Abstract
The governing equations of Maxwell electrodynamics in multi-dimensional spaces are derived from the variational principle of least action, which is applied to the action function of the electromagnetic field. The Hamiltonian approach for the electromagnetic field in multi-dimensional pseudo-Euclidean (flat) spaces has also been developed and investigated. Based on the two arising first-class constraints, we have generalized to multi-dimensional spaces a number of different gauges known for the three-dimensional electromagnetic field. For multi-dimensional spaces of non-zero curvature the governing equations for the multi-dimensional electromagnetic field are written in a manifestly covariant form. Multi-dimensional Einstein’s equations of metric gravity in the presence of an electromagnetic field have been re-written in the true tensor form. Methods of scalar electrodynamics are applied to analyze Maxwell equations in the two and one-dimensional spaces.
Highlights
The main goal of this communication is to develop the logically closed and noncontradictory version of electrodynamics in the multi-dimensional space
The notation ek stands for the electric charge of the k-th particle, while mk means the mass of the same particle, and Aα is the covariant component of the four-dimensional vector potential Aof the electromagnetic field
As a conclusion of this section, we want to emphasize the fact that our action function, which is chosen in the form of Equation (9), allows one to derive the equations of motion for a system of electrically charged, point particles which move in the electromagnetic field
Summary
The main goal of this communication is to develop the logically closed and noncontradictory version of electrodynamics in the multi-dimensional (or n-dimensional) space. Our current Maxwell theory of electromagnetic fields and corresponding Hamiltonian approach can be used only for three-dimensional (geometrical) spaces. The notation ek stands for the electric charge of the k-th particle, while mk means the mass of the same particle, and Aα is the covariant component of the four-dimensional vector potential Aof the electromagnetic field. This formula, Equation (6), is written for the four-dimensional pseudo-Euclidean (flat) space-time. A separate but closely related problem is the gauge invariance of the free electromagnetic field Another interesting problem is to investigate the explicit form of multi-dimensional Maxwell equations in the presence of multi-dimensional gravitational fields. All new results obtained in the course of our current analysis will be used later to develop the modern united theory of electromagnetic and gravitational fields
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