Analysis of the equations of mathematical physics and foundations of field theories with the help of skew-symmetric differential forms

TL;DR

This paper demonstrates how skew-symmetric differential forms, especially evolutionary forms, reveal fundamental features of mathematical physics equations and field theories, elucidating concepts like conservation laws and causality without relying on specific equations.

Contribution

It introduces the use of evolutionary skew-symmetric differential forms with deforming manifolds to analyze physical structures and processes, providing new insights into the foundations of field theories.

Findings

Reveals physical meaning of conservation laws and causality

Describes discrete transitions and quantum steps

Links mathematical physics with field theory through differential forms

Abstract

In the paper it is shown that, even without a knowledge of the concrete form of the equations of mathematical physics and field theories, with the help of skew-symmetric differential forms one can see specific features of the equations of mathematical physics, the relation between mathematical physics and field theory, to understand the mechanism of evolutionary processes that develop in material media and lead to emergency of physical structures forming physical fields. This discloses a physical meaning of such concepts like "conservation laws", "postulates" and "causality" and gives answers to many principal questions of mathematical physics and general field theory. In present paper, beside the exterior forms, the skew-symmetric differential forms, whose basis (in contrast to the exterior forms) are deforming manifolds, are used. Mathematical apparatus of such differential forms(which were named evolutionary ones) includes nontraditional elements like nonidentical relations and degenerate transformations and this enables one to describe discrete transitions, quantum steps, evolutionary processes, and generation of various structures.