nLab quantization of the 2-sphere

Idea

The 2-sphere with its standard volume form is canonically a Poisson manifold and in fact a symplectic manifold. As such it may be regarded as a simple example of a phase space of a mechanical system. The quantization of this system yields the corresponding quantum mechanical system. This turns out to be that of the degrees of freedom of a rotor or spinor in 3-dimensional Euclidean space (just the “internal” rotor/spin degrees of freedom, no translational motion).

Formalizations

• in geometric quantization: geometric quantization of the 2-sphere

• in deformation quantization: deformation quantization of the 2-sphere