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

## TL;DR

This paper derives the Born rule in Everettian Quantum Mechanics by linking quantum probabilities to a complex Pythagorean theorem, proposing a continuum of universes to interpret measurement outcomes.

## Contribution

It introduces a novel geometric interpretation of quantum probabilities as projection measures, solving the probability and preferred basis problems in Many Worlds Interpretation.

## Key findings

- Born rule corresponds to projection factors as measure contractions
- A continuum of identical universes explains probability distributions
- The complex Pythagorean theorem underpins the measure of quantum states

## 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.