Generalization of the Born rule

TL;DR

This paper introduces a new quantum principle that generalizes the Born rule, linking particle positions at different times and explaining both statistical and quasi-classical phenomena within a unified framework.

Contribution

It proposes a novel principle extending the Born rule, establishing a correlation between particle positions over time, and unifying quantum statistics with classical trajectories.

Findings

The new principle implies particles follow quasi-classical trajectories.

The Born rule is shown to be a special case of the new principle.

The framework unifies statistical and macroscopic quantum phenomena.

Abstract

A new formulation of quantum mechanics is proposed based on a new 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 quasi-classical trajectories. It is also shown that the Born rule is equivalent to a particular case of the new principle. These features allow the principle to provide a unified explanation of the results of the statistical experiments and of the quasi-classical macroscopic evolution. There is a strong analogy between the new 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.