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Everett’s Missing Postulate and the Born Rule

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Abstract

Everett’s Relative State Interpretation has gained increasing interest due to the progress of understanding the role of decoherence. In order to fulfill its promise as a realistic description of the physical world, two postulates are formulated. In short they are (1) for a system with continuous coordinates \({\mathbf {x}}\), discrete variable j, and state \(\psi _j({\mathbf {x}})\), the density \(\rho _j({\mathbf {x}})=|\psi _j({\mathbf {x}})|^2\) gives the distribution of the location of the system with the respect to the variables \({\mathbf {x}}\) and j; (2) an equation of motion for the state \(i\hbar \partial _t \psi = H\psi\). The first postulate connects the mathematical description to the physical reality, which has been missing in previous versions. The contents of the standard (Copenhagen) postulates are derived, including the appearance of Hilbert space and the Born rule. The approach to probabilities earlier proposed by Greaves replaces the classical probability concept in the Born rule. The new quantum probability concept, earlier advocated by Deutsch and Wallace, is void of the requirement of uncertainty.

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Notes

  1. Relational Quantum Mechanics [2, 3] is in some respects similar but differs in that observers should not describe themselves in terms of a ‘universal’ wavefunction.

  2. Classical probability refers to situations in which there is uncertainty about a definite outcome.

  3. They consider an agent that is located in different identical rooms but has not yet observed the result. The agent’s mind has not yet split; thus, it is not a classical uncertainty situation.

  4. The system can be a reasonably isolated system or the whole universe. The latter needs a quantum gravity formulation to be adequately addressed, which is beyond the present study.

  5. The separation is here meant to be the process of preparation of experiments with all the complications that such a process entails. For example, the preparation of beam particles and targets in collision experiments.

  6. The postulation of Hilbert space in S1 seems equally backward. A normalization requirement is implied by the Born rule, which in turn implies that the state belongs to a Hilbert space. This possibility puts the derivation of the Born rule by Gleason’s theorem [28] in a new light.

  7. Vaidman [30] has defined ‘worlds’ as having different macroscopic structure. This definition might not be generally appropriate. However, successful measurement results give rise to different macroscopic structures, so the branches that are discussed here are precisely Vaidman ‘worlds’.

  8. Rovelli wrongly claimed [3] that “decoherence depends on which observation P will make”, where P is a second observer. In a laboratory setting, it is the detector system that create decoherence and defines which quantity is measured.

  9. Zurek [35] proved under the assumption of non-disturbing measurement that the measured basis states have to be orthogonal in order for the measurement apparatus to differentiate them.

  10. There is also an objective Bayesian view of probabilities, where objective refers to that we all might agree on the probability value. Here we also classify such probabilities to be subjective.

  11. The \(\langle \, \rangle\) notation can be read as a classical probabilistic expectation value or a quantum (presence) average.

  12. Allori et al. [56] have formulated a ‘primitive ontology,’ which is an ontology formulated entirely in ordinary 3-space. Their quantity \(m({\mathbf {x}})\) is the sum of all single-particle densities multiplied with respective mass. This mass density does not contain the information that the world consists of individual particles that build up individual items with electromagnetic properties etc. Instead, we have to acknowledge that configuration space is the proper physical space, as it is the space of many particles in 3d-space. If densities are physical, relativity implies that currents are too. In turn, currents imply that spins are physical.

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Acknowledgements

I wish to acknowledge Ben Mottelson, David Wallace, Robert Geroch and Lev Vaidman for stimulating discussions and useful suggestions

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    Per Arve

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Arve, P. Everett’s Missing Postulate and the Born Rule. Found Phys 50, 665–692 (2020). https://doi.org/10.1007/s10701-020-00338-4

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