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The book represents a systematical development of several rarely used or new mathematical techniques to construct new wave equations for elementary particles. On this ground, it develops the quantum mechanics of the particles with additional electromagnetic structures in presence of external electromagnetic fields, and, on the curved space-time background, it details many new exactly solvable problems in the field. In Volume II, the wave equations for particles of spin 0, 1/2, 1 with additional intrinsic structure (such as polarizability, anomalous magnetic moment, quadrupole electric moment, Darwin–Cox structure) are solved for several special types of external electric and magnetic fields, in Minkowski flat space and in spaces with simple non-Euclidean geometry: hyperbolic Lobachevsky and spherical Riemann models. The main attention is focused on new and additional effects which are due to the more general structure of the wave equations in presence of external fields. The following problems are considered: reflection of spin 0 and spin 1 particles by an effective (geometrical) medium; Schrodinger and Dirac particles in electric field on the background of Lobachevsky and Riemann models (LR-models); Dirac-K¨ahler boson in LR-models; spinless particle with polarizability in Coulomb and magnetic fields; spin 1/2 particle with anomalous magnetic moment in homogeneous magnetic and electric fields, and in Coulomb field; spin 1 particle (ordinary and with polarizability) in external magnetic field; spin 1 particle with anomalous magnetic and quadrupole moment in magnetic and electric fields; spinless Darwin–Cox particle with intrinsic structure in external electric and magnetic fields in Euclid, Lobachevsky and Riemann models; fermion with two masses in presence of magnetic field.

The book represents the systematical development of several rarely used or new mathematical techniques to construct new wave equations for elementary particles. On this ground, it develops the quantum mechanics of the particles with additional electromagnetic structures in the presence of external electromagnetic fields, and, concerning the curved space-time background, it details many new solvable problems in the field. Volume I of this book is devoted to studying the mathematical formalism of the generalized theory of relativistic wave equations, which permits the construction of new equations for particles with an intrinsic spectrum of the spin, mass, and a number of additional electromagnetic characteristics, which lead to additional physical effects in the presence of external fields (electromagnetic and gravitational). The main theoretical tools are: The theory of the Lorentz group; the wave equations and the Gel'fand–Yaglom canonical basis; the Lagrangian formulation of the theory; the minimal polynomials approach; the projective operators technique; and the definiteness of energy and charge. Primary attention is focused on the possibility of constructing different relativistic wave equations for particles with a fixed value of spin and mass, proving that such wave equations describe substantially different objects. These equations possess some intrinsic electromagnetic structures in addition to the electric charge, and this leads to physical effects in the presence of external – electromagnetic and gravitational – fields. Extended models for particles of spins 0, 1/2, 1, 3/2, and 2 are developed.

This book is devoted to investigating the spinor structures in particle physics and in polarization optics. In fact, it consists of two parts joined by the question: Which are the manifestations of spinor structures in different branches of physics. It is based on original research. The main idea is the statement that the physical understanding of geometry should be based on physical field theories. The book contains numerous topics with the accent on field theory, quantum mechanics and polarization optics of the light, and on the spinor approach.