Abstract. Hurwitz pairs and triples are discussed in connection with algebra, complex analysis, and field theory. Introduction. This paper is the third part of our study of Hurwitz pairs. In the first part [11], we gave an outline of a field theory defined by Hurwitz pairs. We introduced a field equation of Dirac type and soliton equations by means of Hurwitz pairs. In the second paper [12] the concepts of a Hurwitz pair and the corresponding supercomplex structure were considered in a slightly more general setting of weighted Hurwitz pairs. Then, as applications, several results were obtained for field equations with mass and for Beltrami equations generating quasiconformal mappings in the plane
A complex analytical method of solving the generalised Dirac-Maxwell system has recently been proposed by two of us for a certain class of complex Riemannian metrics. The Dirac equation without the field potential in such a metric appeared to be equivalent to the Dirac-Maxwell system including the field potentials produced by the currents of a particle in question. The method proposed is connected with applying the Fourier transform with respect to the electric charge treated as a variable, with the consideration of the mass as an eigenvalue, and with solving suitable convolution equations. In the present research an explicit calculation based on linearization of the spinor connections is given. The conditions for the motion are interpreted as a starting point to seek selection rules for curved space-times corresponding to actually existing particles. Then the same method is applied to solids. Namely, by a suitable transformation of the configuration space in terms of elements of the interaction matrix corresponding to the Coulomb, exchange, and dipole integrals, the interaction term in the hamiltonian becomes zero, thus leading to experimentally verificable formulae for the autocorrelation time
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