ZM theory III: Classical oscillators and semi-classical Bohr-Sommerfeld quantization

We consider the description of classical oscillatory motion in ZM theory, and explore the relationship of ZM theory to semi-classical Bohr-Sommerfeld quantization. The treatment illustrates some features of ZM theory, especially the inadequacies of classical and semi-classical treatments due to non-analyticity of the mapping of classical trajectories onto the ZM clock field. While the more complete ZM formalism is not developed here, the non-analyticities in the classical treatment resemble issues in the comparison of classical and quantum formalisms. We also show that semi-classical quantization is valid for a periodic manifold in ZM theory, though the quantum number $n=0$ is allowed, as it would be in quantum mechanics for a periodic manifold. Still, this suggests a connection to the first-order success of Bohr theory in describing the phenomenology of atomic quantum states. The approximate nature of the semi-classical treatment of three dimensional atomic orbits is, however, also apparent in relation to ZM theory. These observations are preliminary to a discussion of ZM theory in relation to quantum mechanics and quantum field theory in subsequent papers.