with two polarizations and Hamiltonian |cp|

TL;DR

This paper demonstrates that Maxwell's source-free equations are equivalent to the time-dependent Schrödinger equation for a massless photon with two polarizations, using the square-root Hamiltonian, and explores their relation to photon wave functions.

Contribution

It shows the equivalence of Maxwell's equations to the Schrödinger equation for photons and clarifies the role of the square-root Hamiltonian in relativistic quantum mechanics.

Findings

Maxwell's equations decompose into two Schrödinger equations for transverse fields.

The transverse field components obey the square-root Hamiltonian for massless particles.

Photon wave functions can be derived from Maxwell's equations in radiation gauge.

Abstract

It was pointed out in a previous paper that although neither the Klein-Gordon equation nor the Dirac Hamiltonian produces sound solitary free-particle relativistic quantum mechanics, the natural square-root relativistic Hamiltonian for a nonzero-mass free particle does achieve this. Failures of the Klein-Gordon and Dirac theories are reviewed: the solitary Dirac free particle has, inter alia, an invariant speed well in excess of c and staggering spontaneous Compton acceleration, but no pathologies whatsoever arise from the square-root relativistic Hamiltonian. Dirac's key misapprehension of the underlying four-vector character of the time-dependent, configuration-representation Schroedinger equation for a solitary particle is laid bare, as is the invalidity of the standard "proof" that the nonrelativistic limit of the Dirac equation is the Pauli equation. Lorentz boosts from the particle rest frame point uniquely to the square-root Hamiltonian, but these don't exist for a massless particle. Instead, Maxwell's equations are dissected in spatial Fourier transform to separate nondynamical longitudinal from dynamical transverse field degrees of freedom. Upon their decoupling in the absence of sources, the transverse field components are seen to obey two identical time-dependent Schroedinger equations (owing to two linear polarizations), which have the massless free-particle diagonalized square-root Hamiltonian. Those fields are readily modified to conform to the attributes of solitary-photon wave functions. The wave functions' relations to the potentials in radiation gauge are also worked out. The exercise is then repeated without the considerable benefit of the spatial Fourier transform.