Conformally Invariant Vector-Tensor Field Theories Consistent with Conservation of Charge in a Four-Dimensional Space

TL;DR

This paper classifies all conformally invariant vector-tensor field theories in four-dimensional space that conserve charge and are compatible with flat space, revealing only two possible classes of Lagrangians.

Contribution

It identifies the only two classes of Lagrangians satisfying conformal invariance, charge conservation, and flat space compatibility in four-dimensional vector-tensor theories.

Findings

One class yields the Bach tensor density.

The other class corresponds to Maxwell's equations.

Only these two classes satisfy the specified assumptions.

Abstract

In this paper I shall construct, in a four-dimensional space, all vector-tensor field theories that are conformally invariant, consistent with conservation of charge, and flat space compatible. By the last assumption I mean that the Lagrangian of the theory in question, is well defined and differentiable, when evaluated for either a flat metric tensor (and) or vanishing vector field. Under these assumptions there exists only two classes of Lagrangians. One is represented by the Lagrangian which yields the Bach tensor density multiplied by a constant, while the other is represented by a constant multiple of the usual Lagrangian that yields Maxwell's equations. Thus under the aforementioned assumptions, the field equation obtained by varying the vector field, must be Maxwell's.