Abstract
AIP Conf.Proc.672:183-196,2003 Phase Space is the framework best suited for quantizing superintegrable
systems--systems with more conserved quantities than degrees of freedom. In
this quantization method, the symmetry algebras of the hamiltonian invariants
are preserved most naturally, as illustrated on nonlinear $\sigma$-models,
specifically for Chiral Models and de Sitter $N$-spheres. Classically, the
dynamics of superintegrable models such as these is automatically also
described by Nambu Brackets involving the extra symmetry invariants of them.
The phase-space quantization worked out then leads to the quantization of the
corresponding Nambu Brackets, validating Nambu's original proposal, despite
excessive fears of inconsistency which have arisen over the years. This is a
pedagogical talk based on hep-th/0205063 and hep-th/0212267, stressing points
of interpretation and care needed in appreciating the consistency of Quantum
Nambu Brackets in phase space. For a parallel discussion in Hilbert space, see
T Curtright's contribution in these Proceedings [hep-th 0303088].