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

TL;DR

This paper constructs all conformally invariant scalar-vector-tensor field theories in four dimensions that are compatible with charge conservation and flat space, showing they can be derived from second-order Lagrangians extending Maxwell's equations.

Contribution

It provides a complete classification of conformally invariant scalar-vector-tensor theories consistent with charge conservation, demonstrating they originate from second-order Lagrangians.

Findings

All such theories can be derived from a linear combination of six conformally invariant Lagrangians.

Five of these Lagrangians are at most second-order, while one is third-order but differs from a second-order Lagrangian by a divergence.

The vector equations extend Maxwell's equations with additional first-order terms that vanish for constant scalar fields.

Abstract

In a four-dimensional space I shall construct all of the conformally invariant, scalar-vector-tensor field theories that are 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 constant scalar field, (and) or vanishing vector potential. The Lagrangian of any such field theory can be chosen to be a linear combination of six conformally invariant scalar-vector-tensor Lagrangians, with the coefficients being scalar functions of the scalar field. Five of these generating Lagrangians are at most of second-order, while the sixth one is of third-order. However, the third-order Lagrangian differs from a non-conformally invariant second-order Lagrangian by a divergence. Consequently, all of the conformally invariant scalar-vector-tensor field theories that are consistent with conservation of charge, and flat space compatible, can be obtained from a second-order Lagrangian. The vector equation of any such theory is at most of second-order, and is an extension of Maxwell's equations, incorporating two other first-order terms that vanish when the scalar field is constant. Hence in regions where the scalar field is constant, the vector equation reduces to Maxwell's.