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Analysis of Wallace’s Proof of the Born Rule in Everettian Quantum Mechanics II: Concepts and Axioms

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Abstract

Having analyzed the formal aspects of Wallace’s proof of the Born rule, we now discuss the concepts and axioms upon which it is built. Justification for most axioms is shown to be problematic, and at times contradictory. Some of the problems are caused by ambiguities in the concepts used. We conclude the axioms are not reasonable enough to be taken as mandates of rationality in Everettian Quantum Mechanics. This invalidates the interpretation of Wallace’s result as meaning it would be rational for Everettian agents to decide using the Born rule.

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Notes

  1. For simplicity, we assume the system remains in state \(\left| i \right\rangle \), but this is not necessary.

  2. Reflecting its stochastic approach, Consistent Histories is usually presented using density operators.

  3. With operations \(P_1\wedge P_2=P_1P_2\), \(P_1\vee P_2=P_1+P_2-P_1P_2\), and \(\lnot P=\mathbb {1}-P\).

  4. The terminology varies. Wallace uses consistency for additivity of probabilities, and decoherence for non-interference, which gets mixed with the related, but not equivalent, concept from Sect. 2.3.1.

  5. Branching-Decoherence Theorem, in Wallace’s terminology.

  6. In Gell-Mann and Hartle’s terminology, medium decoherence.

  7. If necessary, different families of cells can be used at each time.

  8. In [52, p. 175] it says conjunction, but Wallace clearly means disjunction.

  9. As noted in [36], it should be \(\{0\}\) instead of \(\emptyset \), and there is a missing hypothesis, that \(E\wedge F=\{0\}\).

  10. M is null for\(\psi \)andU iff, whenever acts \(V_1\) and \(V_2\) are identical on \(M^\perp \), \(V_1U\sim _\psi V_2U\), i.e. the agent at \(\psi \) is indifferent to what happens in M after U. The reason should be that \(U\psi \) has no branches in M, but there are problems with this concept [36].

  11. Of course, this brings the whole decision theoretic approach into question.

  12. There is nothing unphysical in a horned horse, even if in our world no such species has evolved.

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  1. Departamento de Matemática, Universidade Federal da Bahia, Salvador, BA, Brazil

    André L. G. Mandolesi

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Mandolesi, A.L.G. Analysis of Wallace’s Proof of the Born Rule in Everettian Quantum Mechanics II: Concepts and Axioms. Found Phys 49, 24–52 (2019). https://doi.org/10.1007/s10701-018-0226-4

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