An invitation to the principal series

TL;DR

This paper explores scalar unitary representations of de Sitter space's isometry group in two dimensions, linking them to a quantum mechanical model and discussing implications for de Sitter vacua and potential generalizations.

Contribution

It demonstrates that the isometry generators on a massive scalar field in dS2 reproduce the DFF quantum model and adapts this model for principal series representations, highlighting classical and quantum aspects.

Findings

Generators reproduce DFF model in dS2

Modified DFF model accommodates principal series

Hamiltonian formulation avoids classical Lagrangian issues

Abstract

Scalar unitary representations of the isometry group of $d$-dimensional de Sitter space $SO(1,d)$ are labeled by their conformal weights $\Delta$. A salient feature of de Sitter space is that scalar fields with sufficiently large mass compared to the de Sitter scale $1/\ell$ have complex conformal weights, and physical modes of these fields fall into the unitary continuous principal series representation of $SO(1,d)$. Our goal is to study these representations in $d=2$, where the relevant group is $SL(2,\mathbb{R})$. We show that the generators of the isometry group of dS$_2$ acting on a massive scalar field reproduce the quantum mechanical model introduced by de Alfaro, Fubini and Furlan (DFF) in the early/late time limit. Motivated by the ambient dS$_2$ construction, we review in detail how the DFF model must be altered in order to accommodate the principal series representation. We point out a difficulty in writing down a classical Lagrangian for this model, whereas the canonical Hamiltonian formulation avoids any problem. We speculate on the meaning of the various de Sitter invariant vacua from the point of view of this toy model and discuss some potential generalizations.