  Phase Space is the framework best suited for quantizing superintegrable
systems, naturally preserving the symmetry algebras of the respective
hamiltonian invariants. The power and simplicity of the method is fully
illustrated through new applications to nonlinear sigma models, specifically
for de Sitter N-spheres and Chiral Models, where the symmetric quantum
hamiltonians amount to compact and elegant expressions. Additional power and
elegance is provided by the use of Nambu Brackets to incorporate the extra
invariants of superintegrable models. Some new classical results are given for
these brackets, and their quantization is successfully compared to that of
Moyal, validating Nambu's original proposal.
