On the nature of the Born rule

TL;DR

This paper explores the fundamental nature of the Born rule in quantum mechanics, proposing it arises from a Fourier transformation linking quantum and classical descriptions, and emphasizes the role of statistical averaging over measurements.

Contribution

It introduces a novel perspective that the Born rule originates from a Fourier transformation connecting quantum and classical frames, explaining its statistical nature.

Findings

The Born rule results from a Fourier transformation between quantum and classical descriptions.

Statistical averaging over measurements removes dependence on initial and end time coordinates.

The approach links the Born rule to the Ehrenfest theorem and the projection from wave to particle descriptions.

Abstract

A physical experiment comprises along the time trajectory a start, a time evolution (duration), and an end, which is the measurement. In non relativistic quantum mechanics the start of the experiment is defined by the wave function at time 0 taking into account the starting conditions, the evolution is described by the wave function following the Schr\"odinger equation and the measurement by the Born rule. While the Schr\"odinger equation is deterministic, it is the Born rule that makes quantum mechanics statistical with all its consequences. The nature of the Born rule is thereby unknown albeit necessary since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is necessary in order to eliminate the unknown initial and end time coordinate of the experiment in reference to the beginning of the universe.