Classical and Quantum Electrodynamics Concept Based on Maxwell   Equations' Symmetry
  The symmetry studies of Maxwell equations gave new insight on the nature of electromagnetic (EM) field. Tey are reviewed in the work presented. It is drawing the attention on the following aspects.   EM-field has in general case quaternion structure, consisting of four independent field constituents, which differ from each other by the parities under space inversion and time reversal.   There exists physical conserving quantity, which is simultaneously invariant under both Rainich dual and additional hyperbolic dual symmetry transformation of Maxwell equations. It is spin in general case or spirality in the corresponding geometry.   Generalized Maxwell equations for quaternion four-component EM-field are obtained.   Invariants for EM-field, consisting of dually symmetric parts are found.   The main postulate of quantum mechanics: "To any mechanical quantity can be set up in the correspondence the Hermitian matrix by quantization" was proved.   Canonical Dirac quantization method was developed in two aspects. The first aspect is its application the only to observable quantities. The second aspect is the realization along with well known time-local quantization of space-local quantization and space-time-local quantization. It is also shown, that Coulomb field can be quantized in 1D and 2D systems.   The photons in quantized EM-field are main excitations in oscillator structure of EM-field, which is equivalent to spin S = 1 "boson-atomic" 1D lattice structure, consisting of the "atoms" with zeroth rest mass. The photons of the first kind and of the second kind represent themselves respectively neutral chargeless spin 1/2 EM-solitons and charged spinless EM-solitons of Su-Schrieffer-Heeger family. 
Dmitri Yerchuck, Alla Dovlatova, Andrey Alexandrov
math-ph math.MP