Abstract
A new concept of holomorphy in pseudo-Euclidean spaces is briefly presented. The set of extended Cauchy-Riemannn differential equations, which are verified by the holomorphic functions, is obtained. A form of the general pseudo-rotation matrix was developed. The generalized d’Alembert- operator and extended Poisson’s equations are defined. Applying these results to the relativistic space-time, the charge conservation and general Maxwell equations are derived.
Highlights
In a paper [1], published in 1981, Salingaros proposed an extension of the Cauchy-Riemann equations of holomorphy to fields in higher-dimensional spaces
In the present article we introduce a different definition of monogenity/holomorphy applied to vector functions in a pseudo-Euclidean space. This enables us to obtain a set of equations, which applied to the Minkowski space-time, lead to general Maxwell equations and to the charge conservation law
3.2 Charge conservation and Maxwell’s Equations Identifying (2.7) with inhomogeneous wave equations presented by Feynman [6] we find the expressions for charge and current density
Summary
In a paper [1], published in 1981, Salingaros proposed an extension of the Cauchy-Riemann equations of holomorphy to fields in higher-dimensional spaces. He formulated the theory of holomorphic fields by using Clifford algebras [2]. In the present article we introduce a different definition of monogenity/holomorphy applied to vector functions in a pseudo-Euclidean space. This enables us to obtain a set of equations, which applied to the Minkowski space-time, lead to general Maxwell equations and to the charge conservation law. All physical quantities involved in the ongoing presentation are expressed in geometric units [3], i.e. meters
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