Coherence and randomness in quantum theory

Classical ensemble theory is compared to quantum mechanics. The remarkable feature is that classical ensemble theory, when suitable generalized, leads, for a system of one degree of freedom to two independent uncertainty relations. On the contrary classical trajectory theory which deals only with functions of time, contains no uncertainty relations whatsoever. Quantum mechanics appears, from this viewpoint, to occupy an intermediate position between the two, as it leads to a single uncertainty relation. This situation is analyzed in the present paper which has mainly a pedagogical character. Quantum mechanics leads in the language of classical ensemble theory to collective coherent effects as clearly manifested by the change of the structure of the fundamental dynamical operators. The physical interpretation of quantum mechanics has therefore to be thought in terms of an increased coherence or equivalently in terms of an overdetermined ensemble theory and not in terms of supplementary hidden variables. No appeal to perturbation due to the observer or other subjectivistic element is made in this approach to quantum theory. Our paper contains also an extension of Gibbs ensembles, necessary when average values of physical quantities related by unitary but not canonical transformation theory are introduced. It is shown that under such conditions, classical amplitudes appear naturally. Still these classical amplitudes differ fundamentally from quantum amplitudes or wave functions, as they still depend on both coordinates and momenta. This shows again that the characteristic feature of quantum mechanics is a reduction of the number of independent variables expressing the increased coherence of the dynamical description. (orig.)