Second-Order Bi-Scalar-Vector-Tensor Field Equations Compatible with Conservation of Charge in a Space of Four-Dimensions

TL;DR

This paper investigates second-order bi-scalar-vector-tensor field equations in four-dimensional space, focusing on their compatibility with charge conservation and Maxwell's equations, and explores implications for early Universe physics and gauge theories.

Contribution

It introduces restrictions on such field equations derived from a variational principle, discusses limitations in constructing a comprehensive Lagrangian, and suggests applications to cosmology and gauge theories.

Findings

Identifies conditions for vector equations to be consistent with charge conservation.

Shows difficulties in formulating a universal Lagrangian for these field equations.

Proposes potential applications to Higgs-induced electromagnetic fields and gauge charge conservation.

Abstract

The purpose of this paper is to explore, in a space of four-dimensions, the possible forms that second-order, bi-scalar-vector-tensor field equations derivable from a variational principle can assume. In order to restrict this enormous class of field equations I shall first require that the equations governing the vector field (which will be identified with the vector potential of an electromagnetic field) be consistent with the notion of conservation of charge. Secondly I shall require that these vector equations reduce to Maxwell's equations in a flat space when the scalar fields are constant. Unfortunately even with these two powerful restrictions on the form of the field equations I have not been able to construct a Lagrangian which yields all possible field equations of this nature. This situation will lead to a discussion of other ways in which the field equations can be restricted to obtain viable bi-scalar-vector-tensor field equations. Lastly I shall make a few remarks on how the results obtained can be used to show that the Higgs field can generate electromagnetic fields in the early Universe, and how to couple bi-scalar fields to gauge-tensor fields to construct second-order, bi-scalar-Yang-Mills-tensor field theories compatible with the conservation of gauge charge.