Conservation laws in de Sitter space from action principles in five-space

Abstract

By recognizing the resemblance of the de Sitter group algebra to that of the conformal group, the method by which manifestly conformally covariant field equations in sixdimensional space are rewritten in Minkowski space is adapted to fields in flat five-dimensional space, the embedding space of de Sitter space. A quantum action principle based solely on rotational invariance in five-dimensional space is devised, and the resulting commutation relations are shown to correspond to the correct ones in curved four-space. As well as recovering the ten conservation laws associated with de Sitter group invariance, the five extra conservation laws present whenever conformal symmetry holds are determined directly in five-space. The derivation is found to be complicated by a new feature—the Lagrangian density does not transform as a field either for special conformal transformations or for dilations; this is true only for the former transformations in flat space.