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 $\text{SO}(2,d)$.

arXiv comment47+16 pages, 10 figures; references added, acknowledgement corrected

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