Revisiting the conformal invariance of the scalar field: from Minkowski space to de Sitter space

TL;DR

This paper explores the relationship between conformal invariance of scalar fields in Minkowski and de Sitter spaces, clarifying how representations transfer via Weyl correspondence and revealing effects of curvature from a Minkowskian perspective.

Contribution

It demonstrates the realization of conformal scalar field representations on de Sitter space derived from Minkowski space through an intertwining operator, clarifying the conformal invariance link.

Findings

Established the link between Minkowski and de Sitter conformal representations.

Derived the de Sitter scalar representation from Minkowski space using an intertwining operator.

Showed how curvature effects appear to Minkowskian observers.

Abstract

In this article, we clarify the link between the conformal (i.e. Weyl) correspondence from the Minkowski space to the de Sitter space and the conformal (i.e. SO(2,$d$)) invariance of the conformal scalar field on both spaces. We exhibit the realization on de Sitter space of the massless scalar representation of SO$(2,d)$. It is obtained from the corresponding representation in Minkowski space through an intertwining operator inherited from the Weyl relation between the two spaces. The de Sitter representation is written in a form which allows one to take the point of view of a Minkowskian observer who sees the effect of curvature through additional terms.