Constructing conformally invariant equations using the Weyl geometry

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 the Weyl geometry. (i) Weyl space is defined by the metric tensor and the Weyl vector W; it is equivalent to the Riemann space when W is a gradient. (ii) Any homogeneous differential equation written in the 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 the Riemann space. Applications to scalar, vector and spin-2 fields are presented, which demonstrate the efficiency of this 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 Minkowski and de Sitter spaces, respectively.