AI 中文总结
研究指出除量子理论统计结构外，需‘诺特定理结构’描述系统组合及定义完全正定操作，其反映厄米算符双重作用，研究限于有限维情况并建立与完全正定操作的联系。
AI 中文摘要
在本文中，我们认为，除了量子理论的统计结构之外，另一种结构，这里称为“诺特定理结构”，对于描述系统的组成和定义完全正定操作是必要的。诺特定理结构反映了厄米算符一方面作为可观测量，另一方面作为对称变换生成元的双重作用。这个想法在阿尔夫森和舒尔茨的作品中以类似的形式表达过，他们研究了约旦积可以扩展为算子代数的结合积的条件。我们对诺特定理结构和系统组成的研究仅限于有限维情况，并建立了与完全正定操作的联系。在纯操作的情况下，后者可以被表征为保取向映射。
英文摘要
In this paper we argue that, in addition to the statistical structure of quantum theory, another structure, referred to here as the ``Noether structure," is necessary to describe the composition of systems and to define completely positive operations. A Noether structure reflects the dual role of Hermitian operators as observables on the one hand and as generators of symmetry transformations on the other. This idea has been expressed in a similar form in the works of Alfsen and Shultz, who investigated the conditions under which the Jordan product can be extended to an associative product of operator algebras. Our investigations into the Noether structure and the composition of systems are limited to the finite-dimensional case and establish a connection to completely positive operations. In the case of pure operations, the latter can be characterized as orientation-preserving maps.
CommentsContains some corrections, extensions and further references
