A covering probability amplitude formalism for classical and quantum mechanics, and possible new modes of non-classical behaviour

A generalized dynamics is postulated in a product space ${\cal R}^{3}\times {\cal S}^{1}$ with ${\cal R}^3$ representing the configuration space of a one particle system to which is attached the U(1) fibre bundle represented by the manifold ${\cal S}^{1}$. The manifold ${\cal S}^{1}$ is chosen to correspond to the action phase, with the action being the principal function corresponding to the associated classical dynamical system. A Hilbert space representation of the flow equation representing the dynamics in the above product space is then obtained, which is found to yield a probability amplitude formalism in terms of a generalized set of equations of the Schr\"odinger form which constitutes a covering formalism for both quantum and classical mechanics. Being labelled by an integral index $n$, the equation corresponding to $n=1$ is identified with the Schr\"odinger equation, while equations corresponding to $n\rightarrow large$ yield the classical dynamics through the WKB limit. Equations with lower indices n=2,3,4,... predict the existence of new modes of non-classical behaviour, the observational implications of which are discussed. Some preliminary experimental results are presented which point to the existence of these modes of non-classical behaviour.