Ladder operators of the de Sitter algebra

TL;DR

This work identifies ladder operators for the de Sitter algebra $so(d+1,1)$ in $dS_4$ by treating translations $T_i$ as dilatation-raising and special conformal transformations $C_i$ as dilatation-lowering operators, analyzing their action on weight states $| abla, l\rangle$ with the Casimir constraint $c_{so(d+1,1)}=\Delta(\Delta- d)+l(l+d-2)$. By fixing the action of the lowering operator on chosen lowest-weight states within the discrete-series (notably the massless scalar states $|0,0\rangle$ and $|3,0\rangle$ at the late-time boundary), the authors derive normalization relations for the ladder operators and construct explicit towers of states. They find a finite one-step tower arising from $|0,0\rangle$ and an infinite ladder from $|3,0\rangle$, with concrete relations such as $T_i|3,0\rangle=3a|4,1\rangle$ and $C_i|4,1\rangle=-\frac{2}{a}|3,0\rangle$, illustrating how $(\Delta,l)$ evolve along the ladders. The results provide a practical framework to identify lowest/highest-weight sectors for quantized fields on a fixed de Sitter background and point to future work on higher-spin cases and field-theory realizations, as well as connections to alternative ladder-operator constructions in the literature.

Abstract

We identify raising and lowering operators of the de Sitter algebra with focus on their action on states in particular in 4 spacetime dimensions. There isn't a unique solution to the question of how the de Sitter ladder operators act on states. By fixing the action of certain generators one can conclude on the action of the rest. Our main aim is to be able to identify highest and lowest weight states such that we can recognize them for quantized fields on a rigid de Sitter background.