Quantum fractionalism: the Born rule as a consequence of the complex Pythagorean theorem

Abstract: Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces. To show that projection factors can work as probabilities, we postulate the existence of a continuum infinity of identical quantum universes, all with the same quasi-classical worlds. In a measurement, these factors give the relative amounts of worlds with each result, which we associate to frequentist and Bayesian probabilities. This solves the probability problem of Everett's theory, allowing its preferred basis problem to be solved as well, and may help settle questions about the nature of probability.