v2 is is the definitive (improved compare to v1) versionInternational audienceIn this article, we quantize the Maxwell ("massless spin one") de Sitter field in a conformally invariant gauge. This quantization is invariant under the SO0(2,4) group and consequently under the de Sitter group. We obtain a new de Sitter-invariant two-point function which is very simple. Our method relies on the one hand on a geometrical point of view which uses the realization of Minkowski, de Sitter and anti-de Sitter spaces as intersections of the null cone in R6 and a moving plane, and on the other hand on a canonical quantization scheme of the Gupta-Bleuler type

The present work deals with two different but subtilely related kinds of conformal mappings: Weyl rescaling in d > 2 dimensional spaces and SO(2, d) transformations. We express how the difference between the two can be compensated by diffeomorphic transformations. This is well known in the framework of String Theory but in the particular case of d = 2 spaces. Indeed, the Polyakov formalism describes world-sheets in terms of two-dimensional conformal field theory. On the other hand, B. Zumino had shown that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski space leads to an SO(2, 4)-invariant field theory. We extend Zumino's result to relate Weyl and SO(2, d) symmetries in arbitrary conformally flat spaces (CFS). This allows us to assert that a classical SO(2, d)-invariant field does not distinguish, at least locally, between two different d-dimensional CFSs.

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.