In Everett's many worlds interpretation, where quantum measurements are seen as decoherence events, inexact decoherence may let large worlds mangle the memories of observers in small worlds, creating a cutoff in observable world size. I solve a growth-drift-diffusion-absorption model of such a mangled worlds scenario, and show that it reproduces the Born probability rule closely, though not exactly. Thus deviations from exact decoherence may allow the Born rule to be derived in a many worlds approach via world counting, using a finite number of worlds and no new fundamental physics.
In Everett's many worlds interpretation, where quantum measurements are seen as decoherence events, inexact decoherence may let large worlds mangle the memories of observers in small worlds, creating a cutoff in observable world size. I solve a growth-drift-diffusion-absorption model of such a mangled worlds scenario, and show that it reproduces the Born probability rule closely, though not exactly. Thus deviations from exact decoherence may allow the Born rule to be derived in a many worlds approach via world counting, using a finite number of worlds and no new fundamental physics.
In Everett's many-worlds interpretation, where quantum measurements are seen as decoherence events, inexact decoherence may let large worlds mangle the memories of observers in small worlds, creating a cutoff in observable world measure. I solve a growth–drift–diffusion–absorption model of such a mangled worlds scenario, and show that it reproduces the Born probability rule closely, though not exactly. Thus, inexact decoherence may allow the Born rule to be derived in a many-worlds approach via world counting, using a finite number of worlds and no new fundamental physics.
Quantum Mechanics notoriously faces a measurement problem, the problem that the unitary time evolution, encoded in its dynamical equations, together with the kinematical structure of the theory generally implies the non-existence \nof definite measurement outcomes. There have been multiple suggestions to solve this problem, among them the so called many worlds interpretation that originated with the work of Hugh Everett III. According to it, the quantum state and time evolution fully and accurately describe nature as it is, implying that under certain conditions multiple measurement outcomes that are seemingly mutually exclusive can be realized at the same time – but as different 'worlds' contained in a global, quantum mechanical structure, sometimes referred to as 'the multiverse'. The many worlds interpretation has, however, been confronted with serious difficulties over the course of its development, some of which were solved by the advent of decoherence theory. The present thesis critically investi- \ngates the state of play on a key remaining problem of the many worlds interpretation, the problem of the meaning and quantification of probabilities in a quantum multiverse. Recent attempts of deriving the pivotal statistical ingredient of quantum mechanics, Born’s rule, from either principles of decision theory or from quantum mechanics alone, supplemented with a few general premises about \nprobability are analyzed and their premises are scrutinized. It will be argued that, though both approaches yield promising results, they both ultimately fail \nto clearly establish the validity of Born’s rule in the context of the many worlds interpretation. It is hence suggested that further research on this problem is indicated.
Attempts to derive the Born rule, either in the Many Worlds or Copenhagen interpretation, are unsatisfactory for systems with only a finite number of degrees of freedom. In the case of Many Worlds this is a serious problem, since its goal is to account for apparent collapse phenomena, including the Born rule for probabilities, assuming only unitary evolution of the wavefunction. For finite number of degrees of freedom, observers on the vast majority of branches would not deduce the Born rule. However, discreteness of the quantum state space, even if extremely tiny, may restore the validity of the usual arguments.
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.
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.
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.