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Analysis of Wallace's Decision Theoretic Proof of the Born Rule in Everettian Quantum Mechanics

André L. G. Mandolesi

Published Apr 20, 2015• 0 citations
OpenAlex

Abstract

We analyse Wallace’s adaptation of Decision Theory to formally prove the Born Rule in Everettian Quantum Mechanics. His proof was reorganized with some improvements, and many details were filled in. In the process, we detected some inconsistencies in his concepts and axioms, as well as other problems, which, if not fixed, would invalidate the proof.

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arXiv:1704.01173•Apr 4, 2017•1 citation
We analyse Wallace’s adaptation of Decision Theory to formally prove the Born Rule in Everettian Quantum Mechanics. His proof was reorganized with some improvements, and many details were filled in. In the process, we detected some inconsistencies in his concepts and axioms, as well as other problems, which, if not fixed, would invalidate the proof.
D. Wallace has combined the Decoherent Histories formalism and Everettian Quantum Mechanics, in an attempt to solve the preferred basis problem. This in turn serves as a foundation for his proof of the Born rule. But approximations used in decoherence depend on the probabilistic interpretation of Born weights, resulting in a circular reasoning. We contest his arguments that the approximations are valid even without probabilities, and show that, without them, the combination of those theories leads to unacceptable results. There is a proliferation of ill behaved histories, compromising his proof of the Born rule. We propose the development of a new causal histories formalism, which might improve the situation. Histories of small amplitude are discarded, not on probability grounds, but for suffering too much interference, which blurs causal relations. The remaining histories might still suffer small amounts of interference, leaving open the possibility of experimental verification.
D. Wallace has tried to use decoherence to solve the preferred basis problem of Everettian Quantum Mechanics, and this solution lays the foundation for his proof of the Born rule. But this is a circular argument, as approximations used in decoherence usually rely on the probabilistic interpretation of the Hilbert space norm. He claims the norm can measure approximations even without probabilities, but this assumption has not been properly justified. Without it, the combination of the Everettian and decoherent histories formalisms leads to strange consequences, such as a proliferation of small amplitude histories with lots of macroscopic quantum jumps. Still, this erratic behavior may provide a way to justify the approximations, in a new histories formalism, in which macroscopic causal relations play a central role. Small histories, suffering too much interference, may lose causality, being thus discarded as invalid. The remaining branches can present some small interference, opening the possibility of experimental verification.