Glimpses of the Octonions and Quaternions History and Today's Applications in Quantum Physics

Abstract

Before we dive in this essay into the accessibility stream of nowadays indicatory applications of octonions and quaternions to computer and other sciences and to quantum physics (see for example [50-53], [41], [33]) and to Clifford algebras (see for example [17,16], 18) let us focus for a while on the crucially relevant events for today's revival on interest to nonassociativities while the role of associative quaternions in eight periodicity constructive classification of associative Clifford algebras is now a text-book knowledge. Our reflections keep wandering back to the Brahmagupta-Fibonacci Two-Square Identity and then via the Euler Four-Square Identity up to the Degen-Graves-Cayley Eight-Square Identity. These glimpses of history incline and invite us to re-tell the story on how about one month after quaternions have been carved on the Brougham bridge octonions were discovered by John Thomas Graves (1806-1870), jurist and mathematician - a friend of William Rowan Hamilton (1805-1865). As for today we just mention en passant quaternionic and octonionic quantum mechanics, generalization of Cauchy-Riemann equations for octonions and Triality Principle and G 2 group in spinor language in a descriptive way in order not to daunt non-specialists. Relation to finite geometries is recalled and the links to the 7Stones of seven sphere, seven "imaginary" octonions' units in out of the Plato's Cave Reality applications are appointed. This way we are welcome back to primary ideas of Heisenberg, Wheeler and other distinguished founders of quantum mechanics and quantum gravity foundations. © 2011 Springer Basel AG.