# Conformally Invariant Scalar-Tensor Field Theories in a Four-Dimensional Space

## TL;DR

This paper systematically constructs all conformally invariant scalar-tensor field theories in four-dimensional flat space, showing they are at most fourth-order and can be derived from second-order Lagrangians.

## Contribution

It provides a complete classification of conformally invariant scalar-tensor theories in flat space, identifying the fundamental Lagrangians and their derivative orders.

## Key findings

- All such theories are at most fourth-order in derivatives.
- They can be expressed as linear combinations of four fundamental Lagrangians.
- Third-order Lagrangian differs from a second-order one by a divergence.

## Abstract

In a four-dimensional space, I shall construct all of the conformally invariant scalar-tensor field theories, which are flat space compatible; i.e., well-defined and differentiable when evaluated for a flat metric tensor and constant scalar field. It will be shown that all such field theories must be at most of fourth-order in the derivatives of the field variables. The Lagrangian of any such field theory can be chosen to be a linear combination of four conformally invariant scalar-tensor Lagrangians, with the coefficients being functions of the scalar field. Three of these "generating" Lagrangians are of second-order, while one of 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, flat space compatible, scalar-tensor field theories, can be obtained from a second-order Lagrangian.

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Source: https://tomesphere.com/paper/1706.04827