Scalar unitary representations of the isometry group of
dd-dimensional
de Sitter space SO(1,d)SO(1,d)
are labeled by their conformal weights \DeltaΔ.
A salient feature of de Sitter space is that scalar fields with
sufficiently large mass compared to the de Sitter scale
1/\ell1/ℓ
have complex conformal weights, and physical modes of
these fields fall into the unitary continuous principal series
representation of SO(1,d)SO(1,d).
Our goal is to study these representations in
d=2d=2,
where the relevant group is SL(2,\mathbb{R})SL(2,ℝ).
We show that the generators of the isometry group of
dS_22
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_22
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.
Group theoretical approach to quantum fields in de Sitter space, I. The principal series
