Constructing conformally invariant equations using Weyl geometry

TL;DR

This paper introduces a novel method using Weyl geometry to systematically construct conformally invariant equations in four-dimensional spaces, ensuring invariance through Weyl covariant derivatives and tensors.

Contribution

The paper presents a new approach leveraging Weyl geometry to generate conformally invariant equations, unifying the construction process in a geometric framework.

Findings

Constructed conformally invariant scalar field equations

Verified conformal invariance of Maxwell equations

Recovered Eastwood-Singer gauge condition

Abstract

A new method for the construction of conformally invariant equations in an arbitrary four dimensional (pseudo-) Riemannian space is presented. This method uses the Weyl geometry as a tool and exploits the natural conformal invariance we can build in the framework of this geometry. Indeed, working in a Weyl space, using the Weyl covariant derivative and the intrinsic Weylian geometrical tensors, all conformally homogeneous operators will be conformally invariant, as will the equations they determine. A Weyl space is defined by two independent objects: the metric tensor $g_{\mu\nu}$ and the Weyl vector $W_{\mu}$. A simple procedure allows us to go from a Weyl space into a Riemann space by imposing the Weyl vector to be a gradient. Under some conditions, the Weylian conformally invariant equations reduce to Riemannian conformally invariant equations. This method is applied to construct some conformally invariant scalar field equations, check the conformal invariance of Maxwell equations and recover the Eastwood-Singer conformal gauge fixing condition.