SciPost Phys. 9, 028 (2020) ·
published 1 September 2020
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· pdf
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.
