Nambu mechanics and its quantization

The algebra of observables inherent in the Nambu formalism [Phys. Rev. D 7, 2405 (1973)] for a generalization of classical Hamiltonian dynamics is investigated. A consistency requirement of time evolution of the Nambu bracket leads to a five-point identity. Two types of algebras are possible at the classical level. Their composition properties under a tensor product are considered and the physical implications are analyzed. A quantum generalization of these algebras is shown to be impossible.