An invitation to the principal series

Abstract

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.

Three-dimensional de Sitter horizon thermodynamics

“…The generators (5.23) furnish the principal series representation with weight ∆ = 1/2 + ip n /2. It is known that the spectrum of Ĥn + Kn is given by the even integers [64][65][66], and is thus unbounded. Consequently, the single mode energy spectrum is given by…”

“…where Qt = n Q(n) t . We represent the raising and lowering operators as We then make the following change of coordinates: which is the conformal quantum mechanics problem studied in [63,64]. We need to patch this solution to the one for |y| > |x| which can be found by exchanging x n ↔ y n in (C.4) and following the same steps.…”

“…. The operator on the left-hand side of(5.22) is precisely that of a conformal quantum mechanics as studied for example in[63][64][65][66], whose sl(2, R) generators are given by…”

AdS3/AdS2 degression of massless particles

“…AdS2 field equations and their solutions attracted recently some attention mainly because of the SYK/JT duality problem and the conformal bootstrap, see e.g. recent[8][9][10] 3. For more discussion in the present context see e.g [11][12][13][14].…”

Wavefunctions in dS/CFT revisited: principal series and double-trace deformations

“…See however, ref [46]. for recent discussion on heavy fields in dS2/CFT1, which made an attempt toward a possible holographic interpretation of the de Alfaro, Fubini and Furla model of conformal mechanics.…”