发表机构
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences; Université Paris Cité; CNRS; University of Białystok（保加利亚科学院数学与信息学研究所; 巴黎西岱大学; 法国国家科学研究中心; 比亚韦斯托克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究德西特群\(\mathrm{SO}_0(1,4)\)的标量离散系列酉不可约表示\(\Pi_{p,0}\)，通过傅里叶型体 - 边界变换，在德西特双曲面上实现其dS协变克莱因结构，建立体与边界物理扇区对应，保持反射正性，确保边界构造一致性。
AI 中文摘要
我们表明，德西特（dS）群\(\mathrm{SO}_0(1,4)\)的标量离散系列酉不可约表示（UIRs）\(\Pi_{p,0}\)（\(p = 1,2,\cdots\)）在德西特双曲面上允许一种dS协变的克莱因实现，赋予其dS不变的非退化克莱因 - 戈登（KG）半双线性形式，其中群作用不可分解且自然地组织成一个古普塔 - 布勒勒三元组。正负范数扇区已存在于底层克莱因空间中，而零扇区仅在中间阶段出现，此时诱导的KG形式退化，其根规范地导致承载UIR\(\Pi_{p,0}\)的物理商空间。我们进一步表明，在“未来”和“过去”共形边界\({\mathcal{I}}^\pm\)处体理论的适当极限产生具有诱导核内积的dS不变边界实现。虽然体负范数扇区没有独立的边界对应物，但边界实现保留了从体继承的物理和规范结构。所得的边界模仍然不可分解，其物理商空间承载离散系列表示\(\Pi_{p,0}\)。对映对称性在\({\mathcal{I}}^+\)和\({\mathcal{I}}^-\)上的实现之间提供了自然关系，确保了边界构造及其几何解释的一致性。分析的核心是一个傅里叶型体 - 边界变换，它提供了体和边界物理扇区的dS协变识别，在\(\Pi_{p,0}\)的体和边界实现之间建立了一一交织对应，同时保持反射正性。
英文摘要
We show that scalar discrete-series unitary irreducible representations (UIRs) $Π_{p,0}$ ($p=1,2,\cdots$) of the de Sitter (dS) group $\mathrm{SO}_0(1,4)$ admit a dS-covariant Krein realization on the dS hyperboloid, endowed with a dS-invariant non-degenerate Klein-Gordon (KG) sesquilinear form, in which the group action is indecomposable and organizes naturally into a Gupta-Bleuler triplet. The positive- and negative-norm sectors are already present in the underlying Krein space, whereas a null sector emerges only at an intermediate stage, where the induced KG form becomes degenerate and its radical leads canonically to the physical quotient carrying the UIR $Π_{p,0}$. We further show that suitable limits of the bulk theory at the ``future'' and ``past'' conformal boundaries ${\mathcal{I}}^\pm$ give rise to dS-invariant boundary realizations endowed with induced kernel inner products. While the bulk negative-norm sector admits no independent boundary counterpart, the boundary realization retains the physical and gauge structures inherited from the bulk. The resulting boundary module nevertheless remains indecomposable, with its physical quotient carrying the discrete-series representation $Π_{p,0}$. The antipodal symmetry provides a natural relation between the realizations on ${\mathcal{I}}^+$ and ${\mathcal{I}}^-$, ensuring the consistency of the boundary construction and its geometric interpretation. At the heart of the analysis lies a Fourier-type bulk-boundary transform that provides a dS-covariant identification of the bulk and boundary physical sectors, establishing a one-to-one intertwining correspondence between the bulk and boundary realizations of $Π_{p,0}$ while preserving reflection positivity.
Comments33 pages, 4 figures, 1 table, version accepted for publication in PRD
