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arXiv:quant-ph/0605162 · Foundations of Physics

Typicality vs. Probability in Trajectory-Based Formulations of Quantum Mechanics

Bruno Galvan

Published Sep 19, 2007• 12 citations
arXivHTMLPDF
DOI
10.1007/s10701-007-9174-0

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An alternative formulation of quantum mechanics is proposed based on a principle that can be considered a generalization of the Born rule. The principle is composed of a mathematical expression and an associated interpretation, and establishes a correlation between the positions of a particle at two different times. Under reasonable conditions for the wave function, this correlation implies that the particles follow quasiclassical trajectories. It is also shown that the Born rule is equivalent to a particular case of the evolved principle. These features allow the principle to provide a unified explanation of the results of the statistical experiments and of the quasiclassical macroscopic evolution. There is a strong analogy between the quantum principle and a probabilistic principle which is necessary to derive empirical predictions from the mathematical formalism of probability theory. This principle is referred to by some authors as Cournot's principle, while other authors use the equivalent notion of typicality. In this paper probability theory and quantum mechanics are formulated in such a way as to explicitly include the two principles and to emphasize the very similar conceptual structure of the two theories.
This paper defines, on the Galilean space-time, the group of asymptotically Euclidean transformations (AET), which are equivalent to Euclidean transformations at space-time infinity, and proposes a formulation of nonrelativistic quantum mechanics which is invariant under such transformations. This formulation is based on the asymptotic quantum measure, which is shown to be invariant under AET's. This invariance exposes an important connection between AET's and Feynman path integrals, and reveals the nonmetric character of the asymptotic quantum measure. The latter feature becomes even clearer when the theory is formulated in terms of the coordinate-free formalism of asymptotically Euclidean manifold, which do not have a metric structure. This mathematical formalism suggests the following physical interpretation: (i) Particles evolution is represented by trajectories on an asymptotically Euclidean manifold; (ii) The metric and the law of motion are not defined a priori as fundamental entities, but they are properties of a particular class of reference frames; (iii) The universe is considered as a probability space in which the asymptotic quantum measure plays the role of a probability measure. Points (ii) and (iii) are used to build the asymptotic measurement theory, which is shown to be consistent with traditional quantum measurement theory. The most remarkable feature of this measurement theory is the possibility of having a nonchaotic distribution of the initial conditions (NCDIC), an extremely counterintuitive but not paradoxical phenomenon which allows to interpret typical quantum phenomena, such as particle diffraction and tunnel effect, while still providing a description of their motion in terms of classical trajectories.