de Sitter群及其表示：一扇洞察de Sitterian基本系统概念的窗口
The de Sitter group and its representations: a window on the notion of de Sitterian elementary systems
AI总结：
本文回顾了de Sitter时空中基本系统的构造，基于dS群的酉不可约表示，探讨了从经典到量子的过渡及热力学解释，并构建了与Minkowski表述平行的一致量子场论。
摘要：
我们回顾了在de Sitter（dS）时空中（“自由”）基本系统的构造，遵循Wigner意义，作为dS（相对论）群的酉不可约表示（UIR's）的关联。本研究强调了在表述此类系统时产生的概念问题，并以数学上严谨的方式讨论了已知结果。特别关注：从经典理论到量子理论的“平滑”过渡；在曲率消失时的物理内容，从局部（“切”）Minkowski观察者的角度；以及（在量子层面上的）热力学解释，即Gibbons-Hawking温度的意义。我们回顾了与描述dS时空及存在于其上的基本系统经典相空间物理相关的dS群的三种分解。我们回顾了源自这些群分解的（射影）dS UIR's的构造。承载UIR's的（射影）Hilbert空间（在某种受限意义上）等同于dS基本系统的量子（“单粒子”）态空间。采用基于Wightman-Gärding公理及复化Riemann流形中解析性要求的成熟Fock程序，我们着手进行dS时空中基本系统的一致量子场论（QFT）表述。该dS QFT表述与相应的Minkowski表述高度相似，而通常的谱条件被某种几何Kubo-Martin-Schwinger（KMS）条件取代，该条件等价于相关“真空”态的精确热力学表现。
英文摘要：
We review the construction of ("free") elementary systems in de Sitter (dS) spacetime, in the Wigner sense, as associated with unitary irreducible representations (UIR's) of the dS (relativity) group. This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: "smooth" transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local ("tangent") Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. We review three decompositions of the dS group physically relevant for the description of dS spacetime and classical phase spaces of elementary systems living on it. We review the construction of (projective) dS UIR's issued from these group decompositions. (Projective) Hilbert spaces carrying the UIR's (in some restricted sense) identify quantum ("one-particle") states spaces of dS elementary systems. Adopting a well-established Fock procedure, based on the Wightman-Gärding axioms and on analyticity requirements in the complexified Riemannian manifold, we proceed with a consistent quantum field theory (QFT) formulation of elementary systems in dS spacetime. This dS QFT formulation closely parallels the corresponding Minkowskian one, while the usual spectral condition is replaced by a certain geometric Kubo-Martin-Schwinger (KMS) condition equivalent to a precise thermal manifestation of the associated "vacuum" states.
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