Non-linear quantum mechanics and semi-classical field theories

Yaneer Bar‐Yam

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.

Karel Koštál, B. Rudolf

Jacques Moret‐Bailly, J -L Monnot

The stochastic electrodynamics (SED) or neoclassical electrodynamics (NED) explains a lot of optical experiments from the hypothesis that a stochastic electromagnetic field exists, equivalent to the half quantum per mode of quantum electrodynamics; we show that, in a semi-classical theory, this field must be introduced for coherence; we propose a non-linear classical model of the interaction between an atom and an electromagnetic field, to build a semi-classical theory.

Stefano Evangelisti

We use the semi-classical approach to study the non-equilibrium dynamics of the O(3) non-linear sigma model. For a class of quenches defined in the text, we obtain the order-parameter dynamical correlator in the thermodynamic limit. In particular we predict quench-dependent relaxation times and correlation lengths. The approach developed for the O(3) non-linear sigma model can also be applied to the transverse field Ising chain, where the semi-classical results can be directly compared to both the exact and the numerical ones, revealing the limits of the method.

B. K. Agarwal