A note on the representations of $\text{SO}(1,d+1)$

Abstract

$\text{SO}(1,d+1)$ is the isometry group of $(d+1)$-dimensional de Sitter spacetime ${\text{dS}}_{d+1}$ and the conformal group of ${\mathbb{R}}^d$. This note gives a pedagogical introduction to the representation theory of $\text{SO}(1,d+1)$, from the perspective of de Sitter quantum field theory and using tools from conformal field theory. Topics include (1) the construction and classification of all unitary irreducible representations (UIRs) of $\text{SO}(1,2)$ and $\text{SL}(2,\mathbb{R})$, (2) the construction and classification of all UIRs of $\text{SO}(1,d+1)$ that describe integer-spin fields in ${\text{dS}}_{d+1}$, (3) a physical framework for understanding these UIRs, (4) the definition and derivation of Harish-Chandra group characters of $\text{SO}(1,d+1)$, and (5) a comparison between UIRs of $\text{SO}(1,d+1)$ and $\text{SO}(2,d)$.

47+16 pages, 10 figures; references added, acknowledgement corrected

References in corpus (5)

• Lectures on Celestial Amplitudes
• Analyticity and Unitarity for Cosmological Correlators
• From dS to AdS and back
• Light-cone continuous-spin field in AdS space
• Manifest Duality for Partially Massless Higher Spins

Cited by in corpus (10)

• Lectures in Quantum Gravity
• Physical Representations of Corner Symmetries
• Real-time observables in de Sitter thermodynamics
• De Sitter Horizon Edge Partition Functions
• RG flows in de Sitter: c-functions and sum rules
• Semi-Classical Limit of Quantum Gravity on Corners
• A Match Made in Heaven: Linking Observables in Inflationary Cosmology
• New Exceptional EFTs in de Sitter Space from Generalised Energy Conservation
• Gravitons on Nariai Edges
• Soft unification of exceptional effective field theories in de Sitter space