Hidden unique possibilities of mathematical physics equations (Formalism of skew-symmetric forms)

TL;DR

This paper reveals hidden properties of mathematical physics equations, showing they can describe discrete phenomena like quantum jumps and structure emergence through skew-symmetric forms, which are not apparent from the equations themselves.

Contribution

It demonstrates that mathematical physics equations possess hidden discrete properties linked to their integrability, enabling the description of phenomena like structure formation and quantum jumps.

Findings

Mathematical physics equations can describe discrete transitions and structure emergence.

Hidden properties are related to the equations' integrability and degrees of freedom.

Skew-symmetric differential forms reveal these hidden features.

Abstract

It is shown that mathematical physics differential equations have properties that allow describing processes such as the structures emergence, discrete transitions, quantum jumps. The peculiarity is that such properties are hidden. They do not follow directly from the mathematical physics equations but are realized discretely in the solving process. This is due to the mathematical physics equations integrability, which, as shown, can be realized only discretely in the presence of any degrees of freedom. In this case, a transition occurs from the original coordinate space with a solution that is not a function (the solution derivatives do not compose a differential) to integrable structures with a solution that is a discrete function. It is the double solutions and spatial transitions that can describe the processes of the emergence of any structures or phenomena. Due to hidden properties, the mathematical physics equations have unique possibilities in describing physical processes and phenomena that cannot be described in the framework of other mathematical formalisms. Such results were obtained using skew-symmetric differential forms.