In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe–Ritter to the sphere \({{\mathbb {S}}}^n\). We determine the resulting Osterwalder–Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group \(G^c = \mathop {{\mathrm{O}}{}}\nolimits _{1,n}({{\mathbb {R}}})^{\uparrow }\) and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain \(\Xi \), known as the crown of the hyperboloid, containing a half-sphere \({{\mathbb {S}}}^n_+\) and the hyperboloid \({{\mathbb {H}}}^n\) as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of \(G^c\) in spaces of holomorphic functions on \(\Xi \). We connect this analysis with the boundary components which are the de Sitter space and a bundle over the space of future pointing lightlike vectors.

中文翻译：

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球体的反射正性

在本文中，由于Dimock和Jaffe-Ritter到球面\（{{\ mathbb {S}}} ^ n \），我们专门研究了反射正Hilbert空间的构造。我们确定产生的Osterwalder–Schrader Hilbert空间，该结构可以看作是从欧几里德到相对论量子场论的一步。我们证明了这个过程引起了不可逆的Lorentz群\（G ^ c = \ mathop {{\ mathrm {O}} {}} \ nolimits _ {1，n}（{{\ mathbb {R}}}）^ {\ uparrow} \），由此获得的表示是该组的不可约的unit球表示。一个关键工具是某个复杂的域\（\ Xi \），称为双曲面的冠，包含一个半球\（{{\ mathbb {S}}} ^ n _ + \）和双曲面\（{{\ mathbb {H}}} ^ n \）是完全实的子流形。当我们研究\（\ Xi \）上全纯函数空间中 \（G ^ c \）的单位表示时，该域在这两个流形之间提供了桥梁。我们将此分析与边界分量相连，这些边界分量是de Sitter空间和将来指向的类似光的向量空间上的束。