Analysis of Wallace's Proof of the Born Rule in Everettian Quantum Mechanics: Formal Aspects

TL;DR

This paper revisits Wallace's formal proof of the Born rule in Everettian Quantum Mechanics, simplifying its presentation, analyzing its structure, and identifying potential issues to clarify the decision-theoretic approach to quantum probabilities.

Contribution

It offers a clearer, reorganized presentation of Wallace's proof, with simplified notation, detailed explanations, and an analysis of its formal structure and potential problems.

Findings

The proof's structure is clarified and made more accessible.

Potential issues in the original proof are identified and discussed.

A simplified notation and detailed explanations aid understanding.

Abstract

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D.Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D.Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace's proof, reorganized to simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections.