非结合代数与量子物理--一个历史的观点
Non-Associative Algebras and Quantum Physics -- A Historical Perspective
摘要：
我们回顾帕斯夸尔约旦和其他研究人员，最值得注意的是劳伦斯Biedenharn试图推广量子力学通过从关联矩阵或算子代数非结合代数。我们开始与约旦的工作从20世纪30年代初导致约旦代数和第一次尝试将替代环的八元数到物理。约旦的工作八元数从1932年至1952年将涵盖，讨论方面的例外约旦代数和如何表达概率时，工作的八元数和例外约旦代数。从20世纪50年代起，乔丹和其他人也考虑了单边分配系统，如近场，近环，准场和特殊的西格尔系统（最后两个例子不一定是结合的）。由于非线性算子的集合形成了一个近环，甚至是一个近代数，这对于在量子力学的研究中从线性到非线性的尝试可能是有意义的。此外，在20世纪60年代末引入的想法，使用非幂结合代数制定一个最小长度的理论将被覆盖。劳伦斯Biedenharn的和约旦的想法有关的非权力联想octonionic矩阵代数将简要提及，一个很长的部分是专门总结霍斯特Rühaak的博士论文从1968年矩阵代数的octonion。最后，将描述最近尝试在物理学中使用非结合代数。
英文摘要：
We review attempts by Pascual Jordan and other researchers, most notably Lawrence Biedenharn to generalize quantum mechanics by passing from associative matrix or operator algebras to non-associative algebras. We start with Jordan's work from the early 1930ies leading to Jordan algebras and the first attempt to incorporate the alternative ring of octonions into physics. Jordan's work on the octonions from 1932 till 1952 will be covered, discussing aspects of the exceptional Jordan algebra and how to express probabilities when working with the octonions and the exceptional Jordan algebra. From the 1950ies onwards Jordan and others also considered one-sided distributive systems like near-fields, near-rings, quasi-fields and exceptional Segal systems (the last two examples being not necessarily associative). As the set of non-linear operators forms a near-ring and even a near-algebra, this may be of interest for attempts to pass from a linear to a non-linear setting in the study of quantum mechanics. Moreover, ideas introduced in the late 1960ies to use non-power-associative algebras to formulate a theory of a minimal length will be covered. Lawrence Biedenharn's and Jordan's ideas related to non-power-associative octonionic matrix algebras will be briefly mentioned, a long section is devoted to a summary of Horst Rühaak's PhD thesis from 1968 on matrix algebras over the octonions. Finally, recent attempts to use non-associative algebras in physics will be described.
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