Non-Associative Algebras and Quantum Physics -- A Historical Perspective.

We review attempts by Pascual Jordan and other researchers, most notably Lawrence Biedenharn to generalize quantum mechanics by passing from associative matrix or operator algebrasto non-associative algebras. We start with Jordan's work from the early 1930ies leading to Jordan algebrasand the first attempt to incorporate the alternative ring of octonionsinto physics. Jordan's work on the octonionsfrom 1932 till 1952 will be covered, discussing aspects of the exceptional Jordan algebraand how to express probabilities when working with the octonionsand 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-ringand 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 algebrasto formulate a theory of a minimal length will be covered. Lawrence Biedenharn's and Jordan's ideas related to non- power-associative octonionicmatrix algebras will be briefly mentioned, a long section is devoted to a summary of HorstRuhaak's PhD thesis from 1968 on matrix algebras over the octonions. Finally, recent attempts to use non-associative algebrasin physics will be described.