An Invitation to the Principal Series

Abstract

The unitary irreducible representations of the isometry group of d-dimensional de Sitter space SO(1,d) can be distinguished by their conformal dimensions $\Delta$, the eigenvalue of the dilatation operator near the origin. Scalar fields with sufficiently large mass compared to the de Sitter scale $1/L$ have complex conformal weights and physical modes of these fields fall into the continuous principal series representation of $SO(1,d)$. In $d=2$ and in global coordinates, we show that the generators of the isometry group of dS$_2$ acting on a massive scalar field reduces exactly to the quantum mechanical model introduced by de Alfaro, Fubini and Furlan (DFF) in the early/late time limit. In its original presentation, the DFF model describes a single degree of freedom on the positive semi-axis subject to a repulsive potential that diverges at the origin. The Hilbert space of this model furnishes the discrete highest weight representation of $SO(1,2)$. To accommodate the principal series representation, the potential must be made attractive, but this comes at the expense of the failure of self-adjointness of the operators, leading to the speculation that DFF can not accommodate a unitary principal series representation. Motivated by the ambient dS$_2$ construction, we explain in detail how this model must be completed in order to allow for the principal series representation and verify that all operators remain Hermitian and self-adjoint. While the conformal dimensions are complex, the representations are nevertheless completely unitary. By studying this model in detail, we explore some features one must face in the search of a dual quantum field theory that contains states in the principal series.