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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 oneparticle system to which is attached the U(1) fibre bundle represented by themanifold ${\cal S}^{1}$. The manifold ${\cal S}^{1}$ is chosen to correspond tothe action phase, with the action being the principal function corresponding tothe associated classical dynamical system. A Hilbert space representation ofthe flow equation representing the dynamics in the above product space is thenobtained, which is found to yield a probability amplitude formalism in terms ofa generalized set of equations of the Schr\"odinger form which constitutes acovering formalism for both quantum and classical mechanics. Being labelled byan integral index $n$, the equation corresponding to $n=1$ is identified withthe Schr\"odinger equation, while equations corresponding to $n\rightarrowlarge$ yield the classical dynamics through the WKB limit. Equations with lowerindices n=2,3,4,... predict the existence of new modes of non-classicalbehaviour, the observational implications of which are discussed. Somepreliminary experimental results are presented which point to the existence ofthese modes of non-classical behaviour.

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