The algebrodynamics: quaternionic analysis, complex string and the unique worldline

We present an original version of noncommutative analysis over matrix algebras, the algebra of biquaternions (B) in particular. Any B-differentiable function gives rise to a shear-free congruence (SFNC) on the vector B-space❈M and its Minkowski subspace M. Making use of the Kerr-Penrose correspondence between SFNC and twistor functions, we obtain general solution to the equations of B-differentiability (that is, generalization of the Cauchy- Riemann equations in complex analysis) and demonstrate that the source of any SFNC is, generically, a complex string in ❈M. Each singular point of the SFNC caustic belongs to the complex cone of a corresponding point on the generating string. We describe symmetries and SFNC-associated gauge and spinor fields, the two kinds of electromagnetic-like fields among them. Examples of known and novel solutions for fields and singular loci of SFNC are presented. Finally, we describe the conservative algebraic dynamics of an ensemble of identical point particles on the “unique Worldline” and discuss the connections of the procedure with Wheeler-Feynman’s “one-electron Universe” conception.