## Equivalence of Maxwell's Source-Free Equations to the Time-Dependent Schrödinger Equation for a Solitary Particle with Two Polarizations and Hamiltonian |cp|

**Authors:** Steven Kenneth Kauffmann

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 Schrödinger 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 Schrödinger equations (owing to two linear polarizations), which have the massless freeparticle
diagonalized square-root Hamiltonian. Those fields are readily modfied 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.

**Comments:** 17 pages, Also archived as arXiv:1004.1820 [physics.gen-ph].

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### Submission history

[v1] 25 Apr 2010

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