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

TL;DR

This paper reviews the historical development and recent efforts in applying non-associative algebras, such as Jordan algebras and octonions, to generalize quantum mechanics and explore foundational physics concepts.

Contribution

It provides a comprehensive historical overview and discusses recent research on non-associative algebras in quantum physics, highlighting novel algebraic frameworks for physical theories.

Findings

Jordan's work on octonions and exceptional Jordan algebra analyzed

Non-associative algebras used to formulate minimal length theories

Recent research explores non-associative algebras in modern physics

Abstract

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 R\"uhaak's PhD thesis from 1968 on matrix algebras over the octonions. Finally, recent attempts to use non-associative algebras in physics will be described.