Exterior Differential Forms in Field Theory

TL;DR

The paper explores how exterior differential forms relate to conservation laws in field theory, emphasizing the transition from forms associated with material systems to those describing physical fields, revealing the origin of physical structures.

Contribution

It clarifies the role of non-closed exterior forms in material systems and their transition to closed forms, explaining the emergence of physical structures in field theory.

Findings

Physical structures are generated by material systems during evolution.

Transition via degenerate transform links conservation laws of material systems to physical fields.

Analysis of thermodynamics and electromagnetic equations supports the theoretical framework.

Abstract

A role of the exterior differential forms in field theory is connected with a fact that they reflect properties of the conservation laws. In field theory a role of the closed exterior forms is well known. A condition of closure of the form means that the closed form is the conservative quantity, and this corresponds to the conservation laws for physical fields. In the present work a role in field theory of the exterior forms, which correspond to the conservation laws for the material systems is clarified. These forms are defined on the accompanying nondifferentiable manifolds, and hense, they are not closed. Transition from the forms, which correspond to the conservation laws for the material systems, to those, which correspond to the conservation laws for physical fields (it is possible under the degenerate transform), describe a mechanism of origin of the physical structures that format physical fields. In the work it is shown that the physical structures are generated by the material systems in the evolutionary process. In Appendices we give an analysis of the principles of thermodinamics and equations of the electromagnetic field. A role of the conservation laws in formation of the pseudometric and metric spaces is also shown.