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arXiv · 1610.08081

Octonions in random matrix theory

Peter J. Forrester
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Abstract

The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random by symmetry considerations. Only for $N=2$ is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for $N=3$. We then proceed to consider the matrix structure $X^\dagger X$, when $X$ has random octonion entries. Analytic results are obtained from $N=2$, but are observed to break down in the $3 \times 3$ case.

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Peter J. Forrester. 2016-10-25. Octonions in random matrix theory. https://doi.org/10.1098/rspa.2016.0800

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