Constructing conformally invariant equations by using Weyl geometry

TL;DR

This paper introduces a systematic Weyl-to-Riemann method for constructing conformally invariant equations in arbitrary Riemann spaces, demonstrated through applications to various fields including a new spin-2 field equation.

Contribution

The paper presents a novel, practical method to derive conformally invariant equations using Weyl geometry, extending previous equations and removing gauge restrictions.

Findings

A new conformally invariant spin-2 field equation is derived.

The method successfully reproduces known equations in Minkowski and de Sitter spaces.

The approach simplifies constructing conformally invariant equations across different fields.

Abstract

We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. This method that we call "Weyl-to-Riemann" is based on two features of Weyl geometry. i) A Weyl space is defined by the metric tensor and the Weyl vector $W$, it becomes equivalent to a Riemann space when $W$ is gradient. ii) Any homogeneous differential equation written in a Weyl space by means of the Weyl connection is conformally invariant. The Weyl-to-Riemann method selects those equations whose conformal invariance is preserved when reducing to a Riemann space. Applications to scalar, vector and spin-2 fields are presented, which demonstrates the efficiency of the present method. In particular, a new conformally invariant spin-2 field equation is exhibited. This equation extends Grishchuk-Yudin's equation and fixes its limitations since it does not require the Lorenz gauge. Moreover this equation reduces to the Drew-Gegenberg and Deser-Nepomechie equations in respectively Minkowski and de Sitter spaces.