Nambu mechanics viewed as a Clebsch parameterized Poisson algebra -- toward canonicalization and quantization

Abstract: In his pioneering paper [Phys. Rev. E 7, 2405 (1973)], Nambu proposed the idea of multiple Hamiltonian systems. The explicit example examined there is equivalent to the so(3) Lie-Poisson system, which represents noncanonical Hamiltonian dynamics with a Casimir; the Casimir corresponds to the second Hamiltonian of Nambu's formulation. The vortex dynamics of ideal fluid, while it is infinite dimensional, has a similar structure, in which the Casimir is the helicity. These noncanonical Poisson algebras are derived by the reduction, i.e., restricting the phase space to some submanifold embedded in the canonical phase space. We may reverse the reduction to canonicalize some Nambu dynamics, i.e., view the Nambu dynamics as the subalgebra of a larger canonical Poisson algebra. Then, we can invoke the standard corresponding principle for quantizing the canonicalized system. The inverse of the reduction, i.e., representing the noncanonical variables by some canonical variables may be said "Clebsch parameterization" following the fluid mechanical example.