Spinning fields on S$^d$ and dS$_d$, UIRs and Ladder operators

TL;DR

This paper constructs ladder operators for tensor fields on spheres and de Sitter space, connecting different unitary irreducible representations using conformal Killing vectors, extending previous results to spinning fields and exceptional representations.

Contribution

It introduces a unified method to generate ladder operators for spin 0, 1, 2 fields on S^d and dS_d, generalizing previous work and including exceptional UIRs.

Findings

Ladder operators connect different UIRs of SO(d+1) and SO(d,1).

The approach recovers known conformal primary transformations for scalar fields.

Extension to spin-2 fields yields conformal-like operators similar to recent findings.

Abstract

We construct, for spin $0,1,2$ tensor fields on S$^d$, a set of ladder operators that connect the distinct UIRs of SO$(d+1)$. This is achieved by relying on the conformal Killing vectors of S$^d$. For the case of spinning fields, the ladder operators generalize previous expressions with a compensating transformation necessary to preserve the transversality condition. We then extend the results to the Exceptional/Discrete UIRs of SO$(d,1)$, again relying on the conformal Killing vectors of de Sitter space. Our construction recovers the conventional conformal primary transformations for the scalar fields when the mass term leads to conformal coupling. A similar approach for the spin-2 field leads to the conformal-like operators found recently.