Internal connection between the field theory equations. Fundamentals of the field theory

TL;DR

This paper reveals a fundamental correspondence between key field theory equations and closed exterior forms of specific degrees, uncovering internal connections and foundational links to mathematical physics equations.

Contribution

It establishes a novel correspondence between field equations and exterior forms, clarifying the internal structure and foundational basis of field theory.

Findings

Dirac and Schrödinger equations relate to zero-degree forms

Maxwell's equations relate to second-degree forms

Einstein's equations relate to second and third-degree forms

Abstract

It is shown that there is a correspondence between field theory equations such as the Dirac, Shr\H{o}dinger, Maxwell, Einstein equations and closed exterior forms of a certain degree. In this case, the Dirac and Shr\H{o}dinger equations for the wave function correspond to closed exterior forms of zero degree. The Shr\H{o}dinger equation for the state functional corresponds to closed exterior forms of the first degree. The Maxwell's equations based on exterior forms of second degree. Einstein's equation for the gravitational field consists of covariant tensors, which correspond to closed exterior forms of the second degree. However, the covariant tensors of the Einstein equation are derived from the covariant tensors, which correspond to closed exterior forms of third degree. Such a correspondence between the field theory equations and closed exterior forms of a certain degree reveals the internal connection between the field theory equations. At the same time, it was shown that closed exterior forms, on which the field theory equations for physical fields are based, are associated with the equations of mathematical physics for material media, such as thermodynamic, gas-dynamic, electromagnetic, cosmological equations, etc. This indicates the connection between the field theory equations and the mathematical physics equations and reveals the foundations of field theory.