Coherent states for n‐dimensional Euclidean groups E(n) and their application

The n‐dimensional Euclidean group—the group of rotations and translations of Rn —is the natural canonical group for quantizing a system whose configuration space is the sphere Sn−1, n≥2. Since every unitary irreducible representation of E(n) is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations. Instead, coherent states are defined based on unitary, reducible representations of the group, and it will be shown that, with such states, quantum dynamics on the sphere admits a path‐integral representation involving novel first‐order classical actions defined on E(n) as well as integrations over the entire group manifold at each time slice. Irreducibility of the quantization is achieved through a final limiting process.