主系列导引
An invitation to the principal series
AI总结：
本文研究dS$_2$中$SL(2,\mathbb{R})$的连续主系列表示，将其与DFF量子力学模型联系起来，并给出容纳该表示所需的修改及Hamiltonian表述。
摘要：
d维de Sitter空间的等距群$SO(1,d)$的标量幺正表示由其共形权重$Δ$标记。de Sitter空间的一个显著特征是，与de Sitter尺度$1/\ell$相比质量足够大的标量场具有复共形权重，这些场的物理模式归入$SO(1,d)$的幺正连续主系列表示。我们的目标是研究$d=2$时的这些表示，此时相关群为$SL(2,\mathbb{R})$。我们证明，作用在大质量标量场上的dS$_2$等距群生成元在早时/晚时极限下重现de Alfaro、Fubini和Furlan（DFF）提出的量子力学模型。受环境dS$_2$构造的启发，我们详细回顾了为容纳主系列表示必须如何修改DFF模型。我们指出为该模型写出经典Lagrangian存在困难，而正则Hamiltonian表述则避免了任何问题。我们从这个玩具模型的角度推测了各种de Sitter不变真空的含义，并讨论了一些可能的推广。
英文摘要：
Scalar unitary representations of the isometry group of $d$-dimensional de Sitter space $SO(1,d)$ are labeled by their conformal weights $Δ$. A salient feature of de Sitter space is that scalar fields with sufficiently large mass compared to the de Sitter scale $1/\ell$ have complex conformal weights, and physical modes of these fields fall into the unitary continuous principal series representation of $SO(1,d)$. Our goal is to study these representations in $d=2$, where the relevant group is $SL(2,\mathbb{R})$. We show that the generators of the isometry group of dS$_2$ 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$_2$ 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.
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