AI 中文总结
该研究针对德西特背景下的自由标量场，通过结合赤道反射构造希尔伯特积，发现满足共形幺正性边界时可得到AdS等距的正能表示，建立了无需半径解析延拓的AdS与dS的新颖表示论关系。
AI 中文摘要
德西特（De Sitter）幺正性与反射正性在逻辑上相互独立。我们针对固定德西特背景下的自由标量场，仅关注单粒子部分，研究二者的关系。受威滕（Witten）的德西特配对以及布索（Bousso）、马洛尼（Maloney）和斯特罗明格（Strominger）的径向伴随的启发，我们将不变希尔伯特积与赤道反射相结合。在标准主系列实现中，该形式完全消失，其奥斯特瓦尔德-施拉德（Osterwalder-Schrader）商不含任何态；对于互补系列，反射后的乘积成为单位球上非零的反射核。尼布（Neeb）和奥拉夫松（Ólafsson）的结果表明，当低权重满足标量共形幺正性边界时，该核恰好是半正定的。根据尼布和奥拉夫松的结论，带有反射内积的所得希尔伯特空间提供了AdS等距的正能表示。我们从物理学角度确认，该范围与克莱巴诺夫-维滕（Klebanov-Witten）的替代量子化窗口一致。这种对应关系表明，从表示论视角看，存在一种无需常规半径解析延拓的AdS与dS之间的新颖关系。
英文摘要
De Sitter unitarity and reflection positivity are logically independent. We study their relation for a free scalar field on a fixed de Sitter background, restricting attention to the one-particle sector. Motivated by Witten's de Sitter pairing and the radial adjoint of Bousso, Maloney and Strominger, we compose the invariant Hilbert product with equatorial reflection. In the standard principal-series realization, this form vanishes identically and its Osterwalder-Schrader quotient contains no states. For the complementary series, the reflected product becomes a nonzero reflected kernel on the unit ball. A result of Neeb and Ólafsson shows that this kernel is positive semidefinite precisely when the lower weight satisfies the scalar conformal unitarity bound. The resulting Hilbert space with the reflected inner product furnishes a positive-energy representation of AdS isometries according to Neeb and Ólafsson. We identify that the range coincides with the Klebanov-Witten alternative-quantization window from physics point of view. This correspondence suggests a novel relation from the representation viewpoint between AdS and dS without the conventional analytic continuation of radius.
