Geometrical Interpretation of Momentum and Crystal Momentum of Classical and Quantum Ferromagnetic Heisenberg Chains

A rotationally invariant translation-generating functional for the continuum model of a ferromagnetic chain is found, resolving a long-standing paradox. The condition of rotational invariance leads to a topological quantization in the continuum classical model that corresponds to spin quantization in the equivalent discrete quantum model. The relation between the classical continuum model and the discrete quantum models is made precise.