Canonical Groups for Quantization on the Two-Dimensional Sphere and One-Dimensional Complex Projective Space

Using Isham's group-theoretic quantization scheme, we construct the canonical groups of the systems on the two-dimensional sphere and one-dimensional complex projective space, which are homeomorphic. In the first case, we take SO(3) as the natural canonical Lie group of rotations of the two-sphere and find all the possible Hamiltonian vector fields, and followed by verifying the commutator and Poisson bracket algebra correspondences with the Lie algebra of the group. In the second case, the same technique is resumed to define the Lie group, in this case SU (2), of CP'.We show that one can simply use a coordinate transformation from S2 to CP1 to obtain all the Hamiltonian vector fields of CP1. We explicitly show that the Lie algebra structures of both canonical groups are locally homomorphic. On the other hand, globally their corresponding canonical groups are acting on different geometries, the latter of which is almost complex. Thus the canonical group for CP1 is the double-covering group of SO(3), namely SU(2). The relevance of the proposed formalism is to understand the idea of CP1 as a space of where the qubit lives which is known as a Bloch sphere.