Within the many worlds interpretation (MWI) it is believed that, as time passes on, the linearity of the Schrödinger equation together with decoherence generate an exponentially growing tree of branches where "everything happens", provided the branches are defined for a decohering basis. By studying an example, using exact numerical diagonalization of the Schrödinger equation to compute the decoherent histories functional, we find that this picture needs revision. Our example shows decoherence for histories defined at a few times, but a significant fraction (often the vast majority) of branches shows strong interference effects for histories of many times. In a sense made precise below, the histories independently sample an equilibrated quantum process, and, remarkably, we find that only histories that sample frequencies in accordance with Born's rule remain decoherent. Our results suggest that there is more structure in the many worlds tree than previously anticipated, influencing arguments of both proponents and opponents of the MWI.
Within the many worlds interpretation (MWI) it is believed that, as time passes on, the linearity of the Schrödinger equation together with decoherence generate an exponentially growing tree of branches where "everything happens", provided the branches are defined for a decohering basis. By studying an example, using exact numerical diagonalization of the Schrödinger equation to compute the decoherent histories functional, we find that this picture needs revision. Our example shows decoherence for histories defined at a few times, but a significant fraction (often the vast majority) of branches shows strong interference effects for histories of many times. In a sense made precise below, the histories independently sample an equilibrated quantum process, and, remarkably, we find that only histories that sample frequencies in accordance with Born's rule remain decoherent. Our results suggest that there is more structure in the many worlds tree than previously anticipated, influencing arguments of both proponents and opponents of the MWI.
While the Born rule is traditionally introduced as a separate postulate of quantum mechanics, we show it emerges naturally from a modified Schrödinger equation that includes "small-signal truncation". This parallels the way quantum decoherence gives rise to the branches of the multiverse in the "Many-Worlds Interpretation", eliminating the need for a separate measurement postulate. Our approach thus offers a unified framework in which both the emergence of multiple branches of the multiverse and their statistical properties follow from a single fundamental law of motion. We model "small-signal truncation" in a somewhat stylized manner, but we argue that the precise details of its underlying physical mechanism do not affect the emergence of the Born rule.
Abstract This introductory chapter summarizes the principal positive theses of the book (Section 1), namely, that the unitary evolution of the quantum state, for sufficiently large systems of particles in thermal environments, yields a multiplicity of branching structures (worlds) obeying approximately classical equations, a result that follows from decoherence theory with no special assumptions or additions to the quantum formalism; and that further, the branching of worlds has a natural interpretation in terms of probability, as quantified by the Born rule in terms of ratios of squared norms of branch amplitudes. In particular, an agent who knows the branching structures of the quantum state that will result from his actions is rationally compelled to order his preferences among actions by the expected utilities of those branching structures as defined by the Born rule. He is, moreover, entitled to view a branching process as involving uncertainty as to which branch he will find himself. But independent of the truth of these claims, the theory is still testable: an agent who does not know the branching structure of the state, and who is indeed undecided between a many-worlds and a one-world theory, is rationally bound to update her credences in such theories by conditionalisation on observed statistical evidence, just as she is bound to do in the case of conventional quantum mechanics. Section 2 summarizes some of the arguments and counter-arguments for these claims as developed in subsequent chapters. Section 3 provides some more mathematical and historical background to the Everett interpretation, including a brief introduction to the decoherent histories formalism.
This paper constructs a logically self-consistent emergent structural frameworkfor the Many-Worlds Interpretation (MWI) based on decoherence theory, quantum information geometry, and self-referential dynamics. Within this framework,a ”world” is rigorously defined as a quasi-classical branch formed in Hilbert spacethrough environmental decoherence and information-geometric boundaries. Its independence is quantitatively characterized by the branch separation degree Dab defined by the quantum relative entropy between branches. When Dab exceeds a critical value Dc determined jointly by the energy scale of environmental fluctuationsand the decoherence time, the branches separate irreversibly in the informationgeometric sense, constituting independent ”worlds”. Furthermore, by introducingthe concept of a self-consistent observer, this paper demonstrates the dynamic selection mechanism of cosmological laws within the string theory landscape: only thosevacuum branches that can support self-consistent observers (satisfying conditionssuch as the memory decoherence time being much greater than the cognitive operation time) have their wavefunction weights effectively preserved during evolution,thereby acquiring physical reality. Finally, combining Zurek’s envariance argumentwith the requirement of observer memory stability, we show that the Born ruleP(i) = |⟨i|ψ⟩|2 is not an additional assumption but a necessary consequence forthe stable existence of self-consistent observers. This framework provides an operational mathematical foundation for the many-worlds interpretation, transforms theanthropic principle into a dynamic selection problem, and offers an explanation forthe origin of probability based on quantum information principles.
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.
We present the case for a fixed, finite number of discrete, non-interacting, spatiotemporally finite, decohered spacetimes emerging from Everett’s Universal Wave Function, which we refer to as “Many Discrete Worlds” (MDW). No universes “split” in MDW. We argue that a Many Worlds Interpretation (MWI) branching structure that emerges after decoherence is equivalent to individual, weighted universes, each of which is divided into an immense number of discrete, identical copies, the number being proportional to the individual weighting. This ensures that repeated experiments within any such universe will demonstrate consistency with the Born rule. Each of these universes should be considered as complete, containing every decohered outcome over the entire extent of its spacetime, including every event/interaction occurring beyond any cosmological particle horizon for the entire duration of the given universe. We show that a countably infinite number of interactions needs an uncountably infinite number of universes, and show why measures such as the Lebesgue measure will fail in that case, with the result that the Born rule would not be demonstrable. This leads to the conclusion that the number of universes in the multiverse must be finite and, as a surprising corollary, that the universes themselves are finite, both in space and duration.
We present the case for a fixed, finite number of discrete, non-interacting, spatiotemporally finite, decohered spacetimes emerging from Everett’s Universal Wave Function, which we refer to as “Many Discrete Worlds” (MDW). No universes “split” in MDW. We argue that a Many Worlds Interpretation (MWI) branching structure that emerges after decoherence is equivalent to individual, weighted universes, each of which is divided into an immense number of discrete, identical copies, the number being proportional to the individual weighting. This ensures that repeated experiments within any such universe will demonstrate consistency with the Born rule. Each of these universes should be considered as complete, containing every decohered outcome over the entire extent of its spacetime, including every event/interaction occurring beyond any cosmological particle horizon for the entire duration of the given universe. We show that a countably infinite number of interactions needs an uncountably infinite number of universes, and show why measures such as the Lebesgue measure will fail in that case, with the result that the Born rule would not be demonstrable. This leads to the conclusion that the number of universes in the multiverse must be finite and, as a surprising corollary, that the universes themselves are finite, both in space and duration.