IN NATURE, there is a deep connection between exceptional mathematical structures and the laws of micro- and macro-physics --- Quaternions and Octonions have played an important role in the recent development of pure and applied physics.
Quaternions Discovered by Hamilton in 1843, Quaternions' main use in the 19th century consisted in expressing physical theories in "Quaternionic notation". An important work where this was done was Maxwell's treatise on electricity and magnetism. Towards the end of the century, the value of their use in electromagnetic theories led to a heated debate dubbed "The Great Quaternionic War".
In a 1936 paper, Birkhoff and von Neumann presented a propositional calculus for Quantum Mechanics and showed that a concrete realisation leads to the general result that a Quantum Mechanical system may be represented as a vector space over the Real, Complex, and Quaternionic fields. Since then this area has remained active, aiming to extend Complex Quantum Mechanics (CQM) by generalising the complex unit in CQM to Quaternions and to find observable effects of QQM. Jordan Algebras were proposed by Jordan, Neumann and Wigner in formulating non-associative Quantum Mechanics, where quantisation is achieved through associators rather than commutators. This formulation allows mixing of space-time and internal symmetries. Another attractive feature of Jordan Algebra is that critical dimensions of 10 and 26 arise naturally, suggesting a connection to string theory.
Away from physics, Quaternions have recently been used for robotic control, computer graphics, vision theory, spcecraft orientation and geophysics. The space shuttle's flight software uses Quaternions in its guidance navigation and flight control computations.
Octonians In the early days of Quantum Mechanics, there were several attempts to introduce new algebraic structures in physics. In recent years, there has been remarkable activity in the field of physical applications of non-associative algebras: Octonian formulations of Yang-Mills gauge theories and string theory, Octonian description of quarks and leptons, and Octonian supergravity.