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In this work we look at a hypercomplex algebra prominent in quantum chromo (color) dynamics, introduced by S. Okubo, generated by the eight \(3\times 3\) traceless Hermitian matrices of M. Gell-Mann. We go beyond the familiar algebras of W.K. Clifford (geometric algebra), W.R. Hamilton (quaternions and biquaternions) and J.T. Graves and A. Cayley (octonions) and introduce several properties of the eight dimensional Okubo algebra, known to be a division algebra, not unital, not associative, not alternative but flexible with a positive definite norm that is associative and compositional. We give an easy to interpret full multiplication table, show how two units can generate the whole algebra, study several sub-algebras and look at powers and exponentials of Okubo algebra units.

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Contemplating Susumu Okubo’s Algebra of Quantum Color Dynamics

  • 会议论文

  • 发表日期：

    2026年

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  • Eckhard Hitzer

摘要

In this work we look at a hypercomplex algebra prominent in quantum chromo (color) dynamics, introduced by S. Okubo, generated by the eight \(3\times 3\) traceless Hermitian matrices of M. Gell-Mann. We go beyond the familiar algebras of W.K. Clifford (geometric algebra), W.R. Hamilton (quaternions and biquaternions) and J.T. Graves and A. Cayley (octonions) and introduce several properties of the eight dimensional Okubo algebra, known to be a division algebra, not unital, not associative, not alternative but flexible with a positive definite norm that is associative and compositional. We give an easy to interpret full multiplication table, show how two units can generate the whole algebra, study several sub-algebras and look at powers and exponentials of Okubo algebra units.