Relating probability amplitude mechanics to standard statistical models

Mei Xiaochun

Based on the concept of ensemble, it is proved in the manuscript that the probability amplitude function can also been used to describe the classical statistical system. The motion equations of probability amplitude functions of classical statistical system are obtained, which are similar to that in quantum mechanics. It is also proved in the meantime that microsystems that obey quantum mechanics can also be described in the forms of classical statistical mechanics and deterministic classical mechanics. The quantum revised formula of classical interaction force and the revised Liouville's equations are also obtained. In this way, the transformation of two descriptive forms between classical mechanics and quantum mechanics is achieved. A new and rational interpretation about quantum mechanics, including the meaning of wave function, the potential barrier penetration of microparticles, the Uncertainty relation, the wave-particle duality, the essence of spin and the Bell's inequality and so on, is provided in the paper. Based on the transformation, two fundamental hypotheses are put forward as the basic postulates of classical statistical physics. The equal probability hypothesis in the equilibrium states is given up. The united description of nonequilibrium state mechanics and equilibrium state statistical mechanics is reached, and the so-called reversibility paradox can also be eliminated.

Barry Loewer

Andrei Yu. Khrennikov

We consider transition from the classical statistical model to the quantum statistical model through ignorance (of huge volume) of information in process of construction of a wave function { a complex probability amplitude. Our approach clarifles relation between classical and quantum statistical models (and hence relation between classical and quantum information theories). In particular, it can be considered as a step toward demystiflcation of quantum theory. The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities.In our approach it is forbidden to consider abstract context independent probabilities: \flrst context and then probability. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. Our formalism can be applied to various domains outside quantum physics, e.g., cognitive sci- ences, psychology, economics, chemistry: roughly speaking in any domain in that it is impossible to extract the whole information about an ontic (or to say realistic) model and hence the epis- temological (experimental or to say observational) model is constructed through huge reduction of information volume. We introduced a notion of prespace { space of states containing complete information about systems under consideration. Classical phase space and quantum Hilbert space are obtained as projections of such a prespace.

Jan Naudts

Alexander Semenovich Holevo

Frank J. Tipler

Many-Worlds quantum mechanics differs from standard quantum mechanics in that in Many-Worlds, the wave function is a relative density of universes in the multiverse amplitude rather than a probability amplitude. This means that in Many-Worlds, the Born frequencies are approached rather than given a priori. Thus in Many-Worlds the rate of approach to the final frequencies can be calculated and compared with observation. I use Many-Worlds to derive the rate of approach in the double slit experiment, and show that it agrees with observation. Standard quantum theory has never been used to derive an approach formula because it cannot be so used, as has been tacitly acknowledged for 70 years.