Conformal Transformation in Gravity

TL;DR

This paper explores how conformal transformations in gravity lead to a generalized action that includes scalar fields, examines the symmetry properties at the quantum level, and discusses the implications for finiteness and equivalence of different frames.

Contribution

It derives the most general conformally transformed gravitational action, unifies general relativity and scalar-tensor theories, and analyzes quantum anomalies affecting symmetry and finiteness.

Findings

The generalized action encompasses GR and scalar-tensor theories.

Quantum loop corrections break classical conformal symmetry.

The theory is not finite on shell at one-loop order due to anomalies.

Abstract

The conformal transformation in the Einstein - Hilbert action leads to a new frame where an extra scalar degree of freedom is compensated by the local conformal-like symmetry. We write down a most general action resulting from such transformation and show that it covers both general relativity and conformally coupled to gravity scalar field as the particular cases. On quantum level the equivalence between the different frames is disturbed by the loop corrections. New conformal-like symmetry in anomalous and, as a result, the theory is not finite on shell at the one-loop order.