Yang–Mills-like field theories built on division quaternion and octonion algebras

Abstract

Yang–Mills-like field theories are developed using both division quaternion and octonion algebras. Quaternionic field theory is the same as the Yang–Mills gauge theory. The octonionic field theory is an expansion from a similar field theory built with split octonions algebra that uses a complex-form basis as proposed by Gunaydin and Gursey (J Math Phys 14:1651, 1973), now with division octonions. This proposed octonionic field theory is the last possible field theory built using the division algebra of octonions, as permitted by the Hurwitz theorem. The interpretation of this “octonionic field” is not straightforward, except that it is similar to the quaternionic Yang–Mills field. A possible interpretation for this octonionic field theory for spin-1/2 particles is proposed in terms of quantum mechanics.