The role of conformal symmetry in gravity and the standard model

TL;DR

This paper explores conformal symmetry in gravity with affine connections, extending transformation laws and invariance properties, and discusses its potential extension to the standard model with a dilaton field.

Contribution

It generalizes conformal transformations to affine connections and analyzes their invariance properties, also examining the extension of conformal symmetry to matter fields in the standard model.

Findings

Conformal symmetry is exact in classical Einstein gravity.

Conformal invariant Lagrangians exist for matter fields in any dimension.

Gauge invariance of these Lagrangians is restricted to four dimensions.

Abstract

In this paper we consider conformal symmetry in the context of manifolds with general affine connection. We extend the conformal transformation law of the metric to a general metric compatible affine connection, and find that it is a symmetry of both the geodesic equation and the Riemann tensor. We derive the generalised Jacobi equation and Raychaudhuri equation and show that they are both conformally invariant. Using the geodesic deviation~(Jacobi) equation we analyse the behaviour of geodesics in different conformal frames. Since we find that our version of conformal symmetry is exact in classical pure Einstein's gravity, we ask whether one can extend it to the standard model. We find that it is possible to write conformal invariant lagrangians in any dimensions for vector, fermion and scalar fields, but that such lagrangians are only gauge invariant in four dimensions. Provided one introduces a dilaton field, gravity can be conformally coupled to matter.