Branch-counting in the Everett interpretation of quantum mechanics

Abstract

A defence is offered of a version of the branch-counting rule in the Everett interpretation (otherwise known as many worlds interpretation) of quantum mechanics that both depends on the state and is continuous in the norm topology on Hilbert space. The well-known branch-counting rule, for realistic models of measurements, in which branches are defined by decoherence theory, fails this test. The new rule hinges on the use of decoherence theory in defining branching structure, and specifically decoherent histories theory. On this basis ratios of branch numbers are defined, free of any convention. They agree with the Born rule and deliver a notion of objective probability similar to naive frequentism, save that the frequencies of outcomes are not confined to a single world at different times, but spread over worlds at a single time. Nor is it ad hoc : it is recognizably akin to the combinatorial approach to thermodynamic probability, as introduced by Boltzmann in 1879. It is identical to the procedure followed by Planck, Bose, Einstein and Dirac in defining the equilibrium distribution of the Bose–Einstein gas. It also connects in a simple way with the decision-theory approach to quantum probability.

Defending Many Worlds via Case Discrimination: An Attempt to Showcase the Conceptual Incoherence of Anti-Realist Interpretations and Relational Quantum Mechanics

“…Similar to how Wallace himself criticized the lack of objective truthmakers in antirealist interpretations, one might object that a decision-theoretic account of probabilities likewise requires an objective modal structure by which it is packed. Wallace talks of branch weights presumably doing that kind of work [39], but I would rather suggest the use of local relative frequencies, similar to Simon Saunders' recent proposal of an 'equi-amplitude rule' in terms of branch-counting [53]; while global frequencies are indeed ill-defined within an 'Emergent Multiverse' in which allegedly all states are realized, local frequencies might hold relatively to respective decohered branches. For instance, if |α 2 |<|β 2 | holds for equation ( 17) within a particular branch, then relatively fewer sub-branches will exhibit |+ > q than |− > q .…”

“…In other words, relative frequencies in terms of ratios can be given: "There may be no true number of the relevant microstates, in each case, but there may yet be true ratios." [53] (as Saunders argues, instead of Vaidmanian 'degrees of existence', this may rather be an instance of [relative!] numbers of existence.).…”

The Relation between Wavefunction and 3D Space Implies Many Worlds with Local Beables and Probabilities

“…However, since linear combinations of ontic vectors |φ from the same H α also belong to H α , they are quasiclassical, and maybe they should be counted as worlds too. This happens, for example, if we try to prove the Born rule by finding a finite number of orthonormal vectors for the macrostates that add up to |Ψ , as in Equation (18), and counting them, as in [12,22]. If the basis (|φ k ) k∈{1,...,n} from Equation (18) depends on |Ψ , this implies that we have to interpret all such possible orthogonal systems as consisting of words.…”

“…However, Theorem 1 shows that in the continuous case, if we use the same basis, in agreement with Postulate 3, this works. Therefore, Theorem 1 can be understood as the continuous limit of the proposal from [22], necessarily amended with Postulate 3.…”

Probability in many-worlds theories

“…However this approach has led to criticisms [22,23], and because of its focus on the future rather than the past, it seems that a decision-theoretic picture alone cannot explain why we find ourselves in a world in which the historical record of quantum experiments agrees with the Born rule. Very recently, Saunders has proposed an approach [24] based on dividing the state into equal-amplitude branches based on the decoherence structure and following similar counting arguments to those in statistical mechanics.…”