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

Zimo Sun

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

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

Abstract

is the isometry group of -dimensional de Sitter spacetime and the conformal group of . This note gives a pedagogical introduction to the representation theory of , 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 and , (2) the construction and classification of all UIRs of that describe integer-spin fields in , (3) a physical framework for understanding these UIRs, (4) the definition and derivation of Harish-Chandra group characters of , and (5) a comparison between UIRs of and .

Paper Structure

This paper contains 37 sections, 4 theorems, 266 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and conventions
  3. UIRs of ${\text{SO}}(1,2)$
  4. A direct construction
  5. A CFT-type construction
  6. Representation spaces
  7. Consequences of unitarity
  8. A discrete basis of ${\cal F}_\Delta$
  9. Discrete series
  10. Shadow transformations
  11. Summary
  12. Understand the two constructions from QFT
  13. Mode expansions
  14. ${\mathfrak{so}}(1,2)$ action on ${\cal H}_\Delta$
  15. The continuous basis
  16. ...and 22 more sections

Key Result

Theorem 5.3

Every admissible representation $\pi$ of a linear connected reductive group $G$ whose decomposition $\pi |_{{\text{K}}}=\bigoplus_{\tau} n_\tau \, \tau$ satisfies $n_\tau\le C \dim \tau$ has a Harish-Chandra character.

Figures (7)

  • Figure 3.4: Intertwining operator $\partial_x^{2N-1}: {\cal F}_{1-N}\to {\cal F}_N$. $L_{AB}$ denote the action of ${\mathfrak{so}}(1,2)$ on ${\cal F}_N$ and ${\cal F}_{1-N}$.
  • Figure 3.5: The relation between ${\cal F}_3$ and its shadow ${\cal F}_{-2}$. The upper red line represents the space ${\cal F}_{3}$ and each dot denotes the basis $\psi^{(3)}_k$. It contains two invariant subspaces ${\cal F}^\pm_3$ corresponding to the discrete series representations ${\cal D}^\pm_3$. The lower blue line represents the space ${\cal F}_{-2}$ and each dot denotes the basis $\psi^{(-2)}_k$. It contains a 5-dimensional invariant space $P_3$. The upward arrows denote the intertwining operators $\partial_x^5$. It annihilates $P_3$ and maps $\psi_k^{(-2)}$ to $\psi_k^{(3)}$ for $|k|\ge 3$.
  • Figure 3.6: The Penrose diagram of $\text{dS}_2$. The two sides of the Penrose diagram are identified and hence every constant time slice is a circle. The Wick rotation of the $t\le 0$ part of $\text{dS}_2$ is represented by the region bounded by dashed lines. The Euclidean vacuum $|E\rangle$ can be computed by Euclidean path integral in this region.
  • Figure 3.7: The stereographic map between $S^1$ and $\mathbb R$.
  • Figure 4.1: A sequence of intertwining maps for the exceptional representations
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Definition 5.1
  • Definition 5.2
  • Theorem 5.3
  • ...and 7 more