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It was pointed out in a previous paper that although neither the Klein-Gordonequation nor the Dirac Hamiltonian produces sound solitary free-particlerelativistic quantum mechanics, the natural square-root relativisticHamiltonian for a nonzero-mass free particle does achieve this. Failures of theKlein-Gordon and Dirac theories are reviewed: the solitary Dirac free particlehas, inter alia, an invariant speed well in excess of c and staggeringspontaneous Compton acceleration, but no pathologies whatsoever arise from thesquare-root relativistic Hamiltonian. Dirac's key misapprehension of theunderlying four-vector character of the time-dependent,configuration-representation Schroedinger equation for a solitary particle islaid bare, as is the invalidity of the standard "proof" that thenonrelativistic limit of the Dirac equation is the Pauli equation. Lorentzboosts from the particle rest frame point uniquely to the square-rootHamiltonian, but these don't exist for a massless particle. Instead, Maxwell'sequations are dissected in spatial Fourier transform to separate nondynamicallongitudinal from dynamical transverse field degrees of freedom. Upon theirdecoupling in the absence of sources, the transverse field components are seento obey two identical time-dependent Schroedinger equations (owing to twolinear polarizations), which have the massless free-particle diagonalizedsquare-root Hamiltonian. Those fields are readily modified to conform to theattributes of solitary-photon wave functions. The wave functions' relations tothe potentials in radiation gauge are also worked out. The exercise is thenrepeated without the considerable benefit of the spatial Fourier transform.

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